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I am fairly new to statistics as a Phd student. I am trying to understand how dichotomizing a continuous variable can lead to distinct effects on two dependent variables.

So in a cross-sectional sample, I have one predictor (continuous) and two outcomes of interest (both continuous). Using linear regression, I found that predictor is significantly associated with outcome 1 but not outcome 2. Given that my predictor is a validated scale score, with specific cut-offs to create 2 categories, I then dichotimized the predictor to create a binary predictor_category variable. Using linear regression, I found that predictor_category has a significant effect on outcome 2 but not outcome 1.

I want to understand why this is happening? Why is the continuous predictor associated only with outcome 1 and why predictor_category is only associated with outcome 2? Even though predictor_category is derived from the continuous predictor.

I would appreciate any thoughts on this.

Edit: Adding two scatter plots below. I have centered my continuous predictor.

enter image description here

enter image description here

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  • $\begingroup$ @Tim: OP does not mention groups and apparently has only a single predictor, so I don't think this is Simpson's paradox. $\endgroup$
    – Stephan Kolassa
    Commented Aug 8, 2023 at 11:10
  • $\begingroup$ Per Peter Flom's answer, dichotomizing a continuous - or even only ordinal - predictor is usually very bad practice. "Specific cut-offs", to be honest, does not fill me with more confidence. That said, can you edit your post to include two scatterplots, one each for each outcome against the predictor? Perhaps you could even post the data itself if it's not too large? $\endgroup$
    – Stephan Kolassa
    Commented Aug 8, 2023 at 11:13
  • $\begingroup$ Nice, thank you. Is there any overplotting? Especially the first plot suggests that (the very regular pattern at the bottom left). The large predictor values will exert a lot of leverage. I would suggest you add a smoother with confidence bands, which will likely show that there is not a lot of signal in your data, but a lot of noise. (One piece of evidence for that: just based on your scatterplots, I would not be able to say whether the correlation is positive or negative. So: little signal.) I would say that Peter has it exactly right: don't dichotomize, and read Gelman's paper. $\endgroup$
    – Stephan Kolassa
    Commented Aug 8, 2023 at 11:35
  • $\begingroup$ Incidentally, to deal with overplotting (if any), I would suggest you jitter your points a little. Or take a look at sunflowerplots. Good plots are important, put some in your thesis! $\endgroup$
    – Stephan Kolassa
    Commented Aug 8, 2023 at 11:43
  • $\begingroup$ @StephanKolassa Thank you very much for your replies. This has been helpful for me. I will move forward with the continuous predictor as you and Peter Flom suggest. Also thank you for the tip on jittering and sunflower plots! $\endgroup$
    – magg
    Commented Aug 8, 2023 at 12:53

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If you want a mathematical statistics type answer, with proofs and stuff, maybe someone else can provide it.

On the most basic, intuitive, level, I'll say "When you ask different questions, you get different answers." Dichotomizing a continuous variable makes it a new thing, so you are asking a different question.

It is almost always a bad idea to do this during the analysis. This has been discussed here many times.

Another issue here is that looking at "significant" vs. "not significant" is not a good practice. Andrew Gelman has an article about this, The Difference Between “Significant” and “Not Significant” is not. Itself Statistically Significant.

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