Simple Mean-Difference Inference for Continuous Responses

Fits the simplest possible treatment-effect estimator for a continuous response: the unadjusted difference in sample means between the treated and control arms, \(\hat\beta_T = \bar y_T - \bar y_C\), with no covariate adjustment. Inference is by Welch's unequal-variance t-test: standard error \(\sqrt{s_T^2/n_T + s_C^2/n_C}\) (per-arm sample variances, not pooled) with Satterthwaite-Welch degrees of freedom — see $compute_asymp_confidence_interval() for the exact formula. This class has no likelihood tier (likelihood_tier = "none") and provides asymptotic Wald, randomization, and bootstrap (including Bayesian bootstrap) confidence intervals and p-values. Warm starts are disabled for this class, since the simple mean difference is a closed-form estimator (no iterative fit to warm-start).

Initialize a simple mean-difference inference object.

Computes a \(1-\alpha\) level confidence interval for the simple (unadjusted) mean-difference treatment effect \(\hat\beta_T = \bar y_T - \bar y_C\), using Welch's unequal-variance formula: standard error \(\widehat{\mathrm{SE}}(\hat\beta_T) = \sqrt{s_T^2/n_T + s_C^2/n_C}\) (sample variances \(s_T^2\), \(s_C^2\) computed separately per arm, not pooled) with Satterthwaite-Welch degrees of freedom \(\mathrm{df} = (s_T^2/n_T + s_C^2/n_C)^2 \big/ \left(\frac{(s_T^2/n_T)^2}{n_T-1} + \frac{(s_C^2/n_C)^2}{n_C-1}\right)\); the interval is \(\hat\beta_T \pm t_{\mathrm{df}, 1-\alpha/2}\,\widehat{\mathrm{SE}}(\hat\beta_T)\). Requires at least 2 observations per arm; otherwise the standard error and interval are NA. See InferenceAsymp for the shared asymptotic confidence-interval contract this delegates to.

Computes a two-sided Welch's t-test p-value testing \(H_0: \beta_T = \code{delta}\), from the same Welch unequal-variance standard error and Satterthwaite-Welch degrees of freedom used by $compute_asymp_confidence_interval() — see that method's documentation for the full formula. See InferenceAsymp for the shared asymptotic two-sided p-value contract this delegates to.

Computes the simple (unadjusted) mean-difference point estimate \(\hat\beta_T = \bar y_T - \bar y_C\), the difference in sample means between the treated and control arms. NA if either arm has zero observations. See InferenceMLEorKMSummaryTable for the shared estimate-contract this participates in.

Recomputes the simple mean-difference estimate under subject/block bootstrap weights (used by the Bayesian bootstrap and related weighted-resampling machinery — see InferenceNonParamBootstrap). The weighted point estimate is \(\hat\beta_T = \bar y_T^w - \bar y_C^w\), weighted arm means \(\bar y_T^w = \sum_i r_i y_i \mathbb{1}[w_i=1] / \sum_i r_i \mathbb{1}[w_i=1]\) (and analogously for control), where \(r_i\) are the expanded row weights. Unless estimate_only = TRUE, the standard error treats the bootstrap weights as frequency weights. Within each arm, with total weight \(W = \sum r_i\) and weighted residual sum of squares \(Q = \sum r_i(y_i - \bar y_w)^2\), the variance contribution is \(Q / [W(W - 1)]\); the Satterthwaite-Welch degrees of freedom likewise use \(W - 1\). This reduces exactly to the ordinary Welch formula for unit weights. Rows with non-finite or non-positive weight, or a non-finite response, are dropped before computing; if no rows survive, returns NA with all cached variance components set to NA.

Value

A two-sided p-value.

The setting-appropriate (see description) numeric estimate of the treatment effect

References

Welch, B. L. (1947). "The Generalization of 'Student's' Problem when Several Different Population Variances are Involved." Biometrika, 34(1-2), 28-35, doi:10.1093/biomet/34.1-2.28 , for the unequal-variance t-test and its Satterthwaite-Welch degrees-of-freedom approximation used here.

Super class

Inference -> InferenceAllSimpleAverageDiff

Methods

Public methods

• InferenceAllSimpleAverageDiff$compute_rand_two_sided_pval()

• InferenceAllSimpleAverageDiff$new()

• InferenceAllSimpleAverageDiff$compute_asymp_confidence_interval()

• InferenceAllSimpleAverageDiff$compute_asymp_two_sided_pval()

• InferenceAllSimpleAverageDiff$compute_estimate()

• InferenceAllSimpleAverageDiff$compute_estimate_with_bootstrap_weights()

• InferenceAllSimpleAverageDiff$clone()

+ 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()

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InferenceAllSimpleAverageDiff$compute_rand_two_sided_pval()

Uses the randomization-CI layer's two-sided p-value contract (InferenceRandCI's version, not InferenceRand's): for incidence responses this dispatches to the Zhang exact randomization test where applicable rather than refusing outright, matching this class's pre-migration old-ladder behavior (see InferenceIncidRiskDiff's identical rationale). Previously bound to InferenceRand's version instead, which silently regressed Zhang dispatch after migration – see inference_all_abstract_rand_ci.R's compute_rand_two_sided_pval for why it's now safe to splice this in outside the old inheritance chain.

Usage

InferenceAllSimpleAverageDiff$compute_rand_two_sided_pval(
  r = 501,
  delta = 0,
  transform_responses = "none",
  na.rm = TRUE,
  show_progress = TRUE,
  permutations = NULL,
  type = NULL,
  args_for_type = NULL,
  zero_one_logit_clamp = .Machine$double.eps
)

Arguments

r

Number of randomization vectors.

delta

Null treatment effect value.

delta

Null treatment effect value.

transform_responses

Response transformation to apply during the test. For survival responses the default "log" multiplies the recorded times of the units treated under each reference allocation by \(e^\delta\), event and censoring times alike, with censoring indicators unchanged – the rank-based AFT residual construction (Tsiatis 1990; Wei, Ying and Lin 1990; Jin, Lin, Wei and Ying 2003); see compute_rand_confidence_interval() for the assumptions.

na.rm

Whether to remove non-finite simulated statistics.

show_progress

Whether to show progress.

permutations

Optional pre-generated assignment draws.

type

Optional incidence-specific exact randomization type.

args_for_type

Optional arguments keyed by type.

zero_one_logit_clamp

The clamping amount for exact 0 and 1 values when logging

---

InferenceAllSimpleAverageDiff$new()

Usage

InferenceAllSimpleAverageDiff$new(
  des_obj,
  model_formula = NULL,
  verbose = FALSE,
  max_resample_attempts = 50L,
  smart_cold_start_default = NULL
)

Arguments

des_obj

A DesignSeqOneByOne object whose entire n subjects are assigned and response y is recorded within.

model_formula

Optional formula for covariate adjustment. If NULL (default), the formula from the design object is used and its pre-computed design matrix is reused. If a formula is provided, a new design matrix is constructed from the design's imputed covariates.

verbose

Whether to print progress messages. Default FALSE.

max_resample_attempts

Maximum number of times a single bootstrap replicate may be redrawn when the drawn sample fails validity screening. If all attempts fail the replicate is recorded as NA, silently reducing the effective B. Must be a positive integer. Default 50L.

smart_cold_start_default

Whether to use smart cold start values.

---

InferenceAllSimpleAverageDiff$compute_asymp_confidence_interval()

Usage

InferenceAllSimpleAverageDiff$compute_asymp_confidence_interval(alpha = 0.05)

Arguments

alpha

Confidence level.

---

InferenceAllSimpleAverageDiff$compute_asymp_two_sided_pval()

Usage

InferenceAllSimpleAverageDiff$compute_asymp_two_sided_pval(delta = 0)

Arguments

delta

Null treatment effect value.

delta

Null treatment effect value.

---

InferenceAllSimpleAverageDiff$compute_estimate()

Usage

InferenceAllSimpleAverageDiff$compute_estimate(estimate_only = FALSE)

Arguments

estimate_only

If TRUE, skip variance component calculations.

estimate_only

If TRUE, skip variance calculations.

---

InferenceAllSimpleAverageDiff$compute_estimate_with_bootstrap_weights()

Usage

InferenceAllSimpleAverageDiff$compute_estimate_with_bootstrap_weights(
  subject_or_block_weights,
  estimate_only = FALSE
)

Arguments

subject_or_block_weights

Row weights for the bootstrap sample.

estimate_only

If TRUE, skip variance component calculations.

estimate_only

If TRUE, skip variance calculations.

---

InferenceAllSimpleAverageDiff$clone()

The objects of this class are cloneable with this method.

Usage

InferenceAllSimpleAverageDiff$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples

seq_des = DesignSeqOneByOneBernoulli$new(n = 6, response_type = "continuous")
seq_des$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] 0
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] 1
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 = InferenceAllSimpleAverageDiff$new(seq_des)
seq_des_inf$compute_estimate()
#> [1] 2.9825
seq_des_inf$compute_asymp_confidence_interval()
#>      2.5%     97.5% 
#> -2.810951  8.775951 
seq_des_inf$compute_asymp_two_sided_pval()
#> [1] 0.181089