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I am analyzing data with a binary outcome and a variety of continuous and categorical (including dichotomous) predictor variables. My approach is to perform a binary logistic regression and to treat any predictor with more than 20 unique values as continuous. Several arguments against categorization, especially well-documented on Frank Harrell's site are a good reason to not categorize.

However, at a recent meeting where I discussed my analyses approach, a faculty suggested that I would get more accurate risk estimates for the data if I categorized the variables which have a skewed distribution and outliers. Their logic was that the tail of the skewed distribution and the outliers in that tail will have a detrimental effect on the risk estimates generated by the logistic regression and that categorization will address this by erasing the effect of the tail and the skew.

I have several predictor variables that definitely have skewed distributions and some outliers. Is the claim that a variable with a skewed distribution (with outliers) is more likely to produce inaccurate risk estimates compared to the categorized version of the same variable, true? How do skews and outliers in the tail affect logistic regression estimates?

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    $\begingroup$ The logic is flawed. Categorization would be a nonlinear transformation, tantamount to a wholesale change in the model. It potentially could cure a nonlinear relationship between the response and the independent variables, but would lose precision. A better reason for addressing skewness would be the presence of high-leverage data, but there are other solutions for that besides categorization (and the leverage might not even be a problem and the categorization might not reduce the leverage, anyway). What evidence is there that the outliers have a "detrimental effect" on the fit? $\endgroup$ – whuber Dec 21 '11 at 3:22
  • $\begingroup$ Thanks very much for your comment. I tried to search for evidence and references that support that logic after the meeting but did not find any. Hence I asked my question here. Do you perhaps know of references that show that skewed data don't influence effect estimates in a binary logistic regression? You mentioned other solutions for high leverage data in the tail. Apart from dropping them, could you please elaborate on the alternatives? I would really like to strengthen my arguments against categorization as the only route. $\endgroup$ – Ariel Dec 21 '11 at 16:25
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    $\begingroup$ Logistic regression is so similar to OLS regression that you can use your experience with the latter to inform your thinking about the former, Ariel. One way to assess high-leverage points is with a sensitivity analysis: if the estimates change little after removing high-leverage points, they're ok; otherwise, the results depend materially on the high-leverage points, implying you have some hard choices to make (and some careful thinking to do). Also, categorization might not matter: if you do it and get the same or similar results, you're in good shape no matter what. $\endgroup$ – whuber Dec 21 '11 at 16:36
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    $\begingroup$ Again, very helpful. Thanks! I will look at sensitivity analyses and compare categorized and non-categorized results. Sounds like a convincing way to make sure that my results are not being influenced just because of the nature of the variable has changed. If the risk estimates are similar and in addition, the CIs in an analyses that uses continuous forms of the predictor are narrower, this might indicate the gain in power by using continuous forms. $\endgroup$ – Ariel Dec 21 '11 at 18:03
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    $\begingroup$ I would suggest you try some spline terms with your continuous skewed variables. That might help. The linearity assumption with logistic regression is quite strong! $\endgroup$ – kjetil b halvorsen Jan 2 '17 at 13:10
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I would get more accurate risk estimates for the data if I categorized the variables which have a skewed distribution and outliers.

That suggestion is not summarily true. It may be true in some cases. It may be detrimental in others. Categorizing a predictor means dividing it into quartiles or user-defined thresholds. Categorized predictors have the advantage of fitting more flexible trend lines to the data. The disadvantage is that they predict the same risk in all groups within a specific category and they borrow no information across adjacent groups. Categorized predictors have the additional disadvantage of increasing the number of predictors and hence the risk of overfitting.

Categorizing a risk predictor introduces the sensitivity to the definition of the thresholds. It can be difficult to prespecify clinically relevant thresholds. Thresholds defined on quartiles of the sample do not tend to generalize well to other validation samples.

Biologically, however, we would be concerned that extreme values of predictors signify biologic trends or interactions that are not captured in the risk model. For instance, blood pressures or BMI several standard deviations above the mean are no longer consistent with the additive effect on risk in intermediate ranges, but reflect exponentially growing risks for diabetes, hypertension, chronic kidney disease, and MI or stroke.

For this reason, we can use rigorous testing or inspection to assess linearity and add supplemental terms if the model fit is dramatically improved. Rather than categorizing predictors, a hybrid alternative approach is to use polynomial terms like the both the linear and log transformed values of a skewed predictor, or we may use piecewise linear, quadratic, or cubic splines to fit trends that are curvilinear but that borrow information across groups and predict non-constant risk in participants with different values.

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