Solved – Interpretation of binomial glm coefficients when response variable is not binary

binomial distributiongeneralized linear modelpredictive-models

I have modelled a, ticket :{sold , unsold} process using a glm with family = binomial as such:

> head(bob_new)
  ticketCount ticketsRemain artistRating artistVotes                      artistName             paintingTitle       date      day
1           9            21         4.38         616 Stella Mandrak-Pagani #TeamAjaz        Winter OWL in snow 2016-12-01 Thursday
2          10            10         4.23         401                       Meg Burns      Simi Cherry Blossoms 2016-12-01 Thursday
3          15            21         4.57         481                  Veronica Stach Where the Wild Things Are 2016-12-01 Thursday
4          21            13         4.35         100            Christine "Chri" Lee          Lust in the Wind 2016-12-01 Thursday
5          17             0         4.32         113         Nicole Pinder #TeamAjaz             Seagull Beach 2016-12-01 Thursday
6          24             1         4.48         657                Monique Ra Brent       Aurora on the River 2016-12-01 Thursday
     month   percent
1 December 0.3000000
2 December 0.5000000
3 December 0.4166667
4 December 0.6176471
5 December 1.0000000
6 December 0.9600000

> mod_new <- glm(cbind(ticketCount , ticketsRemain) ~ day + artistRating * artistVotes , data = bob_new, family = binomial)
> summary(mod_new)

Call:
glm(formula = cbind(ticketCount, ticketsRemain) ~ day + artistRating * 
    artistVotes, family = binomial, data = bob_new)

Deviance Residuals: 
   Min      1Q  Median      3Q     Max  
-5.611  -3.360  -1.961   1.374  15.680  

Coefficients:
                           Estimate Std. Error z value Pr(>|z|)    
(Intercept)              -2.962e+00  1.574e-01 -18.820  < 2e-16 ***
dayMonday                -3.492e-01  2.858e-02 -12.218  < 2e-16 ***
daySaturday               2.053e-02  2.658e-02   0.773 0.439783    
daySunday                 1.091e-01  2.812e-02   3.879 0.000105 ***
dayThursday              -8.847e-02  2.728e-02  -3.244 0.001180 ** 
dayTuesday               -4.237e-01  2.762e-02 -15.343  < 2e-16 ***
dayWednesday             -3.749e-01  2.875e-02 -13.037  < 2e-16 ***
artistRating              3.692e-01  3.522e-02  10.482  < 2e-16 ***
artistVotes               2.247e-03  4.153e-04   5.409 6.33e-08 ***
artistRating:artistVotes -4.686e-04  9.229e-05  -5.078 3.81e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 60681  on 3992  degrees of freedom
Residual deviance: 59649  on 3983  degrees of freedom
  (85 observations deleted due to missingness)
AIC: 66608

Number of Fisher Scoring iterations: 5

Now I am confused as to how one can interpret this output; when i learned the logistic glm see :https://stats.idre.ucla.edu/r/dae/logit-regression/
I understood that we have for covariates the log odds of success changing by $\beta$ for one unit of change in that covariate, and for factor variables we have an overall change in log odds. From here we can obtain an estimated probability of success curve : $\pi = \frac{e^{X\beta}}{1 + e^{X\beta}}$
Now this makes sense to me, when we have data with a binary random variable. such as admit:

##   admit gre  gpa rank
## 1     0 380 3.61    3
## 2     1 660 3.67    3
## 3     1 800 4.00    1
## 4     1 640 3.19    4
## 5     0 520 2.93    4
## 6     1 760 3.00    2

However how do we proceed when we have as response variable : cbind(ticketCount , ticketsRemain) (refer to > head(bob_new) above).

What type of predictions could I achieve here? In the UCLA admit example we can predict probability of admit.

I appreciate any explanation or references!

Best Answer

This is one of the ways how can you provide data for logistic regression in R (see also here). In this case you are modeling probability of selling ticket given the predictors. You are still estimating probabilities. Moreover, your data is still (conditionally) binomial, you are predicting $k$ successes vs $n-k$ failures provided as $(k, n-k)$ tuples -- this is just another way of representing the same data.

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