Introduction How to test a deterministic relation with a case study

value of the independent concept changes, the value of the dependent concept will always change in a predicted way. The proposition is formulated as follows: “If A is higher then B is higher” There are two ways to use the case study for testing a deterministic rela- tion. The preferred way is to use a longitudinal single case study. In the longitudinal single case study the independent concept in the single case changes “naturally” with time, and the corresponding dependent concept for each moment in time is measured either in real-time or post hoc. In the comparative case study two or more cases are selected, which have different values of the independent concept, but are otherwise similar, and the dependent concept in each case is measured.

6.1.2 Candidate cases

The issues regarding the universe for case selection are the same for all types of theory-testing case studies. See 5.1.2 for a discussion of this topic.

6.1.3 Case selection

In a longitudinal single case study a single case is selected that shows a rela- tively large variation in the value of the independent variable over time, whereas other variables that may also influence the effect should have stayed the same as much as possible. This would a allow mul- tiple tests over time of the proposition within the same case, and b give information about the range of values of the independent concept in which the proposition is correct. In a comparative case study two or more cases are selected that are as similar as possible but that have different values of the independent concept. Regarding the number of cases that must be selected, the general line of thought is first to select the minimum number of cases that is enough for doing the study. For a longitudinal case study one case is enough for the tests. For a comparative case study two cases are enough if the deterministic relation that is tested is continuously increasing or decreasing. For other deterministic relations e.g. parabolic, the minimum number of cases depends on the specific relation. After the first test, a replication strategy must be formulated based on the outcome of the test, and other cases can be selected for additional tests. The number of replications usually is limited only by resource constraints. A theory can always be developed further.

6.1.4 Hypothesis

If the proposition states that the value of the dependent concept increases or decreases deterministically with an increase of the value of the independent concept, then the hypothesis in a longitudinal case study can be formulated as follows: Hypothesis: For each pair of measurement points in time, the value of the dependent variable at one point of the pair differs in the predicted direction from the dependent variable’s value at the other point of the pair. If the predicted relation between the independent and dependent variable is a continuously increasing or continuously decreasing relation or the predicted relation can be considered as a set of continuously increasing or decreasing separate relations, then for each separate continuously increasing or decreasing relation, the hypothesis can be formulated as follows: Hypothesis: The rank order of the measurement points in time, according to the observed values of the independent variable, is exactly the same as the rank order of the measurement points according to the observed values of the dependent variable. If, for instance, ■ the five measurement points were times t 1 , t 2 , t 3 , t 4 , and t 5, , and ■ A 1 , A 2 , A 3 , A 4 , and A 5 are the values of the independent vari- able at these points, and ■ B 1 , B 2 , B 3 , B 4 , and B 5 are the values of the dependent variable at these points, and ■ the independent variable is ranked according to increasing value for example, A 4 , A 5 , A 3 , A 1 , A 2 , then, for a predicted continuously increasing or decreasing relation between the independent and the dependent variable, it is predicted that the dependent variable has exactly the same rank order B 4 , B 5 , B 3 , B 1 , B 2 , or the exact reverse order depending on whether the proposition states that the value of the dependent variable increases or decreases deterministically with an increase of the value the independent variable. If the proposition states that the value of the dependent variable increases or decreases deterministically and continuously increasing or decreasing with an increase of the value the independent variable, then the hypothesis in a comparative case study can be formulated as follows: Hypothesis: The rank order of cases, according to the observed values of the independent variable, is exactly the same as the rank order of the cases according to the observed values of the dependent variable.