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This is the question I am trying to answer : I’m working on a thesis project and have a variable that is normally distributed and related to happiness. I am considering to split my sample into depressed and non-depressed and run the correlation and the rest of the analyses that way. I think it will help clarify the differences between happy people and non-happy people. a. Would you advise me to do this? Why/Why not? b. What would happen to the correlation if I left it continuous or dichotomized it?

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    $\begingroup$ Correlation with what? $\endgroup$
    – Peter Flom
    Commented Apr 6, 2014 at 14:30
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    $\begingroup$ possible duplicate of How to choose between ANOVA and ANCOVA in a designed experiment? $\endgroup$ Commented Apr 6, 2014 at 15:19
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    $\begingroup$ Also see here. $\endgroup$ Commented Apr 6, 2014 at 20:31
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    $\begingroup$ Similar advice to the advice here may be found in relation to dichotomizing variables (or dividing into other small numbers of categories) on many other questions. The almost universal advice is that it's inadvisable, and the list of reasons is quite lengthy, but bias in estimates is usually first on the list. Here's one example which has a list of references. $\endgroup$
    – Glen_b
    Commented Apr 6, 2014 at 22:33

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a) No; Information, precision, and power loss; completely arbitrary choice of cutpoints;

b) Different analysts would obtain different answers to the question depending on the mood they were in at the time.

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If your happiness variable is really normally distributed, you might consider yourself lucky. I tend to get a minor ceiling effect; I suspect it's a consequence of self-enhancement. Depending on your population (especially if it's clinical), you may also have to worry about "faking bad" creating negative skew, or about extreme response style fattening the tails (especially if it's Chinese; see @chl's answer to "Factor analysis of questionnaires composed of Likert items").

Anyway, you're looking to clarify differences between happy and non-happy, depressed and non- depressed...These are different dichotomies, and they are both false dichotomies at that. Dichotomizing a normally distributed sample will not clarify real differences; it will obscure them. After all, obscuring real differences is literally what you're proposing to do if you agree that there are meaningful differences between "happy" and "happiest". Unless there's a particularly strong reason for grouping a continuous variable, "Just say 'no'."

To demonstrate, consider a simulated example in R. Following classic "theory" based on an absurd gun enthusiast magazine as satirized by Lennon (1968), imagine that happiness is ⅓ "a warm gun" and ⅔ Gaussian error (and please remember this is satire). With set.seed(1) for reproducibility, this code simulates a random sample: gun.warmth=rgamma(999,1,1);happiness=gun.warmth+2*rnorm(999) and false.dichotomy=happiness>0 creates a dichotomized version of happiness for an ideal case where zero is meaningful (e.g., a measure standardized to population norms). Since you haven't specified that your other variable is also Gaussian, I'm choosing to make the situation somewhat worse than it might be (though it certainly could be worse) by modeling gun warmth with a gamma distribution (most probably haven't been fired recently, and lose excess heat rapidly; let's define zero as ambient temperature). This means the distribution of happiness is actually a mixture, but this is difficult to detect without a larger sample. Skewness is .07, kurtosis is 3.08, the QQ plot looks acceptable, and even Shapiro–Wilk is inconclusive. (Use skew(happiness);kurtosis(happiness) and shapiro.test(happiness);qqnorm(happiness);qqline(happiness) if you want to see for yourself.)

Using cor(data.frame(gun.warmth,happiness,false.dichotomy)) demonstrates that dichotomization attenuates both the correlation of happiness with itself $(r=.76$ when it should be $1)$ and, consequently, with gun warmth $(r=.26$ instead of $.41)$. This is one price of wasting information with dichotomization: downward-biased effect size estimates. Another is standard error $(SE)$ inflation. In this case, the $SE$ increases from .029 to .031. Therefore the effect size estimate is also less significant (in all senses of the word), and its confidence interval is wider, indicating its imprecision. (require(psych);with(corr.test(data.frame(gun.warmth,happiness,false.dichotomy)),r/t[2:3,1] with(cor.test(happiness,gun.warmth),conf.int[2]-conf.int[1])$=.10$, with(cor.test(as.numeric(false.dichotomy),gun.warmth),conf.int[2]-conf.int[1])$=.12$.)

In conjunction, these effects make it harder to tell there's any relationship whatsoever. Hence dichotomization is a bad idea for your purposes because it obscures both the dimensional nature of individual differences on your first variable and its relationship with your second. Just say 'no'! $☺$

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