1.15 Multivariate Outcomes

Most regressions are set up to model univariate observations (be they scalar, boolean, categorical, ordinal, or count). Even multinomial regressions are just repeated categorical regressions. In contrast, this section discusses regression when each observed value is multivariate. To relate multiple outcomes in a regression setting, their error terms are provided with covariance structure.

This section considers two cases, seemingly unrelated regressions for continuous multivariate quantities and multivariate probit regression for boolean multivariate quantities.

Seemingly Unrelated Regressions

The first model considered is the “seemingly unrelated” regressions (SUR) of econometrics where several linear regressions share predictors and use a covariance error structure rather than independent errors (Zellner 1962; Greene 2011).

The model is easy to write down as a regression,

\[ \begin{array}{rcl} y_n & = & x_n \, \beta + \epsilon_n \\[4pt] \epsilon_n & \sim & \mathsf{multivariate\ normal}(0, \Sigma) \end{array} \]

where \(x_n\) is a \(J\)-row-vector of predictors (\(x\) is an \((N \times J)\) matrix), \(y_n\) is a \(K\)-vector of observations, \(\beta\) is a \((K \times J)\) matrix of regression coefficients (vector \(\beta_k\) holds coefficients for outcome \(k\)), and \(\Sigma\) is covariance matrix governing the error. As usual, the intercept can be rolled into \(x\) as a column of ones.

The basic Stan code is straightforward (though see below for more optimized code for use with LKJ priors on correlation).

data {
  int<lower=1> K;
  int<lower=1> J;
  int<lower=0> N;
  vector[J] x[N];
  vector[K] y[N];
}
parameters {
  matrix[K, J] beta;
  cov_matrix[K] Sigma;
}
model {
  vector[K] mu[N];
  for (n in 1:N)
    mu[n] = beta * x[n];
  y ~ multi_normal(mu, Sigma);
}

For efficiency, the multivariate normal is vectorized by precomputing the array of mean vectors and sharing the same covariance matrix.

Following the advice in the multivariate hierarchical priors section, we will place a weakly informative normal prior on the regression coefficients, an LKJ prior on the correlations and a half-Cauchy prior on standard deviations. The covariance structure is parameterized in terms of Cholesky factors for efficiency and arithmetic stability.

...
parameters {
  matrix[K, J] beta;
  cholesky_factor_corr[K] L_Omega;
  vector<lower=0>[K] L_sigma;
}
model {
  vector[K] mu[N];
  matrix[K, K] L_Sigma;

  for (n in 1:N)
    mu[n] = beta * x[n];

  L_Sigma = diag_pre_multiply(L_sigma, L_Omega);

  to_vector(beta) ~ normal(0, 5);
  L_Omega ~ lkj_corr_cholesky(4);
  L_sigma ~ cauchy(0, 2.5);

  y ~ multi_normal_cholesky(mu, L_Sigma);
}

The Cholesky factor of the covariance matrix is then reconstructed as a local variable and used in the model by scaling the Cholesky factor of the correlation matrices. The regression coefficients get a prior all at once by converting the matrix beta to a vector.

If required, the full correlation or covariance matrices may be reconstructed from their Cholesky factors in the generated quantities block.

Multivariate Probit Regression

The multivariate probit model generates sequences of boolean variables by applying a step function to the output of a seemingly unrelated regression.

The observations \(y_n\) are \(D\)-vectors of boolean values (coded 0 for false, 1 for true). The values for the observations \(y_n\) are based on latent values \(z_n\) drawn from a seemingly unrelated regression model (see the previous section),

\[ \begin{array}{rcl} z_n & = & x_n \, \beta + \epsilon_n \\[4pt] \epsilon_n & \sim & \mathsf{multivariate\ normal}(0, \Sigma) \end{array} \]

These are then put through the step function to produce a \(K\)-vector \(z_n\) of boolean values with elements defined by \[ y_{n, k} = \mathrm{I}(z_{n, k} > 0), \] where \(\mathrm{I}()\) is the indicator function taking the value 1 if its argument is true and 0 otherwise.

Unlike in the seemingly unrelated regressions case, here the covariance matrix \(\Sigma\) has unit standard deviations (i.e., it is a correlation matrix). As with ordinary probit and logistic regressions, letting the scale vary causes the model (which is defined only by a cutpoint at 0, not a scale) to be unidentified (see Greene (2011)).

Multivariate probit regression can be coded in Stan using the trick introduced by Albert and Chib (1993), where the underlying continuous value vectors \(y_n\) are coded as truncated parameters. The key to coding the model in Stan is declaring the latent vector \(z\) in two parts, based on whether the corresponding value of \(y\) is 0 or 1. Otherwise, the model is identical to the seemingly unrelated regression model in the previous section.

First, we introduce a sum function for two-dimensional arrays of integers; this is going to help us calculate how many total 1 values there are in \(y\).

functions {
  int sum2d(int[,] a) {
    int s = 0;
    for (i in 1:size(a))
      s += sum(a[i]);
    return s;
  }
}

The function is trivial, but it’s not a built-in for Stan and it’s easier to understand the rest of the model if it’s pulled into its own function so as not to create a distraction.

The data declaration block is much like for the seemingly unrelated regressions, but the observations y are now integers constrained to be 0 or 1.

data {
  int<lower=1> K;
  int<lower=1> D;
  int<lower=0> N;
  int<lower=0,upper=1> y[N,D];
  vector[K] x[N];
}

After declaring the data, there is a rather involved transformed data block whose sole purpose is to sort the data array y into positive and negative components, keeping track of indexes so that z can be easily reassembled in the transformed parameters block.

transformed data {
  int<lower=0> N_pos;
  int<lower=1,upper=N> n_pos[sum2d(y)];
  int<lower=1,upper=D> d_pos[size(n_pos)];
  int<lower=0> N_neg;
  int<lower=1,upper=N> n_neg[(N * D) - size(n_pos)];
  int<lower=1,upper=D> d_neg[size(n_neg)];

  N_pos = size(n_pos);
  N_neg = size(n_neg);
  {
    int i;
    int j;
    i = 1;
    j = 1;
    for (n in 1:N) {
      for (d in 1:D) {
        if (y[n,d] == 1) {
          n_pos[i] = n;
          d_pos[i] = d;
          i += 1;
        } else {
          n_neg[j] = n;
          d_neg[j] = d;
          j += 1;
        }
      }
    }
  }
}

The variables N_pos and N_neg are set to the number of true (1) and number of false (0) observations in y. The loop then fills in the sequence of indexes for the positive and negative values in four arrays.

The parameters are declared as follows.

parameters {
  matrix[D, K] beta;
  cholesky_factor_corr[D] L_Omega;
  vector<lower=0>[N_pos] z_pos;
  vector<upper=0>[N_neg] z_neg;
}

These include the regression coefficients beta and the Cholesky factor of the correlation matrix, L_Omega. This time there is no scaling because the covariance matrix has unit scale (i.e., it is a correlation matrix; see above).

The critical part of the parameter declaration is that the latent real value \(z\) is broken into positive-constrained and negative-constrained components, whose size was conveniently calculated in the transformed data block. The transformed data block’s real work was to allow the transformed parameter block to reconstruct \(z\).

transformed parameters {
  vector[D] z[N];
  for (n in 1:N_pos)
    z[n_pos[n], d_pos[n]] = z_pos[n];
  for (n in 1:N_neg)
    z[n_neg[n], d_neg[n]] = z_neg[n];
}

At this point, the model is simple, pretty much recreating the seemingly unrelated regression.

model {
  L_Omega ~ lkj_corr_cholesky(4);
  to_vector(beta) ~ normal(0, 5);
  {
    vector[D] beta_x[N];
    for (n in 1:N)
      beta_x[n] = beta * x[n];
    z ~ multi_normal_cholesky(beta_x, L_Omega);
  }
}

This simple form of model is made possible by the Albert and Chib-style constraints on z.

Finally, the correlation matrix itself can be put back together in the generated quantities block if desired.

generated quantities {
  corr_matrix[D] Omega;
  Omega = multiply_lower_tri_self_transpose(L_Omega);
}

The same could be done for the seemingly unrelated regressions in the previous section.

References

Zellner, Arnold. 1962. “An Efficient Method of Estimating Seemingly Unrelated Regression Equations and Tests for Aggregation Bias.” Journal of the American Statistical Association 57: 348–68.

Greene, William H. 2011. Econometric Analysis. 7th ed. Prentice-Hall.

Albert, J. H., and S. Chib. 1993. “Bayesian Analysis of Binary and Polychotomous Response Data.” Journal of the American Statistical Association 88: 669–79.