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I'm reading a paper that uses survival regression to examine hospital length of stay, and then assigns variable importance based on the contribution of the deviance for each variable to model fit. They did not give any references for this, and I am having difficulty understanding their conclusion, based on their table.

Their method section states:

Table 2 reports the results of a Weibull parametric survival regression. The exploratory variables were entered in five sequential blocks, based on unadjusted deviance for that block. Diagnostic category was entered first, hospital setting second, age third, race fourth, gender fifth. The Log Likelihood (LL) statistics in Table 2 are reported for each block entry. The deviance statistics, calculated as -2LL, are interpreted as the block’s relative contribution to the model fit. By this criterion, hospital setting made the largest unadjusted contribution to model fit (631.8), followed by age at admission (397.1), diagnosis (371.7), race (97.7) and gender (24).

In their Table 2, they list the deviance statistics as:

  • Diagnosis - 1480.7
  • Hospital - 1263.6
  • Age at Admission - 794.3
  • Race - 195.5
  • Sex - 48.1

I'm asking for a reference, because a) my understanding is that "variable importance" is a complicated topic, and b) I can see how they calculated their deviances from their log-likelihoods, but I cannot see how they proceeded from their deviance value for Diagnosis (1480.7) in their table to what they stated in their results (the paragraph above, where they refer to the value of Diagnosis as 371.1). Every other deviance value matches what I obtain when I look at the log likelihoods statistics in their table.

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a) Correct. Variable importance is not exactly an statistical concept.

b) Moreover, you gotta be careful about the techniques used in medical journals as they worry more about the epidemiological question than the (validity of) statistical methods.

What you need to do here is to conduct a formal variable selection and, if you are Bayesian, calculate the posterior probability of each model to see how much each variable affects this probability. There is no unique way to assess the importance of a variable, though, since "importance" is not uniquely defined.

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  • $\begingroup$ Thanks for your reply! re: variable selection, my chief concern as a clinician is that the authors of the article didn't really include any clinical variables. Do you have any references for any Bayesian methods specific to survival analysis, or would any general method used in regression to assess model fit be appropriate? $\endgroup$ – Woodstock Nov 10 '17 at 23:54
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    $\begingroup$ @Woodstock It doesn't matter that they didn't include "clinical" variables. The point here is that they take "importance of a variable" as a well known thing, which is not. In a Bayesian context, you can quantify the probability of the posterior probability of a model with a variable X vs the probability of a model without that variable. It is a more complicated than this, but the wikipedia article for Bayes factors may give you some idea: en.wikipedia.org/wiki/Bayes_factor. Yet, this just one way the variable is important. $\endgroup$ – VIV Nov 10 '17 at 23:59
  • $\begingroup$ Thanks again, esp. for the reference. I should have thought to check Wikipedia, ha ha. I'm familiar with ANOVA and AIC for model comparisons, so I can examine the Bayes factor further. $\endgroup$ – Woodstock Nov 11 '17 at 1:26

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