Monotonic trend as an example of a qualitative trend

As far as theory or hypothesis testing is concerned, it can be said that most of the theories and hypotheses in psychology refer to qualitative relations and lead to predictions concerning qualitative trends (see the overview by Hager, 1992 [22]). The type of qualitative trend most often encountered is the monotonic trend, which can be predicted when considering psychological hypotheses like 'The higher the degree of imagery the better the retrieval' (Paivio, 1986 [52]) and when considering hypotheses concerning the effectiveness of different cognitive programs or therapies to a, say, comparison group without any intervention. This type of trend will be the focus of interest in the following sections of this article, as it is generally not addressed by text book authors (but see Bortz, 1993, pp. 259-260) [5]. No graphic representations of qualitative trends will be given here, as these graphs can be misleading because the distances between any two levels of the qualitative independent variable cannot be defined. Choosing equal distances leads to graphs which do not differ from graphs of linear trends, while choosing arbitrary distances leads to arbitrary graphic representations. Both of these procedures are neither correct nor incorrect, but they are just arbitrary, and the impression the graphs give depends mainly upon these arbitrary choices. In some empirical papers which I deliberately do not cite here, equal distances have been chosen for the graphical representations leading to straight lines. This led the authors to claim that the trend was 'linear' although the independent variable was qualitative instead of quantitative. Myers and Well (1991, pp. 568-569) [51] state: '... according to the Yerkes-Dodson law, we would expect a quadratic relation between measures of performance and motivation' (italics added). Since motivation usually is considered a qualitative variable the relation to be expected is qualitative (inverted U-shaped or bitonic).

The most important implication of the lack of a functional rule connecting independent and dependent variable is that the relative magnitude of differences or distances , [referred to as in the SP-lin above] between population means can neither be expressed as a function of the values of the independent variable nor can they be predicted from theory. Thus, the only predictions possible refer to the rank order of the parameters chosen. To speak of strict monotonicity means that there is a strictly increasing order of all ranks assigned to the parameters. Expressing this definition in the format of a statistical hypothesis (SH) referring to population means , we get:
 
The minimum number of experimental conditions is J=2 in this case, since a minimum of two ranks have to be assessed or compared. Let's call this minimum number a 'testing instance,' since a psychological hypothesis is testable when this minimum number is accounted for in the experiment. As soon as the experimenter decides to study more than this minimum number, more than one testing instance can be defined, and these testing instances can be linked either conjunctively or disjunctively (whereby the tests referring to one testing instance should always be conjunctively connected to fulfill the criterion of exhaustiveness). As argued above, a conjunctive linkage represents a strict decision rule and leads to more severe tests of the psychological hypothesis, while the disjunctive connection implies (more) lenient decision rules and leads to more lenient or less severe tests.

Based on these considerations, the definition for monotonicity of a trend can be combined with several decision rules. The strictest decision rule demands that all empirical means must follow the predicted order without exception and the differences between them must be statistically significant in order to accept the SH-mon in (8). The latter demand means that if, for example, the adjacent means and , referring to the mean number of words correctly reproduced (, do not differ significantly, they are not considered different, and then should be assigned equal ranks. This demand of statistical significance between any two adjacent means is analoguous to the two-sample situation (J = 2), in which two means are only considered different if the difference is statistically significant; any significant difference, however, implies that different ranks are assigned to the means. Although this demand may not seem cogent, it will be taken here as an additional criterion. Combining the statistical hypothesis in (9) with this strict decision rule results in the following expression SP-mon1:
 

The following examples refer to various more lenient or less strict decision rules which rely on the disjunctive connection and different numbers of testing instances (pairs of means) to be considered. One more lenient decision rule demands an increasing rank order not among all adjacent means (j = j'-1), but for at least two adjacent means. Combining this decision rule with the SH-mon results in the SP-mon2, for which the number of pairs is J-1:
 
The most lenient decision rule allows the consideration of all possible pairs of means to find at least one pair of means which conforms with the hypothesis, leading to the
 
The lenient decision rules just applied do not address the question of how to handle rank inversions, which can occur with those pairs of means which are not in accordance with the predictions. If one or more, but not all pairs are judged to be different according to predictions, the remainder of the pairs can consist of homogeneous means (; equality of ranks) and/or of pairs of means with inverted rank orders (; inversion of ranks). This point will be discussed further below. For the time being it suffices to keep the possibility of rank inversions in mind.

The next question is, which of the hypotheses discussed so far are tested by using some tests proposed in the literature which, by the way, are only rarely addressed in standard textbooks (for an exception, see Bortz, 1993 [5]).