# Adjusting for covariates in ROC curve analysis

This question is about estimating cut-off scores on a multi-dimensional screening questionnaire to predict a binary endpoint, in the presence of correlated scales.

I was asked about the interest of controlling for associated subscores when devising cut-off scores on each dimension of a measurement scale (personality traits) which might be used for alcoholism screening. That is, in this particular case, the person was not interested in adjusting on external covariates (predictors) -- which leads to (partial) area under covariate-adjusted ROC curve, e.g. (1-2) -- but essentially on other scores from the same questionnaire because they correlate one to each other (e.g. "impulsivity" with "sensation seeking"). It amounts to build an GLM which includes on the left-side the score of interest (for which we seek a cut-off) and another score computed from the same questionnaire, while on the right-hand side the outcome may be drinking status.

To clarify (per @robin request), suppose we have $j=4$ scores, say $x_j$ (e.g., anxiety, impulsivity, neuroticism, sensation seeking), and we want to find a cut-off value $t_j$ (i.e. "positive case" if $x_j>t_j$, "negative case" otherwise) for each of them. We usually adjust for other risk factors like gender or age when devising such cut-off (using ROC curve analysis). Now, what about adjusting impulsivity (IMP) on gender, age, and sensation seeking (SS) since SS is known to correlate with IMP? In other words, we would have a cut-off value for IMP where effect of age, gender and anxiety level are removed.

Apart from saying that a cut-off must remain as simple as possible, my response was

About covariates, I would recommend estimating the AUCs with and without adjustment, just to see if the predictive performance increase. Here, your covariates are merely other subscores defined from the same measurement instrument and I never faced such a situation (usually, I adjust on known risk factors, like Age or Gender). [...] Also, since you are interested in prognostic issues (i.e. screening efficacy of the questionnaire), you may also be interested in estimating the positive predictive value (PPV, probability of patients with positive test results who are correctly classified) provided you are able to classify subjects as "positive" or "negative" depending on their subscores on your questionnaire. Note, however, that it is necessary to know the prevalence of this disorder to correctly interpret the PPV in turn...

Do you have a more thorough understanding of this particular situation, with link to relevant papers when possible?

References

1. Janes, H and Pepe, MS (2008). Adjusting for Covariates in Studies of Diagnostic, Screening, or Prognostic Markers: An Old Concept in a New Setting. American Journal of Epidemiology, 168(1): 89-97.
2. Janes, H and Pepe, MS (2008). Accommodating Covariates in ROC Analysis. UW Biostatistics Working Paper Series, Paper 322.
• I am not an expert but I found the sentence "controlling for associated subscores when devising cut-off scores on each dimension of a measurement scale" a bit esoteric. Can you give me one more line of explanation (otherwise I found it difficult to understand the question) ? – robin girard Nov 2 '10 at 12:38
• @robin Yes, basically I meant: we have $j=4$ scores (e.g. anxiety, impulsivity, neuroticism, sensation seeking) and we want to find a cut-off value $t_j$ (i.e. "positive case" if $x_j>t_j$, "negative case" otherwise) for each of them. We usually adjust for other risk factors like gender or age when devising such cut-off (using ROC curve analysis). Now, what about adjusting impulsivity (IMP) on gender, age, and sensation seeking (SS) since SS is known to correlate with IMP? In other words, we would have a cut-off value for IMP where effect of age, gender and anxiety level are removed. – chl Nov 2 '10 at 22:54
• If the end goal is to predict a binary value, given [correlated] answers to survey questions, this sounds a lot like a standard binary classification problem. Would it be appropriate to think of it this way? Or is it very important to come up with "cutoff values" (which I don't know anything about)? – DavidR Oct 7 '11 at 18:15
• @DavidR Well, the idea is to decide of a cut-off value (read, "subject at risk above a certain value"), which comes with a lot of caveats from a statistical perspective but most clinicians are used to or prefer to work this way. (Sorry about not having noticed your comment earlier!) – chl Nov 27 '11 at 11:10

The way that you've envisioned the analysis is really not the way I would suggest you start out thinking about it. First of all it is easy to show that if cutoffs must be used, cutoffs are not applied on individual features but on the overall predicted probability. The optimal cutoff for a single covariate depends on all the levels of the other covariates; it cannot be constant. Secondly, ROC curves play no role in meeting the goal of making optimum decisions for an individual subject.

To handle correlated scales there are many data reduction techniques that can help. One of them is a formal redundancy analysis where each predictor is nonlinearly predicted from all the other predictors, in turn. This is implemented in the redun function in the R Hmisc package. Variable clustering, principal component analysis, and factor analysis are other possibilities. But the main part of the analysis, in my view, should be building a good probability model (e.g., binary logistic model).

The point of the Janes, Pepe article on covariate adjusted ROC curves is allowing a more flexible interpretation of the estimated ROC curve values. This is a method of stratifying ROC curves among specific groups in the population of interest. The estimated true positive fraction (TPF; eq. sensitivity) and true negative fraction (TNF; eq. specificity) are interpreted as "the probability of a correct screening outcome given the disease status is Y/N among individuals of the same [adjusted variable list]". At a glance, it sounds like what you're trying to do is improve your diagnostic test by incorporating more markers into your panel.

A good background to understand these methods a little better would be to read about the Cox proportional hazards model and to look at Pepe's book on "The Statistical Evaluation of Medical Tests for Classification and ...". You'll notice screening reliability measures share many similar properties with a survival curve, thinking of the fitted score as a survival time. Just as the Cox model allows for stratification of the survival curve, they propose giving stratified reliability measures.

The reason this matters to us might be justified in the context of a binary mixed effects model: suppose you're interested in predicting the risk of becoming a meth addict. SES has such an obvious dominating effect on this that it seems foolish to evaluate a diagnostic test, which might be based on personal behaviors, without somehow stratifying. This is because [just roll with this], even if a rich person showed manic and depressive symptoms, they'll probably never try meth. However, a poor person would show a much larger increased risk having such psychological symptoms (and higher risk score). The crude analysis of risk would show very poor performance of your predictive model because the same differences in two groups were not reliable. However, if you stratified (rich versus poor), you could have 100% sensitivity and specificity for the same diagnostic marker.

The point of covariate adjustment is to consider different groups homogeneous due to lower prevalence and interaction in the risk model between distinct strata.

• (+1) That's an interesting response, thank you very much. My main concern at the time of this writing was that cutoff values will be somewhat "interdependent". But I'll check Pepe's book (found some handouts here in the meantime). – chl Dec 15 '11 at 10:27
• There are issues with using the same data to develop and evaluate a medical test but the fix is easy. You should consider some kind of cross-validation, or divvying the data into "training" and "validation" subsets. This is typically a valid approach to developing a diagnostic/prognostic/risk-prediction model. – AdamO Dec 15 '11 at 18:43