Fast Quasi-Poisson Regression with Variance Calculation (C++ Backend)

Fits the same Poisson log-link mean model as fast_poisson_regression_cpp (see that page for the full model and optimizer contract; point estimates $\hat\beta$ are identical to what that function would return), but instead of the plain Poisson Fisher-information-based variance, scales it by an estimated overdispersion parameter to obtain quasi-likelihood standard errors that are robust to variance-mean deviations from the strict Poisson assumption $\mathrm{Var}(Y_i) = \mu_i$. The dispersion is the Pearson-statistic-based moment estimator, $$\hat\phi = \frac{1}{n-p}\sum_{i=1}^n \frac{(y_i - \hat\mu_i)^2}{\hat\mu_i},$$ computed only when the residual degrees of freedom $n - p > 0$; the reported coefficient variances are then $\widehat{\mathrm{Var}}(\hat\beta_k) = \hat\phi \, [(X^\top \hat W X)^{-1}]_{kk}$ (the ordinary Poisson Fisher information scaled by $\hat\phi$), matching the standard quasi-Poisson GLM correction (as in stats::glm(family = quasipoisson())). If $n \le p$, or $\hat\phi$ comes out non-finite or non-positive, ssq_b_j/ssq_b_2/dispersion are left at their NA defaults (point estimates b and mu are still returned).

Usage

fast_quasipoisson_regression_with_var_cpp(
  X,
  y,
  j = 2L,
  warm_start_beta = NULL,
  smart_cold_start = FALSE,
  maxit = 100L,
  tol = 1e-8,
  fixed_idx = NULL,
  fixed_values = NULL,
  optimization_alg = "irls",
  warm_start_weights = NULL,
  warm_start_fisher_info = NULL
)

Arguments

X: A numeric matrix of predictor variables. It is assumed that an intercept column (e.g., a column of ones) is already included in X if desired.
y: A numeric vector of the response variable, expected to be nonnegative-integer counts.
j: The 1-indexed coefficient whose variance to compute in ssq_b_j. Defaults to 2.
warm_start_beta: Optional starting values for coefficients $\beta$. If provided, smart_cold_start is ignored.
smart_cold_start: Logical. If TRUE and no warm_start_beta is supplied, use a Poisson-specific heuristic initial guess rather than a zero cold start.
maxit: Maximum number of iterations. Defaults to 100.
tol: Convergence tolerance. Defaults to 1e-8.
fixed_idx: Optional integer indices of coefficients to hold fixed rather than estimate.
fixed_values: Optional values to fix the parameters named by fixed_idx at.
optimization_alg: Optimization algorithm: "irls" (default), "lbfgs", or "newton_raphson".
warm_start_weights: Accepted but unused; see fast_poisson_regression_cpp Details.
warm_start_fisher_info: Optional initial curvature (information) matrix to warm-start the first iteration.

Value

A list containing the following components:

b: A numeric vector of the estimated Poisson regression coefficients $\hat\beta$ (point estimates, unaffected by the dispersion correction).
ssq_b_j: The dispersion-scaled variance of the $j$-th coefficient estimator; NA if j indexes a fixed coefficient or the dispersion estimate is unusable.
ssq_b_2: The dispersion-scaled variance of the second coefficient estimator specifically (typically the treatment effect), regardless of what j is; equal to ssq_b_j when j = 2.
dispersion: The estimated Pearson-based overdispersion parameter $\hat\phi$, or NA if the residual degrees of freedom are not positive.
mu: The fitted means $\hat\mu_i$.
converged: A logical value indicating whether the algorithm converged.
iterations: The number of iterations performed.
gradient_norm: The norm of the score vector at convergence.

Usage

Arguments

Value

See also