# Relative importance of predictors - Standardized coefficients in Ordinal Logistic Regression

I am a 4th year psychology student. I need some help in understanding the coefficients in ORDINAL logistic regression. According to Williams (2009) "Using Heterogeneous Choice Models To Compare Logit and Probit Coefficients Across Groups", the predictor variables and residuals are already standardized to the logit distribution (variance = π*π/3 ), and, therefore, so are the reported coefficients in SPSS. Therefore, in order to compare the relative predictive strength of my variables in the model, I should just be able to directly compare the coefficients (or the odds ratios). However, how do I account for the differences in CI/standard error in my comparisons?

For example,

variable 1: B=.021, std error = .0068, Exp(B) = 1.022, 95% CI = 1.008 to 1.035

variable 2: B=.051, std error = .0174, Exp(B) = 1.052, 95% CI = 1.017 to 1.089

From comparison of Bs - Variable 2 is the stronger predictor, but the std error and CI are much larger. So, what conclusion can I make?

• An addendum: I am thinking of simply converting my variable 1 and Variable 2 to z-scores and comparing their coefficients. Unfortunately, one of the variables has a skewed distribution. Although normality is not an assumption of ordinal regression, I wonder whether it will still bias the betas and the CIs? – Anna van Moorst Sep 4 '16 at 15:02

In general, if your predictors are on different metrics, then the subjective assessment of variable importance can not be easily made by simply comparing the raw sizes of the odds ratios.

If all your predictors are continuous, then I think converting the variables to z-scores would be useful for getting a sense of their relative importance. You mention that you have a skewed numeric predictor. I don't think changes anything too much for whether z-scores are appropriate. Ultimately, you have a separate issue of whether you want to apply a shape transformation (z-scores just change mean and variance). If your variable is highly skewed, then consider a transformation, and then z-score the transformed variable.

Some times, you have binary predictors. In that case, the 0-1 scoring is quite intuitive, especially if you have a few such variables.

If i understand you correctly, and if you have two models to choose from with relatively different Betas, the important is too choose the model based on its accuracy rather than merely strength of Betas... I suggest you compare the models based on other fit indicators such as AIC, BIC as well as the classification accuracy.

The confidence interval and std errors of Individual Betas, are not a good indicator when it comes to what model to choose.

But if you are speaking about two Beta's within the same model, the predictive power is measured through it's coefficient and it's P- value rather than its CI or Std Error of betas.

Also there's a good reponse to similar question of yours here: How to interpret coefficient standard errors in linear regression?

• Thank you for your answer, but my model is an Ordinal regression (cumulative logistic model) not Linear regression - completely different beast, In linear regression, I can make a comparison using Betas, which are standardized coefficients. I can also use the adjusted R^2 as effect size. For Ordinal Regression, I am not entirely sure that the beta coefficients reported are standardized - although Williams (2009) indicates that it is so. – Anna van Moorst Sep 4 '16 at 12:29
• However, e.g. Kaufman (1996) - Comparing Effects in Dichotomous Logistic Regression: A Variety of Standardized Coefficients", indicates that it is not so. Williams also suggests full-standardization as a way of addressing this issue, but he employs a program in Stata, and I don't have access to Stata, nor time to learn it, unfortunately. I am currently looking at whether I can use Kaufman's formula to calculate the standardized coefficients for a series of binary logistic regression models that I broke my original cumulative logistic model into. Maybe that'll give me the response I need. – Anna van Moorst Sep 4 '16 at 12:29
• ... So I've played with the Kaufman's formula for standardized coefficients, and I can't get it to work for cumulative logistic regression. The coefficients that it produces are very different to the coefficients I get if I run my original model using predictors converted to z-scores. I'd expect the coefficients from Kaufman's formula to be at least somewhat in the ballpark of z-score predictors. Instead, they are closer to the non-standardized betas. – Anna van Moorst Sep 4 '16 at 14:53
• Yes.. you are right the coefficient in ordinal or logit regression are not standardised... try this.. there is a formula for standardized logit regression Betas (manual calculation, not sure if this formula is the same as the one you used from Kaufman): www3.nd.edu/~rwilliam/stats3/L04.pdf – RomRom Sep 5 '16 at 11:17