Review data block

Let’s review the data block first.

This will tell us the structure and format of data input to the model.

print_stan_file(stan_file, section = 'data')

## data { 
##   int<lower=0> Nobs; 
##   int<lower=0> Ncen; 
##   vector[Nobs] yobs; 
##   vector[Ncen] ycen; 
## }

The censored & observed data points are provided as separate input vectors.

observed data points

• Nobs: number of observed data points
• yobs: times to observed events

censored data points

• Ncen: number of censored data points
• ycen: times to censored events

Recall that this is a NULL model (with no covariate values), so the number & values of observed covariates are not needed.

Review model block

The stan code contains an implicit constant term, in the linear predictor mu.

print_stan_file(stan_file, section = 'model')

## model { 
##   yobs ~ weibull(alpha, exp(-(mu)/alpha)); 
##   increment_log_prob(weibull_ccdf_log(ycen, alpha, exp(-(mu)/alpha))); 
##  
##   alpha_raw ~ normal(0.0, 1.0); 
##   mu ~ normal(0.0, tau_mu); 
## }

Observe how the ccdf (complementary cumulative distribution function) is used to compute the log probability of the censored observations.

What does the ccdf represent in this scenario?

How does the model address the censoring process?

Review parameters block

Our stan code also contains a reparameterization of the alpha term, in the transformed parameters block.

Observe:

print_stan_file(stan_file, section = 'transformed parameters')

## transformed parameters { 
##   real alpha; 
##   alpha <- exp(tau_al * alpha_raw); 
## }

(recall that tau_al is a constant scaling term set to 10, and alpha_raw is a parameter with a normal(0, 1) prior distribution).

This reparameterization achieves two things :

1. The use of tau_al * alpha_raw is an example of a non-centered parameterization.
   • It would have been mathematically equivalent to define a (non-transformed) parameter alpha with a prior normal(0, 10).
   • However, this parameterization yields a parameter (alpha_raw) which is on a similar scale as other parameters in our model. The exp() transformation makes the difference between these two scales even more dramatic.
   • In general, having all parameters on a similar scale makes the sampling more efficient.
2. The exp() transformation of this parameter additionally allows us to put a prior on log alpha.
   • we want to put a prior on log alpha since alpha enters into our likelihood in the exponent.

This seems like a lot of gymnastics to be doing.

However, it has practical implications for our modeling efficiency.

Observe that, for a single value of alpha_raw (e.g. 0.2), the transformation yields:

alpha_raw <- 0.2
tau_al <- 10
log_alpha <- alpha_raw * tau_al
alpha <- exp(log_alpha)
print(alpha)

## [1] 7.389056

which may seem silly.

However

Consider the resulting distribution of alpha over a range of values for alpha_raw sampled from our normal(0, 1) prior:

alpha_raw <- rnorm(1000, 0, 1)
tau_al <- 10
log_alpha <- alpha_raw * tau_al
alpha <- exp(log_alpha)
ggplot(data.frame(alpha = alpha, alpha_raw = alpha_raw), 
       aes(x = alpha)) + 
    geom_density() + 
    scale_x_log10(labels = scientific)

Notice how alpha ranges from 1e-10 to 1e+10 on a log scale. We have to truncate this dramatically to even consider plotting it on its original scale.

Sampling this parameter space may require different step sizes & different tuning parameter values throughout this distribution.

The alpha_raw scale, by comparison, is a lot friendlier.

ggplot(data.frame(alpha = alpha, alpha_raw = alpha_raw), 
       aes(x = alpha, y = alpha_raw)) + 
    geom_density2d() + 
    scale_x_log10(labels = scientific)

This distribution is centered at 0 and has more consistent behavior throughout its range of values.

What’s important to note here is that while the non-centered parameterization is mathematically equivalent to the standard parameterization, it is (in some ways) a different model. You may get different results from each.

Packages like rstanarm which provide easy wrappers to a variety of standard models implemented in Stan use a non-centered parameterization by default.

More information on non-centered parameterization:

1. discussion on stan-dev list
2. Gelman, 2004. Parameterization and Bayesian Modeling

Simulate data for arbitrary input values

We can use this to simulate a dataset for hypothetical parameter values of alpha & mu.

test_alpha <- 0.8
test_mu <- -3

## sample sizes from TCGA blca data
test_nobs <- 179 
test_ncen <- 230

## test these inputs for arbitrary values of alpha & mu
simulated_data <- 
    sim_data(alpha = test_alpha,
                 mu = test_mu,
                 Nobs = test_nobs,
                 Ncen = test_ncen
                 ) 
head(simulated_data)

## Source: local data frame [6 x 2]
## 
##   os_status os_months
##       <chr>     <dbl>
## 1  DECEASED 11.441454
## 2  DECEASED 17.362348
## 3  DECEASED  1.439942
## 4  DECEASED 23.616051
## 5  DECEASED 37.779830
## 6  DECEASED  9.369964

Which yields the following survival curve:

## plot KM curve from simulated data
simulated_data <- 
    simulated_data %>%
    dplyr::mutate(os_deceased = os_status == 'DECEASED')

autoplot(survival::survfit(Surv(os_months, os_deceased) ~ 1,
                      data = simulated_data
                      ), conf.int = F) + 
    ggtitle('Simulated KM curve')

preparing data for stan

Stan takes data input as a list. The contents of the list should match those of the data block in the stan code.

E.g. looking at the data block -

print_stan_file(stan_file, section = 'data')

## data { 
##   int<lower=0> Nobs; 
##   int<lower=0> Ncen; 
##   vector[Nobs] yobs; 
##   vector[Ncen] ycen; 
## }

our input list to Stan should contain dimensions & values for observed & censored data, separately.

observed_data <- simulated_data %>%
    dplyr::filter(os_status == 'DECEASED')

censored_data <- simulated_data %>%
    dplyr::filter(os_status != 'DECEASED')

stan_data <- list(
    Nobs = nrow(observed_data),
    Ncen = nrow(censored_data),
    yobs = observed_data$os_months,
    ycen = censored_data$os_months
)
rm(censored_data)
rm(observed_data)
str(stan_data)

## List of 4
##  $ Nobs: int 179
##  $ Ncen: int 230
##  $ yobs: num [1:179] 11.44 17.36 1.44 23.62 37.78 ...
##  $ ycen: num [1:230] 3.28 71.29 134.28 11.25 28.51 ...

(wrap this prep-data process in a function gen_stan_data for later)

gen_stan_data <- function(data) {
    observed_data <- data %>%
        dplyr::filter(os_status == 'DECEASED')
    
    censored_data <- data %>%
        dplyr::filter(os_status != 'DECEASED')
    
    stan_data <- list(
        Nobs = nrow(observed_data),
        Ncen = nrow(censored_data),
        yobs = observed_data$os_months,
        ycen = censored_data$os_months
    )
}

test simulated values with stan

Let’s call stan:

recover_simulated <- 
    rstan::stan(stan_file,
                data = stan_data,
                chains = 4,
                iter = 1000,
                seed = 1328025050
                )
print(recover_simulated)

## Inference for Stan model: weibull_survival_null_model.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##                    mean    se_mean           sd          2.5%
## alpha_raw  7.000000e-02 1.0000e-01 1.400000e-01 -2.000000e-02
## mu        -3.120000e+00 9.1000e-01 1.290000e+00 -4.230000e+00
## alpha      6.390000e+00 6.7600e+00 9.560000e+00  7.900000e-01
## lp__      -2.581429e+62 3.1608e+62 4.472291e+62 -1.034708e+63
##                     25%     50%     75%   97.5% n_eff    Rhat
## alpha_raw -2.000000e-02   -0.01    0.08    0.31     2   33.85
## mu        -3.950000e+00   -3.77   -2.71   -0.89     2    7.90
## alpha      8.500000e-01    0.89    6.50   22.95     2  262.44
## lp__      -2.574767e+62 -963.74 -963.07 -962.72     2 1235.65
## 
## Samples were drawn using NUTS(diag_e) at Wed Jun 22 22:22:21 2016.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

What’s wrong with this picture?

(A: poor convergence) (A: in some chains, we see a lot of numerical problems.)

Setting initial values

This step is usually optional, but may be necessary for some models.

In this case, it may be useful to set initial values. Recall the projected range of our transformed parameter alpha?

By default, Stan chooses a random initial value for each parameter on the unconstrained scale between -2 and 2. This random initialization is on the unconstrained support for each parameter. This guarantees that initial values are consistent with the constrained range.

When we pass the initial values in, however, these are on the constrained scale. See the Stan manual for more details about transformations applied to constrained variables.

print_stan_file(stan_file, section = 'parameters')

## parameters { 
##   real alpha_raw; 
##   real mu; 
## }

gen_inits <- function() {
      list(
        alpha_raw = 0.01*rnorm(1),
        mu = rnorm(1)
      )
}

stan code with initial values

Let’s try fitting our stan model again with our initial values function.

recover_simulated2 <- 
    rstan::stan(stan_file,
                data = stan_data,
                chains = 4,
                iter = 1000,
                init = gen_inits
                )
print(recover_simulated2)

## Inference for Stan model: weibull_survival_null_model.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##              mean se_mean   sd    2.5%     25%     50%     75%   97.5%
## alpha_raw   -0.01    0.00 0.01   -0.02   -0.02   -0.01   -0.01    0.00
## mu          -3.87    0.01 0.21   -4.29   -4.01   -3.86   -3.74   -3.45
## alpha        0.88    0.00 0.05    0.78    0.84    0.87    0.91    0.97
## lp__      -963.69    0.05 0.98 -966.23 -964.13 -963.37 -962.96 -962.70
##           n_eff Rhat
## alpha_raw   257 1.02
## mu          258 1.02
## alpha       264 1.02
## lp__        448 1.00
## 
## Samples were drawn using NUTS(diag_e) at Wed Jun 22 22:22:26 2016.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

Now we see fewer numerical problems, and better R-hat values.

checking convergence

Normally I would dig further into this model to check convergence, since R-hat alone isn’t sufficient to determine convergence.

But in this case I’m distracted by the summary of posterior estimates for our parameter values.

Have we recovered our parameter values?

The values we used to simulate our data do not fall into the 95% credible intervals from our model.

I will do some quick checks on convergence to see if this might be explained by lack of convergence.

rstan::traceplot(recover_simulated2, 'lp__')

rstan::traceplot(recover_simulated2, c('alpha','mu'), ncol = 1)

if (interactive())
    shinystan::launch_shinystan(recover_simulated2)

thinking about censoring

What if we were to limit our input data to observed events?

recover_simulated_obs <- 
    rstan::stan(stan_file,
                data = gen_stan_data(
                    simulated_data %>% dplyr::filter(os_status == 'DECEASED')
                    ),
                chains = 4,
                iter = 1000,
                init = gen_inits
                )
print(recover_simulated_obs)

## Inference for Stan model: weibull_survival_null_model.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##              mean se_mean   sd    2.5%     25%     50%     75%   97.5%
## alpha_raw   -0.03    0.00 0.01   -0.04   -0.03   -0.03   -0.02   -0.01
## mu          -2.93    0.01 0.21   -3.35   -3.07   -2.94   -2.79   -2.54
## alpha        0.78    0.00 0.04    0.69    0.75    0.78    0.81    0.87
## lp__      -870.97    0.05 0.98 -873.66 -871.38 -870.69 -870.26 -870.00
##           n_eff Rhat
## alpha_raw   322 1.03
## mu          301 1.03
## alpha       318 1.03
## lp__        417 1.01
## 
## Samples were drawn using NUTS(diag_e) at Wed Jun 22 22:22:33 2016.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

And, to our censored observations?

recover_simulated_cen <- 
    rstan::stan(stan_file,
                data = gen_stan_data(
                    simulated_data %>% dplyr::filter(os_status != 'DECEASED')
                    ),
                chains = 4,
                iter = 1000,
                init = gen_inits
                )
print(recover_simulated_cen)

## Inference for Stan model: weibull_survival_null_model.
## 4 chains, each with iter=1000; warmup=500; thin=1; 
## post-warmup draws per chain=500, total post-warmup draws=2000.
## 
##             mean se_mean   sd   2.5%    25%    50%   75% 97.5% n_eff Rhat
## alpha_raw  -0.19    0.01 0.14  -0.42  -0.31  -0.20 -0.08  0.08   250 1.03
## mu        -12.76    0.43 5.43 -27.16 -15.80 -11.45 -8.73 -5.81   161 1.02
## alpha       0.39    0.04 0.62   0.02   0.05   0.14  0.46  2.23   251 1.02
## lp__       -1.12    0.07 0.86  -3.72  -1.38  -0.80 -0.56 -0.45   151 1.02
## 
## Samples were drawn using NUTS(diag_e) at Wed Jun 22 22:22:38 2016.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).

We see that we have (not surprisingly!) very poor inferences from our censored observations – .. and that we are able to recover our parameter estimates when limiting only to observed data points.

This is our first hint that the inability to recover parameter estimates has something to do with how the data are censored. Is this just introducing error into our inferences?

extracting parameters from the Stanfit object

We use the rstan::extract() function to extract parameters from the stanfit object.

E.g. to extract alpha & mu:

pp_alpha <- rstan::extract(recover_simulated2,'alpha')$alpha
pp_mu <- rstan::extract(recover_simulated2,'mu')$mu

Each of these is a 1xD vector of values, where D = the number of posterior (post-warmup) draws.

In this case, we have 2000: 4 chains * 1000 iterations / 2

simulating data for each posterior draw

We can use hadley’s purrr::map2 to simulate data for each pair of mu*alpha values.

pp_newdata <- 
    purrr::map2(.x = pp_alpha,
                .y = pp_mu,
                .f = ~ sim_data(alpha = .x, 
                                mu = .y,
                                Nobs = test_nobs,
                                Ncen = test_ncen
                                )
                )

We now have a list of D datasets, each containing a simulation according to that draw’s parameter values for mu & alpha.

Let’s plot the time to event in the posterior draws, and compare this to the test dataset we used to fit our model.

ggplot(pp_newdata %>%
           dplyr::bind_rows() %>%
           dplyr::mutate(type = 'posterior predicted values') %>%
           bind_rows(simulated_data %>% dplyr::mutate(type = 'actual data'))
       , aes(x = os_months, group = os_status, colour = os_status, fill = os_status)) +
    geom_density(alpha = 0.5) +
    facet_wrap(~type, ncol = 1)

Pretty similar.

summarizing posterior predictive draws

Next we might ask about the posterior estimates of the survival curve. How would we estimate this?

One way (there may be several) is to: 1. compute the cumulative survival at each observed timepoint for each draw from the posterior 2. aggregate the cumulative survival estimates to discrete units of time 3. summarize the cumulative survival for each interval, over the posterior draws.

This is the method we will use here.

## cumulative survival rate for each posterior draw
pp_survdata <-
    pp_newdata %>%
    purrr::map(~ dplyr::mutate(., os_deceased = os_status == 'DECEASED')) %>%
    purrr::map(~ survival::survfit(Surv(os_months, os_deceased) ~ 1, data = .)) %>%
    purrr::map(fortify)

## summarize cum survival for each unit time (month), summarized at 95% confidence interval
pp_survdata_agg <- 
    pp_survdata %>%
    purrr::map(~ dplyr::mutate(., time_group = floor(time))) %>%
    dplyr::bind_rows() %>%
    dplyr::group_by(time_group) %>%
    dplyr::summarize(surv_mean = mean(surv)
                     , surv_p50 = median(surv)
                     , surv_lower = quantile(surv, probs = 0.025)
                     , surv_upper = quantile(surv, probs = 0.975)
                     ) %>%
    dplyr::ungroup()

Finally, we overlay the posterior predictive simulations of the survival curve with that from our original test dataset.

## km-curve for test data 
test_data_kmcurve <- 
    fortify(
        survival::survfit(
            Surv(os_months, os_deceased) ~ 1, 
            data = simulated_data %>% 
                dplyr::mutate(os_deceased = os_status == 'DECEASED')
            )) %>%
    dplyr::mutate(lower = surv, upper = surv)

ggplot(pp_survdata_agg %>%
           dplyr::mutate(type = 'posterior predicted values') %>%
           dplyr::rename(surv = surv_p50, lower = surv_lower, upper = surv_upper, time = time_group) %>%
           bind_rows(test_data_kmcurve %>% dplyr::mutate(type = 'actual data')),
       aes(x = time, group = type, linetype = type)) + 
    geom_line(aes(y = surv, colour = type)) +
    geom_ribbon(aes(ymin = lower, ymax = upper), alpha = 0.2) +
    xlim(c(0, 200))

Here we see a dramatic difference in survival curve, comparing our original simulated data with our posterior predictive distribution.

Saving as a function

As before, we will want to wrap this in a function so that it can be reused in future steps, e.g. when we work with our TCGA data.

pp_predict_surv <- function(pp_alpha, pp_mu, Nobs, Ncen,
                            level = 0.9,
                            plot = F, data = NULL,
                            sim_data_fun = sim_data
                            ) {
    pp_newdata <- 
        purrr::map2(.x = pp_alpha,
                    .y = pp_mu,
                    .f = ~ sim_data_fun(alpha = .x, mu = .y,
                                    Nobs = Nobs, Ncen = Ncen
                                    )
                    )
    
    pp_survdata <-
        pp_newdata %>%
        purrr::map(~ dplyr::mutate(., os_deceased = os_status == 'DECEASED')) %>%
        purrr::map(~ survival::survfit(Surv(os_months, os_deceased) ~ 1, data = .)) %>%
        purrr::map(fortify)
    
    ## compute quantiles given level 
    lower_p <- 0 + ((1 - level)/2)
    upper_p <- 1 - ((1 - level)/2)
    
    pp_survdata_agg <- 
        pp_survdata %>%
        purrr::map(~ dplyr::mutate(.,
                                   time_group = floor(time))) %>%
        dplyr::bind_rows() %>%
        dplyr::group_by(time_group) %>%
        dplyr::summarize(surv_mean = mean(surv)
                         , surv_p50 = median(surv)
                         , surv_lower = quantile(surv,
                                                 probs = lower_p)
                         , surv_upper = quantile(surv,
                                                 probs = upper_p)
                         ) %>%
        dplyr::ungroup()
    
    if (plot == FALSE) {
        return(pp_survdata_agg)
    } 
    
    ggplot_data <- pp_survdata_agg %>%
           dplyr::mutate(type = 'posterior predicted values') %>%
           dplyr::rename(surv = surv_p50,
                         lower = surv_lower,
                         upper = surv_upper, time = time_group)
    
    if (!is.null(data))
        ggplot_data <- 
            ggplot_data %>% 
            bind_rows(
                fortify(
                    survival::survfit(
                        Surv(os_months, os_deceased) ~ 1, 
                        data = data %>% 
                            dplyr::mutate(
                                os_deceased = os_status == 'DECEASED')
                        )) %>%
                dplyr::mutate(lower = surv,
                              upper = surv, type = 'actual data')
                )
    
    pl <- ggplot(ggplot_data,
                 aes(x = time, group = type, linetype = type)) + 
        geom_line(aes(y = surv, colour = type)) +
        geom_ribbon(aes(ymin = lower, ymax = upper), alpha = 0.2)
        
    pl 
}

PP check on observed-event only model

How well is our actual data to our posterior predicted values when we limit our model to observed events only?

pp_alpha_obs <- extract(recover_simulated_obs, 'alpha')$alpha
pp_mu_obs <- extract(recover_simulated_obs, 'mu')$mu
pl <- pp_predict_surv(pp_alpha = pp_alpha_obs,
                pp_mu = pp_mu_obs,
                Nobs = test_nobs,
                Ncen = 0,
                plot = T,
                data = simulated_data %>% 
                    dplyr::filter(os_status == 'DECEASED')
                )
pl + 
    ggtitle('PP check against simulated data\n(model fit using observed events only)') +
    xlim(NA, 200)

This is an example of how posterior predictive checking can be useful.

At this point we know there must be a problem with either our data simulating process, our model, or both.

My suspicion is that it has something to do with the censoring (runif()) process in the data simulation.