Chi-Square

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The Chi-Square is the non-parametric test used to compare frequencies of nominal data.

Assumptions for any Chi-Square test:


 * (1) Observations are independent (the probability of an observation being sampled does not depend on any other observation's inclusion)


 * (2) Categories are mutually exclusive (no observation can be in more than one category)


 * (3) Categories are exhaustive (there must be a category for every possible observation)

Goodness-of-Fit
Used to answer the question: Does the distribution of one nominal variable fit our expectations?

To accomplish this, it compares the observed frequencies to expected frequencies. The expected frequencies can be all the same or they can be determined by theory.

If there is a "good fit" (the results are significant), then there is no significant difference between the observed frequencies and the expected frequencies.

If there is a "poor fit" (the results are not significant), then there is a significant difference between the observed frequencies and the expected frequencies.


 * Chi_square_goodness_of_fit.jpg example, pretend a study was done to determine which factor is most appealing to a consumer when buying a car: Cost, Style, Performance, Reliability, or Other. The null hypothesis is that there is no difference in preference (the same number of people will choose each factor). The alternative hypothesis is that there is a difference in preference (but not which factor is different).

Test of Independence
Used to answer the question: Are two nominal variables related or independent of eachother?

To accomplish this, it compares the frequencies of one group to the frequencies of another group.

If there is independence (the results are significant), then there is no relationship between the variables.

If there is not independence (the results are not significant), then there is a significant relationship between the variables.


 * Chi_square_test_of_independence.jpg example, are gender (male/female) and socio-economic status (low/middle/upper) independent of eachother? The null hypothesis is that there is no relationship between gender and SES (the variables are independent). The alternative hypothesis is that there is a relationship between gender and SES (the variables are related).

Test of Independence in SPSS

 * 1) Click on 'Analyze' -> 'Descriptives' -> 'Crosstabs'
 * 2) Move the first variable into the 'Rows' box
 * 3) Move the other variable into the 'Columns' box
 * 4) Click on 'Statistics'
 * 5) Click 'Chi Square'
 * 6) Click 'Continue'
 * 7) Click 'Continue' again
 * 8) Click 'OK'
 * 9) Your output should appear