# Logistic regression to predict the fraction of small fish among all fish

My question is regarding to which logistic regression test fits my goal best.

My data set contains 641 rows of which each row is one sample with several independent variables (continuous, nominal and ordinal). However I'm a bit confused on how to classify my response variable. The response variable is constructed as follows:

N-breams (length class 16-40cm) /
(N-breams (length class 16-40cm) + N-breams (length class 40cm+).

This results in a response variable within a range of 0-1. Where the number higher than 0.5 have more breams of length class 16-40cm compared to 40cm+ and vice versa.

In a normal aquatic system the ratio should be higher than 0.5 (or even 1.0), however this isn’t always the case where the ratio is lower than 0.5 (or even 0.0). I'm interested which environmental variables influences this ratio.

So, initially I thought of binomial distribution which looks like this in R (using GLM or GLMM):

glm(y ~ x1 + x2 + x3, family = binomial)

With an output which predicts the probability (0-1) in respect to a significant independent variable. This is the part where I get confused. Since the 0.5 value is a "tipping point" which means that every predicted/fitted value (from the output) lower than 0.5 has more 40cm+ breams than 16-40cm, RIGHT? Or are we talking about chances? So that a 0.5 value is a 50% chance?

Question

So my real question is whether the predicted values are chances (%) or still remain ratio values (but predicted like with the output of a poisson or normal model). I'm almost certain that this regards the latter, but somehow I'm still doubting.

• Output of logistic regression is probabilities stats.stackexchange.com/questions/227009/… , and using rule $\hat y > 0.5$ can be misleading, check: stats.stackexchange.com/questions/127042/… – Tim Sep 28 '16 at 10:56
• Would beta regression be another option? It is available in R. – mdewey Sep 28 '16 at 11:22
• @jwimberley +1 to your suggestion, but one does not need to manually increase the size of the dataset; for example in R, glm function can deal with proportion data if supplied by the weights argument. Mark, this is not a beta regression, it is logistic regression. See here: stats.stackexchange.com/questions/26762 Your Q might be a duplicate. – amoeba Sep 28 '16 at 13:39
• @jwimberley Depends on the software I guess, but in R you don't need to replicate anything manually. See stats.stackexchange.com/questions/26762. – amoeba Sep 28 '16 at 13:41
• @Mark The probability tells you the ratio: if $p_+$ is the probability of being $N_+$ then $R = p_+/(1-p_+)$ is the predicted ratio. Perhaps beta regression gives a more unbiased estimate of $R$; is this the motivation for it? But the probability is still just as good of a descriptor of the population. – jwimberley Sep 28 '16 at 14:33