For the following instructions:

* = A single click of the left mouse button
**= A double-click of the left mouse button

Crosstabulation

Crosstabulation is useful to show the relationship between two or more categorical variables. Usually, continuous data is not used for chi-square analyses since a great deal of information is lost by the process of categorization.

Crosstabs Example

Do people with different work statuses (e.g., full-time, retired, etc.) differ in (a) how exciting life is and (b) happiness?
This amounts to tabulating frequencies for life excitement and happiness, but it must be broken down by work status
Crosstabs gives frequencies for one variable separately for each level of another variable


Computing Crosstabs in SPSS

Choose Statistics, Summarize, Crosstabs
Select categorical variables; put one in Row and the other in Column
Output:
Case Processing Summary shows missing values for each table
Crosstab shows frequencies of one variable for each level of the other


Calculating Percentages

Choose Cells, Row Percentages to show percentages across each row
Choose Cells, Column Percentages to show percentages across each column
Choose Cells, Total Percentages to find percentage of respondent that were in each cell


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)


Controlling for a Third Variable

Controlling for a variable means it is held constant
This allows us to look at crosstabs separately for each value of a third variable
Example: wrkstat by life separately for men and women
In SPSS add sex as a layer in Crosstabs


Bar Charts

Simple
Can only show frequencies of one variable
Choose Graphs, Bar, Simple
Cluster and Stacked
Can show frequencies of one variable broken down by another
Percentage information can also be shown


Crosstabs

Compute percentages of happy for different values of wkstat
Compute percentages of wkstat for different values of life; include expected values and residuals
Compute percentages of wkstat for different values of life, layered by gender
Compute bar charts for wkstat by life

Chi-Square (c²)

Chi-Square Lecture

Chi-Square Example

Research questions - Are there gender differences in happiness? How about in how important it is to have a fulfilling job?
What would you hypothesize?
The hypothesis test for whether the pattern of percentages in one variable differs as a function of another is called the chi-square test


Hypothesis Testing

We test the null hypothesis that nothing interesting is happening (versus alternative hypothesis that findings are interesting)
The null hypothesis can only be rejected if there is a .05 probability that our findings are due to chance
Hypothesis tests determine the extent to which our findings may be due to chance


Computing the Pearson Chi-Square test in SPSS

Chi-Square (c²) Tests of Independence: SPSS can compute the expected value for each cell, based on the assumption that the two variables are independent of each other. If there is a large discrepancy between the observed values and the expected values, the c2 statistic would be large, which suggests a significant difference between observed and expected values. In addition, a probability value is also computed.
• *Statistics, *Summarize, *Crosstabs
• * the desired variable in the list to the left, then * the uppermost of the right arrows to indicate that this variable be the row variable.
• * a second variable, and * the middle right arrow (to indicate the column variable).
• For three or more variables: use the lowest box in this window. * on the third variable under section list, and then * the lowest of the three right arrows.
• * OK when complete.
• You can now conduct a chi-square analysis. * Statistics. Here, many different tests of independence or association are listed. * Chi-square, * Phi and Cramer's V, * Continue, * OK
• To conduct a cross tabulation and chi-square analysis on a subset of a certain variable, select the variables for crosstabulation, choose cell values, and the desired statistics. Then, * Data (in the Menu Bar at the top of the screen). * Select Cases, *If condition is satisfied, *If. Select desired variable from list on the left, * right arrow to paste it in the "active" box, type in selected levels to consider. * Continue when completed.
• Output shows Pearson chi-square and "Asymp. Sig." (significance level)
• If "Asymp. Sig." is less than .05 then the residuals differ as a function of the independent variable
  

The chi-square test essentially tells us whether the results of a crosstab are statistically significant
A chi-square will be significant if the residuals 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


The Chi-Square Formula



What are all those symbols?

c² = chi-square
S = Sigma (sum of...)
f_o = frequency observed
f_e = frequency expected
Degrees of freedom are necessary to compute the significance of the chi-square: df = (#rows - 1)(#columns - 1)


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)