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In social or public health research, we often collect data in the form of multiple item scales, where each item has a binary response e.g.:

In the past 6 months have you:
Q1: Experienced symptom X? Yes / No
Q2: Experienced symptom Y? Yes / No
Q3: Had trouble with Z? Yes / No etc.

Typically we then add the number of "yes" responses to form a total score, meaning if there are $N$ items then scores between $0$ and $N$ are possible.

When analysing these kinds of outcomes in a regression model (with terms for treatment group or other covariates included), the common practice seems to be to model the total scores for each individual in a standard linear regression model, essentially treating them as normally distributed. This seems fundamentally incorrect to me, as there are a number of problems with assuming a normal distribution for these scores, the most obvious being that your model can easily predict impossible negative scores. However, as long as the mean scores in your data tend to fall in the middle of the $[0, N]$ range these problems may not occur.

Are there better ways to treat these kinds of outcomes, i.e. scales made up of multiple binary response outcomes? (e.g. modelling the scores as binomially-distributed seems like a good alternative) What are the advantages and disadvantages of assuming a normal distribution versus other distributions for this kind of data?

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  • $\begingroup$ Add the self-study tag for survey problems like this. $\endgroup$ – Michael R. Chernick Mar 7 '17 at 0:54
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    $\begingroup$ @MichaelChernick, why do you think this is a self-study question? $\endgroup$ – T.E.G. - Reinstate Monica Mar 7 '17 at 4:37
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I can think of two solutions: First, you could use a count regression model on the number of yeses. Poison or negative binomial regression would be starting places, anyway.

Second, you could factor analyze the 3 variables. There are methods to do FA on dichotomous items. Then you could use the factor scores as the DV and do ordinary regression. This would work better if there are more than 3 items.

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  • $\begingroup$ I'm unsure about the relative benefits of Poisson versus standard binomial regression here. To some extent in these kinds of scales you are counting symptoms, but there's also a fixed upper bound on the count because you only ask about a fixed number of symptoms, and because of that the variance probably doesn't look like a Poisson (not proportional to the mean due to the upper bound). That's why I'm drawn to binomial/logistic regression here. $\endgroup$ – Marius Mar 7 '17 at 23:48
  • $\begingroup$ If you do binary logistic regression you are tossing away a lot of info. Maybe ordinal logistic. $\endgroup$ – Peter Flom - Reinstate Monica Mar 8 '17 at 0:13

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