Calculating relative importance of predictors in a poisson glm model

Question

This is the model I'm working with, some variables are int, some are num and some are factors.

Call:
glm(formula = FrecS ~ TenC + Sexo + Ori + Area + Sustrato + Enriq + 
Visit, family = poisson(link = "log"), data = cdv1)

Deviance Residuals: 
  1        2        3        4        5        6        7        8        9       10  
 1.3267  -1.4264  -2.4890  -4.2093  -0.7829   3.4657  -1.6681  -2.5281  -1.2699   1.6312  
 11       12       13       14       15  
 1.1201   0.8502   3.1807  -0.1062  -0.0718  

Coefficients:
                 Estimate Std. Error z value Pr(>|z|)    
(Intercept)      9.852230   0.766363  12.856  < 2e-16 ***
TenC            -0.070482   0.025940  -2.717  0.00659 ** 
SexoM           -0.831511   0.410043  -2.028  0.04257 *  
OriCaptive born  0.195265   0.486110   0.402  0.68791    
OriWild born    -0.800144   0.565062  -1.416  0.15677    
Area            -0.013258   0.001539  -8.616  < 2e-16 ***
SustratoMixto    4.322735   0.617623   6.999 2.58e-12 ***
SustratoNatural  0.835887   0.450573   1.855  0.06357 .  
Enriq           -0.033505   0.007799  -4.296 1.74e-05 ***
Visit           -0.002155   0.000240  -8.978  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 445.76  on 14  degrees of freedom
Residual deviance:  65.89  on  5  degrees of freedom
AIC: 143.32

Number of Fisher Scoring iterations: 6

How can I calculate de relative importance of the predictor variables? I want to be able to tell which of the predictors has a bigger impact on the FrecS variable. I've tried the relaimpo package in R, but it wont run if my model is not gaussian.

Can I simply state that the highest absolute value estimated coefficient is the most important predictor? Can I rank them using the p-value (lowest p-value means greatest importance)?

I've read about Wald z-statistic and Pratt index, but to be honest I'm still quite lost.

Answer 1

There are many possible ways to estimate relative importance as Ulrike Gromping, the developer of RELAIMPO, documents in her papers on approaches to estimating this metric. Her method and accompanying R module is one of the more sophisticated. Your first option is to recognize the nonconfirmatory and proxy nature of all of these approaches -- they are all approximations. Given that, why not ignore the gaussian assumptions of RELAIMPO and run your model through the package?

To @jay 's point, you can't analyze coefficients wrt relative importance that are expressed in the scale of their predictors. With that in mind, another approach would be to employ a widely used practice in classic, OLS regression, standardize the predictors and analyze the absolute values of the resulting coefficients.

Yet another approach would be to take the absolute values of, in your model, the Z-statistics, sum them up and then repercentage each abs parameter with that total. By ranking those relativized percentages, a viable heuristic for relative importance can be easily obtained.

Answer 2

Following could be of help https://github.com/dominance-analysis/dominance-analysis This package is designed to determine relative importance of predictors for both regression and classification models. The determination of relative importance depends on how one defines importance; Budescu (1993) and Azen and Budescu (2003) proposed using dominance analysis (DA) because it invokes a general and intuitive definition of "relative importance" that is based on the additional contribution of a predictor in all subset models. The purpose of determining predictor importance in the context of DA is not model selection but rather uncovering the individual contributions of the predictors.

In case the target is a continuous variable, the package determines the dominance of one predictor over another by comparing their incremental R-squared contribution across all subset models. In case the target variable is binary, the package determines the dominance over another by comparing their incremental Pseudo R-Squared contribution across all subset models.

Answer 3

Agree with @DJohnson and I will expand on his answer a bit.

There has been research on a RELAIMPO-based approach (i.e., dominance analysis) as applied to logit regression (also similar other approaches).

Given that the Poisson and Logit model differ, in terms of the Generalized Linear Model, only in their link function (log as opposed to logit) and probability distribution (Poisson as opposed to Bernoulli), the solutions applied to the logit regression should hold on Poisson regression when changing these two model specifications.

The dominance/RELAIMPO approach here should permit a reasonable way to obtain a rank order and stay within the Poisson model.

The only implementation of a dominance approach which accommodates generalized linear models (not specifically logistics), that I know of, is available in Stata.

References

Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological bulletin, 114(3), 542.

Azen, R., & Traxel, N. (2009). Using dominance analysis to determine predictor importance in logistic regression. Journal of Educational and Behavioral Statistics, 34(3), 319-347.

Tonidandel, S., & LeBreton, J. M. (2010). Determining the relative importance of predictors in logistic regression: An extension of relative weight analysis. Organizational Research Methods, 13(4), 767-781.

Luchman, J. (2015). DOMIN: Stata module to conduct dominance analysis.