Cross tabulation and the Pearson Chi-Square Test

Suppose that you have noticed that a lot of psychology majors are women with many fewer men. It could be that there are just more women enrolled in the university, and so you'd expect more women psych majors than men. Or, it could be that there is something about the psychology major that attracts women (or repels men?).

Both major and gender are categorical variables. Crosstabulation is a statistical technique used to display a breakdown of the data by these two variables (that is, it is a table that has displays the frequency of different majors broken down by gender).

The Pearson chi-square test essentially tells us whether the results of a crosstab are statistically significant. That is, are the two categorical variables independent (unrelated) of one another. So basically, the chi square test is a correlation test for categorical variables.

So for our example, the chi-square test will tell us whether there are more female psychology majors than you would expect by chance (based on total number of males and females and total number of people in different majors).

• When we have categorical variables
  • Do the percentages match up with how we thought they would?
  • Are two (or more) categorical variables independent?
• Can do it with continuous variables if you convert them into categories (we won't cover this), but you typically don't want to do this because you lose a lot of information, and these tests are not as "powerful" as parametric tests

Crosstabulation

Example

Suppose that you are interested in whether there is a relationship between gender and educational level (undergraduate vs. graduate students) at ISU (year 2002). That is, are men and women equally likely to pursue a graduate education relative to an undergraduate education.

Setup our data in a "cross tabulation" of our two variables. The data are observed frequencies (f_o).

The next step in the crosstabulation procedure is to compute the marginals for the rows and columns. This simply means add the frequencies across the rows and down the columns.

So what can we tell from this table?

Overall there are more graduate students than undergraduates, and there are more female than male students at ISU.

However this doesn't answer our question about whether women are more or less likely (i.e. that there is a relationship) to pursue graduate school than men. To find this out we need to do an inferential test, the Chi-square.

Hypothesis Testing with Chi-squared

• We test the null hypothesis that nothing interesting is happening (i.e., there is no relationship) versus alternative hypothesis that findings are interesting (i.e., there is a relationship).
  
• The null hypothesis can only be rejected if there is a .05 or lower probability that our findings are due to chance
  Hypothesis tests determine the extent to which our findings may be due to chance
• A chi-square will be significant if the residuals (the differences between observed frequencies and expected frequencies) for one level of a variable differ as a function of another variable.
• The chi-square value does not tell us the nature of the differences

Example

A manufacturer of watches takes a sample of 200 people. Each person is classified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?

Setup our data in a "cross tabulation" of our two variables. The data are observed frequencies (f_o).

Step 1: State the hypotheses and select an alpha level

H₀: Preference is independent of age
H_a: Preference is related to age
We'll set a = 0.05

• Compute your degrees of freedom
  df = (#Columns - 1) * (#Rows - 1)
• Go to Chi-square statistic table and find the critical value
  For this example, with df = 2, and a = 0.05 the critical chi-squared value is 5.99

Part 1: Obtain row and column totals, also called the marginals (in blue).

		Watch preference
		digital	analog	undecided
Age	under 30	90	40	10	140
	over 30	10	40	10	60
		100	80	20

Part 2: Compute the expected frequencies



For people under 30

• prefering digital watches: f_e = (100*140)/200 = 70
• prefering analog watches: f_e = (80*140)/200 = 56
• undecided watches: f_e = (20*140)/200 = 14

For people over 30

• prefering digital watches: f_e = (100*60)/200 = 30
• prefering analog watches: f_e = (80*60)/200 = 24
• undecided watches: f_e = (20*140)/60 = 6

So let's enter the predicted (expected) values (in green) into our crosstabulation.

Part 3: Compute the Chi-squared statistic

• Find the residuals (f_o - f_e) for each cell
• Square these differences
• Divide the squared differences by fe
• Sum the results

  

  So then add them up

  

• here we reject the H₀ and conclude that there is a relationship between age and watch preference

Computing Crosstabs and Chi-squared in SPSS

Choose Analyze, Descriptive Statistics, Crosstabs
Select your categorical variables put one in Row and the other in Column Click on the Statistics button and then check the chi-square option.

Expected Counts Expected counts are based on marginal percentages Multiply the marginal percentages together to get the expected percentage for that cell, then multiply by N to get expected counts Or, have SPSS compute them -- Choose Cells, Expected Counts Residuals Difference between expected and observed counts Choose Cells, Unstandardized Residuals Standardized Residuals are distributed as z-scores (they were divided by the standard deviation of the residuals)
Output: Here is some sample output looking at a crosstab of final grade and review session attendance from the students.sav file. Crosstab shows frequencies of one variable for each level of the other Count refers to the observed frequencies (from the data) expected counts are the expected frequencies
Output shows Pearson chi-square and "Asymp. Sig." (significance level) for the crosstab above. If "Asymp. Sig." is less than .05 then the residuals differ as a function of the independent variable So here the chi square is not significant (sig is greater than a = 0.05), so we would fail to reject the H0. This means that we are not rejecting the hypothesis that final grade and review session attendance are independent (in other words, there is not a relationship between the two variables).
Clustered bar charts are the most common way to present data from these crosstabulations (or as tables). You can get SPSS to plot your tables by clicking the Display Clustered Bar Charts box on the main cross tabs window.

Assumptions of the Chi-Square

Categories are independent (no overlap)
Must have an expected count of at least 5 in each cell
Remember that large samples mean large chi-squares, thus making it easier to find a significant chi-square (this is called power)

For the following two questions download the file students.sav.

4) Were juniors and seniors more likely than freshmen and sophomores to attend the review sessions? Provide a bar chart showing the breakdown. Assuming an a = 0.05, test whether these variables are independent. Remember to state your hypotheses.

5) Were men more likely than women to do an extra credit assignment? Report the number of people who did and didn't do the extra credit project broken down by gender. Assuming an a = 0.05, test whether gender and extra credit participation are independent. Remember to state your hypotheses.