Next: Recommendations Summarized Up: Qualitative Trends And Trend Previous: Testing strategies for monotonic
Some further thoughts on qualitative trends
Analogous procedures can be devised for comparable qualitative
psychological hypotheses. The details need not be specified here
(see Hager, 1992 [22];
Hager & Hasselhorn, 1995 [25]). Nor will details
on testing statistical predictions concerning bitonic or tritonic
trends be presented here. A hypothesis like the YerkesDodson
law addressed above postulates to a bitonic trend. If one chooses
J = 5 experimental conditions (degrees of motivation) the
respective prediction may take the following form, where the
's refer to some measure of performance:
Another consideration refers to power analysis for planned contrasts to test statistical predictions concerning monotonic (or other qualitative) trends. Power analysis enables the determination of the sample size necessary to detect population effect sizes with prechosen error probabilities. In the t test situation, the effect size is the standardized difference among two population means, the values of which have to be selected for each hypothesis on a pair contrast. Some authors (e.g., Bredenkamp, 1984 [8]) argue that specifying these values for each adjacent pair of population means implicitly leads to an upgrading of a monotonic trend to a strictly linear trend. This belief may lead to the recommendation that the monotonic trend may and should be statistically handled as linear trend (see above). Bredenkamp's argument, however, overlooks the fact that predicted effect sizes for a quantitative trend refer to exact values which are functionally dependent on the values of the quantitative independent variable. If at least one of these exact values is (substantially) larger than predicted the strict definition of linearity is violated. On the other hand, when dealing with qualitative trends effect sizes such as are minimum values, which cannot be predicted, but are chosen according to methodological or economic reasons (see above). If one or more population values are larger than prespecified, this would not disagree with the prediction of a particular qualitative trend as long as the other values are still large enough. Referring to the samples, the empirical effects for pairs of means must be large enough to reach statistical significance which, in turn, allows assignment of different ranks to the means. Thus, the choice of some minimum values for should not be interpreted as upgrading a monotonic trend, especially as this interpretation would violate the criterion of exhaustiveness: The statistical hypothesis then actually tested comprises more information than can be derived from the original psychological hypothesis referring to a qualitative variable and trend (see above). Considering J  1 or J(J1)/2 pair contrasts will always lead to contrasts which, considered as a whole, are not orthogonal to one another. Although Hays (1988, p. 415) [28] demands that planned contrasts or comparisons have to be orthogonal, many other textbook authors do not share this opinion as 'after all, contrasts are tested because they are of psychological import, not because they are independent of each other. ... in many and perhaps most cases the contrasts of interest will not be orthogonal' (Myers, 1972, p. 362 [50]; see also Thompson, 1994 [62]; Winer et al., 1991 [68]). In addition, pair contrasts do not contain the complete information inherent in the sums of squares between in an analysis of variance on the same means. But as long as the only (statistical) information needed for examining a psychological hypothesis in a valid manner consists in knowing whether the means are in the predicted order or not, there is no need for further information. But if there is any interest in further information not (directly) related to the examination of the psychological hypothesis each additional test may be performed which is thought to deliver insightful information. But these additional tests should be separated from those which directly refer to the psychological hypothesis of interest.
Next: Recommendations Summarized Up: Qualitative Trends And Trend Previous: Testing strategies for monotonic Methods of Psychological Research 1996, Vol.1, No.4 © 1997 Pabst Science Publishers 
