Cox Proportional Hazards – Formatting Data for Cox PH with Time-Dependent Covariates

cox-modelproportional-hazardstime-varying-covariate

I was hoping for some guidance on the appropriateness of my modeling approach to the following problem.

Problem: I'd like to know whether the days receiving nutrition support (cumulative days on support; i.e., SuppSum) in Hospital bears any influence on the risk of developing an infection (Status). In addition, I wanted to incorporate 2 sparsely recorded time-depedent covariates into the model, these being blood glucose concentrations (Glu) and systolic blood pressure (SBP). I believe the most appropriate model to help answer this question is a Cox PH with time-dependent covariates.

The generic call to the Cox PH function for this problem (I believe) would look something like this:

coxph = coxph(Surv(Start, Stop, Status) ~ Age + Sex + SuppSum + Glu + SBP, data=data)

N.B.: Data table is included below

In an attempt to format the data into a counting process, I have included some example data below for 2 subjects. Subject 1 gets an infection on Day 12 of their hospital stay, whereas Subject 2 leaves the hospital on Day 7 without developing an infection.

Row| ID Start Stop  Status Age  Sex  Supp SuppSum  frac     Glu  SBP
--------------------------------------------------------------------
1  | 1  0     1     0      29   0    1    1        100.0%   ...  ...
2  | 1  1     2     0      29   0    1    2        100.0%   4.4  121
3  | 1  2     3     0      29   0    1    3        100.0%   ...  133
4  | 1  3     4     0      29   0    1    4        100.0%   5.0  125
5  | 1  4     5     0      29   0    1    5        100.0%   ...  ...
6  | 1  5     6     0      29   0    0    5        83.3%    ...  143
7  | 1  6     7     0      29   0    0    5        71.4%    4.3  ...
8  | 1  7     8     0      29   0    0    5        62.5%    4.5  113
9  | 1  8     9     0      29   0    1    6        66.7%    6.1  ...
10 | 1  9     10    0      29   0    1    7        70.0%    ...  125
11 | 1  10    11    0      29   0    1    8        72.7%    ...  ...
12 | 1  11    12    1      29   0    1    9        75.0%    5.1  ...
--------------------------------------------------------------------
13 | 2  0     1     0      45   1    1    1        100.0%   5.0  ...
14 | 2  1     2     0      45   1    1    2        100.0%   ...  ...
15 | 2  2     3     0      45   1    1    3        100.0%   4.7  ...
16 | 2  3     4     0      45   1    0    3        75.0%    ...  ...
17 | 2  4     5     0      45   1    0    3        60.0%    4.9  ...
18 | 2  5     6     0      45   1    0    3        50.0%    ...  121
19 | 2  6     7     0      45   1    0    3        42.9%    4.3  123

Questions

Have I set up my datatable correctly for a time-dependent Cox PH model (I plan to perform the analysis using R)?
Given that my time-dependent covariates, Glu & SBP, are not collected on each day of the subject's hospital stay, how would R handle this when running the coxph() function?
- Would it simply remove all entries row-wise until only entries with both a Glu & SBP are present?
- If so, would that mean my Start and Stop intervals would be discontinuous?
- Or is there something else that I'd need to consider to ensure my data is formatted correctly?

Thanks in advance for your help!

Edit: 27/03/2018

Thank you everyone for your help. That has helped clarify things for me. I will explore using the LOCF approach, and take a look further into a Joint Modeling approach.

Out of curiosity, if the default behaviour of the coxph() function in R is to remove cases row-wise if there is missing data, my original example table would look like this:

Row| ID Start Stop  Status Age  Sex  Supp SuppSum  frac     Glu  SBP
--------------------------------------------------------------------
2  | 1  1     2     0      29   0    1    2        100.0%   4.4  121
4  | 1  3     4     0      29   0    1    4        100.0%   5.0  125
8  | 1  7     8     0      29   0    0    5        62.5%    4.5  113
--------------------------------------------------------------------
19 | 2  6     7     0      45   1    0    3        42.9%    4.3  123

What then are the consequences of running the time-varying form of the cox model on a dataset such as this? i.e., with discontinuous interval in the start & stop columns?

Does this violate the model? Or simply reduce my power because of a much smaller amount of observations?

Thank you again!

Best Answer

Yes, your data format is ok.
I suggest two methods. 1) If value of Glu or SBP is absent at the measurement, use the most recent available one. Then, perform time-dependent Cox regression.

Example based on your sample table)

Row| ID Start Stop  Status Age  Sex  Supp SuppSum  frac     Glu  SBP
--------------------------------------------------------------------
1  | 1  0     1     0      29   0    1    1        100.0%   ...  ...
2  | 1  1     2     0      29   0    1    2        100.0%   4.4  121
3  | 1  2     3     0      29   0    1    3        100.0%   4.4  133
4  | 1  3     4     0      29   0    1    4        100.0%   5.0  125
5  | 1  4     5     0      29   0    1    5        100.0%   5.0  125

2) Consider Joint model. Joint model make mixed model for repeatedly measured values, then use it as covariate in Survival model. There is merits for handling missing data and Censoring. Below links provide simple tutorial for Joint model.

https://www.r-bloggers.com/joint-models-for-longitudinal-and-survival-data/

http://jmr.r-forge.r-project.org/

Related Solutions

Solved – How to use time dependent covariates with cox regression in R

Its hard without seeing your data, so I'll try making it generic. First of all, the two main ways that a data frame should look like for the use in the survival package:

The bare-bones

ID - a unique variable to identify each unit of analysis (e.g., patient, country, organization)
Event - a binary variable to indicate the occurrence of the event tested (e.g., death, , revolution, bankruptcy)
Time - Time until event or until information ends (right-censoring). The Cox model is best used with continuous time, but when the study is over the course of years (especially regarding countries) monthly spells can do.
(Oftentimes) Some covariates

Lets use a made-up model of trying to find the hazard for countries falling into civil war (the event) over ten years (in monthly spells). Using a single covariate (previousCivilWar) which is not time-dependent:

# the first country was censored before an event and the second 
# experienced the event after 8 years
id time event prevCivilWar 
 1  120     0   0 
 2   96     1   1

Adding time-dependent covariates: Method 1

Covariate - In this case you need to know the original value, and whether it changed and to what - and if so, when (at what spell).
Changing the time variable to start and end - when needed to indicate the time of change for (any of the) covariates

Here we will add the binary variable to indicate >40% poverty is40pov:

id time1 time2 event prevCivilWar is40pov
 1    0    80   0               0       0
 1   80   120   0               0       1
 2    0    24   0               1       0
 2   24    60   0               1       1
 2   60    96   1               1       1

When using time-dependent covariates we need to specify the exact time frame until any change in any covariate occurs. Note that the times need to overlap. If a certain subject has no changes in any covariates, than one row suffices.

Method 2 - best for continuously changing covariates This will include $k$ rows per unique ID as there are spells ($k$ rows if censored, or less if the event happened before). So, if you have a database with information on a certain time frame studied, decide on the spell-length that makes sense to you (which makes theoretical sense): If the covariates change on an hourly basis - make it hourly, etc... Once you have decided on the spell-length (e.g., month) and the total time (e.g., ten years) than each ID will have <=$120$ spells.

If you need, create the longitudinal dataset with empty (NA, 0, or whatever) data, for the time-dependent covariates, and make two extra utility columns for dates/times of each spell. Then you can access the database and fetch the specific values for your covariates at those dates/times and fill it in. It is OK if certain rows have no changes in covariates. You will end up with something like:

# The variable pov is the poverty percent of population and measured monthly
id time1 time2 event prevCivilWar pov
 1    0     1   0               0   0.34
 1    1     2   0               0   0.34
 ...
 1   79    80   0               0   0.43
 ...
 1  119   120   1               0   0.41
 2    0     1   0               1   0.25
 ...
 2   23    24   0               1   0.42
 ...
 2   95    96   1               1   0.58

For more info on time-dependent covariates and coefficients, see Therneau, Crowson and Atkinson's 2016 Vignette.

Cox Model – Proportional Hazards Assumption with Time-Dependent Covariates

If we add time-dependent covariates or interactions with time to the Cox proportional hazards model, then it is not a “proportional hazards” model any longer.

See this presentation: http://ms.uky.edu/~mai/sta635/Cox%20model.pdf

or this lecture notes: http://www.math.ucsd.edu/~rxu/math284/slect7.pdf

But this is a widely known feature.

Best Answer

Related Solutions

Solved – How to use time dependent covariates with cox regression in R

Cox Model – Proportional Hazards Assumption with Time-Dependent Covariates

Related Question