DR-TMLE with one inconsistent nuisance#

This study measures complete-data and randomized missing-outcome DRTMLE when one nuisance converges to a wrong limit. The case table states three cases. The interval is valid for a declared mechanism, conservative for a correctly specified parametric mechanism, and not established when the consistent nuisance converges slower than root-n. This study puts each case on a law where the conditioning terms \(R_{3,n}\) and \(\tilde R_{5,n}\) do not vanish.

The study has no external comparator. It uses the reporting policy.

What was compared#

setting	value
law L	\(W_1, W_2 \sim N(0, 1)\), \(g_0 = \operatorname{expit}(0.6 W_1 + 0.3 W_2)\), \(\bar Q_0(a, W) = \operatorname{expit}(-0.2 + 0.5 a - W_1 + 2 W_2)\)
law M	the covariates and outcome of law L. \(A \sim \operatorname{Bernoulli}(0.5)\) is declared. \(Y\) is recorded with probability \(\pi_0(a, W) = \operatorname{expit}(0.4 + 0.2 a + 0.6 W_1 + 0.3 W_2)\)
truths	\(\psi(1) = 0.5422909862\), \(\psi(0) = 0.4717756171\), ATE \(0.0705153691\), by 120-node Gauss-Hermite quadrature on both laws
fit	`DRTMLE(guard=("Q", "g"), reduction="univariate", g_bounds=(0.01, 0.99))`, in sample except the cross-fitted cells
reduced regressions	cubic polynomials in the scalar design: least squares for the outcome reductions, Newton logistic with ridge `1e-8` for the mechanism reduction
intervals	pointwise 95% Wald from the influence curve

setting	outcome regression	mechanism	case
`q_wrong_g_known`	logistic GLM on \((A, W_1)\), wrong	the declared \(g_0\)	valid (H2)
`q_wrong_g_glm`	the same	logistic GLM on \((W_1, W_2)\), correct	conservative
`q_wrong_g_glm_cf`	the same, cross-fitted over the default folds	the same, cross-fitted	conservative
`q_wrong_g_slow`	the same	the correct GLM plus the logit offset \(n^{-1/3} W_2\)	not established
`q_slow_g_wrong`	the correct GLM plus the logit offset \(3 n^{-1/3}(2 W_1 + W_2)/\sqrt 5\)	logistic GLM on \(W_2\), wrong	not established (mirror)
`mar_q_wrong_pi_slow`	logistic GLM on \((A, W_1)\), wrong	declared \(A\). The observation mechanism is the correct GLM plus the logit offset \(2 n^{-1/3} W_2\)	not established

The offsets are constructed errors, not fitted learners. Each one converges at the rate \(n^{-1/3}\), so each conditioning term is of order \(n^{-1/3}\). Its root-n scaled contribution grows by \(16^{1/6} = 1.59\) from \(n = 1{,}000\) to \(n = 16{,}000\). The design test computes each term’s sign by quadrature (tests/unit/test_drtmle_one_wrong_design.py).

q_wrong_g_glm and q_wrong_g_known read the same draws. The primary scenario is q_wrong_g_known, so the accuracy table reads the valid case only.

Accuracy against known truth#

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
law L, declared mechanism, outcome regression on A and W1 (wrong), n = 4,000	`ate`	average treatment effect	`cleverly` DR-TMLE with guards Q and g and the univariate reduction	-0.000676 to 0.000537	0.9543	1.0229	pass
law L, declared mechanism, outcome regression on A and W1 (wrong), n = 4,000	`ey0`	counterfactual mean under no treatment	`cleverly` DR-TMLE with guards Q and g and the univariate reduction	-0.000180 to 0.000730	0.9513	1.0089	pass
law L, declared mechanism, outcome regression on A and W1 (wrong), n = 4,000	`ey1`	counterfactual mean under treatment	`cleverly` DR-TMLE with guards Q and g and the univariate reduction	-0.000244 to 0.000655	0.9505	1.0195	pass

The accuracy table reads the primary scenario, q_wrong_g_known, the valid case. No canonical implementation is compared, so the page has no agreement table.

Rules#

rule	kind	passes when
C	`conservative`	the 99% bootstrap interval of the SE ratio lies wholly above 1.07, the coverage lower bound clears 0.90, and the bias interval lies inside 0.25 empirical SD
V	`valid`	the coverage lower bound clears 0.90, the bias interval lies inside the margin, and the SE ratio lies inside 0.8 to 1.2
F	`first_order`	the bias interval lies wholly beyond the margin on the side of the declared sign, and the SE ratio lies inside 0.1 to 10
G	`growth`	the 99% bootstrap interval of \(s \cdot (\text{bias}/\text{SD})(16{,}000) - s \cdot (\text{bias}/\text{SD})(1{,}000)\) has a lower bound above 0, with \(s\) the declared sign

Rule C asserts a standard error that is too large. A projected variance would be a result-determining change that re-declares the cell. A red C cell is finite-sample only when its SE-ratio interval contains the planning ratio (1.1251 in sample, 1.138 cross-fitted), its coverage interval contains the predicted coverage (0.972, 0.974), and the diagnosis finds no defect.

Theory properties#

property	cell	role	what was tested	what must hold	measured	result
`one_wrong_inference`	`mar_q_wrong_pi_slow__ey0__first_order`	control	arm-0 mean: law M, randomized missing outcome with the declared treatment probability 1/2, outcome regression on A and W1 (wrong), response mechanism a correct GLM plus the constructed logit offset 2 n^(-1/3) W2	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0082 to -0.0060, margin 0.0033, SE ratio 1.0185	pass
`one_wrong_inference`	`mar_q_wrong_pi_slow__ey1__first_order`	control	arm-1 mean: law M, randomized missing outcome with the declared treatment probability 1/2, outcome regression on A and W1 (wrong), response mechanism a correct GLM plus the constructed logit offset 2 n^(-1/3) W2	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0081 to -0.0060, margin 0.0033, SE ratio 1.0065	pass
`one_wrong_inference`	`q_slow_g_wrong__ate__growth`	control	ATE: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong)	the lower bound of the 99% interval of the signed growth of bias/SD from n = 1,000 to n = 16,000 lies above zero	growth 0.4687, 0.3263 to 0.6210	pass
`one_wrong_inference`	`q_slow_g_wrong_n1000__ate__first_order`	control	ATE: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 1,000	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0245 to -0.0203, margin 0.0064, SE ratio 1.2736	pass
`one_wrong_inference`	`q_slow_g_wrong_n16000__ate__first_order`	control	ATE: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 16,000	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0091 to -0.0080, margin 0.0016, SE ratio 0.9387	pass
`one_wrong_inference`	`q_slow_g_wrong_n4000__ate__first_order`	control	ATE: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 4,000	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0145 to -0.0125, margin 0.0032, SE ratio 0.9449	pass
`one_wrong_inference`	`q_wrong_g_glm__ate__conservative`	positive	ATE: law L, outcome regression on A and W1 (wrong), mechanism a correct logistic GLM, in sample	SE-ratio interval wholly above the calibration band, coverage clears the floor, bias inside the margin	bias -0.000966 to 0.000104, coverage 0.9679 to 0.9809, SE ratio 1.1611, 1.1285 to 1.1940	pass
`one_wrong_inference`	`q_wrong_g_glm__ey0__valid`	positive	arm-0 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct logistic GLM, in sample	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.000176 to 0.000687, coverage 0.9579 to 0.9729, SE ratio 1.0646, 1.0361 to 1.0953	pass
`one_wrong_inference`	`q_wrong_g_glm__ey1__valid`	positive	arm-1 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct logistic GLM, in sample	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.000613 to 0.000262, coverage 0.9500 to 0.9664, SE ratio 1.0468, 1.0183 to 1.0771	pass
`one_wrong_inference`	`q_wrong_g_glm_cf__ate__conservative`	positive	ATE: law L, outcome regression on A and W1 (wrong), mechanism a correct logistic GLM, cross-fitted over the default folds	SE-ratio interval wholly above the calibration band, coverage clears the floor, bias inside the margin	bias -0.000472 to 0.000748, coverage 0.9748 to 0.9877, SE ratio 1.1908, 1.1537 to 1.2304	pass
`one_wrong_inference`	`q_wrong_g_glm_cf__ey0__valid`	positive	arm-0 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct logistic GLM, cross-fitted over the default folds	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.000349 to 0.000645, coverage 0.9565 to 0.9739, SE ratio 1.0762, 1.0412 to 1.1130	pass
`one_wrong_inference`	`q_wrong_g_glm_cf__ey1__valid`	positive	arm-1 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct logistic GLM, cross-fitted over the default folds	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.000217 to 0.000790, coverage 0.9551 to 0.9728, SE ratio 1.0600, 1.0264 to 1.0971	pass
`one_wrong_inference`	`q_wrong_g_known__ate__valid`	positive	ATE: law L, outcome regression on A and W1 (wrong), declared mechanism, the draws of q_wrong_g_glm	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.0011 to 0.000112, coverage 0.9404 to 0.9585, SE ratio 1.0085, 0.9808 to 1.0404	pass
`one_wrong_inference`	`q_wrong_g_known__ey0__valid`	positive	arm-0 mean: law L, outcome regression on A and W1 (wrong), declared mechanism, the draws of q_wrong_g_glm	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.000158 to 0.000761, coverage 0.9440 to 0.9614, SE ratio 0.9993, 0.9725 to 1.0276	pass
`one_wrong_inference`	`q_wrong_g_known__ey1__valid`	positive	arm-1 mean: law L, outcome regression on A and W1 (wrong), declared mechanism, the draws of q_wrong_g_glm	coverage clears the floor, bias inside the margin, SE ratio inside the sanity band	bias -0.000662 to 0.000259, coverage 0.9399 to 0.9580, SE ratio 0.9939, 0.9658 to 1.0247	pass
`one_wrong_inference`	`q_wrong_g_slow__ate__growth`	control	ATE: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2	the lower bound of the 99% interval of the signed growth of bias/SD from n = 1,000 to n = 16,000 lies above zero	growth 0.5166, 0.3810 to 0.6621	pass
`one_wrong_inference`	`q_wrong_g_slow_n1000__ate__first_order`	control	ATE: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 1,000	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0195 to -0.0151, margin 0.0067, SE ratio 1.0999	pass
`one_wrong_inference`	`q_wrong_g_slow_n16000__ate__first_order`	control	ATE: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 16,000	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0083 to -0.0072, margin 0.0017, SE ratio 1.1319	pass
`one_wrong_inference`	`q_wrong_g_slow_n4000__ate__first_order`	control	ATE: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 4,000	bias interval wholly beyond the margin on the predicted negative side, SE ratio between 0.1 and 10	bias -0.0123 to -0.0102, margin 0.0033, SE ratio 1.1415	pass
`one_wrong_reported`	`mar_q_wrong_pi_slow__ate__reported`	diagnostic	ATE: law M, randomized missing outcome with the declared treatment probability 1/2, outcome regression on A and W1 (wrong), response mechanism a correct GLM plus the constructed logit offset 2 n^(-1/3) W2	reported: bias, SE ratio and coverage, with no verdict	bias 0.0032 SD, coverage 0.9180 to 0.9577, SE ratio 1.0012	reported
`one_wrong_reported`	`q_slow_g_wrong_n1000__ey0__reported`	diagnostic	arm-0 mean: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 1,000	reported: bias, SE ratio and coverage, with no verdict	bias 0.5424 SD, coverage 0.8820 to 0.9300, SE ratio 1.1778	reported
`one_wrong_reported`	`q_slow_g_wrong_n1000__ey1__reported`	diagnostic	arm-1 mean: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 1,000	reported: bias, SE ratio and coverage, with no verdict	bias -0.5700 SD, coverage 0.8864 to 0.9336, SE ratio 1.2761	reported
`one_wrong_reported`	`q_slow_g_wrong_n16000__ey0__reported`	diagnostic	arm-0 mean: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 16,000	reported: bias, SE ratio and coverage, with no verdict	bias 0.8021 SD, coverage 0.8198 to 0.8788, SE ratio 0.9643	reported
`one_wrong_reported`	`q_slow_g_wrong_n16000__ey1__reported`	diagnostic	arm-1 mean: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 16,000	reported: bias, SE ratio and coverage, with no verdict	bias -0.8921 SD, coverage 0.8252 to 0.8833, SE ratio 0.9930	reported
`one_wrong_reported`	`q_slow_g_wrong_n4000__ey0__reported`	diagnostic	arm-0 mean: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 4,000	reported: bias, SE ratio and coverage, with no verdict	bias 0.6386 SD, coverage 0.8676 to 0.9185, SE ratio 0.9754	reported
`one_wrong_reported`	`q_slow_g_wrong_n4000__ey1__reported`	diagnostic	arm-1 mean: law L, outcome regression a correct GLM plus the constructed logit offset 3 n^(-1/3) (2 W1 + W2) / sqrt(5), mechanism a GLM on W2 (wrong), at n = 4,000	reported: bias, SE ratio and coverage, with no verdict	bias -0.6917 SD, coverage 0.8490 to 0.9033, SE ratio 0.9742	reported
`one_wrong_reported`	`q_wrong_g_slow_n1000__ey0__reported`	diagnostic	arm-0 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 1,000	reported: bias, SE ratio and coverage, with no verdict	bias 0.4107 SD, coverage 0.9260 to 0.9636, SE ratio 1.0739	reported
`one_wrong_reported`	`q_wrong_g_slow_n1000__ey1__reported`	diagnostic	arm-1 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 1,000	reported: bias, SE ratio and coverage, with no verdict	bias -0.4089 SD, coverage 0.9123 to 0.9535, SE ratio 1.0184	reported
`one_wrong_reported`	`q_wrong_g_slow_n16000__ey0__reported`	diagnostic	arm-0 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 16,000	reported: bias, SE ratio and coverage, with no verdict	bias 0.7519 SD, coverage 0.8709 to 0.9212, SE ratio 1.0355	reported
`one_wrong_reported`	`q_wrong_g_slow_n16000__ey1__reported`	diagnostic	arm-1 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 16,000	reported: bias, SE ratio and coverage, with no verdict	bias -0.6698 SD, coverage 0.8775 to 0.9265, SE ratio 1.0218	reported
`one_wrong_reported`	`q_wrong_g_slow_n4000__ey0__reported`	diagnostic	arm-0 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 4,000	reported: bias, SE ratio and coverage, with no verdict	bias 0.5386 SD, coverage 0.9191 to 0.9586, SE ratio 1.0857	reported
`one_wrong_reported`	`q_wrong_g_slow_n4000__ey1__reported`	diagnostic	arm-1 mean: law L, outcome regression on A and W1 (wrong), mechanism a correct GLM plus the constructed logit offset n^(-1/3) W2, at n = 4,000	reported: bias, SE ratio and coverage, with no verdict	bias -0.5336 SD, coverage 0.9043 to 0.9475, SE ratio 1.0349	reported

Result#

Every cell passed. The table gives the endpoints that each rule reads.

case	setting	measured	rule
valid	`q_wrong_g_known`, ATE	SE ratio 1.0085, coverage 0.9500	V, pass
conservative	`q_wrong_g_glm`, ATE	SE ratio 1.1611 (99% interval 1.1285 to 1.1940), coverage 0.9750	C, pass
conservative	`q_wrong_g_glm_cf`, ATE	SE ratio 1.1908 (99% interval 1.1537 to 1.2304), coverage 0.9820	C, pass
not established	`q_wrong_g_slow`, ATE	bias -0.65, -0.86 and -1.16 SD at n = 1,000, 4,000 and 16,000; coverage 0.935, 0.917 and 0.843	F and G, pass
not established	`q_slow_g_wrong`, ATE	bias -0.88, -1.07 and -1.35 SD; coverage 0.825, 0.789 and 0.687	F and G, pass
not established	`mar_q_wrong_pi_slow`, arm means	bias -0.53 SD at arm 0 and -0.54 SD at arm 1; coverage 0.920 and 0.926	F, pass

The measured-values table below checks each of these numbers against the committed rows.

The C cells measure the conservative case that the projection derivation predicts. Each row also reports a projected SE: the standard error of the influence curve after its projection on the mechanism GLM’s score columns. Its ratio to the empirical SD is 0.994 in sample and 1.012 cross-fitted, so the projection accounts for the excess. The empirical SD of the in-sample GLM fit is 0.87 times that of the declared-mechanism fit on the same draws.

Each first-order control’s bias has the sign that the design test computes from the population conditioning term. Each row also reports a remainder proxy: the conditioning term evaluated on the fit’s own rows, over the empirical SD. On law L the proxy has the sign of the measured bias and the same order. On law M it is -0.90 and -0.83 SD against a measured -0.53 and -0.54. The arms of the controls are reported, not gated.

Some fits do not converge and stay in their cells, as declared. Most of them stall, and a few reach the alternation’s outer cap. The mirror ladder has the most: 132, 106 and 63 of 1,000 fits at n = 1,000, 4,000 and 16,000. The C and V cells have at most 4 of 4,000.

Measured values#

Names beginning margin: are thresholds declared before the run. Everything else is measured from the committed results and checked at the precision printed.

quantity	value	source
`replicates`	4000	primary replications
`n`	4000	observations per primary replication
`independent_tests_total`	3	primary tests against truth
`independent_tests_passed`	3	of those, passing
`paired_tests_total`	0	external comparisons declared
`paired_tests_passed`	0	external comparisons passing
`property_cells_total`	19	repeated-sampling property cells
`property_cells_passed`	19	cells whose own and family verdicts pass
`max_standardized_bias`	0.0246	largest absolute primary bias in empirical standard deviations
`min_coverage`	0.9505	lowest measured primary-study coverage
`properties[one_wrong_inference/q_wrong_g_known__ate__valid]:se_ratio`	1.0085	declared-mechanism ATE SE ratio
`properties[one_wrong_inference/q_wrong_g_known__ate__valid]:coverage`	0.9500	its coverage
`properties[one_wrong_inference/q_wrong_g_glm__ate__conservative]:se_ratio`	1.1611	in-sample GLM-mechanism ATE SE ratio
`properties[one_wrong_inference/q_wrong_g_glm__ate__conservative]:se_ratio_ci_lower`	1.1285	its 99% lower endpoint
`properties[one_wrong_inference/q_wrong_g_glm__ate__conservative]:se_ratio_ci_upper`	1.1940	its 99% upper endpoint
`properties[one_wrong_inference/q_wrong_g_glm__ate__conservative]:coverage`	0.9750	its coverage
`properties[one_wrong_inference/q_wrong_g_glm__ate__conservative]:projected_se_ratio`	0.9941	its projected SE over the empirical SD
`properties[one_wrong_inference/q_wrong_g_glm__ate__conservative]:paired_sd_ratio`	0.8704	its empirical SD over the declared-mechanism fit’s, on the same draws
`properties[one_wrong_inference/q_wrong_g_glm_cf__ate__conservative]:se_ratio`	1.1908	cross-fitted GLM-mechanism ATE SE ratio
`properties[one_wrong_inference/q_wrong_g_glm_cf__ate__conservative]:se_ratio_ci_lower`	1.1537	its 99% lower endpoint
`properties[one_wrong_inference/q_wrong_g_glm_cf__ate__conservative]:se_ratio_ci_upper`	1.2304	its 99% upper endpoint
`properties[one_wrong_inference/q_wrong_g_glm_cf__ate__conservative]:coverage`	0.9820	its coverage
`properties[one_wrong_inference/q_wrong_g_glm_cf__ate__conservative]:projected_se_ratio`	1.0123	its projected SE over the empirical SD
`properties[one_wrong_inference/q_wrong_g_slow_n1000__ate__first_order]:standardized_bias`	-0.6473	slow-mechanism bias in SD, n = 1,000
`properties[one_wrong_inference/q_wrong_g_slow_n4000__ate__first_order]:standardized_bias`	-0.8623	the same, n = 4,000
`properties[one_wrong_inference/q_wrong_g_slow_n16000__ate__first_order]:standardized_bias`	-1.1640	the same, n = 16,000
`properties[one_wrong_inference/q_wrong_g_slow_n1000__ate__first_order]:coverage`	0.9350	slow-mechanism coverage, n = 1,000
`properties[one_wrong_inference/q_wrong_g_slow_n4000__ate__first_order]:coverage`	0.9170	the same, n = 4,000
`properties[one_wrong_inference/q_wrong_g_slow_n16000__ate__first_order]:coverage`	0.8430	the same, n = 16,000
`properties[one_wrong_inference/q_wrong_g_slow__ate__growth]:growth`	0.5166	signed growth of the slow-mechanism bias
`properties[one_wrong_inference/q_wrong_g_slow__ate__growth]:growth_ci_lower`	0.3810	its 99% lower endpoint
`properties[one_wrong_inference/q_slow_g_wrong_n1000__ate__first_order]:standardized_bias`	-0.8794	mirror bias in SD, n = 1,000
`properties[one_wrong_inference/q_slow_g_wrong_n4000__ate__first_order]:standardized_bias`	-1.0696	the same, n = 4,000
`properties[one_wrong_inference/q_slow_g_wrong_n16000__ate__first_order]:standardized_bias`	-1.3481	the same, n = 16,000
`properties[one_wrong_inference/q_slow_g_wrong_n1000__ate__first_order]:coverage`	0.8250	mirror coverage, n = 1,000
`properties[one_wrong_inference/q_slow_g_wrong_n4000__ate__first_order]:coverage`	0.7890	the same, n = 4,000
`properties[one_wrong_inference/q_slow_g_wrong_n16000__ate__first_order]:coverage`	0.6870	the same, n = 16,000
`properties[one_wrong_inference/q_slow_g_wrong__ate__growth]:growth`	0.4687	signed growth of the mirror bias
`properties[one_wrong_inference/q_slow_g_wrong__ate__growth]:growth_ci_lower`	0.3263	its 99% lower endpoint
`properties[one_wrong_inference/mar_q_wrong_pi_slow__ey0__first_order]:standardized_bias`	-0.5319	law M arm-0 bias in SD
`properties[one_wrong_inference/mar_q_wrong_pi_slow__ey1__first_order]:standardized_bias`	-0.5378	law M arm-1 bias in SD
`properties[one_wrong_inference/mar_q_wrong_pi_slow__ey0__first_order]:coverage`	0.9200	its arm-0 coverage
`properties[one_wrong_inference/mar_q_wrong_pi_slow__ey1__first_order]:coverage`	0.9260	its arm-1 coverage
`properties[one_wrong_reported/q_slow_g_wrong_n1000__ey0__reported]:nonconverged_fits`	132	mirror fits that did not converge, n = 1,000
`properties[one_wrong_reported/q_slow_g_wrong_n4000__ey0__reported]:nonconverged_fits`	106	the same, n = 4,000
`properties[one_wrong_reported/q_slow_g_wrong_n16000__ey0__reported]:nonconverged_fits`	63	the same, n = 16,000
`margin:confidence_level`	0.9900	confidence level of every Monte Carlo interval
`margin:alpha`	0.0500	nominal size of the reported intervals
`margin:bootstrap_replicates`	10000	resamples behind every bootstrap interval
`margin:standardized_bias`	0.2500	bias equivalence margin, in empirical standard deviations
`margin:coverage_floor`	0.9000	validity floor the coverage lower endpoint must clear
`margin:se_ratio_sanity_lower`	0.8000	SE-ratio screen, lower limit
`margin:se_ratio_sanity_upper`	1.2000	SE-ratio screen, upper limit
`margin:calibration_se_ratio_upper`	1.0700	the line that a C cell’s SE-ratio interval must clear
`margin:calibration_coverage_lower`	0.9200	calibration-cell coverage band, lower limit. No rule of this study reads it
`margin:calibration_coverage_upper`	0.9800	calibration-cell coverage band, upper limit. No rule of this study reads it
`margin:calibration_noninferiority`	0.0500	largest external-comparison calibration excess bound. This study has no comparator
`margin:calibration_se_ratio_lower`	0.9300	calibration-cell SE-ratio band, lower limit. Rule C reads the upper limit only
`margin:coverage_noninferiority`	-0.0250	smallest external-comparison coverage difference bound. This study has no comparator
`margin:excluded_slope`	-0.2500	slower rate a root-n interval must exclude. No rule of this study reads it
`margin:minimum_power`	0.8000	rejection lower bound a power cell must clear. No rule of this study reads it
`margin:nominal_coverage`	0.9500	nominal coverage those intervals claim
`margin:over_coverage_ceiling`	0.9900	above this, coverage is conservative rather than invalid
`margin:paired_difference`	0.1500	paired similarity margin, in pooled empirical standard deviations
`margin:rmse_noninferiority`	1.1000	largest external-comparison RMSE ratio bound. This study has no comparator
`margin:root_n_slope`	-0.5000	contraction rate root-n asymptotics predict. No rule of this study reads it
`margin:root_n_slope_lower`	-0.6250	accepted root-n slope band, lower limit. No rule of this study reads it
`margin:root_n_slope_upper`	-0.3750	accepted root-n slope band, upper limit. No rule of this study reads it
`margin:type_i_ceiling`	0.1000	the rate a positive cell must bound and a control must exceed. No rule of this study reads it

Limitations#

limit	what it means for use
a binary outcome and two Gaussian covariates	the study does not measure other outcome types or covariate laws
the univariate reduction with `guard=("Q", "g")`	the bivariate reduction and other guards have no witness here
GLM and polynomial learners with constructed offsets	no off-the-shelf flexible learner is measured here. A probe with a gradient-boosting mechanism and a wrong outcome regression gave an ATE bias of −2.28 SD in sample and +2.01 SD cross-fitted at n = 4,000. A data-adaptive mechanism gives the cause and the numbers
no strata, clusters, weights or more arms	those routes inherit the case table without a witness

Reproduction#

The manifest records the seeds, the margins, the estimator configuration, the source hashes and the result hashes. Run python -m tests.canonical.drtmle_one_wrong.regenerate --output <scratch> to regenerate the artifacts under the declared run form. The property results carry every published row. The design test, tests/unit/test_drtmle_one_wrong_design.py, pins the truths, the limits, the predicted signs and the pass probabilities.