PLANNED ORTHOGONAL CONTRASTS

Transcription

1 PLANNED ORTHOGONAL CONTRASTS Please note: This handout is useful background for the workshop, not what s covered in it. Basic principles for contrasts are the same in repeated measures. Planned orthogonal contrasts are a "higher form of life" version of ANOVA. They have something in common with post-hoc comparisons following standard ANOVA, such as the Bonferroni test or the Least Significant Difference test, in that they compare pairs of means. They differ in that they compare only some of all the possible pairs of means. The pairs are chosen BEFORE the analysis is run i.e. they are planned. Each pair is independent of every other pair (i.e. orthogonal) in that the statistical significance of one pair gives no hints about the statistical significance of any other pair. Planned orthogonal contrasts are always found in sets. The number of contrasts is always equal to the number of groups in the ANOVA, minus one. An example is below. PLANNED ORTHOGONAL CONTRASTS: Example Data Set Below are the reading scores of 48 people with defects of their central (foveal) vision. The measure is words read per minute. The first 12 people (Group 1) were trained to read using peripheral vision. The second set of 12 people (Group 2) were trained to read with an A4-sized plastic magnifying sheet. The third set of people (Group 3) were trained to use both techniques simultaneously. The final set of 12 people (Group 4) were a waiting list control group. Reading Person Score Group

2 Reading Person Score Group Table 1: Reading Score Group Mean sd Peripheral Magnifier Combination Control Table 2 below is a standard Analysis of Variance table for Reading Score by group. It is shown to highlight differences between standard ANOVA and planned orthogonal contrasts. Please note that all these tables were produced by the free statistics package, Instat, which generally produces cleaner tables than does SPSS. Table 2: ANOVA TABLE for Reading Score Source df SS MS F Prob. Group Error Total

3 This ANOVA has used the variance in the measures to make all possible comparisons between pairs of means simultaneously. It has compared group 1 with all the other groups, group 2 with all the other groups and so on. It has also compared group 1 with the average of groups 2 & 3, group 2 with the average of groups 1 & 3, and so on. The variance in the measures is therefore used to test a large number of comparisons. A general name for this type of ANOVA is "omnibus ANOVA". If the researcher does not know in advance which groups may be different to which, omnibus ANOVA may be the best choice. Often researchers are particularly interested in some comparisons and less interested in others. In these cases omnibus ANOVA may be inefficient because it "wastes" variance testing comparisons of little or no interest. In our example, the researcher might wish to make three specific comparisons: 1) compare the treatment groups to the control group; 2) compare the effectiveness of the combined treatment to the two individual treatments; 3) compare the relative effectiveness of the two individual treatments. Each one of these comparisons could be tested with its own planned contrast. These contrasts would be independent of each other because the outcome of one would tell us nothing about the outcomes of the others. HOW CONTRASTS WORK The simplest example of a contrast is a t test. In fact, contrasts are glorified versions of t tests. Using our example above, contrast 3 compares the Peripheral Vision group to the Magnification group. If this were the only comparison of interest we would do a t test. The first step would be to set up hypotheses Ho: mu(peripheral) = mu(mag) H1: mu(peripheral) =/= mu(mag) Another way of writing this is Ho: mu(peripheral) - mu(mag) = 0 H1: mu(peripheral) - mu(mag) =/= 0 This second form shows more clearly than the first that each mean has a "weight". The Peripheral group is given a "weight" of +1. The Magnification group is given a "weight" of -1. If the null hypothesis is confirmed then the Peripheral group s population mean, weighted by +1, plus the Magnification group s population mean, weighted by -1, should equal 0. If there were really only two groups in the study, Peripheral and Magnification, then contrast 3 would be the only comparison of interest to the researchers and a t test would complete the analysis. Because there are two other groups, comparison 3 needs to ignore the Combined and Control groups. In contrast terms these two groups are each given a "weight" of 0. To test comparison 3 as a contrast the full null hypothesis is Ho: +1mu(peripheral) + -1mu(mag) + 0mu(combined) + 0mu(control) The weights for comparison 3 are

4 Comparison 2 tests the Combined group compared to the two single treatments. This is done by averaging the means of the two single treatments and comparing them to the Combined group mean. The Control group is not part of this comparison, so has a "weight" of 0. The usual way to take a mean of two numbers is to add them and divide by 2. Arithmetically, this is the same as dividing the first mean by two, the second mean by two, and then adding the results. The weights for comparison (contrast) 2 are then It is usually more convenient to use whole-number weights for contrasts. The same effect is obtained if the weights for contrast 2 are Using this system, the three sets of contrast weights are: Peripheral Magnification Combined Control Contrast Contrast Contrast where contrasts 1, 2 and 3 are 1) compare the treatment groups to the control group; 2) compare the effectiveness of the combined treatment to the two individual treatments; 3) compare the relative effectiveness of the two individual treatments. In SPSS these contrast weights are entered into the Contrast part of the dialogue box. SPSS will run the omnibus ANOVA and print out the contrast results as well. The omnibus ANOVA is irrelevant. The contrasts are the more powerful alternative. They use all the variance to test only the comparisons of interest. When the contrasts are run in Instat the following tables are produced. (Table 3 is identical to Table 2 and is reproduced here for ease of reading.) Table 3: ANOVA TABLE for Reading Score Source df SS MS F Prob. Group Error Total Table 4: Contrasts for factor "Group" Contrast Value se SSQ F Prob > F Contrast Contrast Contrast The omnibus ANOVA is not significant, with p = As it happens, none of the contrasts is significant either, but contrast 2 is close. With a larger sample it may well have been significant.

5 Note 1: The F ratio for each contrast is its sum of squares (SSQ) divided by the mean square for error from Table 3 (385.08). Note 2: Each contrast has degrees of freedom between = 1, so its mean square = its sum of squares (MSE = SSE/df = SSE/1 = SSE). df = 1 because contrasts compare one mean to one other mean. With two means, df between groups is always 1 (2-1 = 1).