# How much does quantizing continuous data affect power?

Background/how this came up:

A student has numeric ratings on a variety of items in questionnaires and sports performance data and would like to model the performance on questionnaire scores. He wanted to do an analysis (don't ask me why) where some of the data are split into quantiles so he can compare high, medium, and low performance groups. I suggested he do a linear regression instead so that he wouldn't lose power by throwing away the continuous data - but I realized that the only papers I've seen on this being bad were about doing median splits.

The question:

Intuitively, it seems that quantizing your data will sacrifice power (based on arguments against median splits)- but how is this affected by the number of quantiles you choose to use? Are there circumstances where using, say, quartiles isn't much worse than doing a regression? For a given sample size, can a minimum number of quantiles be calculated to stay above a given power level?

• The answer certainly depends on the distribution of the continuous variable you're discretizing (by quantile splits, apparently). What kind of distribution does the continuous measurement have? – Macro Mar 2 '12 at 3:06
• What is the conceivable motivation for using quantile groups? I can't imagine how this would either improve the analysis or shed light. You are right to believe this will sacrifice a good deal of power in general. By the way, his definition of "low performance group" will change arbitrarily if new subjects' data are added, if using quantiles to stratify the subjects. – Frank Harrell Mar 2 '12 at 3:54
• Not an answer to your question, but my currently favorite answer to the underlying problem: When one has a continuous predictor variable, traditionally we are taught to either assume its effect as linear (losing power to detect non-linear effects) or quantize it (losing power to detect linear effects). An elegant alternative is generalized additive modelling (gam), which permits the data itself to determine the degree of non-linearity that the model attempts to fit. The mgcv package in R implements gam with cross-validation to avoid over-fitting, and has revolutionized my work. – Mike Lawrence Mar 2 '12 at 4:14
• Even simpler is to use regression splines. – Frank Harrell Mar 2 '12 at 18:02
• One major “reason” to do it is that it seems simpler for someone who has had years of exposure to ANOVA (learning, teaching or simply reading articles using it), does not know much about statistical modeling or the GLM and needs a p-value ASAP (e.g. most psychology students or researchers). Psychologists work very hard to make everything a 2x2x… ANOVA design. – Gala Mar 6 '12 at 8:56

• Estimation of quantiles is different from using quantiles as boundaries to create artificial, arbitrary groupings of inherently continuous variables. Quantile grouping is analogous to classifying a person's wealth based on their net worth and the net worth of their neighbors around them. Looking at it another way, suppose the net worths of people in a sample is $145k,$145.1k, $145.2k,$145.3k. Cutting at the lower quartile would cut at \$145.1k making the analysis only relative but on a scale that is likely to be nothing but noise. – Frank Harrell Mar 2 '12 at 18:01