What is Chi-Square? Used to examine differences in the distributions of nominal data A mathematical comparison between expected frequencies and observed frequencies Theoretical, or Expected, Frequencies : developed on the basis of some hypothesis Observed Frequencies : obtained empirically through direct observation

Assumptions for Chi-Square The samples must have been randomly selected. The data must be in nominal form. The groups for each variable must be completely independent of each other; thus, all cell entries are independent of each other.

Chi-Square with a Single Variable χ 2 Goodness-of-Fit Test : the fit is said to be good when the observed frequencies are within random fluctuation of the expected frequencies and the computed χ 2 statistic is small and insignificant

One-Sample Hypotheses Null Hypothesis : There is no significant difference between the observed and expected frequencies. Alternative Hypothesis : There is a significant difference between the observed and expected frequencies.

Chi-Square with Multiple Variables χ 2 Test of Homogeneity : a test to determine if the frequencies of one variable differ as a function of another variable The independent variable(s) in the χ 2 Test of Homogeneity are called the antecedent variable (s); they are the ones which logically precede the others. The Chi-Square Test can accommodate multiple variables, e.g. 2 x 2 3 X 5 2 x 3 x 5

Two-Sample Hypotheses Null Hypothesis: The frequency distribution of variable Y does not differ as a result of group membership in variable X. Non-Directional Alternative Hypothesis: The frequency distribution of variable Y does differ as a result of group membership in variable X.

The Chi-Square Distribution There is a family of χ 2 distributions, each determined by a single degree of freedom value. For a single variable: df = k – 1 For multiple variables: df = (r – 1)(c – 1) Where r = the number of rows c = the number of columns As the degrees of freedom increase, the sampling distribution approaches the normal distribution.

Computing Chi-Square with a Single Variable To enter the data  Create columns for each variable  Each variable will have value labels  The level of measurement for all variables will be nominal Analyze  Nonparametric  Chi-Square Move the variable(s) of interest to the Test Variable List  Click OK

Output for a Single Variable

Computing Chi-Square with More Than One Variable Analyze  Descriptive Statistics  Crosstabs Move the antecedent (independent) variable(s) to the Row(s) box  Move the dependent variable(s) to the Column(s) box Click Statistics  Check Chi-Square  Click Continue  Click OK

Output for a 2 X 2 Chi-Square