# Chi-square (χ²) Test of Independence

**BRIEF DESCRIPTION**

Whereas the One-sample Chi-square (χ²) goodness-of-fit test compares our sample distribution (observed frequencies) of a single variable with a known pre-defined distribution (expected frequencies) such as the population distribution, normal distribution, or poisson distribution, to test for the significance of deviation, the Chi-square (χ²) Test of Independence compares two categorical variables in a cross-tabulation fashion to determine group differences or degree of association (or non-association i.e. independence). Chi-square (χ²) is a non-parametric procedure.

**SIMILAR STATISTICAL PROCEDURES:**

**Fisher’s Exact Test**(for small sample sizes)**Loglinear models**(a multivariate extension of bivariate Chi-square (χ²))**One-Sample Chi-square (χ²)**(a goodness-of-fit test)

**Dependent variable**: categorical scales (dichotomous and nominal), ordinal, or grouped interval and is normally placed in the*row*of the cross tab.**Independent variable**: categorical scales (dichotomous and nominal), ordinal, or grouped interval and is normally placed in the*column*of the cross tab.

**DATA ASSUMPTIONS**

- The Chi-square (χ²) test makes no assumption about the data distribution as it is non-parametric and thus distribution free (nonparametric tests do not require assumptions about the shape of the underlying distribution).
- Size of expected frequency: When the number of cells is less than 10 and particularly with a small sample, it assumes that the expected value for each cell is five or higher (if not, select the Fishers Exact test). The observed frequencies can be any value, including zero.
- The expected frequencies for each category should be at least 1 and no more than 20% of the categories should have expected frequencies of less than.
- Random sampling from a defined population.
- Independence of observations (cells should be independent): Each observation is generated by a different subject and no subject is counted twice.
- Data: Dichotomous, nominal, ordinal or grouped interval.

**WHERE TO FIND IN SPSS?**

ANALYZE / DESCRIPTIVES / CROSSTABS (STATISTICS and select Chi-square)

**HOW TO REPORT THE FINDINGS?**

If the Chi-square (χ²) Test of Independence is significant (e.g. p‘<‘.05) then it indicates that there is a significant statistical relationship between the two categorical variables.

**Reporting example**: “A chi-square test of independence was performed to examine the relation between sales volume and sales training. The relation between these two variables was significant, χ² (2, N = 150) = 12.40, p‘<‘.05. Sales volume is likely to be higher for salespeople who attended training compared to those who did not”. NOTE: If p‘>‘.05 you don’t report the value (e.g. p‘>‘.05 or p=.129) but only indicate it as NS (non-significant).**Explanation of the above example**: The χ² refers to our Chi-square (χ²) procedure, the 2 in brackets is the degrees of freedom (df), the N=150 is the sample size, the 12.40 is the actual Chi-square (χ²) test value, and the p‘<‘.05 indicates an actual p-value which is less (e.g. .023) than our chosen confidence level of .05.

**FURTHER COMMENTS.**

Reminder to first WEIGHT the cases by the FREQ counts data.

Convert string variables to numeric variables by using the Automatic Recode function.

Convert string variables to numeric variables by using the Automatic Recode function.

It is crucial to review and report the

*effect size*of Chi-square (χ²) tests. See Practical Significance and Effect Size measures for more information on effect size._____________________________________________________________

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