Transforming response measures to remove interactions or other sources of variance

psychologically correct response measure in which the interaction has been removed. As a third example, consider a theory in wh ich drive level (0) and incentive level (I) are assumed to combine additively to yield an intervening variable, E, which RICHARD S. BOGARTZ and lOHN H. To sec the role ofthe response seale in determin- determines response speed according to the WA CK WITZ. University of Illinois. ing the presence or absence of an interaction, law, S = E 2 . If a factorial design is used to assume that the functions in Fig. 1 are .5a n- 1 and man i pul ate 0 and I orthogonally, Urbana. IIl. 61801 .5b n -I. where n is the trial number. Treating trial additivity of drive and incentive will be A general polynomial transformation is number and drive levels as the two factors of a concealed by the nonlinear relation of inter- speed to E. Thus Table I gives an additive given for the dependent variable in 2 by 2 factorial design, we can see that the effect comparison is then (.5a8 - 1 experimental design models and multiple action 1 1 8 24 24 .5b - ) - (.5a - - .5b - 1 ), and will be relation for the cell values of2 0 + I =E, but linear or multljile polynomial regression nonzero if a =F b. However, ifinstead oferror prob- the speed scores, equal to E , will be those models such that selected sources of ability, Ihe measure is log log [2{error probabil- shown in Table 2. The apparent interaction variance such as interactions or ity) ]. the interaction comparison becomes in the table of speed scores would be configuralities are reduced or eliminated. A [log log 2{.5a 7 ) - log log 2{.5b 7 )] - removed and the underlying additivity stopping rule is given for addition of terms pog log [2{.5a 23 )] -log log [2{.5b 23 )),. which would be revealed by a square root in the polynomial based on proportion of is zero for all a and b. The intuitive notion of a transformation of the speed scores. Note that in both the second-and third systematic (nonerror) variance accounted floor effect as the basis for the interaction is he re made rigorous by demonstrating a response mea- example, rescaling of the dependent for by the selected source. sure transformation that seales out the interaction. variable to eliminate interaction was Since the presence or absence of the interaetion achieved using information that is Statistically significant interactions are depends only on the choice of the measurement ordinarily not available, namely the not a1ways psychologically meaningful. seale. it is psyehologically meaningless in the sense Often they merely reflect the particular that it indicates no psychologieal process. functional relation of the dependent choice of a measurement scale for the When numerical rating scales are used, variable to the underlying variable. response. Sometimes, general familiarity distortions produced by, say, end effects or Clearly, for situations such as these, it with the occurrence of such interactions is number preferences can also introduce would be desirable to have a statistical sufficient to avoid problems. Other tim es, psychologically meaningless in teractions. technique for rescaling the dependent ignorance of or inadequate handling of the Figure 2 shows an example of end effects variable that would remove interactions response scale problem can have in which bigger differences in the even when the functional relation of the ramifications throughout an entire area of psychological variable V are required at the dependent variable to the underlying investigation. Interactions that may weil be scale ends than at the scale middle to variable is not known. Ideally, the at tributable to inadequacies of the produce a given difference in the numerical technique would yield a simple, explicit, response measurement scale are often rating response, R. If an A by B factorial usually monotonie transformation of the routinely taken as revealing psychological design with two levels of A and three levels dependent variable to a new scale on which processes or mechanisms. of B were used to study the relation of the those interactions due only to the choiee Ceiling effects and floor effects are psychological variable V to the two of scale would be zero. familiar instances in which the presence of independent variables A and B, means of We re port here, in preliminary, a significant interaction is discounted. 11, 13, and 15 for the three levels of B at abbreviated form, a new statistieal method Figure 1 shows two error probability the first level of A, and of 13, 1 5, and I 7 for transforming the dependent variable in functions, one for low drive and the other for the three levels of B at the second level an analysis of variance (ANOVA), such for high drive. The bigger difference of A would show an additive relation. The that the variance due to any selected between the curves at Trial 8 than at function shown in Fig. 2 indieates that source or sources is reduced or removed. Trial 24 would produce a statistical such additivity at the level of the The transformation is of the form: interaction. It would be discounted as due psychological variable V would be S* = S + a2S2 + ••• + anS n , whereby S, the to a floor effect rather than attributed to misrepresented as interaction by the original score, is transformed to S*. The some meaningful interaction of drive level numerical response R. On the numerical method selects the aj to minimize the with trials. response scale, the values 11, 13, and 23 Table 1 would be obtained at the first level of A Drive and lncentive Levels and the values 13,23, and 29 at the second 6 level of A. The apparent interaction does not reflect a nonadditive relation of A and >- 5 2 3 4 B to the psychological variable. Rather. it :: 4 I 2 3 4 5 reveals the end effects characterizing the CD LOW DRIVE D 2 <t 3 4 5 6 Ss' use of the numerical rating response. g; 3 3 4 5 6 7 Just as in the first example, where a a: Cl. transformation of the response measure a: 2 o Table 2 scaled ou t the interaction, so too, in this ~ I Speed Scores ([2) second example, such a transformation of w o LLLLLLLLLLLLL LLI:t='='='-'-I....I...,,?-LL the numerical rating response measure is 4 a I2 possible. Since the function f relating R to TRIALS 2 3 4 V is strictly monotonie, the inverse I 1 4 9 function f- I exists such that V = f- (R). 16 25 Fig. I. Graphical representation of an D 2 9 16 25 36 This constitutes a rescaling of the 3 16 interaction due to a floor effect. 25 36 49 dependent variable to a new Transforming response measures to remove interactions or other sources of variance l ... Psychon. Sei., 1970, Vol. 19 (2) 87 selected source( s) and provides a stopping rule for how many aj terms are needed. The method is appropriate to all instances of the General Linear Model (Mood & Graybill, 1963), including fixed-effects experimental design models, multiple linear, and multiple polynomial regression models. Our original concern was finding a transformation to addit (...truncated)