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| == 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 ||||||||<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 || 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. ||||||||<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. 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.] '''Reference:''' Everitt, BS and Wykes T.(1999) A Dictionary for Psychologists. Arnold:London. (See page 31). |
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I am bare impressed with the artlice I have just read. I wish the writer of technology.ohmygoh.com can continue to provide so much practical information and unforgettable experience to technology.ohmygoh.com readers. There is not much to state except the following universal truth: The people who are actually going to be interesting aren't going to give you meaningful answers to a/s/l anyway. I will be back.