What density function does "HAR1" as prior on coefficients correspond to?

[EricZarahn]

Hello,

First of all, thank you for the dynsurv package.

I am not having any trouble using the code, which is wonderful! However, I would like to know what the prior density on the coefficients is that corresponds to the "HAR1" selection (as exemplified in the code below from the dynsurv documentation). That is, I don't know right now what "HAR1" corresponds to in terms of a known density function.

My current best guess from reading Wang et al. BMC Medical Research Methodology (2016) is that the prior in this case is a Gaussian with mean 0 and variance given by a scaled (by a variable named "ao" in that paper) inverse gamma with the specified shape and scale parameters. Is that right? If so, is "a0" specified/hard-coded somewhere, or does it also have a prior?

Fit dynamic coefficient model with dynamic hazards (in log scales)

fit3 <- bayesCox(myformula, mydata, model = "Dynamic",
base.prior = list(type = "Const"),
coef.prior = list(type = "HAR1", shape = 2, scale = 1),
gibbs = list(iter = num.iter, burn = 20, thin = 1,
verbose = F, nReport = 20),
control = list(intercept = TRUE))

Many thanks,
Eric

[EricZarahn]

I found in the comments of the bayesCox function that a0 defaults to 100:

##' \item{a0:}{Multiplier for initial variance in time-varying or dynamic
##' models, default 100;}

So the only thing that remains from my initial question is whether the prior on the coefficients is a Gaussian with mean 0 and variance given a0 times an inverse gamma with the specified shape and scale parameters.

Thanks again,
Eric

[wenjie2wang]

Hi Eric,

So the only thing that remains from my initial question is whether the prior on the coefficients is a Gaussian with mean 0 and variance given a0 times an inverse gamma with the specified shape and scale parameters.

The prior specifications for the dynamic model are given as follows:

where ${\tau_j}$'s are jump points, $J$ is the number of jumps, and $a_0$ is a hyperparameter. See Section 3 of Wang et al., (2013) for details.

• Wang, X., Chen, M., & Yan, J. (2013). Bayesian dynamic regression models for interval censored survival data with application to children dental health. Lifetime Data Analysis, 19(3), 297–316.

[EricZarahn]

Thank you for the answer Wenjie.

I have another question if you would be so kind: I am doing simulations using in which I generate data with right-censoring for 2 groups of subjects with hazard rate of 0.03 in group A (reference level of factor) vs. 0.01 group B and then estimate the hazard function using dynsurv::bayesCox.

I am finding that, for a fixed sample size of group A, that the estimated hazard ratio decreases from the simulated value of 3 as I increase the sample size of group B. This phenomenon is surprising to me, as I was expecting the estimated hazard ratio to, if anything, become closer to ground truth as I increased the sample size of either group.

Here are some simulation results for N = 50 realizations of data per condition:

This is the call I used:

  fit <- bayesCox(myformula, mydata, model = "Dynamic",
                   base.prior = list(type = "Const"),
                   coef.prior = list(type = "HAR1", shape = 2, scale = 1),
                   gibbs = list(iter = 1000, burn = 20, thin = 1,
                                verbose = F, nReport = 20),
                   control = list(intercept = TRUE))    

Can you provide any insight?

Many thanks again,
Eric

[wenjie2wang]

The dynamic model is intended for interval-censored data, some of which can be right-censored [t, $\infty$). In general, it would not be a good idea to apply it to right-censored data with exact event times as all the unique event times will be candidates for the potential jumps.

[EricZarahn]

Ok, thank you very much for both answers Wenjie!

Best,
Eric