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Monotonic trend as an example of a qualitative trendAs 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. 259260) [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 In some empirical papers which I deliberately do not cite here, equal distances have been chosen for the graphical representations leading to straight lines.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. 568569) [51] state: '... according to the YerkesDodson 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 Ushaped 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 SPlin 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:
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 SHmon 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
twosample 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 SPmon1:
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 SHmon results in the SPmon2, for
which the number of pairs is J1:
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]).
Next: Some testing strategies aiming Up: Qualitative Trends And Trend Previous: Qualitative Trends And Trend Methods of Psychological Research 1996, Vol.1, No.4 © 1997 Pabst Science Publishers 
