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I'm currently doing some analysis for a retrospective cohort study of biomarkers in cancer patients. I've noticed that some papers have utilized univariate cox regression analysis to generate a hazard ratio with confidence intervals. However in some of these papers, they've generated a single hazard ratio with CI's for categorical variables with more than 2 categories.

reference: http://www.ncbi.nlm.nih.gov/pubmed/25575264

My questions:

(1) How does one even generate a single HR for a variable with more than 2 variables? This doesn't seem to make sense?

(2) How does univariate cox regression compare to cox-mantel/log rank? Are they very similar?

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  • $\begingroup$ Log-rank tests are score tests for the hazard ratios from a Cox regression model with a single categorical predictor. Perhaps referencing one of the papers would help someone answer your first question. $\endgroup$ – Scortchi - Reinstate Monica Jun 30 '15 at 11:58
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    $\begingroup$ For a 3 category predictor in which no ordering of levels is assumed, you must have 2 hazard ratios. And why would unadjusted (univariable) hazard ratios be of any interest? $\endgroup$ – Frank Harrell Jun 30 '15 at 12:35
  • $\begingroup$ I've added the paper of interest. Traditionally in cohort studies in medical literature, univariate analysis is used to screen for variables to be used in multivariate analysis. Although there are pitfalls to this methodology, it remains the standard approach and therefore unavoidable if you want to be published. $\endgroup$ – PyPer User Jun 30 '15 at 12:40
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    $\begingroup$ The variables in question with more than 2 categories seem to be T, N, TNM stage, and the modified Glasgow Prognostic Score (mGPS). Most likely the authors treated all these as numeric rather than categorical variables in some of their tables, and the survival curves suggest that this use of numeric scores might have modeled the data fairly well in this case. $\endgroup$ – EdM Jun 30 '15 at 13:10
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    $\begingroup$ Taking TNM stage as an example, the issue is whether each unit increase of stage adds the same increase in hazard ratio, not whether stage is a continuous variable. Figs. 1A and 3A in the paper suggest that this might not be too bad a model for cancer-specific survival in this case, while Figs 1C and 3C might support your understanding that stages I and II have similar overall survival, both better than stage III. I don't think you can know a priori whether treating these variables as numeric is good or bad; that's a question to answer by examining how well the data are modeled. $\endgroup$ – EdM Jun 30 '15 at 14:24
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(1) One HR for multi-valued categorical variables

In the cited paper on colorectal cancer, some variables were listed as multi-valued categorical but only had single hazard ratios (HRs) presented in Cox proportional-hazards survival analysis. The variables in question were: T (tumor size), N (tumor spread to lymph nodes), TNM stage (a combined assessment of disease severity), and the modified Glasgow Prognostic Score (mGPS). These are ordered clinical classifications presented as integer values, following standard practice (although TNM stage is usually represented by Roman numerals and in some types of cancers there are subtypes a, b, etc, for certain classifications).

A single HR can be obtained for each of these classifications if they are treated as numeric values in the Cox regression. If each extra step along an ordered categorical scale contributes the same multiplicative change in hazard ratio, then there is no harm in treating that categorical variable as numeric.

Analysis as numeric variables is almost certainly why only a single HR is presented for each of these classifications; what is uncertain from my reading of the paper is whether this was a deliberate or an accidental choice. In a cursory reading of the paper I also did not find results of diagnostics or validations of the models that might have helped evaluate whether this was a good choice.

As you note, in colon cancer there is generally a large jump in mortality between TNM stage II and stage III, with much less of a jump between stage I and stage II. That would undercut the numeric analysis of stage. Figure 1C and 3C in the paper, on overall survival, seem to support your contention, but the figures on cancer-specific survival (1A and 3A) seem pretty well fit by an equal-step-per-class model.

Whether the numeric treatment of such an ordered classification is appropriate can best be addressed in light of how well it models the data. As models are all approximations to reality at best, it's possible that the errors introduced by such numeric treatment are balanced by other advantages, provided that the validity of the model is not adversely affected. For example, numeric treatment of such a variable has the advantage of only using one degree of freedom, rather than 2 or more if treated as a multi-level factor.

(2) Cox regression and log-rank tests

As @Scortchi put it simply in a comment: "Log-rank tests are score tests for the hazard ratios from a Cox regression model with a single categorical predictor." I find these lecture notes to be a short explanation directly on point, with useful extensions to different tests. The tests are asymptotically equivalent, although their results of course may differ in any particular data set.

Historical note: I discovered item (1) by accident when I was analyzing a data set to produce the unadjusted univariate hazard ratios that are still typically demanded by clinical journals despite their dubious usefulness. Having a large number of both categorical and numeric predictor variables, I wrote a script to automate the process, taking advantage of the ability of R to deal well with both types of variables. I was initially shocked to obtain only a single HR for each of T, N, and TNM stage. Of course these variables had been entered as numbers and, in my poorly designed script, R interpreted them as numeric variables even though I had been thinking of them as ordered categories. What was perhaps more surprising, until I thought about it, was that these numerically-treated categorical variables bore reasonable relations to outcome. Those HRs for the numeric treatment of the data made it into the final publication (carefully annotated as having been analyzed numerically, of course, along with reports on diagnostics and model validation).

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