# multivariate truncated survival analysis

I have many short time series (1-5 data points) that document the development of morphological traits (length and pigmentation) of some lab critters in response to different dietary supplement. I already know that, depending on dietary supplement, these traits increase at different rates (and depending on family but that's maybe not important for this question).

I now would like to test whether growth or pigmentation rates, across different diets, can affect survival. As I said the morphological information I have for every timepoint; survival is determined by how long the timeseries are (1-5: I believe this is referred to as (right) truncated and not censored data). I suspect that organisms who both grow and pigment too fast have a higher change of dying, depending on what diet they feed on, but I am not sure how to test it.

I have been trying with a GAM-framework, 0/1 as response, and morphological info filled with NA after death. Did not work, of course, but just to give you an idea:

mod = gam(Survival ~  Food + Time_num +
s(Time_num, Length, by=Food, k=5) +
s(Time_num, Pigmentation, by=Food, k=5) +
s(Time_num, Length, Pigmentation, by=Food, k=5) +
s(Family, Individual, bs="re")
, family = "binomial", data=mod_data)


How can I account for truncation of individual time series? I would like to stay at the level of each individual (i.e. no family or treatment by timepoint averaging). R syntax would be appreciated.

EDIT: To clarify, I am interested in the time of survival (numeric 1-5 is sampling timepoints/dates), and how it is influenced by how fast organisms grow and/or become pigmented. The data (for all my ~1000 individuals) looks like this:

Individual 1A, Food A
Time:            1    2    3    4    5
Alive:           1    1    1    0    0
Length:          1.1  1.5  2.6  NA   NA
Pigmentation:    0.3  0.5  0.7  NA   NA

Individual 2A, Food A
Time:            1    2    3    4    5
Alive:           1    1    1    1    1
Length:          1.2  1.5  2.9  3.6  5.8
Pigmentation:    0.3  0.45 0.6  0.7  0.8

Individual 3A, Food B
Time:            1    2    3    4    5
Alive:           1    0    0    0    0
Length:          1.1  NA   NA   NA   NA
Pigmentation:    0.3  NA   NA   NA   NA

etc....


The 0/1 (dead/alive) response could of course be converted to 1-5, so I probably want that as my response in the end.

• Can you describe more clearly what your outcome really is and how it is measured? The post suggests that it is the "length of a series" (expressed in days? hours?) but then suggests that it is a "yes/no" type of outcome, which is confusing. – Isabella Ghement Mar 5 at 16:10
• Also, can you clarify whether or not you are interested in "time to death" for each individual or in "whether or not the individual died"? – Isabella Ghement Mar 5 at 16:12
• Is there any reason not to use a Cox PH model ? – Dan Chaltiel Mar 5 at 17:41

This is maybe not the answer you were looking for, but you should take a look at joint models.

These models are a combination of a survival model to consider events (death as 0/1 i your case) and a mixed model to take into account evolution of some factors over time (length and pigmentation for instance).

There is a R tutorial you can try. Here is what I'd try based on your example, but I'm no expert:

library(JM)
# Mixed-effects model fit
lmeFit.p1 <- lme(Time_num ~ Length + Pigmentation + Length:Pigmentation,
data = mod_data, random = ~ Time_num | id)

# Cox survival model fit
mod_data2 = mod_data #create your survival data
survFit.p1 <- coxph(Surv(Time, death) ~ Food + Time_num, data = mod_data2, x = TRUE)

# Joint model
jointFit.p1 <- jointModel(lmeFit.p1, survFit.p1, timeVar = "Time_num",
method = "piecewise-PH-aGH") #maybe another method

• this definitely got me thinking more in the right direction. I tried your specific example and could not yet get it to work because of differing sample sizes between longitudinal and event processes. I think this may be due to the random effect - I tried with cluster in the cox model without success. I'll spend more effort on this soon – mluerig Mar 6 at 22:50