FAQ/ChiTrend - CBU statistics Wiki

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location: FAQ / ChiTrend

## 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 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 ( 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 Linear 7.96 1 0.005 Lack of Fit 0.29 1 0.130 Total 8.25 2 0.004 (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 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 Expected proportion correct 0.30 0.55 0.80 (Fitting a linear trend)

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).