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Vast amounts of statistical information are available in today s global and economic environment because of continual improvements in computer technology. To compete successfully globally, managers and decision makers must be able to understand the information and use it effectively. A Bayesian and a classical statistician analyzing the same data will generally reach the same conclusion. Jurjevich R. Statistical data analysis provides hands on experience to promote the use of statistical thinking and techniques to apply in order to make educated decisions in the business world.

The statistical software package, SPSS, which is used in this course, offers extensive data-handling capabilities and numerous statistical analysis routines that can analyze small to very large data statistics. Computers play a very important role in statistical data analysis. The computer will assist in the summarization of data, but statistical data analysis focuses on the interpretation of the output to make inferences and predictions.

Collecting the data 3. Defining the problem 2. Analyzing the data 4. Defining the Problem. Reporting the results. An exact definition of the problem is imperative in order to obtain accurate data about it. It is extremely difficult to gather data without a clear definition of the problem. Collecting the Data. We live and work at a time when data collection and statistical computations have become easy almost to the point of triviality. Studying a problem through the use of statistical data analysis usually involves four basic steps.

Paradoxically, the design of data collection, never sufficiently emphasized in the statistical data analysis textbook, have been weakened by an apparent belief that extensive computation can make up for any deficiencies in the design of data collection. One must start with an emphasis on the importance of defining the population about which we are seeking to make inferences, all the requirements of sampling and experimental design must be met.

Two important aspects of a statistical study are Population - a set of all the elements of interest in a study Sample - a subset of the population Statistical inference is refer to extending your knowledge obtain from a random sample from a population to the whole population. Designing ways to collect data is an important job in statistical data analysis.

This is known in mathematics as an Inductive Reasoning. That is, knowledge of whole from a particular. Its main application is in hypotheses testing about a given population. The purpose of statistical inference is to obtain information about a population form information contained in a sample. It is just not feasible to test the entire population, so a sample is the only realistic way to obtain data because of the time and cost constraints.

Data can be either quantitative or qualitative. Qualitative data are labels or names used to iq option usuarios an attribute of each element. For the purpose of statistical data analysis, distinguishing between cross-sectional and time series data is important. Cross-sectional data re data collected at the same or approximately the same point in time. Time series data are data collected over several time periods. Data can be collected from existing sources or obtained through observation and experimental studies designed to obtain new data.

In an experimental study, the variable of iq option usuarios is identified. Then one or more factors in the study are controlled so that data can be obtained about how the factors influence the variables. Quantitative data are always numeric and indicate either how much or how many. A survey is perhaps the most common type of observational study. Analyzing the Data. Statistical data analysis divides the methods for analyzing data into two categories exploratory methods and confirmatory methods.

Exploratory methods are used to discover what the data seems to be saying by using simple arithmetic and easy-to-draw pictures to summarize data. In observational studies, no attempt is made to control or influence the variables of interest. Confirmatory methods use ideas from probability theory in the attempt to answer specific questions. Probability is important in decision making because it provides a mechanism for measuring, expressing, and analyzing the uncertainties associated with future events.

The majority of the topics addressed in this course fall under this heading. Through inferences, an estimate or test claims about the characteristics of a population can be obtained from a sample. The results may be reported in the form of a table, a graph or a set of percentages. Because only a small collection sample has been examined and not an entire population, the reported results must reflect the uncertainty through the use of probability statements and intervals of values.

To conclude, a critical aspect of managing any organization is planning 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 happen in the future. Statistical data analysis helps managers forecast and predict future aspects of a business operation.

However, converting that feeling into a number that can be used effectively is difficult. The most successful managers and decision makers are the ones who can understand the information and use it effectively. Data Processing Coding, Typing, and Editing. Coding the data are transferred, if necessary to coded sheets. Typing the data are typed and stored by at least two independent data entry persons.

For example, when the Current Population Survey and other monthly surveys were taken using paper questionnaires, the U. Census Bureau used double key data entry. Editing the data are checked by comparing the two independent typed data. The standard practice for key-entering data from paper questionnaires is to key in all the data twice.

Ideally, the second time should be done by a different key entry operator whose job specifically includes verifying mismatches between the original and second 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 copyingInversion e. 45 is typed as 123. 54Repetition when a number is repeatedDeliberate error.

Type of Data and Levels of Measurement. Qualitative data, such as eye color of a group of individuals, is not computable by arithmetic relations. They are labels that advise in which category or class an individual, object, or process fall. They are called categorical variables. Quantitative data sets consist of measures that take numerical values for which descriptions such as means and standard deviations are meaningful.

They can be put into an order and further divided into two groups discrete data or continuous data. Discrete data are countable data, for example, the number of defective items produced during a day s production. For example, measuring the height of a person. The first activity in statistics is to measure or count.

Continuous data, when the parameters variables are measurable, are expressed on a continuous scale. Measurement counting theory is concerned with the connection between data and reality. A set of data is a representation i.a model of the reality based on a numerical and mensurable scales. Data are called primary type data if the analyst has been involved in collecting the data relevant to his her investigation. Otherwise, it is called secondary type data. Data come in the forms of Nominal, Ordinal, Interval and Ratio remember the French word NOIR for color black.

Data can be either continuous or discrete. While the unit of measurement is arbitrary in Ratio scale, its zero point is a natural attribute. Both zero and unit of measurements are arbitrary in the Interval scale. The categorical variable is measured on an ordinal or nominal scale. Measurement theory is concerned with the connection between data and reality. Both statistical theory and measurement theory are necessary to make inferences about reality.

Since statisticians live for precision, they prefer 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 proper meaning and the proper correction for them is a very difficult problem It gives biased regression coefficients that need shrinkage, i.the coefficients for remaining variables are too large. It has severe problems in the presence of collinearity. It is based on methods e. F-tests for nested models that were intended to be used to test pre-specified hypotheses. Increasing the sample size does not help very much.

Note also that the all-possible-subsets approach does not remove any of the above problems. Further Reading Derksen, S. Keselman, Backward, forward and stepwise automated subset selection algorithms, British Journal of Mathematical and Statistical Psychology45, 265-282, 1992. An Alternative Approach for Estimating a Regression Line. Further Readings Cornish-Bowden A.Analysis of Enzyme Kinetic DataOxford Univ Press, 1995.A History of Mathematical Statistics From 1750 to 1930Wiley, New York, 1998.

Among others, the author points out that in 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 between these methods was to compare results of estimates and residuals.

Exploring the fuzzy data picture sometimes requires a wide-angle lens to view its totality. At other times it requires a closeup lens to focus on fine detail. Multivariate Data Analysis. The graphically based tools that we use provide this flexibility. Most chemical systems are complex because they involve many variables and there are many interactions among the variables. Therefore, chemometric techniques rely upon multivariate statistical and mathematical tools to uncover interactions and reduce the dimensionality of the data.

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 techniques are used across the whole range of fields of statistical application in medicine, physical and biological sciences, economics and social science, and of course in many industrial and commercial applications.

Principal component analysis used for exploring data to reduce the dimension. Generally, PCA seeks to represent n correlated random variables by a reduced set of uncorrelated variables, which are obtained by transformation of the original set onto an appropriate subspace. The uncorrelated variables are chosen to be good linear combination of the original variables, in terms of explaining maximal variance, orthogonal directions in the data.

Two closely related techniques, principal component analysis and factor analysis, are used to reduce the dimensionality of multivariate data. In these techniques correlations and interactions among the variables are summarized in terms of a small number of underlying factors. The methods rapidly identify key variables or groups of variables that control the system under study.

The resulting dimension reduction also permits graphical representation of the data so that significant relationships among observations or samples can be identified. Other techniques include Multidimensional Scaling, Cluster Analysis, and Correspondence Analysis.Statistical Strategies for small Sample Research, Thousand Oaks, CA, Sage, 1999. Further Readings Chatfield C. Collins, Introduction to Multivariate AnalysisChapman and Hall, 1980.

Krzanowski W.Principles of Multivariate Analysis A User s PerspectiveClarendon Press, 1988. Bibby, Multivariate AnalysisAcademic Press, 1979. The Meaning and Interpretation of P-values what the data say. P-value Interpretation P 0. 01 very strong evidence against H0 0. 05 moderate evidence against H0 0. 10 suggestive evidence against H0 0. 10 P little or no real evidence against H0.

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 decided in advance, the distribution of p is uniform assuming the null hypothesis. We would express this as P p x x. That means the criterion of p 0. 05 achieves a of 0. When a p-value is associated with a set of data, it is a measure of the probability that the data could have arisen as a random sample from some population described by the statistical testing model.

The smaller the p-value, the more evidence you have. A p-value is a measure of how much evidence you have against the null hypothesis. One may combine the p-value with the significance level to make decision on a given test of hypothesis. In 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.

In a statistical hypothesis test, the P value is the probability of observing a test statistic at least as extreme as the value actually observed, assuming that the null hypothesis is true. The value of p is defined with respect to a distribution. Therefore, we could call it model-distributional hypothesis rather than the null hypothesis. In short, it simply means that if the null had been true, the p value is the probability against the null in that case.

The p-value is determined by the observed value, however, this makes it difficult to even state the inverse of p. Further Readings Arsham H.Kuiper s P-value as a Measuring Tool and Decision Procedure for the Goodness-of-fit Test, Journal of Applied StatisticsVol. 3, 131-135, 1988. Accuracy, Precision, Robustness, and Quality. The robustness of a procedure 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 Box s original version, and this includes Bayesian considerations, loss as well as prior. The central limit theorem CLT and the Gauss-Markov Theorem qualify as robustness theorems, but the Huber-Hempel definition does not qualify as a robustness theorem. We must always distinguish between bias robustness and efficiency robustness. One needs to be more specific about what the procedure must be protected against. If the sample mean is sometimes seen as a robust estimator, it is because the CLT guarantees a 0 bias for large samples regardless of the underlying distribution.

This estimator is bias robust, but it is clearly not efficiency robust as 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 couple of other folks is bias robust only if the underlying distribution is symmetric.

In the context 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 iq option usuarios the sampling process no assumption needed about the model. Unbiased design-based estimators are usually referred to as robust estimators because the unbiasedness is true for all possible distributions. It seems clear however, that these estimators can still be of poor quality as the variance that can be unduly large.

However, others people will use the word in other imprecise ways. 2, Advanced Theory of Statistics, also cites Box, 1953; and he makes a less useful statement about assumptions. In addition, Kendall states in one place that robustness means merely that the test size, aremains constant under different conditions. Kendall s Vol. This is what people are using, apparently, when they claim that two-tailed t-tests are robust even when variances and sample sizes are unequal.

I find it easier to use the phrase, There is a robust differencewhich 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 constant as covariates. Influence Function and Its Applications. It is main potential application of the influence function is in comparison of methods of estimation for ranking the robustness.

A commonsense form of influence function is the robust procedures when the extreme values are dropped, trimming. There are a few fundamental statistical tests such as test for randomness, test for homogeneity of population, test for detecting outliner sand then test for normality. For all these necessary tests there are powerful procedures in statistical data 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 over 30.

The concept of influence is the study of the impact on the conclusions and inferences on various fields of studies including statistical data analysis. This is possible by a perturbation analysis. For example, the influence function of an estimate is the change in the estimate when an infinitesimal change in a single observation divided by the amount of the change. It acts as the sensitivity analysis of the estimate.

The influence function has been extended to the what-if analysis, robustness, and scenarios analysis, such as adding or deleting an observation, outliners s impact, and so on. For example, for a given distribution both normal or otherwise, for which population parameters have been 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 as the 90 or 10 data. While in estimating the mean on can invoke the central limit theorem for any sample of size over, say 30.

However, we cannot be sure that the calculated variance is the true variance of the population and therefore greater uncertainty creeps in and one need to sue the influence function as a measuring tool an decision procedure. Further Readings Melnikov Y.Influence Functions and MatricesDekker, 1999. What is Imprecise Probability.

What is a Meta-Analysis. a Especially when Effect-sizes are rather small, the hope is that one can gain good power by essentially pretending to have the larger N as a valid, combined sample. b When effect sizes are rather large, then the extra POWER is not needed for main effects of design Instead, it theoretically could be possible to look at contrasts between the slight variations in the studies themselves. For example, to compare two effect sizes r obtained by two separate studies, you may use.

It seems obvious to me that no statistical procedure can be robust in all senses. where z 1 and z 2 are Fisher transformations of r, and the two n i s in the denominator represent the sample size for each study. If you iq option usuarios trust that all things being equal will hold up. The typical meta study does not do the tests for homogeneity that should be required. there is a body of research data literature that you would like to summarize.

one gathers together all the admissible examples of this literature note some might be discarded for various reasons. most important would be the effect that has or has not been found, much larger in sd units is the treatment group s performance compared to one or more controls. call the values in each of the investigations in 3. mini effect sizes. across all admissible data sets, you attempt to summarize the overall effect size by forming a set of individual effects.

and using an overall sd as the divisor. I, personally, do not like to call the tests robust when the two versions of the t-test, which are approximately equally robust, may have 90 different results when you compare which samples fall into the rejection interval or region. certain details of each investigation are deciphered. thus yielding essentially an average effect size.

in the meta analysis literature. sometimes these effect sizes are further labeled as small, medium, or large. across different factors and variables. but, in a nutshell, this is what is done. I recall a case in physics, in which, after a phenomenon had been observed in air, emulsion data were examined. The theory would have about a 9 effect in emulsion, and behold, the published data gave 15.

As it happens, there was no significant difference practical, not statistical in the theory, and also no error in the data. It was just that the results of experiments in which nothing statistically significant was found were not reported. You can look at effect sizes in many different ways. 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 important ones, and than if there is no significance, the effect was not important. We really need to differentiate between the term statistically significantand the usual word significant. Meta-analysis is a controversial type of literature review in which the results of individual randomized controlled studies are pooled together to try to get an estimate of the effect of the intervention being studied.

It s not easy to do well and there are many inherent problems. Further Readings Lipsey M. It increases statistical power and is used to resolve the problem of reports which disagree with each other. Wilson, Practical Meta-AnalysisSage Publications, 2000. What Is the Effect Size. Therefore, the ES is the mean difference between the control group and the treatment group. Effect size permits the comparative effect of different treatments to be compared, even when based on different samples and different measuring instruments.

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 instrument reliability coefficient. ES is commonly used in meta-analysis and power analysis. Further Readings Cooper H. Hedges, The Handbook of Research SynthesisNY, Russell Sage, 1994. What is the Benford s Law. What About the Zipf s Law.

This can be observed, for instance, by examining tables of Logarithms and noting that the first pages are much more worn and smudged than later pages. Bias Reduction Techniques. This implies that a number in a table of physical constants is more likely to begin with a smaller digit than a larger digit. According to legend, Baron Munchausen saved himself from drowning in quicksand by pulling himself up using only his bootstraps.

The statistical bootstrap, which uses resampling from a given set of data to mimic the variability that produced the data in the first place, has a rather more dependable theoretical basis and can be a highly effective procedure for estimation of error quantities in statistical problems. Bootstrap is to create a virtual population by duplicating the same sample over and over, and then re-samples from the virtual population to form a reference set.

Very often, a certain structure is assumed so that a residual is computed for each case. The purpose is often to estimate a P-level. Then you compare your original sample with the reference set to get the exact p-value. What is then re-sampled is from the set of residuals, which are then added to those assumed structures, before some statistic is evaluated. Jackknife is to re-compute the data by leaving on observation out each time.

Leave-one-out replication gives you the same Case-estimates, I think, as the proper 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 reduced bias. Bias reduction techniques have wide applications in anthropology, chemistry, climatology, clinical trials, cybernetics, and ecology. Further Readings Efron B.The Jackknife, The Bootstrap and Other Resampling PlansSIAM, Philadelphia, 1982.

Tu, The Jackknife and BootstrapSpringer Verlag, 1995. Tibshirani, An Introduction to the BootstrapChapman Hall now the CRC Press1994. Number of Class Interval in Histogram. k the smallest integer greater than or equal to 1 Log n Log 2 1 3. Area Under Standard Normal Curve. To have an optimum you need some measure of quality - presumably in this case, the best way to display whatever information is available in the data.

The sample size contributes to this, so the usual guidelines are to use between 5 and 15 classes, one need more classes if you one has a very large sample. You take into account a preference for tidy class widths, preferably a multiple of 5 or 10, because this makes it easier to appreciate the scale. This assumes you have a computer and can generate alternative histograms fairly readily.

There are often management issues that come into it as well. For example, if your data is to be compared to similar data - such as prior studies, or from other countries - you are restricted to the intervals used therein. Beyond this it becomes a matter of judgement - try out a range of class widths and choose the one that works best. Use narrow classes where the class frequencies are high, wide classes where they are low. The following approaches are common.

If the histogram is very skewed, then unequal classes should be considered. 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. Alternatively. Find the range highest value - lowest value. Divide the range by a reasonable interval size 2, 3, 5, 10 or a multiple of 10. Aim for no fewer than 5 intervals and no more than 15. Structural Equation Modeling. A structural equation model may apply to one group of cases or to multiple groups of cases.

When multiple 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 latent variables may also be included in the model. As an application, how do you test the equality of regression slopes coming from the same sample using 3 different measuring methods. You could use a structural modeling approach. 1 - Standardize all three data sets prior to the analysis because b weights are also a function of the variance of the predictor variable and with standardization, you remove this source.

2 - Model the dependent variable as the effect from all three measures and obtain the path coefficient b weight for each one. 3 - Then fit a model in which the three path coefficients are constrained to be equal. If a significant decrement in fit occurs, the paths are not equal. Further Readings Schumacker R. Lomax, A Beginner s Guide to Structural Equation ModelingLawrence Erlbaum, New Jersey, 1996.

Econometrics and Time Series Models. Further Readings Ericsson N. Irons, Testing Exogeneity, Oxford University Press, 1994. Newbold, Forecasting in Business and Economics, Academic Press, 1989.Time Series Models, Causality and Exogeneity, Edward Elgar Pub. Tri-linear Coordinates Triangle. The same holds for the composition of the opinions in a population. When percents for, against and undecided sum to 100, the same technique for presentation can be used. True equilateral may not be preserved in transmission.

let the initial composition of opinions be given by 1. That is, few undecided, roughly equally as much for as against. Let another composition be given by point 2. This point represents a higher percentage undecided and, among the decided, 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 constant.

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 bound estimate of reliability. Congeneric measures This least restrictive model within the framework of classical test theory requires only that true scores on measures said to be measuring the same phenomenon be perfectly 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 we have the following data By examining the data, I think one cannot do better than looking at the paired t-test and Pearson correlations between 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 as an overall summary, and it is precisely what some editors do want to see presented 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 as planning the Power for a proposed study, it does matter whether the raters to be used will be exactly the same individuals.

A good methodology to apply in such cases, is the Bland Altman analysis. SPSS Commands SAS Commands. When to Use Nonparametric Technique. The data entering the analysis are enumerative - that is, count data representing the number of observations in each category or cross-category. The data are measured and or analyzed using a nominal scale of measurement. The data are measured and or analyzed using an ordinal scale of measurement. The inference does not concern a parameter in the population distribution - as, for example, the hypothesis that a time-ordered set of observations exhibits a random pattern.

The probability distribution of the statistic upon which the the analysis is based is iq option usuarios dependent upon specific information or assumptions about the population s which the sample s are drawn, but only on general assumptions, such as a continuous and iq option usuarios 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, as in types 1 through 3; the type of inference, as in type 4 or the generality of the assumptions made about the population distribution, as in type 5.

For example one may use the Mann-Whitney Rank Test as a nonparametric alternative to Students T-test when one does not have normally distributed data. Mann-Whitney To be used with two independent groups analogous to the independent groups t-test Wilcoxon To be used with two related i.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 Data. - Analysis of complete cases, including weighting adjustments, - Imputation methods, and extensions to multiple imputation, and - Methods that analyze the incomplete data directly without requiring a rectangular data set, such as maximum likelihood and Bayesian methods. Each missing datum is replaced by m 1 simulated values, producing m simulated versions of the complete data. Each version is analyzed by standard complete-data methods, and the results are combined using simple rules to produce inferential statements that incorporate missing data uncertainty.

The focus is on the practice of MI for real statistical problems in modern computing environments. Multiple imputation MI is a general paradigm for the analysis of incomplete data. Further Readings Rubin D.Multiple Imputation for Nonresponse in SurveysNew York, Wiley, 1987.Analysis of Incomplete Multivariate DataLondon, Chapman and Hall, 1997.

Rubin, Statistical Analysis with Missing DataNew York, Wiley, 1987. Interactions in 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 rightthe various products for the numerous interactions that can be generated tend to be highly correlated with each other, and with the original predictors. See the diagram below, which should be viewed with a non-proportional letter.

This is sometimes called the problem of multicollinearityalthough it would more accurately be described as spurious multicollinearity. It is possible, and often to be recommended, to adjust the raw products so as to make them orthogonal to the original variables and to lower-order interaction terms as 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 in the predictor variable. There is a lesson here for the planning of experiments. For example, if you were estimating the linear association between 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 10,20,30,40,50,60,70,80,90,100 all else being equal.

To increase the precision of estimators, increase the range of the input. Another cause of large SE s is a small number of event observations or a small number of non-event observations analogous to small variance in the outcome variable. This is not strictly 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 in that case the stochastic disturbance term will often reflect variables, not included explicitly in the model, that may change slowly as time passes by.

In a linear model representing the variation in a dependent variable Y as a linear function of several explanatory variables, interaction between two explanatory variables X and W can be represented by their product that is, by the variable created by multiplying them together. 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 model; when X and W are quasi- continuous variables, this equation describes a multiple linear regression MLR model. In ANOV contexts, the existence of an interaction can be described as a difference between differences the difference in means between two levels of X at one value of W is not the same as the difference in the corresponding means at another value of W, and this not-the-same-ness constitutes the interaction between X and W; it is quantified by the value of b3.

In MLR contexts, an interaction implies a change in the slope of the regression of Y on X from one value of W to another value of W or, equivalently, a change in the slope of the regression of Y on W for different values of X in a two-predictor regression with interaction, the response surface is not a plane but a twisted surface like a bent cookie tinin Darlington s 1990 phrase. To resolve the problem of multi-collinearity. Variance of Nonlinear Random Functions. The change of slope is quantified by the value of b3.

Algebraically such a model is represented by. Var X E Y 2 Var Y E X 2 E Y 4 - 2 Cov X, Y E X E Y 3. E Y 2 Var X E X 2 Var Y 2 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 in the context of small sample size, allowing us to see statistics in 1, or 2-dimensional space. The Mean and The Median.

Let s suppose that they decide to minimize the absolute amount of driving. If they met at 1 st Street, the amount of driving would be 0 2 6 14 22 blocks. If they met at 3 rd Street, the amount of driving would be 2 0 4 12 18 blocks. Finally, at 15 th Street, 14 12 8 0 34 blocks. So the two houses that would minimize the amount of driving would be 3 rd or 7 th Street. Actually, if they wanted a neutral site, any place on 4 th5 thor 6 th Street would also work. If they met at 7 th Street, 6 4 0 8 18 blocks.

Note that any value between 3 and 7 could be defined as the median of 1, 3, 7, and 15. Now, the person at 15 th is upset at always having to do more driving. So the median is the value that minimizes the absolute distance to the data points. So the group agrees to consider a different rule. In deciding to minimize the square of the distance driving, we are using the least square principle. By squaring, we 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 3 rd Street house 4 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 2 3 - x 2 7 - x 2 15 - x 2.

The value that minimizes the sum of squared values is 6. 5, which is also equal to the arithmetic mean of 1, 3, 7, and 15. With calculus, it s easy to show that this holds in general. Consider a small sample of scores with an even number of cases; for example, 1, 2, 4, 7, 10, and 12. The median is 5. 5, the midpoint of the interval between the scores of 4 and 7. As we discussed above, it is true that the median is a point around which the sum of absolute deviations is minimized. In this example the sum of absolute deviations is 22.

However, it is not a unique point. Any point in the 4 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 quite correct. 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 data lie strictly above it; 100 of the data are greater than or equal to the median.

We will make use of this idea in regression analysis. In 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 data 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 data points x and y, on the above real number- line axis, then the Geometric Mean g of these numbers is a point g such that ox og og oywhere ox means the length of line segment ox, for example. Notice that the vector V1 length is. The variance of V1 is. Var V1 S X i 2 n V1 2 n 4. The standard deviation is. Variance, Covariance, and Correlation Coefficient. Now, consider a second observation 2, 4. Similarly, it can be represented by vector V2 -1, 1.

Cov V1, V2 the dot product n 2 -1 -2 1 2 -4 2 -2. OS1 V1 n Ѕ 8 Ѕ 2 Ѕ 2. The covariance is. 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 between the two vectors. The correlation coefficient is therefore.

Cov V1, V2 OS1 OS2 Cos V1, V2 2 1 Cos 180 -2. 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, as expected from the above figure. The distance between the two-point data sets V1, and V2 is also a dot-product.

Now, construct a matrix whose columns are the coordinates of the two vectors V1 and V2, respectively. V1-V2 V1 2 V2 2 - 2 V1 V2 n Var V1 VarV2 - 2Cov V1, V2. Multiplying the transpose of this matrix by itself provides a new symmetric matrix containing n times the variance of V1 and variance of V2 as its main diagonal elements i.8, 2and n times Cov V1, V2 as its off diagonal element i.

You might like to use a graph paper, and a scientific calculator to check the results of these numerical examples and to perform some additional numerical experimentation for a deeper understanding of the concepts. Further Reading Wickens T.The Geometry of Multivariate StatisticsErlbaum Pub. Suppose you have two positive data points x and y, then the geometric mean of these 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.

What Is a Geometric Mean. The geometric means are used extensively by the U. Bureau of Labor Statistics Geomeans as they call them in the computation of the U. Consumer Price Index. The geomeans are also used in price indexes. The statistical use of geometric mean is for index numbers such as the Fisher s ideal index. If some values are very large in magnitude and others are small, then the geometric mean is a better average.

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