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I am reading P.R. Rosenbaum, Model-based direct adjustment, Journal of the American Statistical Association, 1987 (82), pp. 387-394. However, there is one equation, which I can't understand.

$$ d = \frac{1}{N}\{\sum_{s=1}^{S}\sum_{i=1}^{N_s}\frac{z_{si}r_{si}}{\hat{e}_s} - \sum_{s=1}^{S}\sum_{i=1}^{N_s}\frac{(1-z_{si})r_{si}}{1-\hat{e}_s}\}, $$

where $d$ represents the difference, $\hat{e}_s$ is the probability, $z_{si}$ is the indicator function and $r_{si}$ is a binary response such as yes/no. My question is that How can a binary categorical variable $r_{si}$ be divided by a continuous variable $\hat{e}_s$? For example, how can a categorical variable (sex) male = 1 & female = 0 be divided by 0.23? It does not make any sense to me at all. many thanks in advance.

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  • $\begingroup$ Although the question has already been answered mathematics-wise, I think a philosophical answer would appear to be more relevant in this case. I agree with the question author - in reliance to the reality, such division makes no sense. In general, a binary categorical variable represents an answer to the question "Does it exist?" Whereas a continuous variable represents an answer to the question "What it's measure is in comparison to...". Those are different fundamental concepts. $\endgroup$
    – Maxim V. Pavlov
    Commented Jan 29, 2012 at 21:09
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    $\begingroup$ Your comment, Maxim, perceptively suggests that the difficulty here is due to confusing an outcome with a random variable. An answer to a question (the outcome) is not the same thing as the number used to code that answer (the random variable). That's precisely why the random variable is created: so that one can do arithmetic with it! One might then wonder about the apparent arbitrariness of the random variable: what happens when a different encoding is used (such as male=1, female=-1). The answer is that the formula for $d$ may change but the statistical inference will not. $\endgroup$
    – whuber ♦
    Commented Jan 30, 2012 at 15:56

3 Answers 3

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I can't access the article, so I'm guessing, but the answer to your general question is that yes a binary categorical variable can be divided by a continuous variable. I'm assuming the binary variable takes the form 0/1. It will then be setting the distance to 0 when the response is 0 (no), otherwise it reduces to $\frac{z_{si}}{\hat{e_{s}}}$ (and $\frac{1-z_{si}}{1-\hat{e_{s}}}$). So you can only get a distance other than 0 when the response is 1 (yes).

Thanks to Andy W for the TeX link. :)

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Without reading the paper in detail, I'd assume it is using the common convention of coding a binary variable as 0/1 (e.g. 0=No, 1=Yes ; or 0=Incorrect, 1=Correct).

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  • $\begingroup$ Onestop, thanks for answering my question but 0/1 or no/yes does not change the characteristic of the variable. The variable $r_{si}$ is still a binary "categorical" variable. Am I right? Thus, how can a binary categorical variable be divided by a numerical number? $\endgroup$
    – Tu.2
    Commented Jan 29, 2012 at 17:48
  • $\begingroup$ You can divide 0 or 1 by a numerical number, can't you? $\endgroup$
    – onestop
    Commented Jan 29, 2012 at 17:59
  • $\begingroup$ Yes from a mathematical point of view I understand it. But I still can't image. Such as (female = 1)/ 0.3 = 3.3333. What does 3.333 mean for sex variable? $\endgroup$
    – Tu.2
    Commented Jan 29, 2012 at 18:09
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One way of thinking of this, when the variable is coded 0/1, is thinking of it as a logical variable rather than numerical. In the equation you are worried about, we don't just have a binary variable divided by something else - it is part of a bigger expression with various other products. Including the binary variable means "include this set of products and divisions if the binary variable is 1; make it zero otherwise so this particular term doesn't add to my overall result".

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