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| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
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| So there is a linear trend providing a reasonable fit. | So there is a linear trend providing a reasonable fit. |
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| 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. | 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. |
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
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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. 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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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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Reference:
Everitt, BS and Wykes T.(1999) A Dictionary for Psychologists. Arnold:London. (See page 31).