Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chapter 11 Chi-Square Tests and Strategies When Population Distributions Are Not Normal

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chi-Square Tests Hypothesis testing procedure for nominal variables –Focus on number of people/items in each category (e.g., hair color, political party, gender) Compare how well an observed distribution fits an expected distribution Expected distribution can be based on –A theory –Prior results –Assumption of equal distribution across categories

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chi-Square Statistic Compares observed frequency distribution to expected frequency distribution –Compute difference between observed and expected and square each one –Weight each by its expected frequency –Sum them

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chi-Square Distribution Compare obtained chi-square to a chi-square distribution Does mismatch between observed and expected frequency exceed what would be expected by chance alone?

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chi-Square Test for Goodness of Fit Single nominal variable Degrees of freedom = number of categories minus 1

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Contingency table –Lists number of observations for each combination of categories –To determine expected frequencies… Chi-Square Test for Independence Two nominal variables –Independence means no relation between variables –To determine degrees of freedom…

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Key Assumption for Chi-square Test No one individual can be counted in more than one cell. In other words, each score must not have any special relation to any other score.

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Effect Size for Chi-Square For 2  2 chi-square, effect size is the phi coefficient –Same as the correlation between two nominal variables Small =.10 Medium =.30 Large =.50 For larger tables…

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Data Transformations Assumption of parametric tests, that populations follow a normal curve –Sometimes violated –Ceiling or floor effects Can transform data (e.g., square root) –Makes distribution more normal –Preserves order of scores but not mean, SD

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Data Transformation Example Reaction time data before a square root transformation…

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Data Transformation Example Reaction time data after a square root transformation. –Distribution is closer to a normal curve –Mean and SD are changed, but order or scores is preserved

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Rank-order Tests Another strategy for non-normal distributions is to covert scores to ranks –Can then use special rank-order or “nonparametric” tests –Each parametric test has a corresponding nonparametric test (e.g., Wilcoxon rank-sum test in place if a t test for independent means) Can also use conventional, parametric tests on ranks without much loss of accuracy