Beliefs are defined spil someone’s own understanding. Ter belief, “I am” always right and “you” are wrong. There is nothing that can be done to woo the person that what they believe is wrong.

With respect to belief, Henri Poincarй said, “Doubt everything or believe everything: thesis are two identically convenient strategies. With either, wij dispense with the need to think.” Believing means not wanting to know what is fact. Human beings are most apt to believe what they least understand. Therefore, you may rather have a mind opened by wonder than one closed by belief. The greatest derangement of the mind is to believe ter something because one wishes it to be so.

The history of mankind is packed with unsettling normative perspectives reflected ter, for example, inquisitions, witch hunts, denunciations, and brainwashing mechanisms. The “sacred beliefs” are not only within religion, but also within ideologies, and could even include science. Ter much the same way many scientists attempting to “save the theory.” For example, the Freudian treatment is a zuigeling of brainwashing by the therapist where the patient is ter a suggestive mood totally and religiously believing ter whatever the therapist is making of him/hier and blaming himself/herself ter all cases. There is this meaty lumbering momentum from the Cold War where thinking is still not appreciated. Nothing is so tightly believed spil that which is least known.

The history of humanity is also littered with discarded belief-models. However, this does not mean that someone who didn’t understand what wasgoed going on invented the specimen strafgevangenis had no utility or practical value. The main idea wasgoed the cultural values of any wrong specimen. The falseness of a belief is not necessarily an protestation to a belief. The question is, to what extent is it life-promoting, and life enhancing for the believer?

Opinions (or feelings) are slightly less extreme than beliefs however, they are dogmatic. An opinion means that a person has certain views that they think are right. Also, they know that others are entitled to their own opinions. People respect others’ opinions and ter turn expect the same. Te forming one’s opinion, the empirical observations are obviously strongly affected by attitude and perception. However, opinions that are well rooted should grow and switch like a healthy tree. Fact is the only instructional material that can be introduced te an entirely non-dogmatic way. Everyone has a right to his/hier own opinion, but no one has a right to be wrong ter his/hier facts.

Public opinion is often a sort of religion, with the majority spil its prophet. Moreover, the profit has a brief memory and does not provide consistent opinions overheen time.

Rumors and gossip are even weaker than opinion. Now the question is who will believe thesis? For example, rumors and gossip about a person are those when you hear something you like, about someone you do not. Here is an example you might be familiar with: Why is there no Nobel Prize for mathematics? It is the *opinion* of many that Alfred Nobel caught his wifey te an amorous situation with Mittag-Leffler, the foremost Swedish mathematician at the time. Therefore, Nobel wasgoed afraid that if he were to establish a mathematics prize, the very first to get it would be M-L. The story persists, no matter how often one repeats the plain *fact* that Nobel wasgoed not married.

To understand the difference inbetween feeling and strategic thinking , consider cautiously the following true statement: He that thinks himself the happiest man truly is so, but he that thinks himself the wisest is generally the greatest idiot. Most people do not ask for facts ter making up their decisions. They would rather have one good, soul-satisfying emotion than a dozen facts. This does not mean that you should not feel anything. Notice your feelings. But do not think with them.

Facts are different than beliefs, rumors, and opinions. Facts are the voet of decisions. A fact is something that is right and one can prove to be true based on evidence and logical arguments. A fact can be used to coax yourself, your friends, and your enemies. Facts are always subject to switch. Gegevens becomes information when it becomes relevant to your decision problem. Information becomes fact when the gegevens can support it. Fact becomes skill when it is used ter the successful completion of a structured decision process. However, a fact becomes an opinion if it permits for different interpretations, i.e., different perspectives. Note that what happened te the past is fact, not truth. Truth is what wij think about, what happened (i.e., a proefje).

Business Statistics is built up with facts, spil a house is with stones. But a collection of facts is no more a useful and instrumental science for the manager than a heap of stones is a house.

Science and religion are profoundly different. Religion asks us to believe without question, even (or especially) ter the absence of hard evidence. Indeed, this is essential for having a faith. Science asks us to take nothing on faith, to be wary of our penchant for self-deception, to reject anecdotal evidence. Science considers deep but healthy skepticism a prime feature. One of the reasons for its success is that science has built-in, error-correcting machinery at its very heart.

Learn how to treatment information critically and discriminate te a principled way inbetween beliefs, opinions, and facts. Critical thinking is needed to produce well-reasoned representation of reality ter your modeling process. Analytical thinking requests clarity, consistency, evidence, and above all, a consecutive, focused-thinking .

Boudon R., *The Origin of Values: Sociology and Philosophy of Belief*, Transaction Publishers, London, 2001.

Castaneda C., *The Active Side of Infinity*, Harperperennial Library, 2000.

Goodwin P., and G. Wright, *Decision Analysis for Management Judgment*, Wiley, 1998.

Jurjevich R., *The Hoax of Freudism: A Explore of Brainwashing the American Professionals and Laymen*, Philadelphia, Dorrance, 1974.

Kaufmann W., *Religions te Four Dimensions: Existential and Aesthetic, Historical and Comparative*, Reader’s Digest Press, 1976.

### What is Statistical Gegevens Analysis? Gegevens are not Information!

Vast amounts of statistical information are available te today’s global and economic environment because of continual improvements te pc technology. To rival successfully globally, managers and decision makers vereiste be able to understand the information and use it effectively. Statistical gegevens analysis provides forearms on practice to promote the use of statistical thinking and mechanisms to apply ter order to make educated decisions ter the business world.

Computers play a very significant role te statistical gegevens analysis. The statistical software package, SPSS, which is used ter this course, offers extensive data-handling capabilities and numerous statistical analysis routines that can analyze petite to very large gegevens statistics. The rekentuig will assist te the summarization of gegevens, but statistical gegevens analysis concentrates on the interpretation of the output to make inferences and predictions.

Studying a problem through the use of statistical gegevens analysis usually involves four basic steps.

1. Defining the problem

Two. Collecting the gegevens

Three. Analyzing the gegevens

Four. Reporting the results

**Defining the Problem**

An precies definition of the problem is imperative ter order to obtain accurate gegevens about it. It is enormously difficult to gather gegevens without a clear definition of the problem.

**Collecting the Gegevens**

Wij live and work at a time when gegevens collection and statistical computations have become effortless almost to the point of triviality. Paradoxically, the vormgeving of gegevens collection, never adequately emphasized te the statistical gegevens analysis textbook, have bot weakened by an apparent belief that extensive computation can make up for any deficiencies te the vormgeving of gegevens collection. One voorwaarde begin with an emphasis on the importance of defining the population about which wij are seeking to make inferences, all the requirements of sampling and experimental vormgeving voorwaarde be met.

Designing ways to collect gegevens is an significant job te statistical gegevens analysis. Two significant aspects of a statistical examine are:

Population – a set of all the elements of rente te a investigate

Sample – a subset of the population

Statistical inference is refer to extending your skill obtain from a random sample from a population to the entire population. This is known te mathematics spil an Inductive Reasoning. That is, skill of entire from a particular. Its main application is ter hypotheses testing about a given population.

The purpose of statistical inference is to obtain information about a population form information contained te a sample. It is just not feasible to test the entire population, so a sample is the only realistic way to obtain gegevens because of the time and cost constraints. Gegevens can be either quantitative or qualitative. Qualitative gegevens are labels or names used to identify an attribute of each factor. Quantitative gegevens are always numeric and indicate either how much or how many.

For the purpose of statistical gegevens analysis, distinguishing inbetween cross-sectional and time series gegevens is significant. Cross-sectional gegevens re gegevens collected at the same or approximately the same point ter time. Time series gegevens are gegevens collected overheen several time periods.

Gegevens can be collected from existing sources or obtained through observation and experimental studies designed to obtain fresh gegevens. Te an experimental probe, the variable of rente is identified. Then one or more factors te the investigate are managed so that gegevens can be obtained about how the factors influence the variables. Te observational studies, no attempt is made to control or influence the variables of rente. A survey is perhaps the most common type of observational investigate.

**Analyzing the Gegevens**

Statistical gegevens analysis divides the methods for analyzing gegevens into two categories: exploratory methods and confirmatory methods. Exploratory methods are used to detect what the gegevens seems to be telling by using ordinary arithmetic and easy-to-draw pictures to summarize gegevens. Confirmatory methods use ideas from probability theory te the attempt to reaction specific questions. Probability is significant te decision making because it provides a mechanism for measuring, voicing, and analyzing the uncertainties associated with future events. The majority of the topics addressed te this course fall under this heading.

**Reporting the Results**

Through inferences, an estimate or test claims about the characteristics of a population can be obtained from a sample. The results may be reported ter the form of a table, a graph or a set of percentages. Because only a petite collection (sample) has bot examined and not an entire population, the reported results voorwaarde reflect the uncertainty through the use of probability statements and intervals of values.

To conclude, a critical opzicht of managing any organization is programma for the future. Good judgment, intuition, and an awareness of the state of the economy may give a manager a rough idea or “feeling” of what is likely to toebijten ter the future. However, converting that feeling into a number that can be used effectively is difficult. Statistical gegevens analysis helps managers forecast and predict future aspects of a business operation. The most successful managers and decision makers are the ones who can understand the information and use it effectively.

### Gegevens Processing: Coding, Typing, and Editing

**Coding:** the gegevens are transferred, if necessary to coded sheets.

**Typing:** the gegevens are typed and stored by at least two independent gegevens entry persons. For example, when the Current Population Survey and other monthly surveys were taken using paper questionnaires, the U.S. Census Kantoor used dual key gegevens entry.

**Editing:** the gegevens are checked by comparing the two independent typed gegevens. The standard practice for key-entering gegevens from paper questionnaires is to key te all the gegevens twice. Ideally, the 2nd time should be done by a different key entry technicus whose job specifically includes verifying mismatches inbetween the original and 2nd entries. It is believed that this “double-key/verification” method produces a 99.8% accuracy rate for total keystrokes.

**Types of error:** Recording error, typing error, transcription error (incorrect copying), Inversion (e.g., 123.45 is typed spil 123.54), Repetition (when a number is repeated), Deliberate error.

#### Type of Gegevens and Levels of Measurement

Qualitative gegevens, such spil eye color of a group of individuals, is not computable by arithmetic relations. They are labels that advise ter which category or class an individual, object, or process fall. They are called categorical variables.

Quantitative gegevens sets consist of measures that take numerical values for which descriptions such spil means and standard deviations are meaningful. They can be waterput into an order and further divided into two groups: discrete gegevens or continuous gegevens. Discrete gegevens are countable gegevens, for example, the number of defective items produced during a day’s production. Continuous gegevens, when the parameters (variables) are measurable, are voiced on a continuous scale. For example, measuring the height of a person.

The very first activity te statistics is to measure or count. Measurement/counting theory is worried with the connection inbetween gegevens and reality. A set of gegevens is a representation (i.e., a proefje) of the reality based on a numerical and mensurable scales. Gegevens are called “primary type” gegevens if the analyst has bot involved ter collecting the gegevens relevant to his/hier investigation. Otherwise, it is called “secondary type” gegevens.

Gegevens come ter the forms of Nominal, Ordinal, Interval and Ratio (reminisce the French word NOIR for color black). Gegevens can be either continuous or discrete.

Both zero and unit of measurements are arbitrary te the Interval scale. While the unit of measurement is arbitrary te Ratio scale, its zero point is a natural attribute. The categorical variable is measured on an ordinal or nominal scale.

Measurement theory is worried with the connection inbetween gegevens and reality. Both statistical theory and measurement theory are necessary to make inferences about reality.

Since statisticians live for precision, they choose Interval/Ratio levels of measurement.

### Problems with Stepwise Variable Selection

- It yields R-squared values that are badly biased high.
- The F and chi-squared tests quoted next to each variable on the printout do not have the claimed distribution.
- The method yields confidence intervals for effects and predicted values that are falsely narrow.
- It yields P-values that do not have the zindelijk meaning and the zindelijk correction for them is a very difficult problem
- It gives biased regression coefficients that need shrinkage, i.e., the coefficients for remaining variables are too large.
- It has severe problems te the presence of collinearity.
- It is based on methods (e.g. F-tests for nested models) that were intended to be used to test pre-specified hypotheses.
- Enlargening the sample size does not help very much.

Note also that the all-possible-subsets treatment does not eliminate any of the above problems.

Derksen, S. and H. Keselman, Backward, forward and stepwise automated subset selection algorithms, *British Journal of Mathematical and Statistical Psychology*, 45, 265-282, 1992.

### An Alternative Treatment for Estimating a Regression Line

Cornish-Bowden A., *Analysis of Enzyme Kinetic Gegevens*, Oxford Univ Press, 1995.

Hald A., *A History of Mathematical Statistics: From 1750 to 1930*, Wiley, Fresh York, 1998. Among others, the author points out that ter the beginning of 18-th Century researches had four different methods to solve fitting problems: The Mayer-Laplace method of averages, The Boscovich-Laplace method of least absolute deviations, Laplace method of minimizing the largest absolute residual and the Legendre method of minimizing the sum of squared residuals. The only single way of choosing inbetween thesis methods wasgoed: to compare results of estimates and residuals.

### Multivariate Gegevens Analysis

Exploring the fuzzy gegevens picture sometimes requires a wide-angle objectief to view its totality. At other times it requires a closeup objectief to concentrate on fine detail. The graphically based implements that wij use provide this plasticity. Most chemical systems are ingewikkeld because they involve many variables and there are many interactions among the variables. Therefore, chemometric technics rely upon multivariate statistical and mathematical implements to uncover interactions and reduce the dimensionality of the gegevens.

Multivariate analysis is a branch of statistics involving the consideration of objects on each of which are observed the values of a number of variables. Multivariate technics are used across the entire range of fields of statistical application: te medicine, physical and biological sciences, economics and social science, and of course ter many industrial and commercial applications.

Principal component analysis used for exploring gegevens to reduce the dimension. Generally, PCA seeks to represent n correlated random variables by a diminished set of uncorrelated variables, which are obtained by transformation of the original set onto an adequate subspace. The uncorrelated variables are chosen to be good linear combination of the original variables, ter terms of explaining maximal variance, orthogonal directions ter the gegevens. Two closely related technics, principal component analysis and factor analysis, are used to reduce the dimensionality of multivariate gegevens. Te thesis technics correlations and interactions among the variables are summarized te terms of a puny number of underlying factors. The methods rapidly identify key variables or groups of variables that control the system under examine. The resulting dimension reduction also permits graphical representation of the gegevens so that significant relationships among observations or samples can be identified.

Other technics include Multidimensional Scaling, Cluster Analysis, and Correspondence Analysis.

Chatfield C., and A. Collins, *Introduction to Multivariate Analysis*, Chapman and Hall, 1980.

Hoyle R., Statistical Strategies for puny Sample Research, Thousand Oaks, CA, Sage, 1999.

Krzanowski W., *Principles of Multivariate Analysis: A User’s Perspective*, Clarendon Press, 1988.

Mardia K., J. Wetenschap and J. Bibby, *Multivariate Analysis*, Academic Press, 1979.

### The Meaning and Interpretation of P-values (what the gegevens say?)

This interpretation is widely accepted, and many scientific journals routinely publish papers using this interpretation for the result of test of hypothesis.

For the fixed-sample size, when the number of realizations is determined ter advance, the distribution of p is uniform (assuming the null hypothesis). Wij would express this spil P(p £ x) = x. That means the criterion of p <, 0.05 achieves a of 0.05.

When a p-value is associated with a set of gegevens, it is a measure of the probability that the gegevens could have arisen spil a random sample from some population described by the statistical (testing) prototype.

A p-value is a measure of how much evidence you have against the null hypothesis. The smaller the p-value, the more evidence you have. One may combine the p-value with the significance level to make decision on a given test of hypothesis. Te such a case, if the p-value is less than some threshold (usually .05, sometimes a bit larger like 0.1 or a bit smaller like .01) then you reject the null hypothesis.

Understand that the distribution of p-values under null hypothesis H0 is uniform, and thus does not depend on a particular form of the statistical test. Ter a statistical hypothesis test, the P value is the probability of observing a test statistic at least spil extreme spil the value actually observed, assuming that the null hypothesis is true. The value of p is defined with respect to a distribution. Therefore, wij could call it “model-distributional hypothesis” rather than “the null hypothesis”.

Ter brief, it simply means that if the null had bot true, the p value is the probability against the null ter that case. The p-value is determined by the observed value, however, this makes it difficult to even state the inverse of p.

Arsham H., Kuiper’s P-value spil a Measuring Device and Decision Proces for the Goodness-of-fit Test, *Journal of Applied Statistics*, Vol. 15, No.Trio, 131-135, 1988.

### Accuracy, Precision, Robustness, and Quality

The robustness of a proces is the extent to which its properties do not depend on those assumptions which you do not wish to make. This is a modification of Opbergruimte’s original version, and this includes Bayesian considerations, loss spil well spil prior. The central limit theorem (CLT) and the Gauss-Markov Theorem qualify spil robustness theorems, but the Huber-Hempel definition does not qualify spil a robustness theorem.

Wij voorwaarde always distinguish inbetween bias robustness and efficiency robustness. It seems demonstrable to mij that no statistical proces can be sturdy ter all senses. One needs to be more specific about what the proces voorwaarde be protected against. If the sample mean is sometimes seen spil a sturdy estimator, it is because the CLT ensures a 0 bias for large samples regardless of the underlying distribution. This estimator is bias sturdy, but it is clearly not efficiency sturdy spil its variance can increase endlessly. That variance can even be infinite if the underlying distribution is Cauchy or Pareto with a large scale parameter. This is the reason for which the sample mean lacks robustness according to Huber-Hampel definition. The problem is that the M-estimator advocated by Huber, Hampel and a duo of other folks is bias sturdy only if the underlying distribution is symmetric.

Te the setting of survey sampling, two types of statistical inferences are available: the model-based inference and the design-based inference which exploits only the randomization entailed by the sampling process (no assumption needed about the specimen). Unbiased design-based estimators are usually referred to spil sturdy estimators because the unbiasedness is true for all possible distributions. It seems clear however, that thesis estimators can still be of poor quality spil the variance that can be unduly large.

However, others people will use the word ter other (imprecise) ways. Kendall’s Vol. Two, Advanced Theory of Statistics, also cites Opbergruimte, 1953, and he makes a less useful statement about assumptions. Ter addition, Kendall states ter one place that robustness means (merely) that the test size, a , remains onveranderlijk under different conditions. This is what people are using, evidently, when they keuze that two-tailed t-tests are “sturdy” even when variances and sample sizes are unequal. I, personally, do not like to call the tests sturdy when the two versions of the t-test, which are approximately identically sturdy, may have 90% different results when you compare which samples fall into the rejection interval (or region).

I find it lighter to use the phrase, “There is a sturdy difference”, which means that the same finding comes up no matter how you perform the test, what (justifiable) transformation you use, where you split the scores to test on dichotomies, etc., or what outside influences you hold onveranderlijk spil covariates.

### Influence Function and Its Applications

It is main potential application of the influence function is te comparison of methods of estimation for ranking the robustness. A commonsense form of influence function is the sturdy procedures when the extreme values are dropped, i.e., gegevens trimming.

There are a few fundamental statistical tests such spil test for randomness, test for homogeneity of population, test for detecting outliner(s), and then test for normality. For all thesis necessary tests there are powerful procedures ter statistical gegevens analysis literatures. Moreover since the authors are limiting their presentation to the test of mean, they can invoke the CLT for, say any sample of size overheen 30.

The concept of influence is the investigate of the influence on the conclusions and inferences on various fields of studies including statistical gegevens analysis. This is possible by a perturbation analysis. For example, the influence function of an estimate is the switch te the estimate when an infinitesimal switch ter a single observation divided by the amount of the switch. It acts spil the sensitivity analysis of the estimate.

The influence function has bot extended to the “what-if” analysis, robustness, and screenplays analysis, such spil adding or deleting an observation, outliners(s) influence, and so on. For example, for a given distribution both normal or otherwise, for which population parameters have bot estimated from samples, the confidence interval for estimates of the median or mean is smaller than for those values that tend towards the extremities such spil the 90% or 10% gegevens. While ter estimating the mean on can invoke the central limit theorem for any sample of size overheen, say 30. However, wij cannot be sure that the calculated variance is the true variance of the population and therefore greater uncertainty creeps ter and one need to sue the influence function spil a measuring instrument an decision proces.

Melnikov Y., *Influence Functions and Matrices*, Dekker, 1999.

### What is Imprecise Probability?

### What is a Meta-Analysis?

a) Especially when Effect-sizes are rather petite, the hope is that one can build up good power by essentially pretending to have the larger N spil a valid, combined sample.

b) When effect sizes are rather large, then the reserve POWER is not needed for main effects of vormgeving: Instead, it theoretically could be possible to look at contrasts inbetween the slight variations te the studies themselves.

For example, to compare two effect sizes (r) obtained by two separate studies, you may use:

where z 1 and z Two are Fisher transformations of r, and the two n i ‘s ter the denominator represent the sample size for each probe.

If you truly trust that “all things being equal” will hold up. The typical “meta” explore does not do the tests for homogeneity that should be required

1. there is a bod of research/gegevens literature that you would like to summarize

Two. one gathers together all the admissible examples of this literature (note: some might be discarded for various reasons)

Three. certain details of each investigation are deciphered . most significant would be the effect that has or has not bot found, i.e., how much larger te sd units is the treatment group’s vertoning compared to one or more controls.

Four. call the values te each of the investigations te #Trio .. mini effect sizes.

Five. across all admissible gegevens sets, you attempt to summarize the overall effect size by forming a set of individual effects . and using an overall sd spil the divisor .. thus yielding essentially an average effect size.

6. ter the meta analysis literature . sometimes thesis effect sizes are further labeled spil puny, medium, or large .

You can look at effect sizes ter many different ways .. across different factors and variables. but, te a nutshell, this is what is done.

I recall a case ter physics, te which, after a phenomenon had bot observed ter air, emulsion gegevens were examined. The theory would have about a 9% effect ter emulsion, and behold, the published gegevens talent 15%. Spil it happens, there wasgoed no significant difference (practical, not statistical) ter the theory, and also no error te the gegevens. It wasgoed just that the results of experiments te which nothing statistically significant wasgoed found were not reported.

This non-reporting of such experiments, and often of the specific results which were not statistically significant, which introduces major biases. This is also combined with the totally erroneous attitude of researchers that statistically significant results are the significant ones, and than if there is no significance, the effect wasgoed not significant. Wij truly need to differentiate inbetween the term “statistically significant”, and the usual word significant.

Meta-analysis is a controversial type of literature review ter which the results of individual randomized managed studies are pooled together to attempt to get an estimate of the effect of the intervention being studied. It increases statistical power and is used to resolve the problem of reports which disagree with each other. It’s not effortless to do well and there are many inherent problems.

Lipsey M., and D. Wilson, *Practical Meta-Analysis*, Sage Publications, 2000.

### What Is the Effect Size

Effect size permits the comparative effect of different treatments to be compared, even when based on different samples and different measuring instruments.

Therefore, the ES is the mean difference inbetween the control group and the treatment group. Howevere, by Glass’s method, ES is (mean1 – mean2)/SD of control group while by Hunter-Schmit’s method, ES is (mean1 – mean2)/pooled SD and then adjusted by muziekinstrument reliability coefficient. ES is commonly used ter meta-analysis and power analysis.

Cooper H., and L. Hedges, *The Handbook of Research Synthesis*, NY, Russell Sage, 1994.

Lipsey M., and D. Wilson, *Practical Meta-Analysis*, Sage Publications, 2000.

### What is the Benford’s Law? What About the Zipf’s Law?

This implies that a number ter a table of physical constants is more likely to start with a smaller digit than a larger digit. This can be observed, for example, by examining tables of Logarithms and noting that the very first pages are much more worn and smudged than straks pages.

### Bias Reduction Technics

According to legend, Baron Munchausen saved himself from drowning ter quicksand by pulling himself up using only his bootstraps. The statistical bootstrap, which uses resampling from a given set of gegevens to mimic the variability that produced the gegevens ter the very first place, has a rather more dependable theoretical voet and can be a very effective proces for estimation of error quantities te statistical problems.

Bootstrap is to create a virtual population by duplicating the same sample overheen and overheen, and then re-samples from the virtual population to form a reference set. Then you compare your original sample with the reference set to get the precies p-value. Very often, a certain structure is “assumed” so that a residual is computed for each case. What is then re-sampled is from the set of residuals, which are then added to those assumed structures, before some statistic is evaluated. The purpose is often to estimate a P-level.

Jackknife is to re-compute the gegevens by leaving on observation out each time. Leave-one-out replication gives you the same Case-estimates, I think, spil the zindelijk jack-knife estimation. Jackknifing does a bit of logical folding (whence, ‘jackknife’ — look it up) to provide estimators of coefficients and error that (you hope) will have diminished bias.

Bias reduction mechanisms have broad applications ter anthropology, chemistry, climatology, clinical trials, cybernetics, and ecology.

Efron B., *The Jackknife, The Bootstrap and Other Resampling Plans*, SIAM, Philadelphia, 1982.

Efron B., and R. Tibshirani, *An Introduction to the Bootstrap*, Chapman & Hall (now the CRC Press), 1994.

Shao J., and D. Tu, *The Jackknife and Bootstrap*, Springer Verlag, 1995.

### Area Under Standard Normal Curve

### Number of Class Interval te Histogram

k = the smallest rechtschapen greater than or equal to 1 + Loom(n) / Loom (Two) = 1 + Trio.332Log(n)

To have an ‘optimum’ you need some measure of quality – presumably te this case, the ‘best’ way to display whatever information is available ter the gegevens. The sample size contributes to this, so the usual guidelines are to use inbetween Five and 15 classes, one need more classes if you one has a very large sample. You take into account a preference for neat class widths, preferably a numerous of Five or Ten, because this makes it lighter to appreciate the scale.

Beyond this it becomes a matter of judgement – attempt out a range of class widths and choose the one that works best. (This assumes you have a laptop and can generate alternative histograms fairly readily).

There are often management issues that come into it spil well. For example, if your gegevens is to be compared to similar gegevens – such spil prior studies, or from other countries – you are restricted to the intervals used therein.

If the histogram is very skewed, then unequal classes should be considered. Use narrow classes where the class frequencies are high, broad classes where they are low.

The following approaches are common:

Let n be the sample size, then number of class interval could be

Thus for 200 observations you would use 14 intervals but for 2000 you would use 33.

1. Find the range (highest value – lowest value).

Two. Divide the range by a reasonable interval size: Two, Three, Five, Ten or a = numerous of Ten.

Trio. Aim for no fewer than Five intervals and no more than 15.

### Structural Equation Modeling

A structural equation prototype may apply to one group of cases or to numerous groups of cases. When numerous groups are analyzed parameters may be constrained to be equal across two or more groups. When two or more groups are analyzed, means on observed and potentieel variables may also be included te the specimen.

Spil an application, how do you test the equality of regression slopes coming from the same sample using Three different measuring methods? You could use a structural modeling treatment.

1 – Standardize all three gegevens sets prior to the analysis because b weights are also a function of the variance of the predictor variable and with standardization, you eliminate this source.

Two – Prototype the dependent variable spil the effect from all three measures and obtain the path coefficient ( b weight) for each one.

Three – Then gezond a specimen te which the three path coefficients are constrained to be equal. If a significant decrement ter gezond occurs, the paths are not equal.

Schumacker R., and R. Lomax, *A Beginner’s Guide to Structural Equation Modeling*, Lawrence Erlbaum, Fresh Jersey, 1996.

### Econometrics and Time Series Models

Ericsson N., and J. Irons, Testing Exogeneity, Oxford University Press, 1994.

Granger C., and P. Newbold, Forecasting te Business and Economics, Academic Press, 1989.

Hamouda O., and J. Rowley, (Eds.), Time Series Models, Causality and Exogeneity, Edward Elgar Pub., 1999.

### Tri-linear Coordinates Triangle

The same holds for the “composition” of the opinions ter a population. When percents for, against and undecided sum to 100, the same technology for presentation can be used. See the diagram below, which should be viewed with a non-proportional letterteken. True equilateral may not be preserved te transmission. E.g. let the initial composition of opinions be given by 1. That is, few undecided, harshly identically spil much for spil against. Let another composition be given by point Two. This point represents a higher percentage undecided and, among the determined, a majority of “for”.

### Internal and Inter-rater Reliability

Tau-equivalent: The true scores on items are assumed to differ from each other by no more than a onveranderlijk. For a to equal the reliability of measure, the items comprising it have to be at a least tau-equivalent, if this assumption is not met, a is lower roped estimate of reliability.

Congeneric measures: This least limitary monster within the framework of classical test theory requires only that true scores on measures said to be measuring the same phenomenon be flawlessly correlated. Consequently, on congeneric measures, error variances, true-score means, and true-score variances may be unequal

For “inter-rater” reliability, one distinction is that the importance lies with the reliability of the single rating. Suppose wij have the following gegevens By examining the gegevens, I think one cannot do better than looking at the paired t-test and Pearson correlations inbetween each pair of raters – the t-test tells you whether the means are different, while the correlation tells you whether the judgments are otherwise consistent.

Unlike the Pearson, the “intra-class” correlation assumes that the raters do have the same mean. It is not bad spil an overall summary, and it is precisely what some editors do want to see introduced for reliability across raters. It is both a plus and a minus, that there are a few different formulas for intra-class correlation, depending on whose reliability is being estimated.

For purposes such spil programma the Power for a proposed explore, it does matter whether the raters to be used will be exactly the same individuals. A good methodology to apply ter such cases, is the Bland & Altman analysis.

SPSS Directives: Sluis Directions:

### When to Use Nonparametric Mechanism?

1. The gegevens injecting the analysis are enumerative – that is, count gegevens signifying the number of observations ter each category or cross-category.

Two. The gegevens are measured and /or analyzed using a nominal scale of measurement.

Trio. The gegevens are measured and /or analyzed using an ordinal scale of measurement.

Four. The inference does not concern a parameter te the population distribution – spil, for example, the hypothesis that a time-ordered set of observations exhibits a random pattern.

Five. The probability distribution of the statistic upon which the the analysis is based is not dependent upon specific information or assumptions about the population(s) which the sample(s) are drawn, but only on general assumptions, such spil a continuous and/or symmetric population distribution.

By this definition, the distinction of nonparametric is accorded either because of the level of measurement used or required for the analysis, spil te types 1 through Three, the type of inference, spil ter type Four or the generality of the assumptions made about the population distribution, spil te type Five.

For example one may use the Mann-Whitney Rank Test spil a nonparametric alternative to Students T-test when one does not have normally distributed gegevens.

Mann-Whitney: To be used with two independent groups (analogous to the independent groups t-test)

Wilcoxon: To be used with two related (i.e., matched or repeated) groups (analogous to the related samples t-test)

Kruskall-Wallis: To be used with two or more independent groups (analogous to the single-factor between-subjects ANOVA)

Friedman: To be used with two or more related groups (analogous to the single-factor within-subjects ANOVA)

### Analysis of Incomplete Gegevens

– Analysis of finish cases, including weighting adjustments,

– Imputation methods, and extensions to numerous imputation, and

– Methods that analyze the incomplete gegevens directly without requiring a rectangular gegevens set, such spil maximum likelihood and Bayesian methods.

Numerous imputation (Mihoen) is a general paradigm for the analysis of incomplete gegevens. Each missing datum is substituted by m >, 1 simulated values, producing m simulated versions of the accomplish gegevens. Each version is analyzed by standard complete-data methods, and the results are combined using plain rules to produce inferential statements that incorporate missing gegevens uncertainty. The concentrate is on the practice of Mie for real statistical problems ter modern computing environments.

Rubin D., *Numerous Imputation for Nonresponse te Surveys*, Fresh York, Wiley, 1987.

Schafer J., *Analysis of Incomplete Multivariate Gegevens*, London, Chapman and Hall, 1997.

Little R., and D. Rubin, *Statistical Analysis with Missing Gegevens*, Fresh York, Wiley, 1987.

### Interactions te ANOVA and Regression Analysis

Regression is the estimation of the conditional expectation of a random variable given another (possibly vector-valued) random variable.

The easiest construction is to multiply together the predictors whose interaction is to be included. When there are more than about three predictors, and especially if the raw variables take values that are distant from zero (like number of items right), the various products (for the numerous interactions that can be generated) tend to be very correlated with each other, and with the original predictors. This is sometimes called “the problem of multicollinearity”, albeit it would more accurately be described spil spurious multicollinearity. It is possible, and often to be recommended, to adjust the raw products so spil to make them orthogonal to the original variables (and to lower-order interaction terms spil well).

What does it mean if the standard error term is high? Multicolinearity is not the only factor that can cause large SE’s for estimators of “slope” coefficients any regression models. SE’s are inversely proportional to the range of variability ter the predictor variable. For example, if you were estimating the linear association inbetween weight (x) and some dichotomous outcome and x=(50,50,50,50,51,51,53,55,60,62) the SE would be much larger than if x=(Ten,20,30,40,50,60,70,80,90,100) all else being equal. There is a lesson here for the programma of experiments. To increase the precision of estimators, increase the range of the input. Another cause of large SE’s is a petite number of “event” observations or a puny number of “non-event” observations (analogous to puny variance te the outcome variable). This is not rigorously controllable but will increase all estimator SE’s (not just an individual SE). There is also another cause of high standard errors, it’s called serial correlation. This problem is frequent, if not typical, when using time-series, since te that case the stochastic disturbance term will often reflect variables, not included explicitly te the monster, that may switch leisurely spil time passes by.

Te a linear specimen indicating the variation te a dependent variable Y spil a linear function of several explanatory variables, interaction inbetween two explanatory variables X and W can be represented by their product: that is, by the variable created by multiplying them together. Algebraically such a specimen is represented by:

Y = a +b1X + b2 W + b3 XW + e .

When X and W are category systems. This equation describes a two-way analysis of variance (ANOV) specimen, when X and W are (quasi-)continuous variables, this equation describes a numerous linear regression (MLR) specimen.

Te ANOV contexts, the existence of an interaction can be described spil a difference inbetween differences: the difference ter means inbetween two levels of X at one value of W is not the same spil the difference ter the corresponding means at another value of W, and this not-the-same-ness constitutes the interaction inbetween X and W, it is quantified by the value of b3.

Ter MLR contexts, an interaction implies a switch ter the slope (of the regression of Y on X) from one value of W to another value of W (or, equivalently, a switch te the slope of the regression of Y on W for different values of X): ter a two-predictor regression with interaction, the response surface is not a plane but a twisted surface (like “a leaned cookie tin”, ter Darlington’s (1990) phrase). The switch of slope is quantified by the value of b3. To resolve the problem of multi-collinearity.

### Variance of Nonlinear Random Functions

For example, the variance of XY and X/Y based on a large sample size are approximated by:

[E(Y)] Two Var (X) + [E(X)] Two Var(Y) + Two E(X) E(Y) Cov(X, Y)

### Visualization of Statistics: Analytic-Geometry & Statistics

#### Introduction to Visualization of Statistics

Without the loss of generality, and conserving space, the following presentation is ter the setting of puny sample size, permitting us to see statistics ter 1, or 2-dimensional space.

#### The Mean and The Median

Let’s suppose that they determine to minimize the absolute amount of driving. If they met at 1 st Street, the amount of driving would be 0 + Two + 6 + 14 = 22 blocks. If they met at Trio rd Street, the amount of driving would be Two + 0+ Four + 12 = Legal blocks. If they met at 7 th Street, 6 + Four + 0 + 8 = Legal blocks. Ultimately, at 15 th Street, 14 + 12 + 8 + 0 = 34 blocks.

So the two houses that would minimize the amount of driving would be Trio rd or 7 th Street. Actually, if they dreamed a neutral webpagina, any place on Four th , Five th , or 6 th Street would also work.

Note that any value inbetween Three and 7 could be defined spil the median of 1, Trio, 7, and 15. So the median is the value that minimizes the absolute distance to the gegevens points.

Now, the person at 15 th is upset at always having to do more driving. So the group agrees to consider a different rule. Ter determining to minimize the square of the distance driving, wij are using the least square principle. By squaring, wij give more weight to a single very long commute than to a bunch of shorter commutes. With this rule, the 7 th Street house (36 + 16 + 0 + 64 = 116 square blocks) is preferred to the Trio rd Street house (Four + 0 + 16 + 144 = 164 square blocks). If you consider any location, and not just the houses themselves, then 9 th Street is the location that minimizes the square of the distances driven.

Find the value of x that minimizes:

(1 – x) Two + (Trio – x) Two +(7 – x) Two + (15 – x) Two .

The value that minimizes the sum of squared values is 6.Five, which is also equal to the arithmetic mean of 1, Three, 7, and 15. With calculus, it’s effortless to display that this holds ter general.

Consider a petite sample of scores with an even number of cases, for example, 1, Two, Four, 7, Ten, and 12. The median is Five.Five, the midpoint of the interval inbetween the scores of Four and 7.

Spil wij discussed above, it is true that the median is a point around which the sum of absolute deviations is minimized. Te this example the sum of absolute deviations is 22. However, it is not a unique point . Any point te the Four to 7 region will have the same value of 22 for the sum of the absolute deviations.

Indeed, medians are tricky. The 50% above — 50% below is not fairly juist. For example, 1, 1, 1, 1, 1, 1, 8 has no median. The convention says that, the median is 1, however, about 14% of the gegevens lie rigorously above it, 100% of the gegevens are greater than or equal to the median.

Wij will make use of this idea te regression analysis. Ter an analogous argument, the regression line is a unique line, which minimizes the sum of the squared deviations from it. There is no unique line that minimizes the sum of the absolute deviations from it.

#### Arithmetic and Geometric Means

**Arithmetic Mean:** Suppose you have two gegevens points x and y, on real number- line axis:

The arithmetic mean (a) is a point such that the following **vectorial relation** holds: ox – oa = oa – oy.

**Geometric Mean:** Suppose you have two positive gegevens points x and y, on the above real number- line axis, then the Geometric Mean (g) of thesis numbers is a point g such that |ox| / |og| = |og| / |oy|, where |ox| means the **length of line segment** ox, for example.

#### Variance, Covariance, and Correlation Coefficient

Notice that the vector V1 length is:

|V1| = [(Two) Two + (-2) Two ] Ѕ = 8 Ѕ

The variance of V1 is:

Var(V1) = S X i Two / n = |V1| Two /n = Four

The standard deviation is:

|OS1| = |V1| / n Ѕ = 8 Ѕ / Two Ѕ = Two.

Now, consider a 2nd observation (Two, Four). Similarly, it can be represented by vector V2 = (-1, 1).

The covariance is,

Cov (V1, V2) = the dot product / n = [(Two)(-1) + (-2)(1)]/Two = -4/Two = -2

n Cov (V1, V2) = the dot product of the two vectors V1, and V2

Notice that the dot-product is multiplication of the two lengths times the cosine of the angle inbetween the two vectors. Therefore,

Cov (V1, V2) = |OS1| ´, |OS2| ´, Cos (V1, V2) = (Two) (1) Cos(180 ° ) = -2

The correlation coefficient is therefore:

This is possibly the simplest proof that the correlation coefficient is always bounded by the interval [-1, 1]. The correlation coefficient for our numerical example is Cos (V1, V2) = Cos(180 ° ) = -1, spil expected from the above figure.

The distance inbetween the two-point gegevens sets V1, and V2 is also a dot-product:

= n[Var(V1) + VarV2 – 2Cov(V1, V2)]

Now, construct a matrix whose columns are the coordinates of the two vectors V1 and V2, respectively. Multiplying the transpose of this matrix by itself provides a fresh symmetric matrix containing n times the variance of V1 and variance of V2 spil its main diagonal elements (i.e., 8, Two), and n times Cov (V1, V2) spil its off diagonal factor (i.e., -4).

You might like to use a graph paper, and a scientific rekenmachine to check the results of thesis numerical examples and to perform some extra numerical experimentation for a deeper understanding of the concepts.

Wickens T., *The Geometry of Multivariate Statistics*, Erlbaum Pub., 1995.

### What Is a Geometric Mean

Suppose you have two positive gegevens points x and y, then the geometric mean of thesis numbers is a number (g) such that x/g = y/b, and the arithmetic mean (a) is a number such that x – a = a – y.

The geometric means are used extensively by the U.S. Kantoor of Labor Statistics [“Geomeans” spil they call them] ter the computation of the U.S. Consumer Price Index. The geomeans are also used ter price indexes. The statistical use of geometric mean is for index numbers such spil the Fisher’s ideal index.

If some values are very large ter magnitude and others are petite, then the geometric mean is a better average. Te a Geometric series, the most meaningful average is the geometric mean. The arithmetic mean is very biased toward the larger numbers te the series.

Spil an example, suppose sales of a certain voorwerp increase to 110% te the very first year and to 150% of that te the 2nd year. For plainness, assume you sold 100 items primarily. Then the number sold te the very first year is 110 and the number sold te the 2nd is 150% x 110 = 165. The arithmetic average of 110% and 150% is 130% so that wij would incorrectly estimate that the number sold te the very first year is 130 and the number ter the 2nd year is 169. The geometric mean of 110% and 150% is r = (1.65) 1/Two so that wij would correctly estimate that wij would sell 100 (r) Two = 165 items ter the 2nd year.

Spil another similar example, if a mutual fund goes up by 50% one year and down by 50% the next year, and you hold a unit via both years, you have lost money at the end. For every dollar you began with, you have now got 75c. Thus, the spectacle is different from gaining (50%-50%)/Two (= 0%). It is the same spil switching by a multiplicative factor of (1.Five x 0.Five) Ѕ = 0.866 each year. Te a multiplicative process, the one value that can be substituted for each of a set of values to give the same “overall effect” is the geometric mean, not the arithmetic mean. Spil money tends to multiplicatively (“it takes money to make money”), financial gegevens are often better combined te this way.

Spil a survey analysis example, give a sample of people a list of, say Ten, crimes ranging te seriousness:

Theft. Attack . Arson .. Rape . Murder

Ask each respondent to give any numerical value they feel to any crime ter the list (e.g. someone might determine to call arson 100). Then ask them to rate each crime ter the list on a ratio scale. If a respondent thought rape wasgoed five times spil bad spil arson, then a value of 500 would be assigned, theft a quarter spil bad, 25. Suppose wij now dreamed the “average” rating across respondents given to each crime. Since respondents are using their own base value, the arithmetic mean would be worthless: people who used large numbers spil their base value would “swamp” those who had chosen puny numbers. However, the geometric mean — the nth root of the product of ratings for each crime of the n respondents — gives equal weighting to all responses. I’ve used this te a class exercise and it works nicely.

It is often good to log-transform such gegevens before regression, ANOVA, etc. Thesis statistical mechanisms give inferences about the arithmetic mean (which is intimately connected with the least-squares error measure), however, the arithmetic mean of log-transformed gegevens is the loom of the geometric mean of the gegevens. So, for example, a t test on log-transformed gegevens is truly a test for location of the geometric mean.

Langley R., *Practical Statistics Simply Explained*, 1970, Dover Press.

### What Is Central Limit Theorem?

One of the simplest versions of the theorem says that if is a random sample of size n (say, n >, 30) from an infinite population finite standard deviation , then the standardized sample mean converges to a standard normal distribution or, equivalently, the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to standard deviation of the population divided by square root of sample size n. Ter applications of the central limit theorem to practical problems te statistical inference, however, statisticians are more interested te how closely the approximate distribution of the sample mean goes after a normal distribution for finite sample sizes, than the limiting distribution itself. Adequately close agreement with a normal distribution permits statisticians to use normal theory for making inferences about population parameters (such spil the mean ) using the sample mean, irrespective of the actual form of the parent population.

It is well known that whatever the parent population is, the standardized variable will have a distribution with a mean 0 and standard deviation 1 under random sampling. Moreover, if the parent population is normal, then is distributed exactly spil a standard normal variable for any positive rechtschapen n. The central limit theorem states the remarkable result that, even when the parent population is non-normal, the standardized variable is approximately normal if the sample size is large enough (say, >, 30). It is generally not possible to state conditions under which the approximation given by the central limit theorem works and what sample sizes are needed before the approximation becomes good enough. Spil a general guideline, statisticians have used the prescription that if the parent distribution is symmetric and relatively short-tailed, then the sample mean reaches approximate normality for smaller samples than if the parent population is skewed or long-tailed.

On e voorwaarde examine the behavior of the mean of samples of different sizes drawn from a multiplicity of parent populations. Examining sampling distributions of sample means computed from samples of different sizes drawn from a multiplicity of distributions, permit us to build up some insight into the behavior of the sample mean under those specific conditions spil well spil examine the validity of the guidelines mentioned above for using the central limit theorem ter practice.

Under certain conditions, ter large samples, the sampling distribution of the sample mean can be approximated by a normal distribution. The sample size needed for the approximation to be adequate depends strongly on the form of the parent distribution. Symmetry (or lack thereof) is particularly significant. For a symmetric parent distribution, even if very different from the form of a normal distribution, an adequate approximation can be obtained with puny samples (e.g., Ten or 12 for the uniform distribution). For symmetric short-tailed parent distributions, the sample mean reaches approximate normality for smaller samples than if the parent population is skewed and long-tailed. Te some extreme cases (e.g. binomial with ) samples sizes far exceeding the typical guidelines (say, 30) are needed for an adequate approximation. For some distributions without very first and 2nd moments (e.g., Cauchy), the central limit theorem does not hold.

### What is a Sampling Distribution?

Wij will investigate the behavior of the mean of sample values from a different specified populations. Because a sample examines only part of a population, the sample mean will not exactly equal the corresponding mean of the population. Thus, an significant consideration for those programma and interpreting sampling results, is the degree to which sample estimates, such spil the sample mean, will agree with the corresponding population characteristic.

Ter practice, only one sample is usually taken (ter some cases a puny “pilot sample” is used to test the data-gathering mechanisms and to get preliminary information for programma the main sampling scheme). However, for purposes of understanding the degree to which sample means will agree with the corresponding population mean, it is useful to consider what would toebijten if Ten, or 50, or 100 separate sampling studies, of the same type, were conducted. How consistent would the results be across thesis different studies? If wij could see that the results from each of the samples would be almost the same (and almost keurig!), then wij would have confidence te the single sample that will actually be used. On the other arm, eyeing that answers from the repeated samples were too variable for the needed accuracy would suggest that a different sampling project (perhaps with a larger sample size) should be used.

A sampling distribution is used to describe the distribution of outcomes that one would observe from replication of a particular sampling project.

Know that to estimate means to esteem (to give value to).

Know that estimates computed from one sample will be different from estimates that would be computed from another sample.

Understand that estimates are expected to differ from the population characteristics (parameters) that wij are attempting to estimate, but that the properties of sampling distributions permit us to quantify, probabilistically, how they will differ.

Understand that different statistics have different sampling distributions with distribution form depending on (a) the specific statistic, (b) the sample size, and (c) the parent distribution.

Understand the relationship inbetween sample size and the distribution of sample estimates.

Understand that the variability te a sampling distribution can be diminished by enhancing the sample size.

Note that ter large samples, many sampling distributions can be approximated with a normal distribution.

### Outlier Removal

Sturdy statistical technologies are needed to cope with any undetected outliers, otherwise the result will be misleading. For example, the usual stepwise regression is often used for the selection of an suitable subset of explanatory variables to use ter prototype, however, it could be invalidated even by the presence of a few outliers.

Because of the potentially large variance, outliers could be the outcome of sampling. It’s flawlessly keurig to have such an observation that legitimately belongs to the examine group by definition. Lognormally distributed gegevens (such spil international exchange rate), for example, will frequently exhibit such values.

Therefore, you vereiste be very careful and cautious: before announcing an observation “an outlier,” find out why and how such observation occurred. It could even be an error at the gegevens coming in stage.

Very first, construct the BoxPlot of your gegevens. Form the Q1, Q2, and Q3 points which divide the samples into four identically sized groups. (Q2 = median) Let IQR = Q3 – Q1. Outliers are defined spil those points outside the values Q3+k*IQR and Q1-k*IQR. For most case one sets k=1.Five.

Another alternative is the following algorithm

a) Compute s of entire sample.

b) Define a set of boundaries off the mean: mean + k s , mean – k s sigma (Permit user to come in k. A typical value for k is Two.)

c) Liquidate all sample values outside the boundaries.

Now, iterate N times through the algorithm, each time substituting the sample set with the diminished samples after applying step (c).

Usually wij need to iterate through this algorithm Four times.

Spil mentioned earlier, a common “standard” is any observation falling beyond 1.Five (interquartile range) i.e., (1.Five IQRs) ranges above the third quartile or below the very first quartile. The following SPSS program, helps you te determining the outliers.

Outlier detection te the single population setting has bot treated te detail ter the literature. Fairly often, however, one can argue that the detected outliers are not indeed outliers, but form a 2nd population. If this is the case, a cluster treatment needs to be taken. It will be active areas of research to explore the problem of how outliers can arise and be identified, when a cluster treatment voorwaarde be taken.

Hawkins D., *Identification of Outliers*, Chapman & Hall, 1980.

Rothamsted V., V. Barnett, and T. Lewis, *Outliers te Statistical Gegevens*, Wiley, 1994.

### Least Squares Models

Realize that fitting the “best” line by eye is difficult, especially when there is a lotsbestemming of residual variability te the gegevens.

Know that there is a elementary connection inbetween the numerical coefficients ter the regression equation and the slope and intercept of regression line.

Know that a single summary statistic like a correlation coefficient or does not tell the entire story. A scatter plot is an essential complement to examining the relationship inbetween the two variables.

Know that the specimen checking is an essential part of the process of statistical modelling. After all, conclusions based on models that do not decently describe an observed set of gegevens will be invalid.

Know the influence of disturbance of regression prototype assumptions (i.e., conditions) and possible solutions by analyzing the residuals.

### Least Median of Squares Models

### What Is Sufficiency?

A **sufficient statistic** t for a parameter q is a function of the sample gegevens x1. xn, which contains all information te the sample about the parameter q . More formally, sufficiency is defined ter terms of the likelihood function for q . For a sufficient statistic t, the Likelihood L(x1. xn| q ) can be written spil

Since the 2nd term does not depend on q , t is said to be a sufficient statistic for q .

Another way of stating this for the usual problems is that one could construct a random process commencing from the sufficient statistic, which will have exactly the same distribution spil the total sample for all states of nature.

To illustrate, let the observations be independent Bernoulli trials with the same probability of success. Suppose that there are n trials, and that person A observes which observations are successes, and person B only finds out the number of successes. Then if B places thesis successes at random points without replication, the probability that B will now get any given set of successes is exactly the same spil the probability that A will see that set, no matter what the true probability of success happens to be.

### You Vereiste Look at Your Scattergrams!

All three sets have the same correlation and regression line. The significant moral is ** look at your scattergrams**.

How to produce a numerical example where the two scatterplots vertoning clearly different relationships (strengths) but yield the *same* covariance? Perform the following steps:

1. Produce two sets of (X, Y) values that have different correlation’s,

Two. Calculate the two covariances, say C1 and C2,

Three. Suppose you want to make C2 equal to C1. Then you want to multiply C2 by

Four. Since C = r.S x .S y , you want two numbers (one of them might be 1), a and b such that

Five. Multiply all values of X ter set Two by a, and all values of Y by b: for the fresh variables,

An interesting numerical example showcasing two identical scatterplots but with differing covariance is the following: Consider a gegevens set of (X, Y) values, with covariance C1. Now let V = 2X, and W = 3Y. The covariance of V and W will be Two(Trio) = 6 times C1, but the correlation inbetween V and W is the same spil the correlation inbetween X and Y.

### Power of a Test

Power of a test is the probability of correctly rejecting a false null hypothesis. This probability is one minus the probability of making a Type II error ( b ). Recall also that wij choose the probability of making a Type I error when wij set a and that if wij decrease the probability of making a Type I error wij increase the probability of making a Type II error.

#### Power and Alpha:

**Power and the True Difference inbetween Population Means:** Anytime wij test whether a sample differs from a population or whether two sample come from Two separate populations, there is the assumption that each of the populations wij are comparing has it’s own mean and standard deviation (even if wij do not know it). The distance inbetween the two population means will affect the power of our test.

**Power spil a Function of Sample Size and Variance:** You should notice that what indeed made the difference te the size of b is how much overlap there is te the two distributions. When the means are close together the two distributions overlap a fine overeenkomst compared to when the means are further bijzonder. Thus, anything that effects the extent the two distributions share common values will increase b (the likelihood of making a Type II error).

Sample size has an zijdelings effect on power because it affects the measure of variance wij use to calculate the t-test statistic. Since wij are calculating the power of a test that involves the comparison of sample means, wij will be more interested ter the standard error (the average difference te sample values) than standard deviation or variance by itself. Thus, sample size is of rente because it modifies our estimate of the standard deviation. When n is large wij will have a lower standard error than when n is petite. Ter turn, when N is large well have a smaller b region than when n is petite.

**Pilot Studies:** When the needed estimates for sample size calculation is not available from existing database, a pilot explore is needed for adequate estimation with a given precision.

Cohen J., *Statistical Power Analysis for the Behavioral Sciences*, L. Erlbaum Associates, 1988.

Kraemer H., and S. Thiemann, *How Many Subjects?* Provides basic sample size tables , explanations, and power analysis.

Murphy K., and B. Myors, *Statistical Power Analysis*, L. Erlbaum Associates, 1998. Provides a ordinary and general sample size determination for hypothesis tests.

### ANOVA: Analysis of Variance

Thus, when the variability that wij predict (inbetween the two groups) is much greater than the variability wij don’t predict (within each group) then wij will conclude that our treatments produce different results.

**Levene’s Test:** Suppose that the sample gegevens does not support the homogeneity of variance assumption, however, there is a good reason that the variations ter the population are almost the same, then ter such a situation you may like to use the Levene’s modified test: Te each group very first compute the absolute deviation of the individual values from the median te that group. Apply the usual one way ANOVA on the set of deviation values and then interpret the results.

The Proces for Two Populations Independent Means Test

**Click on the picture to enhance it and THEN print it**

The Proces for Two Dependent Means Test

**Click on the picture to increase it and THEN print it**

The Proces for More Than Two Independent Means Test

**Click on the photo to increase it and THEN print it**

The Proces for More Than Two Dependent Populations Test

**Click on the pic to increase it and THEN print it**

### Orthogonal Contrasts of Means ter ANOVA

(mean1+ mean2)/Two – mean3 is orthogonal. Therefore, to determine if two different contrasts of means from the same proefneming are orthogonal, add the product of the weights to see if they sum to zero. If they do not sum to zero, then the two contrasts are not orthogonal and only one of them could be tested. The orthogonal contrasting permits us to compare each mean against all of the other means. There are several effective methods of orthogonal contrasting for applications ter testing, constructing confidence intervals, and the partial F-test spil the post-analysis statistical activities of the usual ANOVA.

Kachigan S., *Statistical Analysis: An Interdisciplinary Introduction to Univariate & Multivariate Methods*, Radius Press, 1986.

Kachigan S., *Multivariate Statistical Analysis: A Conceptual Introduction*, Radius Press, 1991.

### The Six-Sigma Quality

Sigma is a Greek symbol, which is used ter statistics to represent standard deviation of a population. When a large enough random sample gegevens are close to their mean (i.e., the average), then the population has a puny deviation. If the gegevens varies significantly from the mean, the gegevens has a large deviation. Ter quality control measurement terms, you want to see that the sample is spil close spil possible to the mean and that the mean meets or exceed specifications. A large sigma means that there is a large amount of variation within the gegevens. A lower sigma value corresponds to a puny variation, and therefore a managed process with a good quality.

The Six-Sigma means a measure of quality that strives for near perfection. Six-Sigma is a data-driven treatment and methodology for eliminating defects to achieve six sigmas inbetween lower and upper specification boundaries. Accordingly, to achieve Six-Sigma, e.g., ter a manufacturing process it voorwaarde not produce more than Trio.Four defects vanaf million opportunities. Therefore, a Six-Sigma defect is defined for not meeting the customer’s specifications. A Six-Sigma chance is then the total quantity of chances for a defect.

Six-Sigma is a statistical measure voicing how close a product comes to its quality purpose. One sigma means only 68% of products are acceptable, three sigma means 99.7% are acceptable. Six-Sigma is 99.9997% volmaakt or Three.Four defects vanaf million parts or opportunities. The natural spread is 6 times the sample standard deviation. The natural spread is centered on the sample mean, and all weights ter the sample fall within the natural spread, meaning the process will produce relatively few out-of-specification products. Six-Sigma does not necessarily imply Three defective units vanaf million made, it also represents Three defects vanaf million opportunities when used to describe a process. Some products may have ems of thousands of opportunities for defects vanaf finished voorwerp, so the proportion of defective opportunities may actually be fairly large.

Six-Sigma Quality is a fundamental treatment to delivering very high levels of customer satisfaction through disciplined use of gegevens and statistical analysis for maximizing and sustaining business success. What that means is that all business decisions are made based on statistical analysis, not instinct or past history. Using the Six-Sigma treatment will result te a significant, quantifiable improvement.

Is it truly necessary to go for zero defects? Why isn’t 99.9% (about Four.6 sigma) defect-free good enough? Here are some examples of what life would be like if 99.9% were good enough:

- 1 hour of unsafe drinking water every month
- Two long or brief landings at every American cities airport each day
- 400 letters vanaf hour which never arrive at their destination
- Trio,000 newborns accidentally falling from the palms of nurses or doctors each year
- Four,000 incorrect drug prescriptions vanaf year
- 22,000 checks deducted from the wrong bankgebouw account each hour

Spil you can see, sometimes 99.9% good just isn’t good enough.

Here are some examples of what life would be still like at Six-Sigma, 99.9997% defect-free:

- 13 wrong drug prescriptions vanaf year
- Ten newborns accidentally falling from the arms of nurses or doctors each year
- 1 lost article of mail vanaf hour

Now wij see why the quest for Six-Sigma quality is necessary.

Six-Sigma is the application of statistical methods to business processes to improve operating efficiencies. It provides companies with a series of interventions and statistical implements that can lead to breakthrough profitability and quantum gains ter quality. Six-Sigma permits us to take a real world problem with many potential answers, and translate it to a math problem, which will have only one response. Wij then convert that one mathematical solution back to a real world solution.

Six-Sigma goes beyond defect reduction to emphasize business process improvement te general, which includes total cost reduction, cycle-time improvement, enlargened customer satisfaction, and any other metric significant to the customer and the company. An objective of Six-Sigma is to eliminate any waste te the organization’s processes by creating a road ordner for switching gegevens into skill, reducing the amount of stress companies practice when they are dazed with day-to-day activities and proactively uncovering opportunities that influence the customer and the company itself.

The key to the Six-Sigma process is ter eliminating defects. Organizations often waste time creating metrics that are not adequate for the outputs being measured. Executives can get deceptive results if they force all projects to determine a one size fits all metric te order to compare the quality of products and services from various departments. From a managerial standpoint, having one universal device seems beneficial, however, it is not always feasible. Below is an example of the deceptiveness of metrics.

Te the airline industry, the US Air Traffic Control System Guideline Center measures companies on their rate of on time departure. This would obviously be a critical measurement to customers—the flying public. Whenever an airplane departs 15 minutes or more straks than scheduled, that event is considered spil a defect. Unluckily, the government measures the airlines on whether the plane pulls away from the airport gate within 15 minutes of scheduled departure, not when it actually takes off. Airlines know this, so they pull away from the gate on time but let the plane sit on the runway spil long spil necessary before take off. The result to the customer is still a **late departure**. This defect metric is therefore not an accurate representation of the desires of the customers who are impacted by the process. If this were a good descriptive metric, airlines would be measured by the actual delay experienced by passengers.

This example shows the importance of having the right metrics for each process. The method above creates no incentive to reduce actual delays, so the customer (and ultimately the industry) still suffers. With a Six-Sigma business strategy, wij want to see a picture that describes the true output of a process overheen time, along with extra metrics, to give an insight spil to where the management has to concentrate its improvement efforts for the customer.

**The Six Steps of Six-Sigma Loop Process:** The process is identified by the following five major activities for each project:

- Identify the product or service you provide—What do you do?
- Identify your customer base, and determine what they care about—Who uses your products and services? What is indeed significant to them?
- Identify your needs—What do you need to do your work?
- Define the process for doing your work—How do you do your work?
- Eliminate wasted efforts—How can you do your work better?
- Ensure continuous improvement by measuring, analyzing, and controlling the improved process—How ideally are you doing your customer-focused work?

Often each step can create dozens of individual improvement projects and can last for several months. It is significant to go back to each step from time to time te order to determine actual gegevens maybe with improved measurement systems.

Once wij know the answers to the above questions, wij can start to improve the process. The following case explore will further explain the steps applied ter Six-Sigma to Measure, Analyze, Improve, and Control a process to ensure customer satisfaction.

**The Six Sigma General Process and Its Implementation:** The Six-Sigma means a measure of quality that strives for near perfection. Six-Sigma is a data-driven treatment and methodology for eliminating defects to achieve six-sigma’s inbetween lower and upper specification boundaries. Accordingly, to achieve Six-Sigma, e.g., te a manufacturing process it vereiste not produce more than Three.Four defects vanaf million opportunities. Therefore, a Six-Sigma defect is defined for not meeting the customer’s specifications. A Six-Sigma chance is then the total quantity of chances for a defect. The implementation of the Six Sigma system starts normally with a few days workshop of the top level management of the organization.

Only if the advantages of Six Sigma can be clearly stated and supported of the entire Management, then it makes sense to determine together the very first project surrounding field and the pilot project team.

The pilot project team members participate is a few days Six Sigma workshop to learn the system principals, the process, the instruments and the methodology.

The project team meets to compiles main decisions and identifying key stakeholders te the pilot surrounding field. Within the next days the requirements of the stakeholders are collected for the main decision processes by face-to-face interviews.

By now, the workshop of the top management voorwaarde be ready for the next step. The next step for the project team is to determine which and how the achievements should be measured and then start with the gegevens collection and analysis. Whenever the results are understood well then suggestions for improvement will be collected, analyzed, and prioritized based on the urgency and inter-dependencies.

Spil the main outcome, the project team members will determine which improvements should be realized very first. Ter this phase it is significant that rapid successes are obtained, ter order to even the soil for other Six Sigma projects ter the organization.

The activities vereiste be carried out ter parallel whenever possible by a network activity chart. The activity chart will become more and more realistic by a loop-process while spread the improvement via the organization. More and more processes will be included and employees are trained including Black Belts who are the six sigma masters, and the dependency of outer advisors will be diminished.

The main objective of the Six-Sigma treatment is the implementation of a measurement-based strategy that concentrates on process improvement. The aim is variation reduction, which can be accomplished by Six-Sigma methodology.

The Six-Sigma is a business strategy aimed at the near-elimination of defects from every manufacturing, service and transactional process. The concept of Six-Sigma wasgoed introduced and popularized for reducing defect rate of manufactured electronic boards. Albeit the original purpose of Six-Sigma wasgoed to concentrate on manufacturing process, today the marketing, purchasing, customer order, financial and health care processing functions also embarked on Six Sigma programs.

**Motorola Inc.Case:** Motorola is a role proefje for modern manufacturers. The maker of wireless communications products, semiconductors, and electronic equipment loves a stellar reputation for high-tech, high-quality products. There is a reason for this reputation. A participative-management process emphasizing employee involvement is a key factor ter Motorola’s quality thrust. Ter 1987, Motorola invested $44 million te employee training and education ter a fresh quality program called Six-Sigma. Motorola measures its internal quality based on the number of defects te its products and processes. Motorola conceptualized Six-Sigma spil a quality objective ter the mid-1980. Their target wasgoed Six-Sigma quality, or 99.9997% defect free products—which is omschrijving to Trio.Four defects or less vanaf 1 million parts. Quality is a competitive advantage because Motorola’s reputation opens markets. When Motorola Inc. won the Malcolm Baldridge National Quality Award te 1988, it wasgoed ter the early stages of a project that, by 1992, would achieve Six-Sigma Quality. It is estimated that of $9.Two billion ter 1989 sales, $480 million wasgoed saved spil a result of Motorola’s Six-Sigma program. Shortly thereafter, many US firms were following Motorola’s lead.

### Control Charts, and the CUSUM

Developing quality control charts for variables (X-Chart): The following steps are required for developing quality control charts for variables:

- Determine what should be measured.
- Determine the sample size.
- Collect random sample and record the measurements/counts.
- Calculate the average for each sample.
- Calculate the overall average. This is the average of all the sample averages (X-double buffet).
- Determine the range for each sample.
- Calculate the average range (R-bar).
- Determine the upper control limit (UCL) and lower control limit (LCL) for the average and for the range.
- Plot the chart.
- Determine if the average and range values are te statistical control.
- Take necessary act based on your interpretation of the charts.

Developing control charts for attributes (P-Chart): Control charts for attributes are called P-charts. The following steps are required to set up P-charts:

- Determine what should be measured.
- Determine the required sample size.
- Collect sample gegevens and record the gegevens.
- Calculate the average procent defective for the process (p).
- Determine the control boundaries by determining the upper control limit (UCL) and the lower control limit (LCL) values for the chart.
- Plot the gegevens.
- Determine if the procent defectives are within control.

Control charts are also used ter industry to monitor processes that are far from Zero-Defect. However, among the powerful technologies is the counting of the cumulative conforming items inbetween two nonconforming and its combined mechanisms based on cumulative sum and exponentially weighted moving average smoothing methods.

The general CUSUM is a statistical process control when measurements are multivariate. It is an effective contraption te detecting a shift ter the mean vector of the measurements, which is based on the cross-sectional antiranks of the measurements: At each time point, the measurements, after being appropriately transformed, are ordered and their antiranks are recorded. When the process is in-control under some mild regularity conditions the antirank vector at each time point has a given distribution, which switches to some other distribution when the process is out-of-control and the components of the mean vector of the process are not all the same. Therefore it detects shifts te all directions except the one that the components of the mean vector are all the same but not zero. This latter shift, however, can be lightly detected by a univariate CUSUM.

Breyfogle F., Implementing Six Sigma: Smarter Solutions Using Statistical Methods, Wiley, 1999.

del Castillo E., *Statistical Process and Adjustment Methods for Quality Control*, Wiley, 2002.

Juran J, and A. Godfreym, Juran’s Quality Handbook, McGraw-Hill, 1999.

Xie M., T. Goh , and V. Kuralmani, *Statistical Models and Control Charts for High Quality Processes*, Kluwer, 2002.

### Repeatability and Reproducibility

Barrentine L., *Concepts for R&R Studies*, ASQ Quality Press, 1991.

Wheeler D., and R. Lyday, *Evaluating the Measurement Process*, Statistical Process Control Press, 1990.

### Statistical Muziekinstrument, Grab Sampling, and Passive Sampling Mechanisms

**What is a statistical muziekinstrument?** A statistical muziekinstrument is any process that aim at describing a phenomena by using any muziekinstrument or device, however the results may be used spil a control instrument. Examples of statistical instruments are questionnaire and surveys sampling.

**What is grab sampling technology?** The grab sampling mechanism is to take a relatively petite sample overheen a very brief period of time, the result obtained are usually instantaneous. However, the **Passive Sampling** is a technology where a sampling device is used for an extended time under similar conditions. Depending on the desirable statistical investigation, the Passive Sampling may be a useful alternative or even more suitable than grab sampling. However, a passive sampling mechanism needs to be developed and tested ter the field.

### Distance Sampling

line transect sampling, ter which the distances sampled are distances of detected objects (usually animals) from the line along which the observer travels

point transect sampling, te which the distances sampled are distances of detected objects (usually birds) from the point at which the observer stands

cue counting, te which the distances sampled are distances from a moving observer to each detected cue given by the objects of rente (usually whales)

trapping webs, te which the distances sampled are from the web center to trapped objects (usually invertebrates or petite terrestrial vertebrates)

migration counts, ter which the ‘distances’ sampled are actually times of detection during the migration of objects (usually whales) past a see point

Many mark-recapture models have bot developed overheen the past 40 years. Monitoring of biological populations is receiving enhancing emphasis te many countries. Gegevens from marked populations can be used for the estimation of survival probabilities, how thesis vary by age, hookup and time, and how they correlate with outward variables. Estimation of immigration and emigration rates, population size and the proportion of age classes that come in the breeding population are often significant and difficult to estimate with precision for free-ranging populations. Estimation of the finite rate of population switch and fitness are still more difficult to address te a rigorous manner.

Buckland S., D. Anderson, K. Burnham, and J. Laake, *Distance Sampling: Estimating Abundance of Biological Populations*, Chapman and Hall, London, 1993.

Buckland S., D. Anderson, K. Burnham, J. Laake, D. Borchers, and L. Thomas, *Introduction to Distance Sampling*, Oxford University Press, 2001.

### Gegevens Mining and Skill Discovery

The continuing rapid growth of on-line gegevens and the widespread use of databases necessitate the development of technologies for extracting useful skill and for facilitating database access. The challenge of extracting skill from gegevens is of common rente to several fields, including statistics, databases, pattern recognition, machine learning, gegevens visualization, optimization, and high-performance computing.

The gegevens mining process involves identifying an suitable gegevens set to “mine” or sift through to detect gegevens content relationships. Gegevens mining contraptions include technics like case-based reasoning, cluster analysis, gegevens visualization, fuzzy query and analysis, and neural networks. Gegevens mining sometimes resembles the traditional scientific method of identifying a hypothesis and then testing it using an adequate gegevens set. Sometimes however gegevens mining is reminiscent of what happens when gegevens has bot collected and no significant results were found and hence an ad hoc, exploratory analysis is conducted to find a significant relationship.

Gegevens mining is the process of extracting skill from gegevens. The combination of rapid computers, cheap storage, and better communication makes it lighter by the day to taunt useful information out of everything from supermarket buying patterns to credit histories. For clever marketers, that skill can be worth spil much spil the stuff real miners dig from the ground.

Gegevens mining spil an analytic process designed to explore large amounts of (typically business or market related) gegevens ter search for consistent patterns and/or systematic relationships inbetween variables, and then to validate the findings by applying the detected patterns to fresh subsets of gegevens. The process thus consists of three basic stages: exploration, prototype building or pattern definition, and validation/verification.

What distinguishes gegevens mining from conventional statistical gegevens analysis is that gegevens mining is usually done for the purpose of “secondary analysis” aimed at finding unsuspected relationships unrelated to the purposes for which the gegevens were originally collected.

Gegevens warehousing spil a process of organizing the storage of large, multivariate gegevens sets ter a way that facilitates the retrieval of information for analytic purposes.

Gegevens mining is now a rather vague term, but the factor that is common to most definitions is “predictive modeling with large gegevens sets spil used by big companies”. Therefore, gegevens mining is the extraction of hidden predictive information from large databases. It is a powerful fresh technology with superb potential, for example, to help marketing managers “preemptively define the information market of tomorrow.” Gegevens mining devices predict future trends and behaviors, permitting businesses to make proactive, knowledge-driven decisions. The automated, prospective analyses suggested by gegevens mining budge beyond the analyses of past events provided by retrospective instruments. Gegevens mining answers business questions that traditionally were too time-consuming to resolve. Gegevens mining instruments scour databases for hidden patterns, finding predictive information that experts may miss because it lies outside their expectations.

Gegevens mining mechanisms can be implemented rapidly on existing software and hardware platforms across the large companies to enhance the value of existing resources, and can be integrated with fresh products and systems spil they are brought on-line. When implemented on high voorstelling client-server or parallel processing computers, gegevens mining devices can analyze massive databases while a customer or analyst takes a coffee pauze, then supply answers to questions such spil, “Which clients are most likely to react to my next promotional mailing, and why?”

Skill discovery te databases aims at tearing down the last barrier te enterprises’ information flow, the gegevens analysis step. It is a label for an activity performed ter a broad multiplicity of application domains within the science and business communities, spil well spil for pleasure. The activity uses a large and heterogeneous data-set spil a ondergrond for synthesizing fresh and relevant skill. The skill is fresh because hidden relationships within the gegevens are explicated, and/or gegevens is combined with prior skill to elucidate a given problem. The term relevant is used to emphasize that skill discovery is a goal-driven process te which skill is constructed to facilitate the solution to a problem.

Skill discovery maybe viewed spil a process containing many tasks. Some of thesis tasks are well understood, while others depend on human judgment te an implicit matter. Further, the process is characterized by mighty iterations inbetween the tasks. This is very similar to many creative engineering process, e.g., the development of dynamic models. Te this reference mechanistic, or very first principles based, models are emphasized, and the tasks involved te monster development are defined by:

- Initialize gegevens collection and problem formulation. The initial gegevens are collected, and some more or less precise formulation of the modeling problem is developed.

For other prototype types, like neural network models where data-driven skill is utilized, the modeling process will be somewhat different. Some of the tasks like the conceptual modeling phase, will vanish.

Typical application areas for dynamic models are control, prediction, programma, and fault detection and diagnosis. A major deficiency of today’s methods is the lack of capability to utilize a broad multitude of skill. Spil an example, a black-box prototype structure has very limited abilities to utilize very first principles skill on a problem. this has provided a ondergrond for developing different hybrid schemes. Two hybrid schemes will highlight the discussion. Very first, it will be shown how a mechanistic specimen can be combined with a black-box prototype to represent a pH neutralization system efficiently. 2nd, the combination of continuous and discrete control inputs is considered, utilizing a two-tank example spil case. Different approaches to treat this heterogeneous case are considered.

The hybrid treatment may be viewed spil a means to integrate different types of skill, i.e., being able to utilize a heterogeneous skill base to derive a prototype. Standard practice today is that almost any methods and software can treat large homogeneous data-sets. A typical example of a homogeneous data-set is time-series gegevens from some system, e.g., temperature, pressure, and compositions measurements overheen some time framework provided by the instrumentation and control system of a chemical reactor. If textual information of a qualitative nature is provided by plant personnel, the gegevens becomes heterogeneous.

The above discussion will form the ondergrond for analyzing the interaction inbetween skill discovery, and modeling and identification of dynamic models. Ter particular, wij will be interested ter identifying how concepts from skill discovery can enrich state-of-the- kunst within control, prediction, programma, and fault detection and diagnosis of dynamic systems.

Marco D., *Building and Managing the Meta Gegevens Repository: A Total Lifecycle Guide*, John Wiley, 2000.

Thuraisingham B., *Gegevens Mining: Technologies, Technologies, Contraptions, and Trends*, CRC Press, 1998.

Westphal Ch., T. Blaxton, *Gegevens Mining Solutions: Methods and Implements for Solving Real-World Problems*, John Wiley, 1998.

### Neural Networks Applications

The classical approaches are the feedforward neural networks, trained using back-propagation, which remain the most widespread and efficient technology to implement supervised learning. The main steps are: preprocess the gegevens, the suitable selection of variables, postprocessing of the results, and a final validation of the global strategy. Applications include gegevens mining, and stock market predictions.

Schurmann J., *Pattern Classification: A Unified View of Statistical and Neural Approaches*, John Wiley & Sons, 1996.

### Bayes and Empirical Bayes Methods

Bayes and Laagtij methods can be implemented using modern Markov chain Monte Carlo(MCMC) computational methods. Decently structured Bayes and Laagtij procedures typically have good frequentist and Bayesian spectacle, both te theory and te practice. This te turn motivates their use ter advanced high-dimensional prototype settings (e.g., longitudinal gegevens or spatio-temporal mapping models), where a Bayesian monster implemented via MCMC often provides the only feasible treatment that incorporates all relevant prototype features.

Bernardo J., and A. Smith, *Bayesian Theory*, Wiley, 2000.

Carlin B., and T. Louis, *Bayes and Empirical Bayes Methods for Gegevens Analysis*, Chapman and Hall, 1996.

Congdon P., *Bayesian Statistical Modelling*, Wiley, 2001.

Press S., and J. Tanur, *The Subjectivity of Scientists and the Bayesian Treatment*, Wiley, 2001. Comparing and contrasting the reality of subjectivity te the work of history’s fine scientists and the modern Bayesian treatment to statistical analysis.

### Markovian & Memory Theory

Memory Theory and time series share the additive property and inwards a single term there can be multiplication, but like general regression methods this does not always mean that they are all using M Theory. One may use standard time series methods ter the initial phase of modeling things, but instead proceed spil goes after using M Theory’s Cross-Term Dimensional Analysis (CTDA). Suppose that you postulate a prototype y = ge(x) – bg(z) + ch(u) where f, g, h are some functions and x, z, u are what are usually referred to spil independent variables. Notice the minus sign (-) to the left of b and the + sign to the left of c and (implicitly) to the left of a, where a, b, c are positive constants. The variable y is usually referred to spil a dependent variable. According to M Theory, not only do f, g, and h influence/cause y, but g influences/causes f and h at least to some extent. Te fact, M Theory can formulate this ter terms of probable influence spil well spil deterministic influence. All this generalizes to the case where the functions f, g, h depend on two or more variables, e.g., f(x, w), g(z, t, r), etc.

One can switch sides this process. If one thinks that f influences g and h and y but that h and g only influence y and not f, then express the equation of y ter the above form. If it works, one has found something that mainstream regression and time series may fail to detect. Of course, path analysis and Lisrel and partial least squares also rechtsvordering to have ‘causal’ abilities, but only te the standard regression sense of ‘freezing’ so-called independent variables spil ‘givens’ and not ter the M Theory sense which permits them to vary with y. Ter fact, Bayesian probability/statistics methods and M Theory methods use respectively ratios like y/x and differences like y – x + 1 ter their equations, and te the Bayesian monster x is immobile but ter the M Theory specimen x can vary. If one looks cautiously, one will notice that the Bayesian proefje blows up at x = 0 (because division by 0 is unlikely, visit the The Zero Saga pagina), but also near x = 0 since an artificially enormous increase is introduced – precisely near zonderling events. That is one of the reasons why M Theory is more successful for zonderling and/or very influenced/influencing events, while Bayesian and mainstream methods work fairly well for frequent/common and/or low influence (even independent) and/or low dependence events.

Kursunuglu B., S. Mintz, and A. Perlmutter, *Quantum Gravity, Generalized Theory of Gravitation, and Superstring Theory-Based Unification*, Kluwer Academic/Plenum, Fresh York 2000.

### Likelihood Methods

The decision-oriented methods treat statistics spil a matter of act, rather than inference, and attempt to take utilities spil well spil probabilities into account te selecting deeds, the inference-oriented methods treat inference spil a objective speciaal from any act to be taken.

The “hybrid” row could be more decently labeled spil “hypocritical”– thesis methods talk some Decision talk but walk the Inference walk.

Fisher’s fiducial method is included because it is so famous, but the modern overeenstemming is that it lacks justification.

Now it is true, under certain assumptions, some distinct schools advocate very similar calculations, and just talk about them or justify them differently. Some seem to think this is tiresome or impractical. One may disagree, for three reasons:

Very first, how one justifies calculations goes to the heart of what the calculations actually MEAN, 2nd, it is lighter to instruct things that actually make sense (which is one reason that standard practice is hard to instruct), and third, methods that do coincide or almost so for some problems may diverge sharply for others.

The difficulty with the subjective Bayesian treatment is that prior skill is represented by a probability distribution, and this is more of a commitment than warranted under conditions of partial ignorance. (Uniform or improper priors are just spil bad ter some respects spil anything other sort of prior.) The methods te the (Inference, Inverse) cell all attempt to escape this difficulty by presenting alternative representations of partial ignorance.

Edwards, ter particular, uses logarithm of normalized likelihood spil a measure of support for a hypothesis. Prior information can be included ter the form of a prior support (loom likelihood) function, a plane support represents accomplish prior ignorance.

One place where likelihood methods would deviate sharply from “standard” practice is ter a comparison inbetween a acute and a diffuse hypothesis. Consider H0: X

N(0, 100) [diffuse] and H1: X

N(1, 1) [standard deviation Ten times smaller]. Te standard methods, observing X = Two would be undiagnostic, since it is not ter a sensible tail rejection interval (or region) for either hypothesis. But while X = Two is not onbestendig with H0, it is much better explained by H1–the likelihood ratio is about 6.Two te favor of H1. Ter Edwards’ methods, H1 would have higher support than H0, by the amount loom(6.Two) = 1.8. (If thesis were the only two hypotheses, the Neyman-Pearson lemma would also lead one to a test based on likelihood ratio, but Edwards’ methods are more broadly applicable.)

I do not want to emerge to advocate likelihood methods. I could give a long discussion of their limitations and of alternatives that share some of their advantages but avoid their limitations. But it is undoubtedly a mistake to dismiss such methods lightly. They are practical (presently widely used te genetics) and are based on a careful and profound analysis of inference.

### What is a Meta-Analysis?

A Meta-analysis deals with a set of RESULTs to give an overall RESULT that is (presumably) comprehensive and valid.

a) Especially when Effect-sizes are rather petite, the hope is that one can build up good power by essentially pretending to have the larger N spil a valid, combined sample.

b) When effect sizes are rather large, then the toegevoegd POWER is not needed for main effects of vormgeving: Instead, it theoretically could be possible to look at contrasts inbetween the slight variations te the studies themselves.

If you indeed trust that “all things being equal” will hold up. The typical “meta” investigate does not do the tests for homogeneity that should be required

1. there is a bod of research/gegevens literature that you would like to summarize

Two. one gathers together all the admissible examples of this literature (note: some might be discarded for various reasons)

Three. certain details of each investigation are deciphered . most significant would be the effect that has or has not bot found, i.e., how much larger te sd units is the treatment group’s vertoning compared to one or more controls.

Four. call the values ter each of the investigations te #Trio .. mini effect sizes.

Five. across all admissible gegevens sets, you attempt to summarize the overall effect size by forming a set of individual effects . and using an overall sd spil the divisor .. thus yielding essentially an average effect size.

6. ter the meta analysis literature . sometimes thesis effect sizes are further labeled spil puny, medium, or large .

You can look at effect sizes ter many different ways .. across different factors and variables. but, te a nutshell, this is what is done.

I recall a case te physics, ter which, after a phenomenon had bot observed te air, emulsion gegevens wasgoed examined. The theory would have about a 9% effect ter emulsion, and behold, the published gegevens talent 15%. Spil it happens, there wasgoed no significant (practical, not statistical) ter the theory, and also no error te the gegevens. It wasgoed just that the results of experiments ter which nothing statistically significant wasgoed found were not reported.

This non-reporting of such experiments, and often of the specific results which were not statistically significant, which introduces major biases. This is also combined with the totally erroneous attitude of researchers that statistically significant results are the significant ones, and than if there is no significance, the effect wasgoed not significant. Wij indeed need to inbetween the term “statistically significant”, and the usual word significant.

It is very significant to distinction inbetween statistically significant and generally significant, see Detect Tijdschrift (July, 1987), The Case of Falling Nightwatchmen, by Sapolsky. Te this article, Sapolsky uses the example to point out the very significant distinction inbetween statistically significant and generally significant: A diminution of velocity at influence may be statistically significant, but not of importance to the falling nightwatchman.

Be careful about the word “significant”. It has a technical meaning, not a commonsense one. It is NOT automatically synonymous with “significant”. A person or group can be statistically significantly taller than the average for the population, but still not be a candidate for your basketball team. Whether the difference is substantively (not merely statistically) significant is dependent on the problem which is being studied.

Meta-analysis is a controversial type of literature review ter which the results of individual randomized managed studies are pooled together to attempt to get an estimate of the effect of the intervention being studied. It increases statistical power and is used to resolve the problem of reports which disagree with each other. It’s not effortless to do well and there are many inherent problems.

There is also graphical mechanism to assess robustness of meta-analysis results. Wij should carry out the meta-analysis pulling down consecutively one explore, that is if wij have N studies wij should do N meta-analysis using N-1 studies ter each one. After that wij plot thesis N estimates on the y axis and compare them with a straight line that represent the overall estimate using all the studies.

Topics ter Meta-analysis includes: Odds ratios, Relative risk, Risk difference, Effect size, Incidence rate difference and ratio, Plots and precies confidence intervals.

Glass, *et nu.*, Meta-Analysis te Social Research, McGraw Hill, 1987

Cooper H., and L. Hedges, (Eds.), *Handbook of Research Synthesis*, Russell Sage Foundation, Fresh York, 1994

### Industrial Gegevens Modeling

Montgomery D., and G. Runger, *Applied Statistics and Probability for Engineers*, Wiley, 1998.

Ross Sh., *Introduction to Probability and Statistics for Engineers and Scientists*, Academic Press, 1999.

### Prediction Interval

Since wij don’t actually know s Two , wij need to use t ter evaluating the test statistic. The adequate Prediction Interval for Y is

This is similar to construction of interval for individual prediction ter regression analysis.

### Fitting Gegevens to a Violated Line

y = a + b x, for x less than or equal c

y = a – d c + (d + b) x, for x greater than or equal to c

A elementary solution is a brute force search across the values of c. Once c is known, estimating a, b, and d is trivial through the use of indicator variables. One may use (x-c) spil your independent variable, rather than x, for computational convenience.

Now, just fix c at a fine grid of x values te the range of your gegevens, estimate a, b, and d, and then note what the mean squared error is. Select the value of c that minimizes the mean squared error.

Unluckily, you won’t be able to get confidence intervals involving c, and the confidence intervals for the remaining parameters will be conditional on the value of c.

For more details, see *Applied Regression Analysis*, by Draper and Smith, Wiley 1981, Chapter Five, section Five.Four on use of dummy variables. example 6.

### How to Determine if Two Regression Lines Are Parallel?

Ho: slope(group 1) = slope(group 0) is omschrijving to Ho: b Three =0

Use t-test from variables-in-the equation table to test this hypothesis.

### Constrained Regression Proefje

I agree that it’s originally counter-intuitive (see below), but here are two reasons why it’s true. The variance of the slope estimate for the constrained specimen is s Two / S X i Two ), where X i are actual X values and s Two is estimated from the residuals. The variance of the slope estimate for the unconstrained monster (with intercept) is s Two / S x i Two ), where x i are deviations from the mean, and s Two is still estimated from the residuals). So, the constrained prototype can have a larger s Two (mean square error/”residual” and standard error of estimate) but a smaller standard error of the slope because the denominator is larger.

r Two also behaves very strangely te the constrained specimen, by the conventional formula, it can be negative, by the formula used by most rekentuig packages, it is generally larger than the unconstrained r Two because it is dealing with deviations from 0, not deviations from the mean. This is because, ter effect, constraining the intercept to 0 compels us to act spil if the mean of X and the mean of Y both were 0.

Once you recognize that the s.e. of the slope isn’t indeed a measure of overall gezond, the result starts to make a lotsbestemming of sense. Assume that all your X and Y are positive. If you’re compelled to gezond the regression line through the origin (or any other point) there will be less “wiggle” ter how you can getraind the line to the gegevens than there would be if both “finishes” could stir.

Consider a bunch of points that are ALL way out, far from zero, then if you Force the regression through zero, that line will be very close to all the points, and pass through origin, with LITTLE ERROR. And little precision, and little validity. Therefore, no-intercept specimen is hardly everzwijn suitable.

### Semiparametric and Non-parametric modeling

and the unknown e is interpreted spil error term.

The most plain specimen for this problem is the linear regression proefje, an often used generalization is the Generalized Linear Proefje (GLM)

where G is called the listig function. All thesis models lead to the problem of estimating a multivariate regression. Parametric regression estimation has the disadvantage, that by the parametric “form” certain properties of the resulting estimate are already implied.

Nonparametric mechanisms permit diagnostics of the gegevens without this limitation. However, this requires large sample sizes and causes problems ter graphical visualization. Semiparametric methods are a compromise inbetween both: they support a nonparametric modeling of certain features and profit from the plainness of parametric methods.

Hдrdle W., S. Klinke, and B. Turlach, *XploRe: An Interactive Statistical Computing Environment*, Springer, Fresh York, 1995.

### Moderation and Mediation

### Discriminant and Classification

Wij often need to classify individuals into two or more populations based on a set of observed “discriminating” variables. Methods of classification are used when discriminating variables are:

- quantitative and approximately normally distributed,
- quantitative but possibly nonnormal,
- categorical, or
- a combination of quantitative and categorical.

It is significant to know when and how to apply linear and quadratic discriminant analysis, nearest neighbor discriminant analysis, logistic regression, categorical modeling, classification and regression trees, and cluster analysis to solve the classification problem. Schutsluisje has all the routines you need to for decent use of thesis classifications. Relevant topics are: Matrix operations, Fisher’s Discriminant Analysis, Nearest Neighbor Discriminant Analysis, Logistic Regression and Categorical Modeling for classification, and Cluster Analysis.

For example, two related methods which are distribution free are the k-nearest neighbor classifier and the kernel density estimation treatment. Ter both methods, there are several problems of importance: the choice of smoothing parameter(s) or k, and choice of suitable metrics or selection of variables. Thesis problems can be addressed by cross-validation methods, but this is computationally slow. An analysis of the relationship with a neural nipt treatment (LVQ) should yield quicker methods.

Cherkassky V, and F. Mulier, *Learning from Gegevens: Concepts, Theory, and Methods*, John Wiley & Sons, 1998.

Denison, D., C. Holmes, B. Mallick, and A.Smith, *Bayesian Methods for Nonlinear Classification and Regression*, Wiley, 2002.

### Index of Similarity te Classification

I = 2J / [2ab – j(a + b)]

A rather computationally involved for determining a similarity index (I) is due to Fisher, where I is the solution to the following equation:

e aI + e bisexual = 1 + e (a+b-j)I

The index of similarity could be used spil a “distance” so that the ondergrens distance corresponds to the maximum similarity.

Hayek L., and M. Buzas, *Surveying Natural Populations*, Columbia University Press, NY, 1996.

### Generalized Linear and Logistic Models

Hre is how to obtain degree of freedom number for the Two log-likelihood, ter a logistic regression. Degrees of freedom pertain to the dimension of the vector of parameters for a given prototype. Suppose wij know that a specimen ln(p/(1-p))=Bo + B1x + B2y + B3w fits a set of gegevens. Ter this case the vector B=(Bo,B1, B2, B3) is an factor of Four dimensional Euclidean space, or R Four .

Suppose wij want to test the hypothesis: Ho: B3=0. Wij are imposing a confinement on our parameter space. The vector of parameters vereiste be of the form: B’=B=(Bo,B1, B2, 0). This vector is an factor of a subspace of R Four . Namely, B4=0 or the X-axis. The likelihood ration statistic has the form:

Two log-likelihood = Two loom(maximum unrestricted likelihood / maximum restricted likelihood) =

Two loom(maximum unrestricted likelihood)-2 loom (maximum restricted likelihood)

Which is unrestricted B vector 4-dimensions or degrees of freedom – restricted B vector Trio dimensions or degrees of freedom = 1 degree of freedom which is the difference vector: B”=B-B’=(0,0,0,B4) [one dimensional subspace of R Four .

The standard textbook is *Generalized Linear Models* by McCullagh and Nelder (Chapman & Hall, 1989).

Other SPSS Directions: Sluis Guidelines:

Harrell F, *Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis*, Springer Verlag, 2001.

Hosmer D. Jr., and S. Lemeshow, *Applied Logistic Regression*, Wiley, 2000.

Katz M., *Multivariable Analysis: A Practical Guide for Clinicians*, Cambridge University Press, 1999.

Kleinbaum D., *Logistic Regression: A Self-Learning Text*, Springer Verlag, 1994.

Pampel F., *Logistic Regression: A Primer*, Sage, 2000.

### Survival Analysis

The methods of survival analysis are applicable not only ter studies of patient survival, but also studies examining adverse events ter clinical trials, time to discontinuation of treatment, duration te community care before re-hospitalisation, contraceptive and fertility studies etc.

If you’ve everzwijn used regression analysis on longitudinal event gegevens, you’ve most likely come up against two intractable problems:

**Censoring**: Almost every sample contains some cases that do not practice an event. If the dependent variable is the time of the event, what do you do with thesis “censored” cases?

**Time-dependent covariate**: Many explanatory variables (like income or blood pressure)switch ter value overheen time. How do you waterput such variables te a regression analysis?

Makeshift solutions to thesis questions can lead to severe biases. Survival methods are explicitly designed to overeenkomst with censoring and time-dependent covariates te a statistically keurig way. Originally developed by biostatisticians, thesis methods have become popular te sociology, demography, psychology, economics, political science, and marketing.

Te Brief, survival Analysis is a group of statistical methods for analysis and interpretation of survival gegevens. Even tho’ survival analysis can be used te a broad multiplicity of applications (e.g. insurance, engineering, and sociology), the main application is for analyzing clinical trials gegevens. Survival and hazard functions, the methods of estimating parameters and testing hypotheses that are the main part of analyses of survival gegevens. Main topics relevant to survival gegevens analysis are: Survival and hazard functions, Types of censoring, Estimation of survival and hazard functions: the Kaplan-Meier and life table estimators, Elementary life tables, Peto’s Logrank with trend test and hazard ratios and Wilcoxon test, (can be stratified), Wei-Lachin, Comparison of survival functions: The logrank and Mantel-Haenszel tests, The proportional hazards monster: time independent and time dependent covariates, The logistic regression monster, and Methods for determining sample sizes.

Ter the last few years the survival analysis software available te several of the standard statistical packages has experienced a major increment te functionality, and is no longer limited to the triad of Kaplan-Meier kinks, logrank tests, and plain Cox models.

Hosmer D., and S. Lemeshow, *Applied Survival Analysis: Regression Modeling of Time to Event Gegevens*, Wiley, 1999.

Janssen P., J. Swanepoel, and N. Veraverbeke, The modified bootstrap error process for Kaplan-Meier quantiles, *Statistics & Probability Letters*, 58, 31-39, 2002.

Kleinbaum D., *et hoewel.*, *Survival Analysis: A Self-Learning Text*, Springer-Verlag, Fresh York, 1996.

Lee E., *Statistical Methods for Survival Gegevens Analysis*, Wiley, 1992.

Therneau T., and P. Grambsch, *Modeling Survival Gegevens: Extending the Cox Monster*, Springer 2000. This book provides thorough discussion on Cox PH proefje. Since the very first author is also the author of the survival package ter S-PLUS/R, the book can be used closely with the packages ter addition to Schutsluisje.

### Association Among Nominal Variables

### Spearman’s Correlation, and Kendall’s tau Application

Is Var1 ordered the same spil Var2? Two measures are Spearman’s rank order correlation, and Kendall’s tau.

For more details see, e.g., *Fundamental Statistics for the Behavioral Sciences*, by David C. Howell, Duxbury Pr., 1995.

### Repeated Measures and Longitudinal Gegevens

**McNemar Switch Test:** For the yes/no questions under the two conditions, set up a 2×2 contingency table: McNemar’s test of correlated proportions is z = (f01 – f10)/(f01 + f10) Ѕ .

For those items yielding a score on a scale, the conventional t-test for correlated samples would be adequate, or the Wilcoxon signed-ranks test.

### What Is a Systematic Review?

There are few significant questions te health care which can be informed by consulting the result of a single empirical investigate. Systematic reviews attempt to provide answers to such problems by identifying and appraising all available studies within the relevant concentrate and synthesizing their results, all according to explicit methodologies. The review process places special emphasis on assessing and maximizing the value of gegevens, both te issues of reducing bias and minimizing random error. The systematic review method is most suitably applied to questions of patient treatment and management, albeit it has also bot applied to reaction questions regarding the value of diagnostic test results, likely prognoses and the cost-effectiveness of health care.

### Information Theory

Shannon defined a measure of entropy spil:

that, when applied to an information source, could determine the capacity of the channel required to transmit the source spil encoded binary digits. Shannon’s measure of entropy is taken spil a measure of the information contained te a message. This is unlike to the portion of the message that is stringently determined (hence predictable) by inherent structures.

Entropy spil defined by Shannon is closely related to entropy spil defined by physicists te statistical thermodynamics. This work wasgoed the inspiration for adopting the term entropy ter information theory. Other useful measures of information include mutual information which is a measure of the correlation inbetween two event sets. Mutual information is defined for two events X and Y spil:

where H(X, Y) is the join entropy defined spil:

Mutual information is closely related to the log-likelihood ratio test for multinomial distribution, and to Pearson’s Chi-square test.

The field of Information Science has since expanded to voorkant the utter range of mechanisms and abstract descriptions for the storage, retrieval and transmittal of information.

### Incidence and Prevalence Rates

Prevalence rate (PR) measures the number of cases that are present at a specified period of time. It is defined spil: Number of cases present at a specified period of time divides by Number of persons at risk at that specified time.

Thesis two measures are related when considering the average duration (D). That is, PR = IR . D

Note that, for example, county-specific disease incidence rates can be unstable due to puny populations or low rates. Te epidemiology one can say that IR reflects probability to Become thick at given age, while the PR reflects probability to Be thick at given age.

Other topics ter clinical epidemiology include the use of receiver technicus forms, and the sensitivity, specificity, predictive value of a test.

Kleinbaum D., L. Kupper, and K. Muller, *Applied Regression Analysis and Other Multivariable Methods*, Wadsworth Publishing Company, 1988.

Kleinbaum D., *et alreeds.*, *Survival Analysis: A Self-Learning Text*, Springer-Verlag, Fresh York, 1996.

Miettinen O., *Theoretical Epidemiology*, Delmar Publishers, 1986.

### Software Selection

1) Ease of learning,

Two) Amount of help incorporated for the user,

Three) Level of the user,

Four) Number of tests and routines involved,

Five) Ease of gegevens entry,

6) Gegevens validation (and if necessary, gegevens locking and security),

7) Accuracy of the tests and routines,

8) Integrated gegevens analysis (graphs and progressive reporting on analysis te one screen),

No one software meets everyone’s needs. Determine the needs very first and then ask the questions relevant to the above seven criteria.

### Spatial Gegevens Analysis

Many natural phenomena involve a random distribution of points ter space. Biologists who observe the locations of cells of a certain type ter an organ, astronomers who plot the positions of the starlets, botanists who record the positions of plants of a certain species and geologists detecting the distribution of a zonderling mineral te rock are all observing spatial point patterns te two or three dimensions. Such phenomena can be modelled by spatial point processes.

The spatial linear prototype is fundamental to a number of mechanisms used te photo processing, for example, for locating gold/ore deposits, or creating maps. There are many unresolved problems ter this area such spil the behavior of maximum likelihood estimators and predictors, and diagnostic implements. There are strong connections inbetween kriging predictors for the spatial linear specimen and spline methods of interpolation and smoothing. The two-dimensional version of splines/kriging can be used to construct deformations of the plane, which are of key importance te form analysis.

For analysis of spatially auto-correlated gegevens te of logistic regression for example, one may use of the Moran Coefficient which is available is some statistical packages such spil Spacestat. This statistic tends to be inbetween -1 and +1, tho’ are not restricted to this range. Values near +1 indicate similar values tend to cluster, values near -1 indicate dissimilar values tend to cluster, values near -1/(n-1) indicate values tend to be randomly scattered.

### Boundary Line Analysis

The main application of this analysis is ter the soil electrical conductivity (EC) which stems from the fact that sands have a low conductivity, silts have a medium conductivity and clays have a high conductivity. Consequently, conductivity (measured at low frequencies) correlates strongly to soil grain size and texture.

The boundary line analysis, therefore, is a method of analyzing yield with soil electrical conductivity gegevens. This method isolates the top yielding points for each soil EC range and fits a non-linear line or equation to represent the top-performing yields within each soil EC range. This method knifes through the cloud of EC/Yield gegevens and describes their relationship when other factors are liquidated or diminished. The upper boundary represents the maximum possible response to that limiting factor, (e.g. EC), and points below the boundary line represents conditions where other factors have limited the response variable. Therefore, one may also use boundary line analysis to compare responses among species.

Kitchen N., K Sudduth, and S. Drummond, Soil Electrical Conductivity spil a Crop Productivity Measure for Claypan Soils, *Journal of Production Agriculture*, 12(Four), 607-617, 1999.

### Geostatistics Modeling

Christakos G., *Modern Spatiotemporal Geostatistics*, Oxford University Press, 2000.

### Box-Cox Power Transformation

Among others the Box-Cox power transformation is often used for this purpose.

attempting different values of p inbetween -3 and +Trio is usually sufficient but there are MLE methods for estimating the best p. A good source on this and other transformation methods is

Madansky A., *Prescriptions for working Statisticians*, Springer-Verlag, 1988.

For percentages or proportions (such spil for binomial proportions), Arcsine transformations would work better. The original idea of Arcsin(p Ѕ )is to establish variances spil equal for all groups. The arcsin convert is derived analytically to be the variance-stabilizing and normalizing transformation. The same limit theorem also leads to the square root convert for Poisson variables (such spil counts) and to the arc hyperbolic tangent (i.e., Fisher’s Z) convert for correlation. The Arcsin Test yields a z and the 2×2 contingency test yields a chi-sq. But z Two = chi-sq, for large sample size. A good source is

Rao C., *Linear Statistical Inference and Its Applications*, Wiley, 1973.

How to normalize a set of gegevens consisting of negative and positive values, and make them positive inbetween the range 0.0 to 1.0? Define XNew = (X-min)/(max-min).

Opbergruimte & Cox power transformation is also very effective for a broad diversity of nonnormality:

y(transformed) = y l

where l ranges (te practice) from -3.0 to +Three.0. Spil such it includes, inverse, square root, logarithm, etc. Note that spil l approaches 0, one gets a loom transformation.

### Numerous Comparison Tests

Numerous comparison procedures include topics such spil Control of the family-Wise Error rate, The closure Principle, Hierarchical Families of Hypotheses, Single-Step and Stepwise Procedures, and P-value Adjustments. Areas of applications include numerous comparisons among treatment means, numerous endpoints te clinical trials, numerous sub-group comparisons, etc.

Nemenyi’s numerous comparison test is analogous to Tukey’s test, using rank sums te place of means and using [n Two k(nk+1)/12] Ѕ spil the estimate of standard error (SE), where n is the size of each sample and k is the number of samples (means). Similarly to the Tukey test, you compare (rank sum A – rank sum B)/SE to the studentized range for k. It is also omschrijving to the Dunn/Miller test which uses mean ranks and standard error [k(nk+1)/12] Ѕ .

**Multilevel Statistical Modeling:** The two widely used software packages are MLwiN and winBUGS. They perform multilevel modeling analysis and analysis of hierarchical datasets, Markov chain Monte Carlo (MCMC) methodology and Bayesian approaches.

Liao T., *Statistical Group Comparison*, Wiley, 2002.

### Antedependent Modeling for Repeated Measurements

Many technologies can be used to analyze such gegevens. Antedependence modeling is a recently developed method which models the correlation inbetween observations at different times.

### Split-half Analysis

Notice that this is (like factor analysis itself) an “exploratory”, not inferential mechanism, i.e. hypothesis testing, confidence intervals etc. simply do not apply.

Alternatively, randomly split the sample ter half and then do an exploratory factor analysis on Sample 1. Use those results to do a confirmatory factor analysis with Sample Two.

### Sequential Acceptance Sampling

Sequential acceptance sampling minimizes the number of items tested when the early results demonstrate that the batch clearly meets, or fails to meet, the required standards.

The proces has the advantage of requiring fewer observations, on average, than immobilized sample size tests for a similar degree of accuracy.

### Local Influence

Cook defined local influence te 1986, and made some suggestions on how to use or interpret it, various slight variations have bot defined since then. But problems associated with its use have bot pointed out by a number of workers since the very beginning.

### Variogram Analysis

A variogram summarizes the relationship inbetween the variance of the difference ter pairs of measurements and the distance of the corresponding points from each other.

### Credit Scoring: Consumer Credit Assessment

Accurate assessment of financial exposure is vital for continued business success. Accurate, and usable information are essential for good credit assessment te commercial decision making. The consumer credit environment is ter a state of good switch, driven by developments ter laptop technology, more requesting customers, availability of fresh products and enlargened competition. Banks and other financial institutions are coming to rely more and more on increasingly sophisticated mathematical and statistical devices. Thesis devices are used te a broad range of situations, including predicting default risk, estimating likely profitability, fraud detection, market segmentation, and portfolio analysis. The credit card market spil an example, has switched the retail banking industry, and consumer loans.

Both the devices, the behavioral scoring, and the characteristics of consumer credit gegevens are usually the bases for a good decision. The statistical instruments include linear and logistic regression, mathematical programming, trees, nearest neighbor methods, stochastic process models, statistical market segmentation, and neural networks. Thesis technologies are used to assess and predict consumers credit scoring.

Lewis E., *Introduction to Credit Scoring*, Fair, Isaac & Co., 1994. Provides a general introduction to the issues of building a credit scoring prototype.

### Components of the Rente Rates

**The unspoiled rate:** This is the time value of money. A promise of 100 units next year is not worth 100 units this year.

**The price-premium factor:** If prices go up 5% each year, rente rates go up at least 5%. For example, under the Carter Administration, prices rose about 15% vanaf year for a duo of years, rente wasgoed around 25%. Same thing during the Civil War. Ter a deflationary period, prices may druppel so this term can be negative.

**The risk factor:** A junk unie may pay a larger rate than a treasury note because of the chance of losing the principal. Banks te a poor financial condition voorwaarde pay higher rates to attract depositors for the same reason. Threat of confiscation by the government leads to high rates te some countries.

Other factors are generally minor. Of course, the customer sees only the sum of thesis terms. Thesis components fluctuate at different rates themselves. This makes it hard to compare rente rates across disparate time periods or economic condition. The main questions are: how are thesis components combined to form the index? A ordinary sum? A weighted sum? Ter most cases the index is form both empirically and assigned on fundament of some criterion of importance. The same applies to other index numbers.

### Partial Least Squares

The method aims to identify the underlying factors, or linear combination of the X variables, which best prototype the Y dependent variables.

### Growth Curve Modeling

Sometimes wij simply wish to summarize growth observations te terms of a few parameters, perhaps te order to compare individuals or groups. Many growth phenomena ter nature voorstelling an “S” shaped pattern, with primarily slow growth speeding up before slowing down to treatment a limit.

Thesis patterns can be modelled using several mathematical functions such spil generalized logistic and Gompertz forms.

### Saturated Prototype & Saturated Loom Likelihood

### Pattern recognition and Classification

### What is Biostatistics?

Latest advancement ter human genome marks a major step te the advancement of understanding how the human bod works at a molecular level. The biomedical statistics identifies the need for computational statistical implements to meet significant challenges te biomedical studies. The active areas are:

- Clustering of very large dimensional gegevens such spil the micro-array.
- Clustering algorithms that support biological meaning.
- Network models and simulations of biological pathways.
- Pathway estimation from gegevens.
- Integration of multi-format and multi-type gegevens from heterogeneous databases.
- Information and skill visualization technics for biological systems.

Cleophas T., A. Zwinderman, and T. Cleophas, *Statistics Applied to Clinical Trials*, Kluwer Academic Publishers, 2002.

Zhang W., and I. Shmulevich, *Computational and Statistical Approaches to Genomics*, Kluwer Academic Publishers, 2002.

### Evidential Statistics

1. Should this observation lead mij to believe that condition C is present?

Two. Does this observation justify my acting spil if condition C were present?

Trio. Is this observation evidence that condition C is present?

Wij voorwaarde distinguish among thesis three questions ter terms of the variables and principles that determine their answers. Questions of the third type, concerning the “evidential interpretation” of statistical gegevens, are central to many applications of statistics ter many fields.

It is already recognized that for answering the evidential question current statistical methods are gravely flawed which could be corrected by a applying the the Law of Likelihood. This law suggests how the superior statistical paradigm can be altered so spil to generate suitable methods for objective, quantitative representation of the evidence embodied ter a specific set of observations, spil well spil measurement and control of the probabilities that a investigate will produce powerless or misleading evidence.

Royall R., *Statistical Evidence: A Likelihood Paradigm*, Chapman & Hall, 1997.

### Statistical Forensic Applications

One consequence of the failure to recognize the benefits that an organized treatment can bring is our failure to stir evidence spil a discipline into volume case analytics. Any cursory view of the literature exposes that work has centered on thinking about single cases using narrowly defined views of what evidential reasoning involves. There has bot an overheen emphasis on the formal rules of admissibility rather than the rules and principles of a methodological scientific treatment.

Spil the popularity of using DNA evidence increases, both the public and professionals increasingly regard it spil the last word on a suspect’s guilt or innocence. Spil citizens go about their daily lives, chunks of their identities are scattered ter their wake. It could spil some critics warn, one day place an guiltless person at the toneel of a crime.

The traditional methods of statistical forensic, for example, for facial reconstruction date back to the Victorian Era. Tissue depth gegevens wasgoed collected from cadavers at a petite number of landmark sites on the face. Samples were little, commonly numbering less than ten. Albeit thesis gegevens sets have bot superceded recently by tissue innards collected from the living using ultrasound, the same twenty-or-so landmarks are used and samples are still petite and under-representative of the general population. A number of aspects of identity–such spil age, height, geographic ancestry and even sex–can only be estimated from the skull. Current research is directed at the recovery of volume tissue depth gegevens from magnetic resonance imaging scans of the head of living individuals, and the development of ordinary interpolation simulation models of obesity, ageing and geographic ancestry te facial reconstruction.

Gastwirth J., (Ed.), *Statistical Science ter the Courtroom*, Springer Verlag, 2000.

### Spatial Statistics

Diggle P., *The Statistical Analysis of Spatial Point Patterns*, Academic Press, 1983.

Ripley B., *Spatial Statistics*, Wiley, 1981.

### What Is the Black-Sholes Proefje?

Clewlow L., and C. Strickland, *Implementing Derivatives Models*, John Wiley & Sons, 1998.

### What Is a Classification Tree

There are several methods of determining when to zekering. The simplest method is to split the gegevens into two samples. A tree is developed with one sample and tested with another. The mis-classification rate is calculated by fitting the tree to the test gegevens set and enhancing the number of branches one at a time . Spil the number of knots used switches the mis-classification rate switches. The number of knots which minimize the mis-classification rate is chosen.

Graphical Devices for High-Dimensional Classification : Statistical algorithmic classification methods include mechanisms such spil trees, forests, and neural nets. Such methods tend to share two common traits. They can often have far greater predictive power than the classical model-based methods. And they are frequently so ingewikkeld spil to make interpretation very difficult, often resulting ter a “black opbergruimte” appearance. An alternative treatment is using graphical device to facilitate investigation of the internal workings of such classifiers. The A generalization of the ideas such spil the gegevens photo, and the color histogram permits simultaneous examination of dozens to hundreds of variables across similar numbers of observations. Extra information can be visually incorporated spil to true class, predicted class, and casewise variable importance. Careful choice of orderings across cases and variables can clearly indicate clusters, irrelevant or redundant variables, and other features of the classifier, leading to substantial improvements ter classifier interpretability.

The various programs vary te how they operate. For making splits, most programs use definition of purity. More sophisticated methods of finding the stopping rule have bot developed and depend on the software package.

### What Is a Regression Tree

The Tree-based models known also spil recursive partitioning have bot used te both statistics and machine learning. Most of their applications to date have, however, bot te the fields of regression, classification, and density estimation.

S-PLUS statistical package includes some nice features such spil non-parametric regression and tree-based models.

Breiman L., J. Friedman, R. Olshen and C. Stone, *Classification and Regression Trees*, CRC Press, Inc., Boca Raton, Florida, 1984.

### Cluster Analysis for Correlated Variables

- characterize a specific group of rente,
- compare two or more specific groups,
- detect a pattern among several variables.

Cluster analysis is used to classify observations with respect to a set of variables. The widely used Ward’s method is predisposed to find spherical clusters and may perform badly with very ellipsoidal clusters generated by very correlated variables (within clusters).

To overeenkomst with high correlations, some model-based methods are implemented te the S-Plus package. However, a limitation of their treatment is the need to assume the clusters have a multivariate normal distribution, spil well spil the need to determine ter advance what the likely covariance structure of the clusters is.

Another option is to combine the principal component analysis with cluster analysis.

Baxter M., *Exploratory Multivariate Analysis ter Archaeology*, pp. 167-170, Edinburgh University Press, Edinburgh, 1994.

Manly F., *Multivariate Statistical Methods: A Primer*, Chapman and Hall, London, 1986.

### Capture-Recapture Methods

### Tchebysheff Inequality and Its Improvements

P [|X – m | ³, k s ] £ 1/k Two , for any k >, 1

The symmetric property of Tchebysheff’s inequality is useful, e.g., te constructing control thresholds is the quality control process. However the boundaries are very conservative because of lack of skill about the underlying distribution. This bounds can be improved (i.e., becomes tighter) if wij have some skill about the population distribution. For example, if the population is homogeneous, that is its distribution is unimodal, then,

P [|X – m | ³, k s ] £ 1/(Two.25k Two ), for any k >, 1

The above inequality is known spil the Camp-Meidell inequality.

Efron B., and R. Tibshirani, *An Introduction to the Bootstrap*, Chapman & Hall (now the CRC Press), 1994. Contains a test for multimodality that is based on the Gaussian kernel density estimates and then test for multimodality by using the window size treatment.

Grant E., and R. Leavenworth, *Statistical Quality Control*, McGraw-Hill, 1996.

Ryan T., *Statistical Methods for Quality Improvement*, John Wiley & Sons, 2000. A very good book for a starter.

### Frechet Bounds for Dependent Random Variables

Frechet Bounds is often used te stochastic processes with the effect of dependencies, such spil estimating an upper and/or a lower corded on the queue length te a queuing system with two different but known marginal inter-arrivals times distributions of two types of customers.

### Statistical Gegevens Analysis ter Criminal Justice

McKean J., and Bryan Byers, *Gegevens Analysis for Criminal Justice and Criminology*, Allyn & Bacon, 2000.

Walker J., *Statistics te Criminal Justice: Analysis and Interpretation*, Aspen Publishers, Inc., 1999.

### What is Slim Numerical Computation?

### Software Engineering by Project Management

The software project scheduling and tracking is to create a network of software engineering tasks that will enable you to get the job done on time. Once the network is created, you have to assign responsibility for each task, make sure it gets done, and adapt the network spil risks become reality.

Ricketts I., *Managing Your Software Project: A Student’s Guide*, London, Springer, 1998.