# Is the survival given by a linear combination of a large number of iid normally distributed predictors stable? (simulation)

My question is about survival analysis, but I am quite sure that it may apply to regression in general.

I will stick with a very simple simulation for survival times. For example, let's simulate the survival times for 100 subjects from a Weibull distribution $S \sim W (\alpha, \lambda)$, where 1000 covariates/predictors are introduced through $\lambda = x_i \beta$; censoring times are independently drawn from an exponential distribution $C \sim Exp(\mu)$; and the observed times $(T)$ are set to be $T_i = min(S_i, C_i)$.

X = matrix(rnorm(1000*100, mean = 0, sd = 1),ncol=  1000)
beta = rnorm(1000, sd = 10)
survdata <- data.frame(surv_months = rweibull(n = 100,
a = 0.8 , b = exp(- (X %*% beta) /0.8)),
censor_months = rexp(n = 100, rate = 1/100),
stringsAsFactors = F   ) %>%
dplyr::mutate(os_status = ifelse(surv_months < censor_months,
'DECEASED', 'LIVING'
),
os_months = ifelse(surv_months < censor_months,
surv_months, censor_months
)


And this is what results: the survival time for some individuals is extremely short, while others have extreme long survival.

While I would like to obtain instead is the survival times one can get by simulating with the same function, but with only two features/predictors/covariates instead. For example:

X = matrix(rnorm(2*100, mean = 0, sd = 1),ncol=  2)
beta = rnorm(2, sd = 10)
survdata <- data.frame(surv_months = rweibull(n = 100,
a = 0.8 , b = exp(- (X %*% beta) /0.8)),
censor_months = rexp(n = 100, rate = 1/100),
stringsAsFactors = F   ) %>%
dplyr::mutate(os_status = ifelse(surv_months < censor_months,
'DECEASED', 'LIVING'
),
os_months = ifelse(surv_months < censor_months,
surv_months, censor_months
)


And the plot:

• What exactly is your question? Eg, I don't see a "?" anywhere. – gung - Reinstate Monica Apr 30 '18 at 17:43
• Oh sorry, my question is why when using a large number of normally distributed predictors we don't get a stable linear predictor? – Carlos ST Apr 30 '18 at 18:56
• (I think this is probably appropriate here.) What do you mean by "a stable linear predictor"? Are you wondering why the observed survival times don't converge onto a single number (ie., why they seem to spread out over a wide range of times)? – gung - Reinstate Monica Apr 30 '18 at 19:27
• Hi gung thanks. Yes I mean that when using a large number of covariates, (although I think they should cancel out because they are drawn from a normal distribution), for many individuals it gives a sum of the linear combination that destabilise the survival. – Carlos ST Apr 30 '18 at 21:17