Chi-Square. 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
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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 • χ2Goodness-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 χ2 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 • χ2 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 χ2 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 χ2 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
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