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| == Linear trend test on proportions == | |
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| Time1 Time2 Time3 Correct 3 6 10 Incorrect 9 6 2 |
||||||||<25% style="TEXT-ALIGN: center"> ||<25% style="TEXT-ALIGN: center"> '''Time 1''' ||<25% style="TEXT-ALIGN: center"> '''Time 2'''||<25% style="TEXT-ALIGN: center"> '''Time 3''' || ||||||||<25% style="VERTICAL-ALIGN: top"> Correct ||<25% style="VERTICAL-ALIGN: top"> 3 ||<25% style="VERTICAL-ALIGN: top"> 6 ||<25% style="VERTICAL-ALIGN: top"> 10 || ||||||||<25% style="VERTICAL-ALIGN: top"> Incorrect ||<25% style="VERTICAL-ALIGN: top"> 9 ||<25% style="VERTICAL-ALIGN: top"> 6 ||<25% style="VERTICAL-ALIGN: top"> 2 || |
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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). |
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| Chi-square Degrees of Freedom | Linear-by-linear association = $$r^text{2} (N-1)$$ |
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| Linear 7.96 on 1 p=.005 Lack of fit 0.29 on 1 p=.130 |
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 [[https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/notes/chisqr_assumptions | here]]). |
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| Total 8.25 on 2 (Pearson chi-square) | 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) |
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| So there is a linear trend providing a reasonable fit. | 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 If one has the regular Pearson chi-square (e.g., in the output from statistical software), it can be converted to the 'N - 1' chi-square as follows: 'N -1' 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. ||||||||<25% style="TEXT-ALIGN: center"> '''Model''' ||<25% style="TEXT-ALIGN: center"> '''Chi-square''' ||<25% style="TEXT-ALIGN: center"> '''Df'''||<25% style="TEXT-ALIGN: center"> '''p-value''' || ||||||||<25% style="VERTICAL-ALIGN: top"> Linear ||<25% style="VERTICAL-ALIGN: top"> 7.96 ||<25% style="VERTICAL-ALIGN: top"> 1 ||<25% style="VERTICAL-ALIGN: top"> 0.005 || ||||||||<25% style="VERTICAL-ALIGN: top"> Lack of Fit ||<25% style="VERTICAL-ALIGN: top"> 0.29 ||<25% style="VERTICAL-ALIGN: top"> 1 ||<25% style="VERTICAL-ALIGN: top"> 0.130 || ||||||||<25% style="VERTICAL-ALIGN: top"> Total ||<25% style="VERTICAL-ALIGN: top"> 8.25 ||<25% style="VERTICAL-ALIGN: top"> 2 ||<25% style="VERTICAL-ALIGN: top"> 0.004 || ||||||||<25% style="VERTICAL-ALIGN: top"> ||<25% style="VERTICAL-ALIGN: top"> (Pearson Chi-square) ||<25% style="VERTICAL-ALIGN: top"> ||<25% style="VERTICAL-ALIGN: top"> || 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 | 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. ||||||||<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"> || 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
|
Time 1 |
Time 2 |
Time 3 |
|||
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
If one has the regular Pearson chi-square (e.g., in the output from statistical software), it can be converted to the 'N - 1' chi-square as follows:
- 'N -1' 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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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).