Which determines whether you can causally interpret the results: the specific statistical model chosen to analyze the data, or the design of the study?
The specific statistical model chosen to analyze the data is useful in causally interpreting the results. The design of the study is done just for 'collection of data in an organised manner'.
For example, an experiment can be performed in a completely randomized design (CRD), randomized blocked design (RBD), latin square design (LSD), BIBD, etc.
But the data in all these cases is modelled as some equation by which the response variable is related to the regressor variables. This equation/model can be linear, quadratic, have or not have interaction terms of the regressors and so on.
Hence, the choice 'specific statistical model'.
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