Intro to Statistics for the Behavioral Sciences PSYC 1900 Lecture 17: Nonparametric Tests & Course Summary

Chi-Square Effect Size Most common is Cramer’s Phi. Most common is Cramer’s Phi. Cramer’s squared gives an index of the amount of variance explained (similar to eta sqaured): Cramer’s squared gives an index of the amount of variance explained (similar to eta sqaured): Note that # groups in analyses with more than one classification variable refers to the smallest number of groups for one of the variables. Note that # groups in analyses with more than one classification variable refers to the smallest number of groups for one of the variables.

Definitions Nonparametric and distribution-free tests make fewer assumptions than others. Nonparametric and distribution-free tests make fewer assumptions than others. Do not assume population parameters Do not assume population parameters Do not assume normality Do not assume normality Still usually assume comparable distributions across groups Still usually assume comparable distributions across groups Less sensitive to outliers. More sensitive to medians. Less sensitive to outliers. More sensitive to medians. Rank-randomization tests Rank-randomization tests Class of nonparametrics tests based on the theoretical distribution of randomly assigned ranks Class of nonparametrics tests based on the theoretical distribution of randomly assigned ranks Usually less power than parametric versions. Usually less power than parametric versions.

Mann-Whitney Analogous to t test for two independent groups Analogous to t test for two independent groups Works with ranks rather than raw scores Works with ranks rather than raw scores Rank without regard to group, and compare sums of ranks in each group Rank without regard to group, and compare sums of ranks in each group If H 0 : true, sums of ranks should be approximately equal If H 0 : true, sums of ranks should be approximately equal

Example: Behavior Patterns and Cholesterol Selvin (1991) reports cholesterol of 20 men who are Type A and 20 Type B. Selvin (1991) reports cholesterol of 20 men who are Type A and 20 Type B. High cholesterol risk factor for heart disease High cholesterol risk factor for heart disease Question: Is there a relation between cholesterol and personality type? Question: Is there a relation between cholesterol and personality type?

Raw Data

Problems and Solution Data have several outliers Data have several outliers Not normally distributed Not normally distributed Convert to ranks (assigning tied ranks to tied scores) Convert to ranks (assigning tied ranks to tied scores) Rank without regard to group membership Rank without regard to group membership Sum ranks in each group. Sum ranks in each group.

Ranked Data W S = smaller sum (groups or abs. value) = 317 W S = smaller sum (groups or abs. value) = 317 Critical value with n 1 = n 2 = 20 is 328 Critical value with n 1 = n 2 = 20 is 328 Reject is Ws is LESS than critical value Reject is Ws is LESS than critical value

Conclusions Since 317 < 328, reject H 0 and conclude that the two groups do not have the same average cholesterol level. Since 317 < 328, reject H 0 and conclude that the two groups do not have the same average cholesterol level. Type A personality people have significantly higher cholesterol levels. Type A personality people have significantly higher cholesterol levels. Note that if is highest ranks are congregated in smaller group, use: Note that if is highest ranks are congregated in smaller group, use:

z Approximation We could use z approximation for large n j We could use z approximation for large n j p (z > 2.52) =.0059 p (z > 2.52) =.0059 p (z > +2.52) =.012: Reject H 0 p (z > +2.52) =.012: Reject H 0

Wilcoxon’s Matched-Pairs Signed-Ranks Test Analogous to matched-sample t Analogous to matched-sample t Each subject observed twice Each subject observed twice Compute difference scores Compute difference scores Rank difference scores Rank difference scores

Wilcoxon--cont. Compute sum of ranks of + and - difference scores separately Compute sum of ranks of + and - difference scores separately If difference is 0, ignore and reduce n If difference is 0, ignore and reduce n If H 0 true, sum of + and - ranks approx. equal If H 0 true, sum of + and - ranks approx. equal If tied differences, use tied ranks If tied differences, use tied ranks

Example: Stress and Beta-endorphins Hoaglin et al. (1985) report data on 19 patients collected at 12 hours and again at 10 min. before surgery. Hoaglin et al. (1985) report data on 19 patients collected at 12 hours and again at 10 min. before surgery. Dependent variable = beta-endorphin level. Dependent variable = beta-endorphin level. Beta-endorphins are body’s pain killers. Beta-endorphins are body’s pain killers. Data have several outliers. Data have several outliers.

Wilcoxon--cont. Sum positive ranks = 22 Sum positive ranks = 22 Sum negative ranks = 131 Sum negative ranks = 131 T = smaller sum = 22 T = smaller sum = 22 Critical value for n = 19 is between 46 and 47 for  =.05 Critical value for n = 19 is between 46 and 47 for  =.05 Since 22 < 46, reject null hypothesis Since 22 < 46, reject null hypothesis Beta--endorphins rise before surgery Beta--endorphins rise before surgery

Choice of Analysis Step 1 Step 1 What is the form of the dependent variable? What is the form of the dependent variable? Categorical Categorical Chi-Square Chi-Square Continuous Continuous Go to Step 2 Go to Step 2

Choice of Analysis Step 2 Step 2 What is the form of the independent variable? What is the form of the independent variable? Categorical with 1 iv Categorical with 1 iv 2 Levels: t-test 2 Levels: t-test 3 or more levels: ANOVA 3 or more levels: ANOVA Categorical with 2 or more iv’s Categorical with 2 or more iv’s Factorial ANOVA Factorial ANOVA Continuous with 1 iv Continuous with 1 iv Correlation or bivariate regression Correlation or bivariate regression Continuous with 2 or more iv’s Continuous with 2 or more iv’s Multiple Regression Multiple Regression