# Linear Regression: Treat binomial dependent variable as normal due to large n

Say I wish to do a linear model and my dependent variable $$Y$$ follows a binomial distribution: $$Y_i \sim Bin(n_i,p_i), i = 1,...,N$$. Usually I would use a generalized linear model and perform a logistic regression. But if the $$n_i$$'s are large enough, the $$Y_i$$'s will approximately follow a normal distribution, $$Y_i \sim N(n_ip_i, \ n_ip_i(1-p_i))$$. So in that case shouldn't I be able to simply perform a simple linear regression, without using a generalized linear model?

• Here are the pros and cons to using a linear model stats.stackexchange.com/questions/304437/… Commented Apr 22, 2023 at 20:26
• Another assumption of linear regression is that the conditional responses have approximately the same variance. That's a matter of the $n_ip_i(1-p_i)$ not appreciably changing with $i.$ If that's not the case, watch out!
– whuber
Commented Aug 23, 2023 at 19:24
• There’s also the logistic regression link function that squeezes the predictions into the unit interval. Your linear model will not do that and could make a prediction like $-0.2$.
– Dave
Commented Aug 23, 2023 at 20:50

One of the assumptions of the linear regression is homoscedasticity (constant variance). In the binomial model, the variance obviously depends on $$p_i$$, which is the variable you're trying to model. As long as your curve is reasonably flat, or, in other words, the $$p_i$$'s don't vary too much, this will probably not be an issue. But still, if I were to review your results, I'd probably ask why you hadn't used logistic regression. What are the benefits of using a model which you know is less suited than another? ("All models are wrong, but some are useful.")
• I see, it's a question of available software. If I were you, I'd check that, for your range of $n_i$'s and $p_i$'s, the differences between the binomial and normal distributions and the "variance of the variances" are below some pre-defined thresholds and, if they are, use the approximation. Using such approximations is not unusual in statistics: Pearson's chi-squared test, used on contingency tables, also approximates counts (integers!) with normal distribution. Commented Jul 1, 2020 at 7:09