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| == Linear trend test on proportions == |
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| 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. | 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. The lack of fit is the difference between the Pearson chi-square value and the linear-by-linear one. |
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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 gives 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. |
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| ||||||||<25% style="VERTICAL-ALIGN: top"> Observed proportion correct ||<25% style="VERTICAL-ALIGN: top"> 0.33 ||<25% style="VERTICAL-ALIGN: top"> 0.50 ||<25% style="VERTICAL-ALIGN: top"> 0.83 || ||||||||<25% style="VERTICAL-ALIGN: top"> Expected proportion correct ||<25% style="VERTICAL-ALIGN: top"> 0.30 ||<25% style="VERTICAL-ALIGN: top"> 0.55 ||<25% style="VERTICAL-ALIGN: top"> 0.80 || (Fitting a linear trend) ||<25% style="VERTICAL-ALIGN: top"> ||<25% style="VERTICAL-ALIGN: top"> ||<25% style="VERTICAL-ALIGN: top"> || |
||||||||<70% style="VERTICAL-ALIGN: top"> Observed proportion correct ||<10% style="VERTICAL-ALIGN: top"> 0.33 ||<10% style="VERTICAL-ALIGN: top"> 0.50 ||<10% style="VERTICAL-ALIGN: top"> 0.83 || ||||||||<70% style="VERTICAL-ALIGN: top"> Expected proportion correct ||<10% style="VERTICAL-ALIGN: top"> 0.30 ||<10% style="VERTICAL-ALIGN: top"> 0.55 ||<10% style="VERTICAL-ALIGN: top"> 0.80 || ||||||||<70% style="VERTICAL-ALIGN: top"> (Fitting a linear trend) ||<10% style="VERTICAL-ALIGN: top"> ||<10% style="VERTICAL-ALIGN: top"> ||<10% style="VERTICAL-ALIGN: top"> || |
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| You can also compare linear trends of proportions in [[FAQ/poly| 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). |
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 ( 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.
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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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).