Linear trend test on proportions
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
|
Time 1 |
Time 2 |
Time 3 |
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Correct |
3 |
6 |
10 |
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Incorrect |
9 |
6 |
2 |
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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 ( also called the Mantel-Haenszel statistic).
Linear-by-linear association = $$r^text{2} (N-1)$$
where r is the Pearson correlation of the rows (correct/incorrect) with the columns (group) and N is the total sample size. Bruce Weaver has shown that provided all expected cell counts are greater than 1 the Linear-by-Linear association is the most powerful preferred chi-square for 2x2 tables (see here).
In particular for a 2x2 table Bruce shows that the linear-by-linear chi-square has the special form equal to N(ad-bc)^2 / (mnrs)
where: * N is the total number of observations * a, b, c, and d are the observed counts in the 4 cells * ^2 means "squared"
* m, n, r, s are the 4 marginal totals
For a 2x2 table (only) the regular Pearson chi-square (e.g., in the output from statistical software), can be converted to the 'N - 1' chi-square as follows:
'N -1' chi-square = Linear-by-Linear chi-square = Pearson chi-square x (N -1) / N
The lack of fit is the difference between the Pearson chi-square value and the linear-by-linear one.
Model |
Chi-square |
Df |
p-value |
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Linear |
7.96 |
1 |
0.005 |
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Lack of Fit |
0.29 |
1 |
0.130 |
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Total |
8.25 |
2 |
0.004 |
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(Pearson Chi-square) |
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|
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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 proportions correct give regression estimates of the above linear trend. The Pearson chi-square lack of fit term is (O-E)*(O-E)/E where O are observed table frequencies and E are expected frequencies using the expected proportions from the linear regression.
Observed proportion correct |
0.33 |
0.50 |
0.83 |
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Expected proportion correct |
0.30 |
0.55 |
0.80 |
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(Fitting a linear trend) |
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You can also compare linear trends of proportions in SPSS LOGISTIC.
References:
Agresti, A (2013) Categorical Data Analysis. Third Edition. Wiley:New York. Pages 86-87 mention the above testing for linear trend.
Everitt, BS and Wykes T.(1999) A Dictionary for Psychologists. Arnold:London. (See page 31).