A more powerful form of chi-square specifically tests for a linear trend in proportions across groups. For example, proportion remembered correctly using a memory aid.
Example
||<tablewidth="52%"17%> ||<38%>Time 1||<45%>Time 2|| ||<17%>Time 3
23 |
12 |
|
Old |
<38%>13||<45%>11||
- Time1 Time2 Time3
Correct 3 6 10 Incorrect 9 6 2
Does the proportion correct change linearly over time?
The chi-square testing the presence of a linear trend is outputted by SPSS CROSSTABS as the Linear-by-Linear association term. The lack of fit is the difference between the Pearson chi-square value and the linear-by-linear one.
Chi-square Degrees of Freedom
Linear 7.96 on 1 p=.005 Lack of fit 0.29 on 1 p=.130
Total 8.25 on 2 (Pearson chi-square)
So there is a linear trend providing a reasonable fit.
Denoting the time points by –1,0 and 1 and regressing these on the observed