For a single-case patient study (case profile), I have 20 IVs such as medication intake, amount of sleep etc; and one DV representing the severity of the symptom reported by the patient in each of k=40 days (rows):
I am trying to identify which of these 20 factors (if any) has the most predictive power in determining the DV. Although any of these predictors (or interactions thereof) can be suspected to influence the symptom, some vague hypotheses exist as to which are more likely. Also, there is little covariance between the IVs, since they refer to independent (separate) aspects that are not necessarily correlated.
Since the sample size is 1, I guess linear mixed-effects makes no sense (as there is no "subject" random effect), and that multiple regression is the most suitable analysis tool. However, the following features of this dataset are problematic as they reduce statistical power:
1) I know that roughly 10-20 observations are needed per predictor, which is not the case here, with only twice as many observations as predictors. I suspect this makes false positives likely if I were to run the mult.regr. like that, with so many comparisons;
2) the DV values in each row are of a limited range (most around 7 to 9, scale is 0-10).
The question therefore is how to reconcile the many predictors with the few observations. Assuming the regression cannot be run with the dataset being as it is now, which of these options is more advisable (a) collect more observations (doable, but not convenient), or (b) try to restrict the predictor space, e.g. try to identify and keep only a handful of 'likely' predictors while getting rid of the ones that can be said to have a lower prior probability of influencing the DV
Any other suggestions will be very welcomed.