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I have a sample of 412 young subjects, measured twice in an interval between 20 days and 3 years. I am interested in how two external factor (lets say sunlight and ice-cream) relates to growth. Subjects were exposed to sunlight and ice-cream somewhat randomly, and I have calculated the cumulative exposure (CE) as the sum of sunny days over the time between the measures (CE_sun), and the sum of ice-cream over the test period (CE_ice-cream) So, for example, subject A was tested on June 1 and June 30, and those 30 days were all sunny days with lots of ice-cream, so Subject A has a CE_sun of 30 (June 1 sun, June 2 sun, ...) and a CE_ice-cream of 30 (ice cream every day). Subject B was tested on November 1 and November 30, but since november was gloomy, she only had 5 days of CE, and 0 CE_ice-cream.

So in my linear regression, I have a factor for subject age, a factor for time between measures, and factors for CE_sun and CE_ice_cream. I want to test if CE_sun and CE_ice-cream effects growth. The problem is that that time between tests is highly correlated to CE_sun (r=0.97) and to CE_ice-cream (r=0.898). CE_sun also correlates with CE_ice-cream (r=0.78). So in the linear regression, I am not sure the CE_sun and CE_ice-cream are estimated well. To convince myself, an idea was to define the 100 subjects where "time between tests" and CE_sun and CE_ice-cream are least correlated, and run the linear regression in this subset. In this subset, the correlation between time and CE_sun is R = -0.10, time and CE_ice-cream is r=0.70, and between CE_sun and CE_ice-cream r= -0.77.

Is this approach incorrect?

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In short, yes. You're basically cherry-picking data to find an effect.

If you have significantly correlated predictors, you should consider removing some of them.

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  • $\begingroup$ You may not want to remove them straight away, as you could lose some information held there. You may consider grouping the variables and making them orthogonal via principal components. $\endgroup$ – ERT Jul 23 '18 at 20:27
  • $\begingroup$ This, of course, depends on whether or not you can actually interpret the principal components. $\endgroup$ – Kevin Li Jul 23 '18 at 21:09
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Well seems really obvious to me why time between has such correlation with Sun and Ice-Cream. A quick way to tackle with this is to find the mean of each day in Sun and Ice-Cream $\frac{CESun}{TimeBetween}$ and $\frac{CE Ice-Cream}{TimeBetween}$ for each subject. Using the two new variables, you should not have the same problem and be able to interpret the regression much easier.

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