. Author manuscript; available in PMC: 2021 Jul 1.

Published in final edited form as: Stat Methods Med Res. 2019 Oct 1;29(7):1891–1912. doi: 10.1177/0962280219877520

Table 4:

Sample sizes and empirical power results for an end-of-study comparison of the DTR recommending only treatments indexed by 1 and that which recommends only treatments indicated by −1. δ is the true standardized effect size as defined in equation (9), r is the common probability of response to first-stage treatment, and ρ is the true exchangeable within-person correlation. n is computed using formula (10) with α= 0.05 and β= 0.2. R_⫫ refers to a generative model in which response status is independent of all prior outcomes; R₊ and R₋ refer to generative models in which response is positively or negatively correlated with Y₁, respectively. All violation scenarios assume the R₊ generative model, except working assumption A1(c). Results are the proportion of 3000 Monte Carlo simulations in which we reject H₀ : c^⊤θ = 0 at the 5% level.

					Empirical power
					A1 and A2 satisfied			Violation of A1			Violation of A2
Design	δ	r	ρ	n	R_⫫	R₊	R₋	A1(a)	A1(b)	A1(c)	True AR(1)
I	0.3	0.4	0.0	698	0.798	0.807	0.803	0.798	0.796	^‡	^‡
			0.3	635	0.819	0.817	0.800	0.820	0.804	0.815	0.780^*
			0.6	447	0.815	0.862	0.773^*	0.865	0.817	0.827	0.728^*
			0.8	252	0.835	0.925	0.733^*	^†	^†	0.840	0.721^*
		0.6	0.0	698	0.796	0.799	0.806	0.800	0.791	^‡	^‡
			0.3	635	0.808	0.813	0.792	0.824	0.805	0.807	0.775^*
			0.6	447	0.833	0.856	0.798	0.859	0.831	0.838	0.727^*
			0.8	252	0.827	0.901	0.758^*	^†	^†	0.835	^†
	0.5	0.4	0.0	252	0.799	0.801	0.798	0.798	0.801	^‡	^‡
			0.3	229	0.813	0.815	0.797	0.814	0.811	0.814	0.771^*
			0.6	161	0.824	0.872	0.789	0.868	0.833	0.843	0.742^*
			0.8	91	0.843	0.931	0.734^*^§	0.926	^†	0.839^§	0.725^*
		0.6	0.0	252	0.796	0.797	0.810	0.792	0.802	^‡	^‡
			0.3	229	0.817	0.815	0.808	0.811	0.823	0.823	0.771^*
			0.6	161	0.838	0.859	0.790	0.861	0.832	0.837	0.749^*
			0.8	91	0.835^§	0.896	0.765^*^§	0.896	^†	0.859	^†
II	0.3	0.4	0.0	559	0.801	0.801	0.808	0.778^*	0.803	^‡	^‡
			0.3	508	0.804	0.813	0.831	0.800	0.797	0.798	0.795
			0.6	358	0.817	0.819	0.834	0.807	0.759^*	0.788	0.811
			0.8	201	0.836	0.814	0.836	0.809	^†	0.792	0.806
		0.6	0.0	489	0.804	0.796	0.793	0.736^*	0.810	^‡	^‡
			0.3	445	0.797	0.804	0.818	0.758^*	0.795	0.780^*	0.804
			0.6	313	0.824	0.831	0.844	0.793	0.752^*	0.770^*	0.824
			0.8	176	0.845	^†	^†	0.754^*	^†	0.776^*	0.842
	0.5	0.4	0.0	201	0.801	0.800	0.802	0.768^*	0.794	^‡	^‡
			0.3	183	0.813	0.800	0.819	0.790	0.813	0.796	0.803
			0.6	129	0.814	0.828	0.833	0.810	0.763^*	0.799	0.815
			0.8	73	0.839	0.841	0.852	0.829	^†	0.795	0.804
		0.6	0.0	176	0.807	0.799	0.796	0.733^*	0.808	^‡	^‡
			0.3	160	0.816	0.815	0.821	0.767^*	0.808	0.802	0.812
			0.6	113	0.829	0.830	0.837	0.792	0.765^*	0.770^*	0.817
			0.8	64	0.845^§	^†	^†	0.783^*^§	^†	0.789^§	^†
III	0.3	0.4	0.0	454	0.806	0.813	0.806	0.782^*	0.794	^‡	^‡
			0.3	413	0.815	0.809	0.814	0.789	0.800	0.800	0.775^*
			0.6	291	0.821	0.811	0.818	0.794	0.783^*	0.787^*	0.687^*
			0.8	164	0.824	0.812	0.839	0.812	^†	0.802	0.637^*
		0.6	0.0	419	0.813	0.814	0.817	0.781^*	0.769^*	^‡	^‡
			0.3	381	0.823	0.812	0.808	0.776^*	0.791	0.795	0.771^*
			0.6	268	0.823	0.817	0.844	0.807	0.750^*	0.754^*	0.709^*
			0.8	151	0.820	^†	^†	0.803	^†	0.784^*	^†
	0.5	0.4	0.0	164	0.808	0.804	0.795	0.776^*	0.802	^‡	^‡
			0.3	149	0.822	0.815	0.827	0.811	0.791	0.805	0.789
			0.6	105	0.811	0.810	0.812	0.810	0.798	0.785^*	0.698^*
			0.8	59	0.838	^†	0.823	0.845	^†	0.817^§	0.684^*
		0.6	0.0	151	0.798	0.809	0.803	0.778^*	0.772^*	^‡	^‡
			0.3	138	0.812	0.809	0.814	0.800	0.782^*	0.799	0.778^*
			0.6	97	0.803^§	0.812	0.826^§	0.826^§	0.762^*	0.774^*^§	0.705^*^§
			0.8	55	0.826^§	^†	^†	0.837^§	^†	0.797^§	^†

Statistically significantly less than 0.8 at the 5% level.

^†

Our data generative model could not accommodate this scenario (see appendix C).

^‡

Violation of this working assumption is not applicable when ρ = 0.

^§

Fewer than 3000 simulations generated data in which all treatment sequences were observed.