FAQ/ChiTrend352013-08-28 10:47:50PeterWatson342013-08-28 10:47:17PeterWatson332013-08-28 10:42:20PeterWatson322013-04-12 14:15:55PeterWatson312013-03-08 10:17:15localhostconverted to 1.6 markup302012-07-10 14:55:44PeterWatson292012-07-10 09:38:14PeterWatson282012-07-10 09:37:16PeterWatsonRevert to revision 26.272012-07-09 13:55:53ip-50-63-144-190.ip.secureserver.netGwE3la , [url=http://jzcbymbrdouf.com/]jzcbymbrdouf[/url], [link=http://nqjqbweyftln.com/]nqjqbweyftln[/link], http://rupcxzutoyis.com/262012-07-09 13:27:30PeterWatsonRevert to revision 21.252012-07-08 22:24:11hmx-blitz-08v.sccnj04.client.logicworks.netWWQHxV <a href="http://sbmbenjpdeic.com/">sbmbenjpdeic</a>242012-07-07 19:22:15216.77.162.166asOWPv , [url=http://vqwlqjqjuqyb.com/]vqwlqjqjuqyb[/url], [link=http://jfpvzvcgnoxm.com/]jfpvzvcgnoxm[/link], http://mnutgeaaemzj.com/232012-07-07 13:43:17static-89-237-138-12.mobily.com.saxDoVti <a href="http://gpddyghvqiwj.com/">gpddyghvqiwj</a>222012-07-06 18:50:09ip-90-153-161-203.static.pipenetworks.comI 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.ohmy212009-06-16 11:04:54PeterWatson202009-06-16 11:00:58PeterWatson192008-02-12 15:09:24PeterWatson182008-02-12 15:09:04PeterWatson172008-02-12 15:07:17PeterWatson162007-03-08 11:51:12PeterWatson152007-01-25 12:32:58PeterWatson142006-08-17 14:24:59PeterWatson132006-08-17 14:24:13PeterWatson122006-08-17 14:23:26PeterWatson112006-08-17 14:19:46PeterWatson102006-08-17 14:18:26PeterWatson92006-08-17 14:18:06PeterWatson82006-08-17 14:17:01PeterWatson72006-08-17 14:16:23PeterWatson62006-08-17 14:13:01PeterWatson52006-08-17 14:08:18PeterWatson42006-08-17 14:04:18PeterWatson32006-08-17 09:32:17PeterWatson22006-08-17 09:30:53PeterWatson12006-08-17 09:26:06PeterWatsonLinear trend test on proportionsA 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: 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).