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So I'm wresting with some homework from survival analysis. It's going well I just need a little clarification.

Problem statement: The data set “hepatocelluar” is in the “asaur” package. It contains 17 linical and biomarker measurements on 227 patients, as well as overall survival and time to recurrence, both recorded in months. There are three measures of CXCL17 activity, CXCL17T (intratumoral), CXCL17P (peritumoral), and CXCL17N (nontumoral). There is a particular interest in whether they are related to recurrence-free survival. Which of the three measures of CXCL17 activity is most strongly related to recurrence-free survival (the outcome), after adjustment for age and sex? You may assume linear functional shape for age, and run each predictor separately (with age and sex) assuming linear functional shape.

Question:

When it states to run each predictor separately does that mean fitting three Cox-Regressions with one of the three CXCL17 measures along with age and sex or does it mean I should assume no interaction between the CXCL17 measures and therefore do one Cox-Regression with a "+" between the measures?

library(asaur)
library(survminer)
library(survival)
library(rms)
### Load data
data("hepatoCellular")
hc <- hepatoCellular
hc$Genderf <- factor(hc$Gender)
#Get the time-to-recurrence in years:
hc$RFSy = hc$RFS / 12
### Three separate
hc.cox = coxph(Surv(RFSy, Recurrence) ~ CXCL17P + Age + Genderf, data = hc)
hc.cox2 = coxph(Surv(RFSy, Recurrence) ~ CXCL17T + Age + Genderf, data = hc)
hc.cox3 = coxph(Surv(RFSy, Recurrence) ~ CXCL17N + Age + Genderf, data = hc)
### One regression
hc.cox4 = coxph(Surv(RFSy, Recurrence) ~ CXCL17N + CXCL17T + CXCL17P + 
              Age + Genderf, data = hc)

And then to determine the measure most strongly related to recurrence-free survival, let's take hc.cox4 as an example:

cox4

Isn't the smallest value (CXCL17N) the one most associated with recurrence-free survival since they have a lower risk of death? The p-value is pretty high though and the estimates of the coefficient are pretty small too. Any help with interpreting these results would be appreciated!

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    $\begingroup$ Please add a 'self-study' tag. $\endgroup$ – mkt - Reinstate Monica Mar 27 '18 at 13:23
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    $\begingroup$ You will likely need to do more than compare p-values to get the right answer. Look into Akaike's Information Criterion. $\endgroup$ – Todd D Mar 28 '18 at 2:01
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1st question: The assignment says "run each predictor separately (with age and sex) assuming linear functional shape." Our task is as much a matter of interpreting the assignment as of applying statistical knowledge. But it sounds as if you should run 3 regressions, each using only 1 measure of CXCL17 activity. Presumably this is because including more than one of them in the same model will muddy results; the 3 measures must be too highly correlated.

2nd question: The assignment asks, "Which of the three measures of CXCL17 activity is most strongly related to recurrence-free survival (the outcome), after adjustment for age and sex?" That is, assuming the outcome is coded as 0 = Recurrence-Free Survival, than what's desired is to identify the predictor with the lowest (most highly negative) coefficient...or the one with the lowest exponentiated coefficient (odds ratio or "exp[coef]" far below 1.0). If the outcome is coded in reverse, then look for the largest positive coefficient and an odds ratio that exceeds 1.0 by the greatest amount.

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