Was this the appropriate regression model?

Question

I was recently proof-reading a friend's thesis (for their writing, not stats usage) when I came across a usage of a regression model which I would regard as incorrect. However, I'm pretty new to the game so I wanted to see what other people thought.

In short: They used a linear regression model for a percentage. I think they should have used a logistic regression. I'll explain below.

They were testing the percentage of flowers which had grown from seedlings and whether there were observable differences depending on which soil mixtures were used i.e. did soil A result in a higher percentage of seeds which flowered compared with soil B.

To me, this seems like a binary outcome i.e. yes, it did flower or no, it did not flower. Consequently, a logistic regression model would have been a more appropriate model to use as a percentage is not a quantitative variable in the way that blood pressure is a quantitative variable.

Is this incorrect? I would be very interested in knowing, chiefly because it's bugging me. Thanks!

Answer 1

I agree with @Maarten Buis in broad terms, but more can be said.

The predictors for this problem are not well explained, but appear to be indicators for soil type.

An objection in principle to linear regression with a percent response is that the predicted value can go outside $[0, 1]$ or $[0, 100\%]$, which is absurd scientifically (biologically, in this case) as well as statistically. However, whether it does that within the range of your data is an empirical point and in some cases the linear regression and the logistic regression may yield very similar predictions. However, if you have any values near or at $0$ and/or near or at $100\%$ it is unlikely that a linear fit will be a good idea. A related but not identical point is that nonlinearity within the range is more plausible than linearity. A further related point is that variability can not be constant. As the mean goes to $0$, the variance goes to $0$ too, as the only way the mean can be $0$ is if all values are $0$. A similar point applies to $100\%$: if the mean is $100\%$, then the variance is $0$. That also motivates working with logits.

On your comment that "a percentage is not a quantitative variable in the way that blood pressure is a quantitative variable". It is difficult to know what you are driving at here. A percent scale is often, as it appears to be here, essentially a scale for the mean of a binary variable. It is different from counted or measured variables, but no less valid.

Answer 2

The percentage of a barley crop in different fields and varieties of barely invested with leaf blotch is a classic example used for regression on a percentage variable, see for the original reference here. This is fairly similar to modeling the percentage flowering in different fields (and other experimental conditions).

However, this example was used to illustrate an alternative to standard linear regression: quasi-likelihood regression with a logit link function and different forms of variance functions. A good place to start learning about these models is:

McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC.