Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Parametric & Nonparametric Models for Within-Groups Comparisons• overview• X2 tests• parametric & nonparametric stats• Mann-Whitney U-test• Kruskal-Wallis test• Median testStatistics 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 Wil’s t- / F-test Wil’s Fried’skwg Cochran’s F-test Fried’sM-W -- Mann-Whitney U-Test Wil’s -- Wilcoxin’s TestK-W -- Kruskal-Wallis Test Fried’s -- Friedman’s F-test Mdn -- Median Test McNem -- McNemar’s X2Statistical Tests for BG Designs w/ qualitative variablesPearson’s X² Can be 2x2 or kxk – depending upon the number of categories of the qualitative outcome variable• H0: Populations represented by the design conditions have thesame distribution across conditions/categories of the outcome variable• degrees of freedom df = (#colums - 1) * (#rows - 1)• Range of values 0 to • Reject Ho: If ²obtained > ²critical (of – ef)2 X2 = ef ΣRow Column total total= N22 54 7646 32 78*efRow 1 Row 2 68 86 154 Col 1 Col 2The expected frequency for each cell is computed assuming that the H0: is true – that there is no relationship between the row and column variables.If so, the frequency of each cell can be computed from the frequency of the associated rows & columns. (76*68)/154 (76*86)/154 76(78*68)/154 (78*86)/154 78Row 1 Row 2 68 86 154 Col 1 Col 2Usually the column variable is the grouping variable and the row variable is the DV.(of – ef)2 X2 = ef Σdf = (2-1) * (2-1) = 1X2 1,.05 = 3.84X2 1, .01 = 6.63p = .0002 using online p-value calculatorSo, we would reject H0: and conclude that the two groups have different distributions of responses of the qualitative DV.Parametric tests for BG Designs using ND/Int variablest-tests• H0: Populations represented by the IV conditions have the same mean DV.• degrees of freedom df = N - 2• Range of values -  to • Reject Ho: If | tobtained | > tcritical• Assumptions• data are measured on an interval scale• DV values from both groups come from ND with equal STDANOVA• H0: Populations represented by the IV conditions have the same mean DV.• degrees of freedom df numerator = k-1, denominator = N - k• Range of values 0 to • Reject Ho: If Fobtained > Fcritical• Assumptions• data are measured on an interval scale• DV values from both groups come from ND with equal STDNonparametric tests for BG Designs using ~ND/~Int variablesThe nonparametric BG models we will examine, and the parametric BG models with which they are most similar…2-BG ComparisonsMann-Whitney U test between groups t-test2- or k-BG ComparisonsKruskal-Wallis test between groups ANOVAMedian test between groups ANOVAAs with parametric tests, the k-group nonparametric tests can be used with 2 or k-groups.Let’s start with a review of applying a between groups t-testHere are the data from such a design : Qual variable is whether or not subject has a 2-5 year oldQuant variable is “liking rating of Barney” (1-10 scale) No Toddlertoddler 1+ Toddlerss1 2 s3 6s2 4 s5 8s4 6 s6 9s8 7 s7 10M = 4.75 M = 8.25The BG t-test would be used to compare these group means.When we perform this t-test …As you know, the H0: is that the two groups have the same mean on the quantitative DV, but we also …1. Assume that the quantitative variable is measured on a interval scale -- that the difference between the ratings of “2” and “4” mean the same thing as the difference between the ratings of “8” and “6”.2. Assume that the quant variable is normally distributed.3. Assume that the two samples have the same variability (homogeneity of variance assumption)Given these assumptions, we can use a t-test tp assess the H0: M1 = M2Nonparametric tests for BG Designs using ~ND/~Int variablesIf we want to “avoid” these first two assumptions, we can apply the Mann-Whitney U-testThe test does not depend upon the interval properties of the data, only their ordinal properties -- and so we will convert the values to ranks• lower scores have lower ranks, and vice versa• e.g. #1 values 10 11 13 14 16 ranks 1 2 3 4 5• Tied values given the “average rank” of all scores with that value• e.g. #2 values 10 12 12 13 16 ranks 1 2.5 2.5 4 5• e.g., #3 values 9 12 13 13 13 ranks 1 2 4 4 4Preparing these data for analysis as ranks... No Toddlestoddler 1+ Toddlers rating ranks rating ranks s1 2 1 s3 6 3.5s2 4 2 s5 8 6s4 6 3.5 s6 9 7s8 7 5 s7 10 8  = 11.5  = 24.5The “U” statistic is computed from the summed ranks. U=0 when the summed ranks for the two groups are the same (H0:) All the values are ranked at once -- ignoring which condition each “S” was in.Notice the group with the higher values has the higher summed ranksThere are two different “versions” of the H0: for the Mann-Whitney U-test, depending upon which text you read. The “older” version reads:H0: The samples represent populations with the same distributions of scores.Under this H0:, we might find a significant U because the samples from the two populations differ in terms of their:• centers (medians - with rank data)• variability or spread• shape or skewnessThis is a very “general” H0: and rejecting it provides little info.Also, this H0: is not strongly parallel to that of the t-test (which is specifically about mean differences)Over time, “another” H0: has emerged, and is more commonly seen in textbooks today:H0: The two samples represent populations with the same median (assuming these populations have distributions with identical variability and shape).You can see that this H0:• increases the specificity of the H0: by making assumptions (That’s how it works -