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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 by 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, i.data 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.

Iq option zahlt nicht aus 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 iq option zahlt nicht aus the denominator represent the sample size for each study. If you really 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, i.how 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 iq option zahlt nicht aus 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 iq option zahlt nicht aus 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 not 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 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, 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 iq option zahlt nicht aus.

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. In a Geometric series, the most meaningful average is the geometric mean. The arithmetic mean is very biased toward the larger numbers in the series.

As an example, suppose sales of a certain item increase to 110 in the first year and to 150 of that in the second year. For simplicity, assume you sold 100 items initially. Then the number sold in the iq option zahlt nicht aus year is 110 and the number sold in the second is 150 x 110 165. The arithmetic average of 110 and 150 is 130 so that we would incorrectly estimate that the number sold in the first year is 130 and the number in the second year is 169.

The geometric mean of 110 and 150 is r 1. 65 1 2 so that we would correctly estimate that we would sell 100 r 2 165 items in the second year. As 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 throughout both years, you have lost money at the end. For every dollar you started with, you have now got 75c.

Thus, the performance is different from gaining 50 -50 2 0. It is the same as changing by a multiplicative factor of 1. In 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. 866 each year. As money tends to multiplicatively it takes money to make moneyfinancial data are often better combined in this way. Ask each respondent to give any numerical value they feel to any crime in the list e.

As a survey analysis example, give a sample of people a list of, say 10, crimes ranging in seriousness. someone might decide to call arson 100. Then ask them to rate each crime in the list on a ratio scale. If a respondent thought rape was five times as bad as arson, then a value of 500 would be assigned, theft a quarter as bad, 25. Suppose we now wanted the average rating across respondents given to each crime. Since respondents are using their own base value, the arithmetic mean would be useless people who used large numbers as their base value would swamp those who had chosen small 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 in a class exercise and it works nicely. It is often good to log-transform such data before regression, ANOVA, etc. These statistical techniques give inferences about the arithmetic mean which is intimately connected with the least-squares error measure ; however, the arithmetic mean of log-transformed data is the log of the geometric mean of the data.

So, for instance, a t test on log-transformed data is really a test for location of the geometric mean. Further Reading Langley R.Practical Statistics Simply Explained1970, 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 deviationthen 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.

In applications of the central limit theorem to practical problems in statistical inference, however, statisticians are more interested in how closely the approximate distribution of the sample mean follows a normal distribution for finite sample sizes, than the limiting distribution itself. Sufficiently close agreement with a normal distribution allows statisticians to use normal theory for making inferences about population parameters such as 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 as a standard normal variable for any positive integer 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. As 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 must study the behavior of the mean of samples of different sizes drawn from a variety of parent populations. Examining sampling distributions of sample means computed from samples of different sizes drawn from a variety of distributions, allow us to gain some insight into the behavior of the sample mean under those specific conditions as well as examine the validity of the guidelines mentioned above for using the central limit theorem in practice.

The sample size needed for the approximation to be adequate depends strongly on the shape of the parent distribution. Symmetry or lack thereof is particularly important. Under certain conditions, in large samples, the sampling distribution of the sample mean can be approximated by a normal distribution. For a symmetric parent distribution, even if very different from the shape of a normal distribution, an adequate approximation can be obtained with small samples e.10 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. binomial with samples sizes far exceeding the typical guidelines say, 30 are needed for an adequate approximation. For some distributions without first and second moments e.Cauchythe central limit theorem does not hold. What is a Sampling Distribution. We will study 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 important consideration for those planning and interpreting sampling results, is the degree to which sample estimates, such as the sample mean, will agree with the corresponding population characteristic. In practice, only one sample is usually taken in some cases a small pilot sample is used to test the data-gathering mechanisms and to get preliminary information for planning 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 happen if 10, or 50, or 100 separate sampling studies, of the same type, were conducted. How consistent would the results be across these different studies. If we could see that the results from each of the samples would be nearly the same and nearly correct.

In some extreme cases e.then we would have confidence in the single sample that will actually be used. On the other hand, seeing that answers from the repeated samples were too variable for the needed accuracy would suggest that a different sampling plan 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 plan.

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 we are trying to estimate, but that the properties of sampling distributions allow us to quantify, probabilistically, how they will differ.

Understand that different statistics have different sampling distributions with distribution shape depending on a the specific statistic, b the sample size, and c the parent distribution. Understand the relationship between sample size and the distribution of sample estimates. Understand that the variability in a sampling distribution can be reduced by increasing the sample size. Outlier Removal. Robust statistical techniques are needed to cope with any undetected outliers; otherwise the result will be misleading.

Note that in large samples, many sampling distributions can be approximated with a normal distribution. Because of the potentially large variance, outliers could be the outcome of sampling. It s perfectly correct to have such an observation that legitimately belongs to the study group by definition. For example, the usual stepwise regression is often used for the selection of an appropriate subset of explanatory variables to use in model; however, it could be invalidated even by the presence of a few outliers.

Lognormally distributed data such as international exchange ratefor instance, will frequently exhibit such values. Therefore, you must be very careful and cautious before declaring an observation an outlier, find out why and how such observation occurred. It could even be an error at the data entering stage. First, construct the BoxPlot of your data. Form the Q1, Q2, and Q3 points which divide the samples into four equally sized groups.

Q2 median Let IQR Q3 - Q1. Outliers are defined as those points outside the values Q3 k IQR and Q1-k IQR. For most case one sets k 1. Another alternative is the following algorithm. a Compute s of whole sample. b Define a set of limits off the mean mean k smean - k s sigma Allow user to enter k. A typical value for k is 2. c Remove all sample values outside the limits. Now, iterate N times through the algorithm, each time replacing the sample set with the reduced samples after applying step c.

Usually we need to iterate through this algorithm 4 times. As mentioned earlier, a common standard is any observation falling beyond 1. 5 interquartile range i. 5 IQRs ranges above the third quartile or below the first quartile. The following SPSS program, helps you in determining the outliers. Outlier detection in the single population setting has been treated in detail in the literature.

Quite often, however, one can argue that the detected outliers are not really outliers, but form a second population. If this is the case, a cluster approach needs to be taken. Further Readings Hawkins D. It will be active areas of research to study the problem of how outliers can arise and be identified, when a cluster approach must be taken.Identification of OutliersChapman Hall, 1980. Rothamsted V. Barnett, and T. Lewis, Outliers in Statistical DataWiley, 1994. Least Squares Models.

Realize that fitting the best line by eye is difficult, especially when there is a lot of residual variability in the data. Know that there is a simple connection between the numerical coefficients in 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 whole story.

A scatter plot is an essential complement to examining the relationship between the two variables. Know that the model checking is an essential part of the process of statistical modelling. After all, conclusions based on models that do not properly describe an observed set of data will be invalid. Know the impact of violation of regression model assumptions i.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 data x1. xn, which contains all information in the sample about the parameter q. More formally, sufficiency is defined in terms of the likelihood function for q. For a sufficient statistic t, the Likelihood L x1. xn q can be written as. Since the second term does not depend on qt 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 starting from the sufficient statistic, which will have exactly the same distribution as the full 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 these successes at random points without replication, the probability that B will now get any given set of successes is exactly the same as the probability that A will see that set, no matter what the true probability of success happens to be.

You Must Look at Your Scattergrams. All three sets have the same correlation and regression line. The important moral is look at your scattergrams. How to produce a numerical example where the two scatterplots show clearly different relationships strengths but yield the same covariance. Produce two sets of X, Y values that have different correlation s; 2. Perform the following steps.

Calculate the two covariances, say C1 and C2; 3. Suppose you want to make C2 equal to C1. Then you want to multiply C2 by C1 C2 ; 4. S yyou want two numbers one of them might be 1a and b such that a. Multiply all values of X in set 2 by a, and all values of Y by b for the new variables, C r. An interesting numerical example showing two identical scatterplots but with differing covariance is the following Consider a data set of X, Y values, with covariance C1. Now let V 2X, and W 3Y.

The covariance of V and W will be 2 3 6 times C1, but the correlation between V and W is the same as the correlation between 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.

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15.01.2020 : 10:00 Zolojinn:
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