- Understand the standardized residuals from a chi-square test of homogeneity
Even though we have already drawn a conclusion from our hypothesis test, there is still some information we can glean by looking at the difference between the observed count and the expected count for each cell. The data analysis tool calls this difference the residual for that cell (and the idea is similar to the concept of residuals you saw when looking at the differences between observed values and predicted values in the linear regression context).
Residuals are calculated using the formula: [latex]\text{Residual} = \text{Observed} - \text{Expected}[/latex]
Since the values in our cells may vary quite a bit, it’s a good idea to look at what the data analysis tool calls standardized residuals instead.
standardized residuals
These are sometimes referred to as Standardized Pearson residuals.
Standardized residuals are values that standardize the residuals so that if the null hypothesis is assumed to be true, they can be interpreted as normal [latex]z[/latex]-scores.
In particular, most standardized residuals for a given test will fall between [latex]-2[/latex] and [latex]2[/latex]. We can use these standardized residuals to determine how far off our observed count is from what was expected if the null hypothesis is true (i.e., if the distributions are really the same). The sign of the standardized residual tells us whether we observed more cases in that cell than we expected (a positive residual) or fewer cases than we expected (a negative residual).
[Trouble viewing? Click to open in a new tab.]