Relationship between logistic regression and linear regression

I've encountered a problem where I need to analyze the relationship between a movie's length, a movie's price and it's sale on a video streaming platform. Now I have two choices to quantify sale as my dependent variable:

1. whether or not a user ended up buying the movie
2. selling rate (# of people buying the movie / # of people watched the trailer)

if I use selling rate I essentially would use a linear regression where I have $$selling\ rate= \beta_0 + \beta_1*length + \beta_2*price + \beta_3*length*price$$

But if I'm asked to use option 1 where my response is a binary output, and I assume I need to switch to logistic regression, will my coefficients change and how will them change? How would the standard error change? Will the standard error be an underestimate?

This is just a discussion question and I don't have actual data for it, I'm confused on how is logistic and linear regression related and whether they are interchangeable in some way?

As both scenarios represent a binary choice of buy/didn't, both would best be modeled with logistic regression. Linear regression is designed for continuous outcomes that are theoretically unrestricted in values. That not the case for either scenario.

The second "selling rate" model might be interpreted in two ways. If all who bought also had watched the trailer, then its outcomes are restricted to the range [0,1] and a logistic regression makes sense. If people can buy without watching the trailer, how do you deal in a linear regression with an infinite "selling rate," as you defined it, if no one happened to watch the trailer but at least one bought? The latter interpretation of the second scenario would also best be modeled by logistic regression, with watchedTrailer as a predictor in the model and including all visits to the site in the data set.

It's hard in general to associate linear and logistic regression coefficients, as they represent different things. For linear regression it's the change in value of the outcome per unit change in a predictor, here a change of probability. But probabilities can't be outside of [0,1]. For logistic regression it's the corresponding change in log odds of the probability of the outcome, with values covering [$$-\infty,\infty$$].

That said, sometimes people do formulate a linear probability model, with probability as the outcome variable in a linear regression. If you do use a linear probability model, you can interpret a regression coefficient in terms of the fractional change in probability associated with a unit change in the predictor. If probabilities are in the middle range near 50/50 that can work OK and results might be similar to a logistic regression. But with probabilities near the edges of [0,1] a linear probability model can make predictions of probabilities outside that theoretically allowed range.

So use a logistic regression or other modeling approach that takes the [0,1] limits of probability into account when you encounter these types of scenarios.

• Does it mean that since selling rate takes range [0,1], I can treat it like a probability and model it use a logistic regression? Aug 8 '20 at 16:44
• @RoxS yes. Logistic regression software like glm() in R allows for outcomes to be individual 0/1 results or aggregate counts of yes/no. So if you have aggregated data for each movie (its covariates, the number who saw the trailer, and the number who bought) then you can simply do a logistic regression once you have the data in the proper format. For example, glm() expects outcomes for aggregated data to be in a 2-column matrix of success counts and corresponding failure counts. But do that with the raw counts, to take into account the greater information provided by more counts.
– EdM
Aug 8 '20 at 17:04
• What if people buy the movie without watching the trailer? Aug 8 '20 at 19:47
• @kurtosis the "selling rate" outcome proposed in the OP wouldn't make sense in that situation. In the answer I nevertheless suggested modeling with watchedTrailer as a logistic regression predictor if individuals can buy movies without watching the trailer. That would be most useful with individual-level data, but you might even get some information with aggregated data on numbers/proportions who watched
– EdM
Aug 8 '20 at 19:56
• Beta regression would be an option for the selling rate? Aug 9 '20 at 3:31