Chapter 13 CHI-SQUARE AND NONPARAMETRIC PROCEDURES

Going Forward Your goals in this chapter are to learn: When to use nonparametric statistics The logic and use of the one-way chi square The logic and use of the two-way chi square The names of the nonparametric procedures with ordinal scores

Parametric Versus Nonparametric Statistics

Nonparametric statistics are inferential procedures used with either nominal or ordinal data.

Chi Square Procedures

Chi Square The chi square procedure is the nonparametric procedure for testing whether the frequencies in each category in sample data represent specified frequencies in the population The symbol for the chi square statistic is  2

One-Way Chi Square: The Goodness of Fit Test

One-Way Chi Square The one-way chi square test is computed when data consist of the frequencies with which participants belong to the different categories of one variable

Statistical Hypotheses H 0 : all frequencies in the population are equal H a : all frequencies in the population are not equal

Observed Frequency The observed frequency is the frequency with which participants fall into a category It is symbolized by f o The sum of the f o s from all categories equals N

Formula for Expected Frequencies The expected frequency is the frequency we expect in a category if the sample data perfectly represent the distribution of frequencies in the population described by H 0 The symbol is f e

Assumptions of the One-Way Chi Square 1.Participants are categorized along one variable having two or more categories, and we count the frequency in each category 2.Each participant can be in only one category 3.Category membership is independent 4.We include the responses of all participants in the study 5.The f e must be at least 5 per category

Computing One-Way Chi-Square Statistic Where f o are the observed frequencies and f e are the expected frequencies df = k – 1 where k is the number of categories

The  2 Distribution

“Goodness of Fit” Test The one-way chi square procedure is also called the goodness of fit test That is, how “good” is the “fit” between the data and the frequencies we expect if H 0 is true

The Two-Way Chi Square: The Test of Independence

The two-way chi square procedure is used for testing whether category membership on one variable is independent of category membership on the other variable. Two-Way Chi Square

Computing Two-Way Chi Square Statistic Where f o are the observed frequencies and f e are the expected frequencies df = (number of rows – 1)(number of columns – 1)

Two-Way Chi Square A significant two-way chi square indicates the sample data are likely to represent variables that are dependent (correlated) in the population When a 2 x 2 chi square test is significant, we compute the phi coefficient (  ) to describe the strength of the relationship

Nonparametric Statistics

Nonparametric Tests Spearman correlation coefficient is analogous to the Pearson correlation coefficient for ranked data Mann-Whitney test is analogous to the independent samples t-test Wilcoxon test is analogous to the related-samples t-test

Nonparametric Tests Kruskal-Wallis test is analogous to a one-way between-subjects ANOVA Friedman test is analogous to a one-way within-subjects ANOVA

Example A survey is conducted where respondents are asked to indicate (a) their sex and (b) their preference in pets between dogs and cats. The frequency of males and females making each pet selection is given below. Perform a two-way chi square test. MalesFemales Dogs2411 Cats1554

Example The expected values for each cell are: (39)(35)/104 = (65)(39)/104 = (39)(69)/104 = (65)(69)/104 = MalesFemales Dogs Cats

Example