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NOTE: This question relates to Binomial logistic regression. Thus, the $n$ in the title, refers to the parameter of the Binomial distribution.

Most real-world use cases of logistic regression introduced in statistics courses and employed in machine learning models are about binary logistic regression where $n = 1$.

Given that the logistic regression, in general, is based on a Binomial distribution (as can be seen, for example, in the application of logistic regression models in R, where the R library/package, glm, requires the user to set the argument family to binomial, in order to perform logistic regression.

My question is thus: What are use cases of logistic regression where $n \neq 1$, i.e., $n >1$?

CausalQuestions
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1 Answers1

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Assuming that your $n$ is the number of cases of each type, you end up looking at proportions of each successful rather than $0-1$ and you weight by $n$.

Stealing from https://stackoverflow.com/questions/50078497/weighted-logistic-regression-in-r gives an example of sample data of proportions of successes (prop) plus sample sizes (n) for each value of the independent variable (x)

datf <- data.frame(prop  = c(0.125, 0,  0.667, 1,  0.9),
                   n     = c(8,     1,  3,     3,  10),
                   x     = c(11,    12, 15,    16, 18))

glm(prop ~ x, family=binomial, data=datf, weights=n)

which gives

Call:  glm(formula = prop ~ x, family = binomial, data = datf, weights = n)

Coefficients:
(Intercept)            x

    -9.3533       0.6714


Degrees of Freedom: 4 Total (i.e. Null);  3 Residual
Null Deviance:      17.3 
Residual Deviance: 2.043    AIC: 11.43

you get a reasonable model producing a sensible looking chart like logistic prediction

though you also get a warning about non-integer #successes which you can ignore, and the wrong values for degrees of freedom, deviance and AIC.

Henry
  • 39,459
  • With $n$, I mean the number of trials (Binomial distribution, i.e., number of Bernoulli trials). The chart looks like a logistic regression which allows values other than ${0,1}$, i.e. any real value in interval $[0,1]$. Is that thinking correct? Clarification comments about your answer. In the example, the dataset comes with an additional variable (here sample sizes $n$), that represents the number of trials $n$ used to get the proportion values $prop_i$? If I were to replace each $n_i$ in the dataset with 1, would the fitted model then be a binary logistic regression again? – CausalQuestions Sep 26 '22 at 22:36
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    @CausalQuestions I think that is what I have provided in my example, but note that the values in prop are proportion of successes rather than numbers of successes, so in the first case when $x=11$ you have $1$ binomial success out of $n=8$ attempts shown as a proportion of $0.125$. – Henry Sep 26 '22 at 22:41