== 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. 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]]). 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. ||||||||<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.]] '''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).