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I'm constructing a Cox regression model where patients were randomised to treatment or placebo, and then had certain continuous covariates measured at multiple time points through the course of the study. Risk of developing arthritis is being modelled.

Dependent:
Arthritis status

Independents:
Treatment (ordinal)
CII (continuous, measured at 3 separate timepoints)
GlyCII (continuous, measured at 3 separate timepoints)
HOClCII (continuous, measured at 3 separate timepoints)

As these covariates change over time, I've created time-segmented variables as shown here (SPSS syntax):

COMPUTE T_COV_CII = (T_ < 3) * CII.0 + (T_ >= 3 & T_ < 9) * CII.6 + (T_ >= 9) * CII.12.

Where T_COV_CII is my new variable which uses the measurement at 0 months (CII.0) when time is less than 3 months, the measurement at 6 months (CII.6) when time is between 3 and 9 months, and the measurement at 12 months (CII.12) when time is later than 9 months. As suggested here:http://www-01.ibm.com/support/knowledgecenter/SSLVMB_20.0.0/com.ibm.spss.statistics.help/idh_coxt.htm

Here are my hazard ratios:
Treatment                3.32
T_COV_CII             344.0
T_COV_GlyCII        0.000
T_COV_HOClCII     0.9

I know how to interpret hazard ratios for time-independent variables like my treatment, but how to I interpret hazard ratios of my time-segmented variables? Thanks for your help.

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The interpretation is essentially the same as with non-timevarying covariates. The model is $h(t|Z(t)) = h_0(t) \exp(\beta Z(t))$, so $\exp(\beta)$ measures the hazard ratio of having a "current" value of $Z$ that is one unit larger. This effect is modeled as being time-invariant. Given the extreme hazard ratios in your model, your predictors likely have a small dynamic range (so in reality you never see a one unit increase).

I also want to note that the incorporation of time-dependent covariates the way you do it is only really proper for extrinsic covariates, not within-subject biomarkers. There are a lot of subtle issues when the same underlying process affects the covariates and the event occurrence. There is an extensive literature on joint modeling of longitudinal and time-to-event outcomes dedicated to doing it properly.

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  • $\begingroup$ Thank you, that is very useful. Firstly yes you are right, the scale of my variables is mostly under 1 unit so I'll probably use a log2 transformation to interpret the change in hazard given a doubling of the variable. With regards to your second point, I'm trying to use a joint model as described here: r-bloggers.com/joint-models-for-longitudinal-and-survival-data but I'm unsure how to incorporate multiple covariates instead of just the one that this webpage shows. I don't suppose you could offer any advice? Thanks again. $\endgroup$ – Hefin Nov 22 '15 at 15:45

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