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I'm working on my bachelor's thesis and have an analysis where the dependent variable (number of months of parental leave of fathers) has a very skewed distribution, as follows: 1089 times the value 0, 18 times the value 1, 89 times the value 2, 29 times the value 3, 11 times the value 4, and so on, with all further values occurring less than 10 times.

Now, the same variable from the same data set has already been analyzed in several papers that got published in scientific journals, and they all used several variants of linear regression on the untransformed data.

My question: Is this approach really valid? From all I have learned in my introductory statistics classes, you need a normally distributed dependent variable for linear regression. And these data are clearly non-normal and cannot be transformed to be normal either. What other methods could be used instead? Might negative binomial regression be an option? Or is linear regression OK to use after all?

Thanks, Stefanie

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  • $\begingroup$ Those are counts, so some form of count regression would be appropriate. However, in my experience, data like that aren't typically well-behaved from the perspective of standard count distributions, so you might want to use ordinal logistic regression. $\endgroup$ – gung - Reinstate Monica May 6 '16 at 16:11
  • $\begingroup$ You could also investigate zero-inflated models or hurdle models. Each exists in Poisson or negative binomial flavours. $\endgroup$ – mdewey May 6 '16 at 16:15
  • $\begingroup$ Thanks, gung and mdewey, I will have a look at those. So do I see it correctly that linear regression is definitely not an option and should not have been used in the scientific articles I have read? $\endgroup$ – Stefanie Müller May 6 '16 at 16:35
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(Zero-inflated) Negative binomial regression would seem like a logical regression model to use. With the type of data you describe, linear regression will tend to be problematic in some respects (e.g. the error model is just wrong, as a result you may get negative months predicted for some records, the confidence intervals do not respect that negative months are not possible, hypothesis tests may not have the specified level etc.), while if the median count is pretty high (let's say 20 or 40) and just a few zeros occur, linear regression will often work pretty well.

The zero-inflated part would distinguish those taking any "meaningful" (i.e. not (rounded to?) zero) leave versus those taking at least something (rounded?) to 1 month. I am speculating here regarding to the rounding, since I would have assumed many would take at least a few days and that the real unit of time taken off would be working days or half-working-days - or is this in any case specific parental leave that usually comes in a unit of months (or weeks) as opposed to taking available vacation time/personal days?

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    $\begingroup$ Thanks, Björn. The values are actually not rounded, all the zeros are 'real' zeros. The parental leave I'm looking at is always taken for whole months. $\endgroup$ – Stefanie Müller May 6 '16 at 16:48
  • $\begingroup$ @Stefanie There is a slight issue as count data can (in theory) take on any value up to infinity and I assume leave is capped at some value. In practice this may not be too serious if there are few cases taking lots of leave. If (say) the values range from zero to twelve you could consider some form of ordinal regression model as gung suggested $\endgroup$ – mdewey May 6 '16 at 17:14
  • $\begingroup$ Thanks again, mdewey, I think ordinal regression looks very promising. Can I use it on the data 'as is', or do I have to categorize it first? $\endgroup$ – Stefanie Müller May 6 '16 at 17:25
  • $\begingroup$ @StefanieMüller it depends on how many categories there are but it is best to use as fine a categorisation as you can. $\endgroup$ – mdewey May 7 '16 at 12:48

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