LME/Multiple regression with many predictors and limited DV range

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.

• So the total sample size is around 40 and you have 20 predictors ? – Robert Long May 20 '19 at 17:54
• Sample size, if defined as the number of subjects, is 1; 40 is the number of observations. There are 20 predictors, yes, although as I say their number can be vastly reduced if need be (down to, say, 2 or 3) – z8080 May 20 '19 at 20:23
• Check out lasso. It does variable selection and regularization. If using R, it is available in the glmnet package. – Heteroskedastic Jim May 21 '19 at 15:55
• Wow, sounds useful, will try it, thanks! – z8080 May 21 '19 at 16:37

My first advice is definitely to collect more data. Before collecting any data (ie in the first place) it is important to conduct a power analysis so that you know what sample size is needed.

What follows is predicated on the statements that there is little covariance between the covariates, and that you are only interested in determining which covariates have the most predictive power.

The first thing I would do is to plot the data. That is, 20 bivariate plots of the outcome variable against each covariate. This may provide good evidence of variables that have the strongest predictive power. If all the relationships do not appear to be non-linear then compute the correlation coefficients and you could simply stop there - the one(s) with the highest correlation will have most predictive power. If you see noticeably non-linear associations then you could transform the covariates in question to attain something that resembles a linear association.

Dimension reduction techniques such as Principle Components Analysis seem to be out of the question due to small covariance between covariates, however if there appears to be no association, linear or nonlinear, in any of the bivariate plots then you can simply ignore those variables for what follows in the next paragraphs. If these coincide with those that you already identified as having a lower prior probability of influencing the outcome, that will be good news. Also, if any of the covariates are mediators - that is, if they lie on the causal path between any other covariate and the outcome - do not include them. One way to identify mediators is with the use of a causal diagram, or directed acyclic graph (DAG).

You cannot fit a mixed model because you have only one subject.

I would then proceed with fitting the full model in a multiple linear regression. The ratio of 10-20 observations per covariate is a rule of thumb and the main issue is with statistical power. With so few observations the ability to detect significant effects is very much diminished. But you can still fit the model. If you standardize the variables first, then you simply look for largest regression coefficients.

Since the outcome variable is ordinal, you could then repeat using ordinal regression, and compare the results. Hopefully they will identify the same covariate(s) as being most important.

• Thank you Robert this is immensely helpful. I will try your suggestions one by one, but just a quick point that collecting more data is not an option, since this is a case-study of a 'unique' patient; unless what you meant is collect more observations from the same patient, which as I said is conceivable. Thanks again. – z8080 May 21 '19 at 15:57
• You are very welcome. Yes, I meant to collect more data from the same patient. – Robert Long May 21 '19 at 16:09