Estimating and testing sequential causal effects based on alternative G-formula: an observational study of the influence of early diagnosis on survival of cardia cancer

Figures & data

Table 1. Frequencies or means (standard deviations) of covariates and outcome across the diagnosing and treating hospitals for 157 cardia cancer patients.

	Cardia cancer: frequency or mean (standard deviation) on levels of $z_1$ and $z_2$
	$z_1=0$	$z_1=1$	$z_2=0$	$z_2=1$
$N$	69	88	39	118
$x_{11}=0$$=1$	14 55	26 62	10 29	30 88
$x_{12}=0$$=1$	43 26	39 49	27 12	55 63
$x_{13}$	68.04 (11.37)	65.38 (13.22)	71.90(9.20)	64.78(12.93)
$x_2=1$$=2$$=3$$=4$	7 7 21 34	12 13 17 46	3 3 9 24	16 17 29 56
One year survival $y=0$ $=1$	43 26	58 30	31 8	70 48

Hospital types:$z_1=0, 1$ for small and large diagnosing hospitals; $z_2=0, 1$ for small and large treating hospitals.

Stationary covariates: $x_{11}=0, 1$ for female and male; $x_{12}=0, 1$ for rural and urban residential areas; $x_{13}$ for age.

Time-dependent covariate: $x_2=1, 2, 3, 4$ for cancer stages.

Outcome: $y=0, 1$ death and survival within one year after diagnosis.

Table 2. Point effects, blip effects and optimal SCEs of diagnosing and treating hospitals on one-year survival of cardia cancer: estimate, p-value and 95 % CI.

	Estimate, p-value and 95 % CI in column for causal effects
			${\vartheta }_1$	${\vartheta }_2$
Point effect			$-$0.020 0.799 $(-0.171, 0.132)$	0.194 0.009 $(0.048, 0.340)$
Blip effect	${\gamma }_{10}$	${\gamma }_{11}$	${\gamma }_1$	${\gamma }_2$
	$-0.158$ 0.090 $(-0.341, 0.024)$	$-0.109$ 0.179 $(-0.267, 0.050)$	$-$0.121 0.142 $(-0.283, 0.040)$	0.194 0.009 $(0.048, 0.340)$
Optimal SCE	${\Gamma }_{10}$	${\Gamma }_{11}$	${\Gamma }_1$	${\Gamma }_2$
	0.194 0.009 $(0.048, 0.340)$	0.194 0.009 $(0.048, 0.340)$	0.194 0.009 $(0.048, 0.340)$	0.194 0.009 $(0.048, 0.340)$

Two point effects: ${\vartheta }_1$ of large diagnosing hospital; ${\vartheta }_2$ of large treating hospital.

Four blip effects: ${\gamma }_{10}$ of large diagnosing hospital among women; ${\gamma }_{11}$ of large diagnosing hospital among men; ${\gamma }_1$ of large diagnosing hospital in the population; ${\gamma }_2$ of large treating hospital.

Four optimal SCEs: ${\Gamma }_{10}$ under optimal regime of diagnosing and treating hospital among women; ${\Gamma }_{11}$ under optimal regime of diagnosing and treating hospital among men; ${\Gamma }_1$ under optimal regime of diagnosing and treating hospital in the population; ${\Gamma }_2$ under optimal regime of treating hospital in population. The estimated optimal treating hospital is ${\widehat{O}}_2=1$ (the large treating hospital). The estimated optimal diagnosing hospital is ${\widehat{O}}_1=0$ (small diagnosing hospital).

Table 3. Comparison of our method with available methods in Sec. 5: estimate, p-value and 95 % confidence interval (95% CI) for causal effects of diagnosing and treating hospitals on one-year survival of cardia cancer.

	Estimate, p-value and 95 % CI in column of the causal effect
	${\gamma }_{10}$	${\gamma }_{11}$	${\gamma }_1$	${\mathrm{{\rm Y}}}_1$	${\Gamma }_1$
Method (i)	$-0.158$ 0.090 $(-0.341, 0.024)$	$-0.109$ 0.179 $(-0.267, 0.050)$	$-$0.121 0.142 $(-0.283, 0.040)$	$-$0.121 0.142 $(-0.283, 0.040)$	0.194 0.009 $(0.048, 0.340)$
Method (ii)			$-$0.066 0.605 $(-0.316, 0.184)$	$-$0.101 0.392 $(-0.333, 0.131)$	0.200 0.048 $(0.002, 0.399)$
Method (iii)			0.463 0.259 $(-0.342, 1.267)$	$-$0.167 0.087 $(-0.359, 0.024)$	0.249 0.012 $(-0.056, 0.443)$
Method (iv)	$-0.242$ 0.078 $(-0.509, 0.027)$	$-0.092$ 0.369 $(-0.291, 0.108)$	$-$0.127 0.162 $(-0.305, 0.051)$	$-$0.127 0.162 $(-0.305, 0.051)$

Four estimation methods: method (i) our method; method (ii) the parametric method based on the well-known G-formula; (iii) the marginal structural model based on inverse probability of treatment weighting; method (iv) the G-estimation based on SNMM or optimal SNMM; Empty cells imply that they are not easily estimable by the method.

Causal effects: ${\gamma }_{10}$ blip effect of large diagnosing hospital among women, ${\gamma }_{11}$ blip effect of large diagnosing hospital among men; ${\gamma }_1$ the blip effect of large diagnosing hospital in the population; ${\mathrm{{\rm Y}}}_1$ the optimal blip effect of large diagnosing hospital in the population; ${\Gamma }_1$ the SCE under the optimal regime of diagnosing and treating hospitals in the population. The estimated optimal treating hospital is ${\widehat{O}}_2=1$ (the large treating hospital). The estimated optimal diagnosing hospital is ${\widehat{O}}_1=0$ (small diagnosing hospital).