Regression Models

Regression Models

Linear Regression

Stan

data {
  int N;
  int K;
  matrix[N, K] X;
  vector[N] y;
}
parameters {
  real alpha;
  vector[K] beta;
  real sigma;
}
model {
  alpha ~ normal(0, 10);
  beta ~ normal(0, 5);
  sigma ~ exponential(1);
  y ~ normal(alpha + X * beta, sigma);
}
generated quantities {
  array[N] real y_rep;
  for (n in 1:N)
    y_rep[n] = normal_rng(alpha + X[n] * beta, sigma);
}

JAGS

model {
  for (i in 1:N) {
    y[i] ~ dnorm(mu[i], tau)
    mu[i]  N;
  int K;
  matrix[N, K] X;
  array[N] int y;
}
parameters {
  real alpha;
  vector[K] beta;
}
model {
  alpha ~ normal(0, 2.5);
  beta ~ normal(0, 2.5);
  y ~ bernoulli_logit(alpha + X * beta);
}

JAGS

model {
  for (i in 1:N) {
    y[i] ~ dbern(p[i])
    logit(p[i])  phi;  // Overdispersion
}
model {
  phi ~ exponential(1);
  y ~ neg_binomial_2_log(alpha + X * beta, phi);
}

Robust Regression (Student-t Errors)

Stan

parameters {
  real alpha;
  vector[K] beta;
  real sigma;
  real nu;  // Degrees of freedom
}
model {
  nu ~ gamma(2, 0.1);  // Prior on df
  y ~ student_t(nu, alpha + X * beta, sigma);
}

QR Decomposition (For Correlated Predictors)

transformed data {
  matrix[N, K] Q = qr_thin_Q(X) * sqrt(N - 1.0);
  matrix[K, K] R = qr_thin_R(X) / sqrt(N - 1.0);
  matrix[K, K] R_inv = inverse(R);
}
parameters {
  vector[K] theta;
  real sigma;
}
model {
  y ~ normal(Q * theta, sigma);
}
generated quantities {
  vector[K] beta = R_inv * theta;
}

Prior Recommendations

| Parameter | Weakly Informative | Reference | |-----------|-------------------|-----------| | Intercept | normal(0, 10) | Scale of outcome | | Coefficients | normal(0, 2.5) | Gelman et al. | | SD (sigma) | exponential(1) | Half-normal alternative | | Logistic coef | normal(0, 2.5) | ~4 logit units = extreme |

Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

• Author: choxos
• Source: choxos/BiostatAgent
• License: MIT

Install and usage instructions live in the source repository linked above.