Infering relative importance of features in driving a dependent variable

Question

In an auctions website I maintain, auctions are listed most-recent-first. There are 20 auctions per page. A user can click next in the footer to view older auctions.

It's early days so there's currently no other way to search an auction. The most-recent-first view is the only discovery mechanism. I do provide a filter along cities, but that too in most-recent-first ordering.

I need to infer which features are driving bid submission in this auction website (so that I can improve bid submission rates).

My dataset comprises auctions that were all alive for 7 whole days. My initial plan was to apply logistic regression to this dataset with unique_bids_per_day as the dependent variable. I had a very useful discussion about that here.

In a nutshell, I was advised that if the dependent variable followed the Bernoulli or Binomial distribution, then logistic/binomial regression could be useful. So I did a quick analysis to check distribution of unique_bids across the 7 days an auction is live. Results suggest that most bids come within 24 hours of auction submission (or creation). I.e.

This is unsurprising given how the website is organized (described at the start).

So wouldn't this mean that unique_bids_per_day is not following a binomial distribution (the probability of getting a bid is not uniformly distributed over the life of an auction)? And if that is the case, that would jeopardize using logistic regression in this type of scenario. So then what should I do to infer which features are driving bids? Would be great to get an illustrative answer.

Note: features are categorical and numeric both

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This is the head of the data (summarized; the actual data has more features). unique_clicks_per_day is actually unique_bids_per_day.

This is the natural log of days_since_submission. Looks slightly bi-modal:

Answer 1

I am not well versed in Matlab but I am confident that you can run a generalized linear model in the program. A quick Google search led me to this site with what looks like relevant code for such a model.

I'll walk you through a Poisson regression using R. The basic properties and output of the analysis should be the same.

First, I simulated some truncated data (I tried to give it some similar properties to your data set):

> head(dat)
  total_clicks descrp
1            4     23
2            1     19
3            3     22
4            7     21
5            0     14
6            1     15 

My goal here is to determine whether total clicks over the 7 days can be predicted by the number of words in the description.

Here is the model and output from R:

> fit<-glm(total_clicks~descrp, family = 'poisson', data=dat)
> summary(fit)

Call:
glm(formula = total_clicks ~ descrp, family = "poisson", data = dat)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.7788  -0.7558  -0.1032   0.6439   2.5157  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)   
(Intercept)  0.50713    0.26060   1.946  0.05165 . 
descrp       0.03516    0.01250   2.812  0.00492 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 116.82  on 99  degrees of freedom
Residual deviance: 108.88  on 98  degrees of freedom
AIC: 402.55

Number of Fisher Scoring iterations: 5

Briefly, the model suggests there is a positive relation between the number of words in a description and the number of unique clicks received. However, this value does not have an obvious interpretation. We have to convert it back to the units of measure (instead of taking the log - which is the link function for the Poisson regression - we exponentiate this coefficient)

In R code this is how you exponentiate:

> exp(fit$coefficients)
(Intercept)      descrp 
   1.660519    1.035782 

This gives you a value for the intercept (which is the estimated frequency of clicks when your predictor or predictors equal 0) and the slope. Some call this latter exponentiated coefficient the incident rate ratio (IRR) others refer to it as the event rate ratio (ERR). Regardless of the terminology used the interpretation is the same. For each additional word in a description the click rate increases by a factor of 1.036. Or another way of thinking of this value that may be more intuitive is that each additional word is related to a 3.6% increase in the frequency of clicks (on average).