# Using logistic regression to infer relative importance of features

I need to ascertain what features drive users to submit more bids for items in my auction website.

I have a dataset where # of bids is the dependent variable, and there's several measurements of features in other columns. Some of these columns would be highly correlated to driving # of bids.

I'm planning on running a logistic regression on this data set. I'll classify the # of bids columns as high or low depending on whether bids attracted were above median or below median.

My question is simply this: is running logistic regression in a scenario like this correct? OR is logistic regression ill-suited for measurements of this sort (in which case, another question would be what's a better technique to use).

• Why categorise the counts? Why not use Poisson regression on them. Categorisation wastes information. Aug 2, 2017 at 11:33
• @HassanBaig You should look into the vast literature on generalized linear models (GLMs). There are numerous books available on the topic, a classic being the book by McMullagh and Nelder. Understanding this class of statistical models will help you understand when you should use one type of models versus another. Aug 2, 2017 at 12:04
• @JasonMorgn: I've read "Intro to Statistical Learning" once (James, Witten, Hastie, Tibshirani). I'll give GLM by McCullagh and Nelder a shot. Aug 2, 2017 at 12:14
• I personally prefer the book by Dobson & Barnett. Theory is important but your focus is on application so you need a textbook with concrete examples. Aug 2, 2017 at 12:40

The way you're applying logistic regression won't be of much value. Logistic or binomial regression would only be useful on your scenario if you were to regress a proportion, not a categorised variable. A typical regression analysis in companies like eBay, FB, or Google would be to use Binomial regression to predict/explain the conversion rate, i.e. the ratio of "conversions" per "views".

If you had a metric similar to "views" that, combined with your bids metric, would be modelled as a discrete random variable that followed the Bernoulli or Binomial distribution, then logistic regression would be very much useful to regress the probability of conversion.

Without such metric, you can still use a simple linear model, a log-log model, or a Poisson model (as mentioned in the comments section), though they will probably be less beneficial than a logit model.

• I can't cleanly calculate views since multiple auctions appear as a list together. One can then scroll all the way down and skip to the next page of auctions. But what I do have is the time period, in seconds, the auction lived for. Do you suppose I could use that with the bids metric? Aug 2, 2017 at 12:09
• You don't need a bid-per-view match so I think you're good. In Binomial regression, it's common to group input samples in a way that each group tells you that y bids occurred among n views. There are many ways to create those groups (it can be temporal). So yes, it sounds like your data is sufficient in order to derive the proportion of grouped bids per views e.g. within a certain time period. NB: be sure to check for the presence of autocorrelation in views and bids between temporal groups. Aug 2, 2017 at 12:20
• Follow up question: my data-set isn't a time series, it's cross-sectional (the x-axis isn't t, it's bids/unit_of_time. So under normal circumstances, would it be autocorrelation proof, or no? Aug 2, 2017 at 13:18
• This will depend entirely on your grouping. If you were to aggregate bids and views on a daily period, then you would likely observe some significant autocorrelation. If you were to aggregate them on a weekly period, you would most likely observe less or none. If you were to aggregate them in a way that would make up for seasonality (e.g. group 1: sum of bids per Monday, group 2: sum of bids per Tuesday, etc.) then you would most likely have none. Either way, you can always either check for the presence of autocorrelation using an ACF plot or the Ljung-Box test. Aug 2, 2017 at 13:24
• Continued: And for the cases where mild autocorrelation exists, you can easily make up for it by adding a "time" variable into your model (e.g. a variable of integers in incremental order). There won't be a need for a time series model unless you actually want to forecast bids as a function of time but that's a whole different story. Aug 2, 2017 at 13:27

The comments and answers are very helpful in terms of addressing the OP's explicitly stated 'simple question' regarding an appropriate modeling framework but, unless I'm missing something, none of them provide direction about how to evaluate the relative (not relevant) importance of features in the resulting model(s). This concern was articulated both in the title as well as the very first sentence of this question. So, my reading is that the OP really has two questions: the first is manifest and has already been answered, while the second was not explicitly stated, is therefore latent and remains unanswered. This response will attempt to address this second, latent question.

Many heuristics for evaluating relative importance of features have been proposed in the literature. Ulrike Gromping's papers and R module -- RELAIMPO -- are probably the single best source for both a thorough literature review as well as evaluatory software (RELAIMPO stands for 'relative importance of regressors', e.g., see here ... https://prof.beuth-hochschule.de/groemping/software/relaimpo/).

Some heuristics involve adding up the values of metrics output from the model and then repercentaging to create a relative ranking of features that sum up to 1 (or 100%). For example, one widely used method is to standardize continuous regressors onto a unit scale (mean=0, std dev=1) followed by evaluating, repercentaging and ranking the absolute values of the resulting beta coefficients from the model. Another related method is not to standardize features but to sum up the F- or t-test values (the absolute values in the case of the t) output from the model and repercentaging those. The logic of this latter approach is that the beta coefficients are expressed in the units or scale of the underlying, unstandardized feature while the F- and t-values are standardized metrics.

Both of these approaches are limited by two assumptions: first, that a single pass at the data can be fully informative about the behavior of the model and, second, that one of the four types of estimable functions is correct -- usually the type 3 assumption of independence of the regressors (e.g., see this SAS document, Four Types of Estimable Functions, https://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#introglmest_toc.htm). Workarounds concerning the first assumption of reliance on a single pass at the data leverage iterative and approximating tools such as the bootstrap-, jackknife- or random forest-type algorithms to obtain more precise and robust 'central tendencies' wrt model parameters. Second, rotation through the four different approaches to estimating functions can be employed as an additional control on these central tendencies. See Gromping's papers for greater detail about how these latter methodologies are to be employed.

Personally, I like the iterative and approximating approaches as they enable a type of empirical 'simulation' of differing data landscapes, enabling more robust estimates. And while I can see the value in rolling through the four types of estimable functions insofar as it is exhaustive and thorough, it still seems like unnecessary overkill from a purely computational and resource utilization perspective.

Given that, the key questions become ones of evaluating the asymptotic accuracy and performance of both the iterative, approximating algorithms as well as inclusion of tests for the four types of functions. Chen and Xie's paper, A Split-and-Conquer Approach for Analyzing Extraordinarily Large Data, addresses the statistical question of the accuracy of these iterative algorithms and concludes that they are, for the most part, delivering quite comparably accurate results (see here ... http://www3.stat.sinica.edu.tw/sstest/oldpdf/A24n49.pdf). Regarding the second concern with adding the four types of functions, my own comparison of RELAIMPO-based relative importance with methods that do not test for the four types suggests that there is little to be gained from their addition.

• Thank you for this great answer on relative importance of features. Heuristic/empirical approaches are particularly helpful in nonparametric estimation scenarios. The beauty with GLM is that the mathematical properties of the model are usually sufficient to assess importance and causal inference without reverting to heuristic approaches. The issue with explaining all that in a single answer would be the fact that every such model, be it Poisson or Binomial or log-log, has different structure and interpretation approach. Aug 4, 2017 at 14:31
• [continued] Some of these approaches such as the infamous Odds Ratio have been extensively covered in many community answers and I think anything but providing credible references to the OP would take us off topic. I find the textbooks suggested in the comment section to be a great resource. Aug 4, 2017 at 14:35
• I should add that, as someone with professional hands on experience on the OP's scenario, I could tell from the start what he was looking for. It's true that if you isolate the title from the body of the question, the term "relative importance" can be misleading. "What features drive users to submit more bids" is right on the spot, but features acting as drivers has to do with parametric causal inference and not relative importance per se. Aug 4, 2017 at 14:39
• I still think this is a great answer because the native way of interpreting a GLM can be time consuming and difficult to automate - if at possible when it comes to handling a great number of features. In this R&D scenario where a full automation is required, the references provided by @DJohnson's can be very useful (though in the Industry this is mostly handled with Deep Learning). Aug 4, 2017 at 14:48
• @digio Thank you for your several comments. You make one point with which I don't agree, "The mathematical properties of the model are usually sufficient to assess importance and causal inference without reverting to heuristic approaches." In my view, the power of iterative, approximating tools is that 1), they explicitly acknowledge the limitations of any and all models possessing an error term, including GLM. 2), Regardless of the "mathematical properties" of GLM models, as the underlying data evolves across the finite data samples, so will the estimates resulting from the model evolve. Aug 5, 2017 at 11:34