what is chi-square? used to examine differences in the distributions of nominal data a mathematical...
TRANSCRIPT
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
• χ2 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 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
Employed Not Employed
Male
Female
O
OO
OE
E
E
E
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
Output for a Single VariablePolitical Party
11 10.0 1.0
9 10.0 -1.0
20
Democrat
Republican
Total
Observed N Expected N Residual
Test Statistics
.200
1
.655
Chi-Square a
df
Asymp. Sig.
Political Party
0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 10.0.
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-SquareGender * Political Party Crosstabulation
Count
3 7 10
8 2 10
11 9 20
Male
Female
Gender
Total
Democrat Republican
Political Party
Total
Chi-Square Tests
5.051b 1 .025
3.232 1 .072
5.300 1 .021
.070 .035
4.798 1 .028
20
Pearson Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
Linear-by-LinearAssociation
N of Valid Cases
Value dfAsymp. Sig.
(2-sided)Exact Sig.(2-sided)
Exact Sig.(1-sided)
Computed only for a 2x2 table