Mean-Difference IVWC Inference for KK Matching-on-the-Fly Designs
Source:R/inference_all_KK_mean_diff_IVWC.R
InferenceAllKKMeanDiffIVWC.RdFits a compound (inverse-variance-weighted combination, "IVWC") mean-difference
estimator of the treatment effect for continuous responses under a
DesignSeqOneByOne-family KK matching-on-the-fly
design (see DesignSeqOneByOneKK14 and
DesignSeqOneByOneKK21). Such a design
produces two structurally different kinds of subjects: subjects successfully
matched into pairs during the sequential design, and unmatched "reservoir"
subjects randomized independently. This estimator combines both:
$$\hat\beta_T = w^* \bar d + (1 - w^*)\, \bar r, \qquad
w^* = \frac{\widehat{\mathrm{Var}}(\bar r)}{\widehat{\mathrm{Var}}(\bar r) +
\widehat{\mathrm{Var}}(\bar d)},$$
where \(\bar d\) is the mean within-pair (treated minus control) difference
among matched subjects and \(\bar r\) is the treated-minus-control difference
in means among reservoir subjects, weighted inversely by their estimated
variances (see $compute_asymp_confidence_interval() for the full
variance formula and the fallback behavior when only one of the two
sub-estimates is usable). Inference is Wald-only: this class has no
likelihood tier (likelihood_tier = "none") and provides asymptotic
Wald, randomization, and bootstrap (including Bayesian bootstrap) confidence
intervals and p-values, but no score/likelihood-ratio/gradient tests.
Initialize KK IVWC mean-difference inference.
Computes the compound IVWC (inverse-variance-weighted
compound) mean-difference point estimate \(\hat\beta_T\): the
inverse-variance-weighted combination \(w^* \bar d + (1-w^*)\, \bar
r\) of the matched-pair mean within-pair difference \(\bar d\) and the
reservoir treated-minus-control mean difference \(\bar r\), falling
back to whichever of the two is usable if the other is not (see
$compute_asymp_confidence_interval() for the full weighting
formula and usability conditions).
Computes a \(1-\alpha\) level frequentist confidence interval for the compound IVWC (inverse-variance-weighted compound) mean-difference estimator \(\hat\beta_T\).
Computes a two-sided Wald p-value for the compound
IVWC mean-difference estimator \(\hat\beta_T\) testing
\(H_0: \beta_T = \code{delta}\), using the same asymptotically-normal
point estimate and standard error (\(z = (\hat\beta_T -
\code{delta})/\widehat{\mathrm{SE}}(\hat\beta_T)\)) that
$compute_asymp_confidence_interval() inverts to form its interval
— see that method's documentation for the full inverse-variance-weighted
combination formula. This class has no likelihood tier
(likelihood_tier = "none"), so no score, likelihood-ratio, or
gradient test is available here; this is a plain Wald test, not a
likelihood-backed one.
Value
The setting-appropriate (see description) numeric estimate of the treatment effect
A (1 - alpha)-sized frequentist confidence interval for the treatment effect
The approximate frequentist p-value
Details
The point estimate combines two sub-estimates depending on which are
usable: the mean within-pair difference among matched subjects,
\(\bar d\), with estimated variance \(\widehat{\mathrm{Var}}(\bar
d)\), and the treated-minus-control difference in means among reservoir
(unmatched) subjects, \(\bar r\), with estimated variance
\(\widehat{\mathrm{Var}}(\bar r)\). When both are usable (at least 2
matched pairs and at least 2 treated/2 control reservoir subjects, with
finite positive variance estimates), they are combined by classical
inverse-variance weighting,
$$\hat\beta_T = w^* \bar d + (1 - w^*)\, \bar r, \qquad w^* =
\frac{\widehat{\mathrm{Var}}(\bar r)}{\widehat{\mathrm{Var}}(\bar r) +
\widehat{\mathrm{Var}}(\bar d)},$$
with combined variance the standard inverse-variance-pooled form
\(\widehat{\mathrm{Var}}(\hat\beta_T) = \left(\widehat{\mathrm{Var}}(\bar
r)^{-1} + \widehat{\mathrm{Var}}(\bar d)^{-1}\right)^{-1} =
\widehat{\mathrm{Var}}(\bar r)\,\widehat{\mathrm{Var}}(\bar d) \big/
\left(\widehat{\mathrm{Var}}(\bar r) + \widehat{\mathrm{Var}}(\bar
d)\right)\). If only one of the two sub-estimates is usable (e.g. the
reservoir is empty or degenerate, or no pairs matched), \(\hat\beta_T\)
and its variance fall back to that sub-estimate alone. The compound
estimator is treated as asymptotically normal, so the interval is
\(\hat\beta_T \pm z_{1-\alpha/2}\sqrt{\widehat{\mathrm{Var}}(\hat\beta_T)}\)
(or a \(t\)-based critical value, depending on
private$compute_z_or_t_ci_from_s_and_df's degrees-of-freedom
resolution).
Legacy status
Legacy class. Not fully tested in
comprehensive_tests.R; prefer a more actively maintained KK
continuous-response inference class (e.g.
InferenceContinKKOLSIVWC) for new
analyses unless this specific unadjusted mean-difference estimator is
required.
References
Kapelner, A., and Krieger, A. M. (2014). "Matching on-the-fly: Sequential allocation with higher power and efficiency." Biometrics, 70(2), 378-388, doi:10.1111/biom.12148 , for the KK matching-on-the-fly design this estimator targets, and for the inverse-variance combination of matched-pair and reservoir estimates.
Super class
Inference -> InferenceAllKKMeanDiffIVWC
Methods
+ inherited public methods from Inference
Inference$capabilities()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$get_treatment()Inference$is_nonestimable()Inference$set_optimization_alg()Inference$set_seed()Inference$supports()
InferenceAllKKMeanDiffIVWC$new()
Usage
InferenceAllKKMeanDiffIVWC$new(
des_obj,
verbose = FALSE,
harden = TRUE,
model_formula = NULL,
smart_cold_start_default = NULL
)Examples
seq_des = DesignSeqOneByOneKK14$new(n = 6, response_type = "continuous")
seq_des$add_one_subject_to_experiment_and_assign(MASS::biopsy[1, 2 : 10])
#> [1] 0
seq_des$add_one_subject_to_experiment_and_assign(MASS::biopsy[2, 2 : 10])
#> [1] 0
seq_des$add_one_subject_to_experiment_and_assign(MASS::biopsy[3, 2 : 10])
#> [1] 1
seq_des$add_one_subject_to_experiment_and_assign(MASS::biopsy[4, 2 : 10])
#> [1] 0
seq_des$add_one_subject_to_experiment_and_assign(MASS::biopsy[5, 2 : 10])
#> [1] 0
seq_des$add_one_subject_to_experiment_and_assign(MASS::biopsy[6, 2 : 10])
#> [1] 1
seq_des$add_all_subject_responses(c(4.71, 1.23, 4.78, 6.11, 5.95, 8.43))
seq_des_inf = InferenceAllKKMeanDiffIVWC$
new(seq_des)
seq_des_inf$compute_estimate()
#> [1] 2.105
seq_des_inf$compute_asymp_confidence_interval()
#> 2.5% 97.5%
#> -1.884097 6.094097
seq_des_inf$compute_asymp_two_sided_pval()
#> [1] 0.3010192