Stratified identity-link MSM#

This study validates the identity-link marginal structural model (MSM) of ordinary point-treatment TMLE with baseline strata, strata=["V"]. The fit solves one score block \(I(V=s) H_s / P_n(V=s)\) for each stratum in one pooled fluctuation. Each stratum’s coefficients are the projection of the outcome regression on the working model under the law given \(V = s\). MSM projections states the construction.

What was tested#

setting	declaration
law	L1 of `tests/studies/stratified_alternating_law.py`, the baseline-strata law, with a bounded outcome \(Y \sim \operatorname{Beta}(24 Q, 24 (1 - Q))\). \(E[Y \mid A, W, V]\) is L1’s \(Q\), so every truth is exact
working model	`MSM.linear(modifiers=("W",), interaction=False)`: \((1, a, W)\) with uniform weights
fit	in sample, a quasi-binomial outcome GLM on \((A, W, V)\) and a main-terms logistic treatment regression on \((W, V)\), with `g_bounds=(0.01, 0.99)`
primary scenario	`stratified_msm_identity`: the nine stratum coefficients, \(n = 2000\), 1,000 replications
Monte Carlo inference	exact 99% intervals around every rate, and 99% intervals around every primary endpoint

The comparator is R tmle3 Param_MSM at commit ed72f8a. tmle3 has no stratified MSM, so the runner calls Param_MSM once per stratum subset, as the marginal tmle3_msm runner builds it, and transforms the arm-indicator coefficients to the \((1, a, W)\) basis. The two implementations differ in one declared way: tmle3 fits its nuisances inside each stratum subset with Lrnr_glm, and cleverly pools the nuisances over the strata and solves one block for each stratum. A subset curve is on the subset’s scale, so the runner’s standard error is \(\operatorname{sd} / \sqrt{n_s}\), the number the stratified fit’s embedded curve gives.

Accuracy against known truth#

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=0]`	point-treatment MSM projection intercept coefficient, in stratum V=0	`cleverly` identity-link MSM TMLE with baseline strata	-0.000735 to 0.000059	0.9430	0.9763	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=0]`	point-treatment MSM projection intercept coefficient, in stratum V=0	R `tmle3` `Param_MSM`, once per stratum subset	-0.000742 to 0.000053	0.9410	0.9756	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=1]`	point-treatment MSM projection intercept coefficient, in stratum V=1	`cleverly` identity-link MSM TMLE with baseline strata	-0.000820 to 0.000312	0.9460	0.9894	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=1]`	point-treatment MSM projection intercept coefficient, in stratum V=1	R `tmle3` `Param_MSM`, once per stratum subset	-0.000814 to 0.000319	0.9480	1.0067	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=2]`	point-treatment MSM projection intercept coefficient, in stratum V=2	`cleverly` identity-link MSM TMLE with baseline strata	-0.000316 to 0.000973	0.9410	0.9975	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=2]`	point-treatment MSM projection intercept coefficient, in stratum V=2	R `tmle3` `Param_MSM`, once per stratum subset	-0.000337 to 0.000954	0.9470	1.0332	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=0]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=0	`cleverly` identity-link MSM TMLE with baseline strata	-0.000439 to 0.000227	0.9450	0.9719	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=0]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=0	R `tmle3` `Param_MSM`, once per stratum subset	-0.000432 to 0.000234	0.9470	0.9799	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=1]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=1	`cleverly` identity-link MSM TMLE with baseline strata	-0.000349 to 0.000374	0.9610	1.0260	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=1]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=1	R `tmle3` `Param_MSM`, once per stratum subset	-0.000345 to 0.000378	0.9700	1.0691	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=2]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=2	`cleverly` identity-link MSM TMLE with baseline strata	-0.000567 to 0.000187	0.9360	0.9732	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=2]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=2	R `tmle3` `Param_MSM`, once per stratum subset	-0.000563 to 0.000193	0.9480	1.0220	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=0]`	point-treatment MSM projection treatment coefficient, in stratum V=0	`cleverly` identity-link MSM TMLE with baseline strata	-0.000134 to 0.000880	0.9480	0.9931	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=0]`	point-treatment MSM projection treatment coefficient, in stratum V=0	R `tmle3` `Param_MSM`, once per stratum subset	-0.000130 to 0.000885	0.9450	0.9940	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=1]`	point-treatment MSM projection treatment coefficient, in stratum V=1	`cleverly` identity-link MSM TMLE with baseline strata	-0.000514 to 0.000660	0.9490	0.9875	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=1]`	point-treatment MSM projection treatment coefficient, in stratum V=1	R `tmle3` `Param_MSM`, once per stratum subset	-0.000523 to 0.000649	0.9510	0.9939	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=2]`	point-treatment MSM projection treatment coefficient, in stratum V=2	`cleverly` identity-link MSM TMLE with baseline strata	-0.000651 to 0.000411	0.9460	0.9825	pass
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=2]`	point-treatment MSM projection treatment coefficient, in stratum V=2	R `tmle3` `Param_MSM`, once per stratum subset	-0.000627 to 0.000438	0.9450	0.9850	pass

Agreement with the canonical implementation#

Comparator-backed: a paired comparison with tmle3-msm-stratified on the same samples. Each row reports estimate equivalence within the declared margin, the RMSE ratio and the coverage difference, and names its comparison kind.

law	estimand	what was compared	paired difference	share of margin used	RMSE ratio bound	coverage difference	comparison kind	calibration resolution	result
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=0]`	point-treatment MSM projection intercept coefficient, in stratum V=0	0.000006	0.0082	1.0032	0.0020	related target	0.0043 vs 0.0500	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=1]`	point-treatment MSM projection intercept coefficient, in stratum V=1	-0.000006	0.0060	1.0056	-0.0020	related target	0.0173 vs 0.0500	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[(intercept)][V=2]`	point-treatment MSM projection intercept coefficient, in stratum V=2	0.000020	0.0172	1.0070	-0.0060	related target	0.0675 vs 0.0500 >	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=0]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=0	-0.000007	0.0119	1.0047	-0.0020	related target	0.0037 vs 0.0500	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=1]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=1	-0.000004	0.0060	1.0077	-0.0090	related target	0.0519 vs 0.0500 >	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[W][V=2]`	point-treatment MSM projection baseline-covariate coefficient, in stratum V=2	-0.000006	0.0081	1.0047	-0.0120	related target	0.0476 vs 0.0500	inconclusive
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=0]`	point-treatment MSM projection treatment coefficient, in stratum V=0	-0.000005	0.0048	1.0027	0.0030	related target	0.0028 vs 0.0500	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=1]`	point-treatment MSM projection treatment coefficient, in stratum V=1	0.000010	0.0089	1.0059	-0.0020	related target	0.0038 vs 0.0500	equivalent
baseline-strata law with a bounded outcome Y ~ Beta(24 Q, 24 (1 - Q))	`msm[a][V=2]`	point-treatment MSM projection treatment coefficient, in stratum V=2	-0.000025	0.0260	1.0037	0.0010	related target	0.0063 vs 0.0500	equivalent

Repeated-sampling properties#

property	cell	role	what was tested	what must hold	measured	result
`interval_calibration`	`identity_marginal_a__correctly_specified`	positive	treatment coefficient of the identity MSM (1, a, W), marginal: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9278 to 0.9551, SE ratio 0.9350 to 1.0150, empirical efficiency ratio 0.9853 to 1.0695, reported efficiency ratio 0.9990 to 1.0011	pass
`interval_calibration`	`identity_marginal_a__noise_control`	control	treatment coefficient of the identity MSM (1, a, W), marginal: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8131 to 0.8563, SE ratio 0.6893 to 0.7446, empirical efficiency ratio 1.3429 to 1.4509, reported efficiency ratio 0.9990 to 1.0011	pass
`interval_calibration`	`identity_marginal_a__shrunken_se_control`	control	treatment coefficient of the identity MSM (1, a, W), marginal: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8037 to 0.8478, SE ratio 0.6551 to 0.7111, empirical efficiency ratio 0.9849 to 1.0689, reported efficiency ratio 0.6994 to 0.7008	pass
`interval_calibration`	`identity_marginal_intercept__correctly_specified`	positive	intercept coefficient of the identity MSM (1, a, W), marginal: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9406 to 0.9652, SE ratio 0.9791 to 1.0599, empirical efficiency ratio 0.9441 to 1.0221, reported efficiency ratio 0.9993 to 1.0017	pass
`interval_calibration`	`identity_marginal_intercept__noise_control`	control	intercept coefficient of the identity MSM (1, a, W), marginal: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8294 to 0.8709, SE ratio 0.6972 to 0.7582, empirical efficiency ratio 1.3197 to 1.4341, reported efficiency ratio 0.9993 to 1.0017	pass
`interval_calibration`	`identity_marginal_intercept__shrunken_se_control`	control	intercept coefficient of the identity MSM (1, a, W), marginal: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8115 to 0.8549, SE ratio 0.6844 to 0.7426, empirical efficiency ratio 0.9428 to 1.0232, reported efficiency ratio 0.6995 to 0.7012	pass
`interval_calibration`	`identity_marginal_w__correctly_specified`	positive	baseline-covariate coefficient of the identity MSM (1, a, W), marginal: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9417 to 0.9661, SE ratio 0.9698 to 1.0526, empirical efficiency ratio 0.9508 to 1.0317, reported efficiency ratio 0.9995 to 1.0017	pass
`interval_calibration`	`identity_marginal_w__noise_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), marginal: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8010 to 0.8454, SE ratio 0.6678 to 0.7261, empirical efficiency ratio 1.3785 to 1.4981, reported efficiency ratio 0.9994 to 1.0018	pass
`interval_calibration`	`identity_marginal_w__shrunken_se_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), marginal: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8173 to 0.8601, SE ratio 0.6799 to 0.7375, empirical efficiency ratio 0.9499 to 1.0303, reported efficiency ratio 0.6996 to 0.7012	pass
`interval_calibration`	`identity_v0_a__correctly_specified`	positive	treatment coefficient of the identity MSM (1, a, W), in stratum V = 0: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9316 to 0.9582, SE ratio 0.9347 to 1.0142, empirical efficiency ratio 0.9857 to 1.0701, reported efficiency ratio 0.9984 to 1.0016	pass
`interval_calibration`	`identity_v0_a__noise_control`	control	treatment coefficient of the identity MSM (1, a, W), in stratum V = 0: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.7911 to 0.8364, SE ratio 0.6584 to 0.7115, empirical efficiency ratio 1.4054 to 1.5182, reported efficiency ratio 0.9983 to 1.0016	pass
`interval_calibration`	`identity_v0_a__shrunken_se_control`	control	treatment coefficient of the identity MSM (1, a, W), in stratum V = 0: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7937 to 0.8387, SE ratio 0.6548 to 0.7108, empirical efficiency ratio 0.9847 to 1.0689, reported efficiency ratio 0.6989 to 0.7011	pass
`interval_calibration`	`identity_v0_intercept__correctly_specified`	positive	intercept coefficient of the identity MSM (1, a, W), in stratum V = 0: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9327 to 0.9591, SE ratio 0.9348 to 1.0190, empirical efficiency ratio 0.9830 to 1.0714, reported efficiency ratio 0.9995 to 1.0036	pass
`interval_calibration`	`identity_v0_intercept__noise_control`	control	intercept coefficient of the identity MSM (1, a, W), in stratum V = 0: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8068 to 0.8506, SE ratio 0.6714 to 0.7299, empirical efficiency ratio 1.3725 to 1.4915, reported efficiency ratio 0.9995 to 1.0035	pass
`interval_calibration`	`identity_v0_intercept__shrunken_se_control`	control	intercept coefficient of the identity MSM (1, a, W), in stratum V = 0: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7995 to 0.8440, SE ratio 0.6540 to 0.7125, empirical efficiency ratio 0.9835 to 1.0721, reported efficiency ratio 0.6996 to 0.7025	pass
`interval_calibration`	`identity_v0_w__correctly_specified`	positive	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 0: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9278 to 0.9551, SE ratio 0.9461 to 1.0233, empirical efficiency ratio 0.9780 to 1.0574, reported efficiency ratio 0.9979 to 1.0033	pass
`interval_calibration`	`identity_v0_w__noise_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 0: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8115 to 0.8549, SE ratio 0.6787 to 0.7365, empirical efficiency ratio 1.3581 to 1.4739, reported efficiency ratio 0.9979 to 1.0032	pass
`interval_calibration`	`identity_v0_w__shrunken_se_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 0: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7963 to 0.8411, SE ratio 0.6623 to 0.7169, empirical efficiency ratio 0.9770 to 1.0571, reported efficiency ratio 0.6985 to 0.7022	pass
`interval_calibration`	`identity_v1_a__correctly_specified`	positive	treatment coefficient of the identity MSM (1, a, W), in stratum V = 1: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9305 to 0.9573, SE ratio 0.9387 to 1.0184, empirical efficiency ratio 0.9810 to 1.0640, reported efficiency ratio 0.9969 to 1.0013	pass
`interval_calibration`	`identity_v1_a__noise_control`	control	treatment coefficient of the identity MSM (1, a, W), in stratum V = 1: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.7901 to 0.8354, SE ratio 0.6627 to 0.7188, empirical efficiency ratio 1.3897 to 1.5075, reported efficiency ratio 0.9969 to 1.0012	pass
`interval_calibration`	`identity_v1_a__shrunken_se_control`	control	treatment coefficient of the identity MSM (1, a, W), in stratum V = 1: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8005 to 0.8449, SE ratio 0.6582 to 0.7120, empirical efficiency ratio 0.9825 to 1.0623, reported efficiency ratio 0.6979 to 0.7009	pass
`interval_calibration`	`identity_v1_intercept__correctly_specified`	positive	intercept coefficient of the identity MSM (1, a, W), in stratum V = 1: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9255 to 0.9533, SE ratio 0.9440 to 1.0247, empirical efficiency ratio 0.9771 to 1.0601, reported efficiency ratio 0.9984 to 1.0038	pass
`interval_calibration`	`identity_v1_intercept__noise_control`	control	intercept coefficient of the identity MSM (1, a, W), in stratum V = 1: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8178 to 0.8605, SE ratio 0.6884 to 0.7471, empirical efficiency ratio 1.3402 to 1.4553, reported efficiency ratio 0.9984 to 1.0038	pass
`interval_calibration`	`identity_v1_intercept__shrunken_se_control`	control	intercept coefficient of the identity MSM (1, a, W), in stratum V = 1: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7979 to 0.8425, SE ratio 0.6604 to 0.7170, empirical efficiency ratio 0.9774 to 1.0611, reported efficiency ratio 0.6989 to 0.7027	pass
`interval_calibration`	`identity_v1_w__correctly_specified`	positive	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 1: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9473 to 0.9704, SE ratio 0.9723 to 1.0507, empirical efficiency ratio 0.9544 to 1.0308, reported efficiency ratio 0.9993 to 1.0058	pass
`interval_calibration`	`identity_v1_w__noise_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 1: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8037 to 0.8478, SE ratio 0.6726 to 0.7292, empirical efficiency ratio 1.3751 to 1.4912, reported efficiency ratio 0.9994 to 1.0056	pass
`interval_calibration`	`identity_v1_w__shrunken_se_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 1: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8121 to 0.8553, SE ratio 0.6812 to 0.7358, empirical efficiency ratio 0.9538 to 1.0300, reported efficiency ratio 0.6996 to 0.7040	pass
`interval_calibration`	`identity_v2_a__correctly_specified`	positive	treatment coefficient of the identity MSM (1, a, W), in stratum V = 2: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9233 to 0.9515, SE ratio 0.9264 to 1.0042, empirical efficiency ratio 0.9925 to 1.0766, reported efficiency ratio 0.9937 to 1.0005	fail
`interval_calibration`	`identity_v2_a__noise_control`	control	treatment coefficient of the identity MSM (1, a, W), in stratum V = 2: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8021 to 0.8463, SE ratio 0.6742 to 0.7302, empirical efficiency ratio 1.3648 to 1.4795, reported efficiency ratio 0.9937 to 1.0005	pass
`interval_calibration`	`identity_v2_a__shrunken_se_control`	control	treatment coefficient of the identity MSM (1, a, W), in stratum V = 2: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7885 to 0.8340, SE ratio 0.6484 to 0.7032, empirical efficiency ratio 0.9930 to 1.0758, reported efficiency ratio 0.6955 to 0.7003	pass
`interval_calibration`	`identity_v2_intercept__correctly_specified`	positive	intercept coefficient of the identity MSM (1, a, W), in stratum V = 2: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9305 to 0.9573, SE ratio 0.9462 to 1.0239, empirical efficiency ratio 0.9762 to 1.0558, reported efficiency ratio 0.9943 to 1.0038	pass
`interval_calibration`	`identity_v2_intercept__noise_control`	control	intercept coefficient of the identity MSM (1, a, W), in stratum V = 2: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.7937 to 0.8387, SE ratio 0.6543 to 0.7087, empirical efficiency ratio 1.4101 to 1.5271, reported efficiency ratio 0.9945 to 1.0040	pass
`interval_calibration`	`identity_v2_intercept__shrunken_se_control`	control	intercept coefficient of the identity MSM (1, a, W), in stratum V = 2: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8021 to 0.8463, SE ratio 0.6614 to 0.7175, empirical efficiency ratio 0.9753 to 1.0566, reported efficiency ratio 0.6962 to 0.7026	pass
`interval_calibration`	`identity_v2_w__correctly_specified`	positive	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 2: both nuisances are correctly specified with an independently computed efficiency bound	SE ratio and coverage intervals both inside their calibration bands, with both efficiency-ratio intervals inside their bands	coverage 0.9311 to 0.9578, SE ratio 0.9466 to 1.0276, empirical efficiency ratio 0.9692 to 1.0513, reported efficiency ratio 0.9891 to 1.0009	pass
`interval_calibration`	`identity_v2_w__noise_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 2: one efficiency-bound unit of independent noise is added to each estimate	the empirical efficiency ratio must rise above the band	coverage 0.8016 to 0.8459, SE ratio 0.6653 to 0.7209, empirical efficiency ratio 1.3808 to 1.4947, reported efficiency ratio 0.9890 to 1.0009	pass
`interval_calibration`	`identity_v2_w__shrunken_se_control`	control	baseline-covariate coefficient of the identity MSM (1, a, W), in stratum V = 2: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8010 to 0.8454, SE ratio 0.6624 to 0.7199, empirical efficiency ratio 0.9683 to 1.0508, reported efficiency ratio 0.6924 to 0.7007	pass

The property grid calibrates each coefficient of the stratified fit, marginal and in each stratum, against its exact efficiency bound under the bounded outcome. Each cell has a shrunken-SE control and a noise control.

Two results are red, both in stratum 2, which holds about 400 of the 2,000 rows. The interval_calibration/identity_v2_a__correctly_specified cell has an SE-ratio interval of 0.926 to 1.004 against a floor of 0.93. On the primary draws the same coefficient reads 0.983 in cleverly and 0.985 in R. The paired msm[W][V=2] row is inconclusive on its calibration leg alone. The estimates differ by 6e-6 on average with the same spread, and R reports a standard error 2.2% above its own spread. That is consistent with the declared nuisance difference: R fits a Gaussian Lrnr_glm inside each subset.

The owner X8-identity-small-stratum in the roadmap records the reading. The study declared gated and moved to reporting before a re-run at the same seeds.

Measured values and declared margins#

quantity	value	source
`replicates`	1000	paired replications
`n`	2000	observations per primary replication
`independent_tests_passed`	18	truth tests passing
`independent_tests_total`	18	truth tests reported
`paired_tests_passed`	8	paired comparisons passing
`paired_tests_total`	9	paired comparisons reported
`property_cells_passed`	35	property cells passing
`property_cells_total`	36	property cells reported
`max_standardized_bias`	0.0706	largest primary standardized bias
`min_coverage`	0.9360	lowest primary coverage
`max_margin_utilization`	0.0260	largest paired similarity-margin share
`margin:confidence_level`	0.9900	Monte Carlo confidence level
`margin:alpha`	0.0500	nominal test size
`margin:nominal_coverage`	0.9500	nominal interval coverage
`margin:bootstrap_replicates`	10000	bootstrap replications
`margin:standardized_bias`	0.2500	standardized-bias margin
`margin:coverage_floor`	0.9000	primary coverage floor
`margin:over_coverage_ceiling`	0.9900	descriptive overcoverage threshold
`margin:se_ratio_sanity_lower`	0.8000	primary SE-ratio lower screen
`margin:se_ratio_sanity_upper`	1.2000	primary SE-ratio upper screen
`margin:calibration_se_ratio_lower`	0.9300	calibration SE-ratio lower bound
`margin:calibration_se_ratio_upper`	1.0700	calibration SE-ratio upper bound
`margin:calibration_coverage_lower`	0.9200	calibration coverage lower bound
`margin:calibration_coverage_upper`	0.9800	calibration coverage upper bound
`margin:type_i_ceiling`	0.1000	type-I upper bound
`margin:paired_difference`	0.1500	paired similarity margin in pooled SDs
`margin:rmse_noninferiority`	1.1000	RMSE-ratio noninferiority bound
`margin:coverage_noninferiority`	-0.0250	coverage-difference noninferiority bound
`margin:calibration_noninferiority`	0.0500	calibration-excess noninferiority bound
`margin:minimum_power`	0.8000	minimum power lower bound
`margin:root_n_slope`	-0.5000	expected root-n slope
`margin:root_n_slope_lower`	-0.6250	root-n slope lower bound
`margin:root_n_slope_upper`	-0.3750	root-n slope upper bound
`margin:excluded_slope`	-0.2500	slower rate the interval must exclude
`margin:efficiency_ratio_lower`	0.9000	efficiency-ratio lower bound
`margin:efficiency_ratio_upper`	1.1000	efficiency-ratio upper bound
`margin:shrunken_se_factor`	0.7000	negative-control SE multiplier
`bound:msm[(intercept)][V=0]_standard_error`	0.0048	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[(intercept)][V=1]_standard_error`	0.0069	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[(intercept)][V=2]_standard_error`	0.0079	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[W][V=0]_standard_error`	0.0040	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[W][V=1]_standard_error`	0.0045	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[W][V=2]_standard_error`	0.0045	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[a][V=0]_standard_error`	0.0062	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[a][V=1]_standard_error`	0.0071	exact bounded-outcome efficiency-bound standard error at primary n
`bound:msm[a][V=2]_standard_error`	0.0064	exact bounded-outcome efficiency-bound standard error at primary n

Limits#

The study covers one finite-support law with one three-level stratum and a bounded outcome.
The fits are in sample. A cross-fitted stratified MSM is not measured here.
Linked and continuous-dose stratified MSMs are measured in the stratified incremental MSM study.

Reproduction#

The fixture README, manifest, replications, paired decisions, and property results carry the protocol, provenance, and every published row.