Truncated or Censored Data

Data in which measurements have been truncated or censored can be coded in Stan following their respective probability models.

Truncated distributions

Truncation in Stan is restricted to univariate distributions for which the corresponding log cumulative distribution function (CDF) and log complementary cumulative distribution (CCDF) functions are available. See the reference manual section on truncated distributions for more information on truncated distributions, CDFs, and CCDFs.

Truncated data

Truncated data are data for which measurements are only reported if they fall above a lower bound, below an upper bound, or between a lower and upper bound.

Truncated data may be modeled in Stan using truncated distributions. For example, suppose the truncated data are \(y_n\) with an upper truncation point of \(U = 300\) so that \(y_n < 300\). In Stan, this data can be modeled as following a truncated normal distribution for the observations as follows.

data {
  int<lower=0> N;
  real U;
  array[N] real<upper=U> y;
}
parameters {
  real mu;
  real<lower=0> sigma;
}
model {
  y ~ normal(mu, sigma) T[ , U];
}

The model declares an upper bound U as data and constrains the data for y to respect the constraint; this will be checked when the data are loaded into the model before sampling begins.

This model implicitly uses an improper flat prior on the scale and location parameters; these could be given priors in the model using distribution statements.

Constraints and out-of-bounds returns

If the sampled variate in a truncated distribution lies outside of the truncation range, the probability is zero, so the log probability will evaluate to \(-\infty\). For instance, if variate y is sampled with the statement.

y ~ normal(mu, sigma) T[L, U];

then if any value inside y is less than the value of L or greater than the value of U, the distribution statement produces a zero-probability estimate. For user-defined truncation, this zeroing outside of truncation bounds must be handled explicitly.

To avoid variables straying outside of truncation bounds, appropriate constraints are required. For example, if y is a parameter in the above model, the declaration should constrain it to fall between the values of L and U.

parameters {
  array[N] real<lower=L, upper=U> y;
  // ...
}

If in the above model, L or U is a parameter and y is data, then L and U must be appropriately constrained so that all data are in range and the value of L is less than that of U (if they are equal, the parameter range collapses to a single point and the Hamiltonian dynamics used by the sampler break down). The following declarations ensure the bounds are well behaved.

parameters {
  real<upper=min(y)> L;           // L < y[n]
  real<lower=fmax(L, max(y))> U;  // L < U; y[n] < U

For pairs of real numbers, the function fmax is used rather than max.

Unknown truncation points

If the truncation points are unknown, they may be estimated as parameters. This can be done with a slight rearrangement of the variable declarations from the model in the previous section with known truncation points.

data {
  int<lower=1> N;
  array[N] real y;
}
parameters {
  real<upper=min(y)> L;
  real<lower=max(y)> U;
  real mu;
  real<lower=0> sigma;
}
model {
  L ~ // ...
  U ~ // ...
  y ~ normal(mu, sigma) T[L, U];
}

Here there is a lower truncation point L which is declared to be less than or equal to the minimum value of y. The upper truncation point U is declared to be larger than the maximum value of y. This declaration, although dependent on the data, only enforces the constraint that the data fall within the truncation bounds. With N declared as type int<lower=1>, there must be at least one data point. The constraint that L is less than U is enforced indirectly, based on the non-empty data.

The ellipses where the priors for the bounds L and U should go should be filled in with a an informative prior in order for this model to not concentrate L strongly around min(y) and U strongly around max(y).

Censored data

Censoring hides values from points that are too large, too small, or both. Unlike with truncated data, the number of data points that were censored is known. The textbook example is the household scale which does not report values above 300 pounds.

Estimating censored values

One way to model censored data is to treat the censored data as missing data that is constrained to fall in the censored range of values. Since Stan does not allow unknown values in its arrays or matrices, the censored values must be represented explicitly, as in the following right-censored case.

data {
  int<lower=0> N_obs;
  int<lower=0> N_cens;
  array[N_obs] real y_obs;
  real<lower=max(y_obs)> U;
}
parameters {
  array[N_cens] real<lower=U> y_cens;
  real mu;
  real<lower=0> sigma;
}
model {
  y_obs ~ normal(mu, sigma);
  y_cens ~ normal(mu, sigma);
}

Because the censored data array y_cens is declared to be a parameter, it will be sampled along with the location and scale parameters mu and sigma. Because the censored data array y_cens is declared to have values of type real<lower=U>, all imputed values for censored data will be greater than U. The imputed censored data affects the location and scale parameters through the last distribution statement in the model.

Integrating out censored values

Although it is wrong to ignore the censored values in estimating location and scale, it is not necessary to impute values. Instead, the values can be integrated out. Each censored data point has a probability of \[\begin{align*} \Pr[y_{\mathrm{cens},m} > U] &= \int_U^{\infty} \textsf{normal}\left(y_{\mathrm{cens},m} \mid \mu,\sigma \right) \,\textsf{d}y_{\mathrm{cens},m} \\ &= 1 - \Phi\left(\frac{U - \mu}{\sigma}\right), \end{align*}\]

where \(\Phi()\) is the standard normal cumulative distribution function. This probability is equivalent to the likelihood contribution of knowing that \(y_{\mathrm{cens},m}>U\). With \(M\) censored observations, the likelihood on the log scale is \[\begin{align*} \log \prod_{m=1}^M \Pr[y_{\mathrm{cens},m} > U] &= \log \left( 1 - \Phi\left(\left(\frac{U - \mu}{\sigma}\right)\right)^{M}\right) \\ &= M \times \texttt{normal}\mathtt{\_}\texttt{lccdf}\left(U \mid \mu, \sigma \right), \end{align*}\]

where normal_lccdf is the log of complementary CDF (Stan provides <distr>_lccdf for each distribution implemented in Stan).

The following right-censored model assumes that the censoring point is known, so it is declared as data.

data {
  int<lower=0> N_obs;
  int<lower=0> N_cens;
  array[N_obs] real y_obs;
  real<lower=max(y_obs)> U;
}
parameters {
  real mu;
  real<lower=0> sigma;
}
model {
  y_obs ~ normal(mu, sigma);
  target += N_cens * normal_lccdf(U | mu, sigma);
}

For the observed values in y_obs, the normal model is used without truncation. The likelihood contribution from the integrated out censored values can not be coded with distribution statement, and the log probability is directly incremented using the calculated log cumulative normal probability of the censored observations.

For the left-censored data the CDF (normal_lcdf) has to be used instead of complementary CDF. If the censoring point variable (L) is unknown, its declaration should be moved from the data to the parameters block.

data {
  int<lower=0> N_obs;
  int<lower=0> N_cens;
  array[N_obs] real y_obs;
}
parameters {
  real<upper=min(y_obs)> L;
  real mu;
  real<lower=0> sigma;
}
model {
  L ~ normal(mu, sigma);
  y_obs ~ normal(mu, sigma);
  target += N_cens * normal_lcdf(L | mu, sigma);
}
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