30 SPSS UPDATE 7-9 1.29 Tukey's Test for Nonadditivity In factorial designs with only one observation per cell there is no within-cell error and thus no direct estimate of the experimental error. Freguently, the highest-order interaction is assumed to be part of the experimental error and its mean sguare is used to provide a denominator for F tests on the remaining model terms. One method of checking the tenability of this no- interaction assumption is provided by Tukey's test for nonadditivity (Tukey(1949)). SPSS-MANOVA can perform Tukey's test by using the fact that Tukey's sum of sguares for nonadditivity is the linear » knear component of interaction in the metric of the estimates of the man efleeis (sce Winei(19/1) page 395). Tukey's test reguites (Wo separate tuns: 4 The first run obtains ninin effect parametet estimates using an additive main eflcets model. 2 The second run uses the parameter estimates from the first run as the metric in polynomial contrasts for the factors; the desiga specifies a linear x linear single-degrec-of-freedom interaciron tern wlučli actually provides (he sam o£ sauintev tor Hikey test. To illustrate this procedure consider the data in Table 1.29 taken from Winer(1971), page 474. These data comprise a 3 X 4 factorial with one observation per cell. Table 1.29 4 B 1 2 3 4 l 1 : A 2 3 First, estimates of main effects are computed by using the following MANOVA specifications. ; 'MANOVA Y BY A(1,3) B(1,4)/ H PRINT-PARAMETERS ( NEGSUM ) / š DESIGN: A, B/ ; The PRINT- FARAMETERS(NEGSUM]) results in the printing of the estimate of the last main effect as the negative sum of the previous estimates. The default deviation'contrast must be used to get these estimates. Figure 1.29a displays the estimates. | Figure 1.29a H ] ESTIMATES FOR Y s, CONSTANT ; PARAMETER COEFF. STD. ERR. T-VALUE SIG. OF T LOWER .95 CL UPPER .95 CL . iš 1 11.0000000000 -B49BA 12.94366 »000 8.92054 13.07946 to 4A ; I K Ka E sad PARAMETER COEFF. STD. ERR. T-VALVE SIG. OF T LOWER .95 CL UPPER .95 CL . ij v 2 3 .0000000000 1.20185 2.49615 .047 05919 5..94081 P UE: 3 -2 .0000000000 1.20185 -1.66410 147 —4.94081 .94081 ia 4 -1.0000000000 . a . z dia B i PARAMETER COEFF. STD. ERR. T-VALUE SIG. OF T LOWER .95 CL UPPER .95 CL s o. 4 —6.0000000000 1.47196 —4.07620 007 -9.60174 -2.. 39826 5 -5 .0000000000 1.47196 -3,39683 .015 -8.60174 -1.39826 ; 6 2. 0000000000 1.47196 1.35873 .223 -1.60174 5.60174 7 9,0000000000 . s : ; : ge ; 3 In the second run, orthogonal polynomial contrasts for each factor are reguested. The metric IKI for each factor consists of the parameter estimates for that factor's categories produced by the initial run: ab CONTRAST( A)