Intro to Parametric & Nonparametric StatisticsSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Intro to Parametric & Nonparametric Statistics•Kinds & definitions of nonparametric statistics•Where parametric stats come from•Consequences of parametric assumptions•Organizing the models we will cover in this class•Common arguments for using nonparametric stats•Common arguments against using nonparametric stats•Using ranks instead of values to compute statisticsThere are two kinds of statistics commonly referred to as “nonparametric”...Statistics for quantitative variables w/out making “assumptions about the form of the underlying data distribution”• univariate stats -- median & IQR • univariate stat tests -- 1-sample test of median• bivariate -- analogs of the correlations, t-tests & ANOVAs Statistics for qualitative variables• univariate -- mode & #categories • univaraite stat tests -- goodness-of-fit X²• bivariate -- Pearson’s Contingency Table X²Have to be careful!!  X² tests are actually parametric (they assume an underlying normal distribution – more later)Defining nonparametric statistics ...Nonparametric statistics (also called “distribution free statistics”) are those that can describe some attribute of a population, test hypotheses about that attribute, its relationship with some other attribute, or differences on that attribute across populations , across time or across related constructs, that require no assumptions about the form of the population data distribution(s) nor require interval level measurement.Now, about that last part…… that require no assumptions about the form of the population data distribution(s) nor require interval level measurement. This is where things get a little dicey. Today we get just a taste , but we will examine this very carefully after you know the relevant models …Most of the statistics you know have a fairly simple “computational formula”.As examples...Here are formulas for two familiar parametric statistics: The mean ... M =  X / NThe standard S =  ( X - M ) 2deviation ...  NBut where to these formulas “come from” ???As you’ve heard many times, “computing the mean and standard deviation assumes the data are drawn from a population that is normally distributed.”What does this really mean ???formula for the normal distribution: e - ( x -  )² / 2  ² ƒ(x) = --------------------   2πFor a given mean () and standard deviation (), plug in any value of x to receive the proportional frequency of that value in that particular normal distribution.The computational formula for the mean and std are derived from this formula.First …Since the computational formulas for the mean and the std are derived based upon the assumption that the normal distribution formula describes the data distribution…if the data are not normally distributed …then the formulas for the mean and the std don’t provide a description of the center & spread of the population distribution.Same goes for all the formulae that you know !!Pearson’s corr, Z-tests, t-tests, F-tests, X2 tests, etc…..Second …Since the computational formulas for the mean and the std use +, -, * and /, they assume the data are measured on an interval scale (such that equal differences anywhere along the measured continuum represent the same difference in construct values, e.g., scores of 2 & 6 are equally different than scores of 32 & 36)if the data are not measured on an interval scale …then the formulas for the mean and the std don’t provide a description of the center & spread of the population distribution.Same goes for all the formulae that you know !!Pearson’s corr, Z-tests, t-tests, F-tests, X2 tests, etc…..Normally distributed dataZ scoresLinear trans. of ND Known σ1-sample Z testsLinear trans. of NDKnown σX2 tests ND2df = k-1 or (k-1)(j-1)F testsX2 / X2df = k – 1 & N-k2-sample Z testsLinear trans. of NDKnown σ1-sample t tests Linear trans. of ND Estimated σdf = N-12-sample t testsLinear trans. of ND Estimated σdf = N-1r testsbivNDOrganizing nonparametric statistics ...Nonparametric statistics (also called “distribution free statistics”) are those that can describe some attribute of a population,, test hypotheses about that attribute, its relationship with some other attribute, or differences on that attribute across populations, across time, or across related constructs, that require no assumptions about the form of the population data distribution(s) nor require interval level measurement. describe some attribute of a populationtest hypotheses about that attributeits relationship with some other attributedifferences on that attribute across populationsacross time, or across related constructsunivariate statsunivariate statistical teststests of associationbetween groups comparisonswithin-groups comparisonsStatistics We Will Consider Parametric Nonparametric DV Categorical Interval/ND Ordinal/~NDunivariate stats mode, #cats mean, std median, IQRunivariate tests gof X2 1-grp t-test 1-grp Mdn testassociation X2 Pearson’s r Spearman’s r2 bg X2 t- / F-test M-W K-W Mdnk bg X2 F-test K-W Mdn2wg McNem Crn’s t- / F-test Wil’s Fried’skwg Crn’s F-test Fried’sM-W -- Mann-Whitney U-Test Wil’s -- Wilcoxin’s Test Fried’s -- Friedman’s F-test K-W -- Kruskal-Wallis TestMdn -- Median Test McNem -- McNemar’s X2 Crn’s – Cochran’s TestThings to notice…X2 is used for tests of association between categorical variables & for between groups comparisons with a categorical DVk-condition tests can also be used for 2-condition situationsThese WG-comparisons can only be used with binary DVsCommon reasons/situations FOR using Nonparametric stats• & a caveat to considerData are not normally distributed • r, Z, t, F and related statistics are rather “robust” to many violations of these assumptions Data are not measured on an interval scale.• Most psychological data are measured “somewhere between” ordinal and interval levels of measurement. The good news is that the “regular stats” are pretty robust to this influence, since the rank