Modified Poisson Regression Inference for Incidence Responses
Source:R/inference_incidence_modified_poisson.R
InferenceIncidModifiedPoisson.RdFits Zou's (2004) modified Poisson regression for binary (incidence)
responses: \(\log E[Y_i \mid w_i, x_i] = \beta_0 + \beta_T w_i +
x_i^\top \gamma\), fit by maximizing the ordinary Poisson log-likelihood
treating the binary \(Y_i\) as if it were Poisson-distributed (a valid
estimating equation for the conditional mean regardless of the true
outcome distribution, exactly as
InferencePropFractionalLogit's
quasi-binomial fit is for fractional responses). \(\hat\beta_T\) is a
log risk ratio: \(\exp(\hat\beta_T)\) is the estimated treatment
relative risk, the same estimand as
InferenceIncidLogBinomial's
log-binomial model, but modified Poisson never produces a fit failure from
the \([0,1]\)-probability constraint that a genuine binomial log-link
model can hit. Standard errors use Zou's (2004) robust/sandwich
correction, not the ordinary (model-based) Poisson Fisher information: the
misspecified Poisson working model assumes \(\mathrm{Var}(Y_i) = \mu_i\),
but \(Y_i\) is actually Bernoulli with \(\mathrm{Var}(Y_i) = p_i(1 -
p_i) \le \mu_i\), so the naive Fisher-information SE materially
overstates the true sampling variance. The sandwich correction is computed
in R from quantities fast_poisson_regression_with_var_cpp
already returns (mu and the unrestricted Fisher information
fisher_information), not a change to the underlying kernel.
Changed 2026-10: earlier versions of this class used the naive
Fisher-information SE directly, which measured at roughly 0.26
Type-I error against a nominal 5
Bernoulli design, model_formula = ~., asymptotic Wald test); every
standard error, confidence interval, and p-value this class has ever
returned differs from this version's.
likelihood_tier = "full" metadata is set for component-composition
purposes, but private$supports_likelihood_tests() is hard
FALSE — only Wald inference is exposed
(get_supported_testing_types_impl() returns "wald" only), not
likelihood-ratio/score/gradient tests. Fits with implausible coefficients
or fitted linear predictors (checked via
private$is_modified_poisson_fit_reasonable(), capped by
max_abs_reasonable_coef/max_abs_reasonable_linear_predictor)
are cached as nonestimable rather than returned.
Computes the randomization distribution of the treatment effect estimate under the sharp null.
Whether compute_rand_two_sided_pval() is actually
usable on this instance right now – FALSE exactly when it
would stop(): an incidence-response instance with no
custom randomization statistic and a design not eligible for
design-randomization-based incidence inference (see
private$should_use_design_randomization_for_incidence()).
TRUE for every other case, including every non-incidence
response type. Public, self-contained (only reads already-set
instance state, no side effects), so InferenceSuite can
check this before attempting the sentinel instead of relying on
the stop() being silently swallowed into a pval = NA
"ok" row – the single source of truth for both this check and
compute_rand_two_sided_pval()'s own guard, so the two can
never drift apart.
Value
When debug = FALSE (default), a numeric vector of length r. When
debug = TRUE, a list with: values, errors (list of character
vectors, one per iteration), warnings (list of character vectors, one per
iteration), num_errors, num_warnings,
prop_iterations_with_errors, prop_iterations_with_warnings, and
prop_illegal_values.
A single logical.
References
Zou, G. (2004). "A Modified Poisson Regression Approach to Prospective Studies with Binary Data." American Journal of Epidemiology, 159(7), 702-706, doi:10.1093/aje/kwh090 .
See also
InferenceIncidLogBinomial
for the genuine log-binomial alternative with the same log-risk-ratio
estimand. Comparable Python API:
statsmodels
discrete models (Poisson family on binary data). See also:
Poisson
regression (Wikipedia).
Super class
Inference -> InferenceIncidModifiedPoisson
Methods
+ inherited public methods from Inference
Inference$capabilities()Inference$compute_asymp_confidence_interval()Inference$compute_asymp_two_sided_pval()Inference$compute_estimate()Inference$compute_exact_confidence_interval()Inference$compute_exact_two_sided_pval_for_treatment_effect()Inference$duplicate()Inference$get_analysis_data()Inference$get_covariates()Inference$get_design_object()Inference$get_model_formula()Inference$get_nonestimable_reason()Inference$get_nonestimable_stage()Inference$get_optimization_alg()Inference$get_response()Inference$get_response_type()Inference$get_treatment()Inference$initialize()Inference$is_nonestimable()Inference$set_optimization_alg()Inference$set_seed()Inference$supports()
InferenceIncidModifiedPoisson$approximate_randomization_distribution_beta_hat_T()
Usage
InferenceIncidModifiedPoisson$approximate_randomization_distribution_beta_hat_T(
r = 501,
delta = 0,
transform_responses = "none",
show_progress = TRUE,
permutations = NULL,
debug = FALSE,
zero_one_logit_clamp = .Machine$double.eps
)Arguments
rNumber of randomization vectors. Default 501.
deltaThe null difference. Default 0.
transform_responsesType of transformation. Default "none".
show_progressShow progress bar. Default TRUE.
permutationsPre-computed permutations. Default NULL.
debugIf
TRUE, return a list with the distribution values and per-iteration diagnostics including error messages, warning messages, counts of each, and summary proportions for iterations with errors, warnings, and illegal (non-finite) values. Runs serially. DefaultFALSE.zero_one_logit_clampThe clamping amount for exact 0 and 1 values when logging
Examples
# \donttest{
seq_des = DesignSeqOneByOneBernoulli$new(n = 10, response_type = 'incidence')
for (i in 1:10) {
seq_des$add_one_subject_to_experiment_and_assign(data.frame(x1 = rnorm(1)))
}
seq_des$add_all_subject_responses(rbinom(10, 1, 0.5))
inf = InferenceIncidModifiedPoisson$new(seq_des)
inf$compute_estimate()
#> [1] 0.6606907
# }