Verifiable identification condition for nonignorable nonresponse data with categorical instrumental variables

Figures & data

Table 1. Results of S1–S4: Bias, root mean square error (RMSE), and coverage rate (CR,%) with $95\mathrm{\%}$ confidence interval are reported.

Scenario	Parameter	${\kappa }_2$	Method	Bias	RMSE	CR
S1	$E\left[y\right]$	1.0	CC	0.053	0.066	73.5
			FI	0.000	0.043	95.4
		0.5	CC	0.039	0.053	80.9
			FI	−0.001	0.059	97.1
		0.1	CC	0.034	0.049	83.0
			FI	0.021	0.136	99.8
	β	1.0	FI	0.001	0.163	95.2
		0.5	FI	0.003	0.330	98.6
		0.1	FI	−0.146	0.865	100
S2	$E\left[y\right]$	1.0	CC	0.146	0.152	5.7
			FI	−0.004	0.051	94.8
		0.5	CC	0.130	0.136	7.7
			FI	−0.008	0.086	86.2
		0.1	CC	0.127	0.133	9.4
			FI	−0.007	0.105	92.4
	β	1.0	FI	0.008	0.148	95.4
		0.5	FI	0.044	0.365	100
		0.1	FI	0.033	0.448	100
S3	$E\left[y\right]$	–	CC	0.100	0.102	0.3
		–	FI	0.001	0.022	95.3
	β	–	FI	−0.023	0.279	95.0
S4	$E\left[y\right]$	–	CC(Logistic)	0.341	0.355	5.4
		–	FI(Logistic)	0.005	0.079	95.4
		–	CC(Cauchy)	0.296	0.312	10.7
		–	FI(Cauchy)	0.007	0.080	94.3
	β	–	FI(Logistic)	0.006	0.050	94.7
		–	FI(Cauchy)	0.011	0.063	93.8

Note: CC: complete case; FI: fractional imputation.

Figure 1. Residual plots of respondents' outcome in ACTG175 data.

Table 2. Estimated parameters: Estimates and standard errors for the target parameters are reported.

Parameter	Model	Estimate	SE	Parameter	Model	Estimate	SE
${\alpha }_0$	Logistic	0.464	0.104		Logistic	0.125	0.156
	Cauchy	0.417	0.260		Cauchy	0.108	0.139
	$T_2$	0.341	0.081	${\alpha }_1$	$T_2$	0.091	0.113
	$T_5$	0.306	0.069		$T_5$	0.082	0.102
	$T_{10}$	0.295	0.066		$T_{10}$	0.080	0.099
${\alpha }_2$	Logistic	0.255	0.192		Logistic	0.093	0.107
	Cauchy	0.244	0.207		Cauchy	0.083	0.097
	$T_2$	0.196	0.148	${\alpha }_3$	$T_2$	0.069	0.079
	$T_5$	0.169	0.126		$T_5$	0.062	0.070
	$T_{10}$	0.160	0.120		$T_{10}$	0.060	0.068
β	Logistic	−0.032	0.314		Logistic	276.70	13.476
	Cauchy	−0.030	0.387		Cauchy	276.51	14.107
	$T_2$	−0.027	0.235	$E\left[y\right]$	$T_2$	276.57	13.437
	$T_5$	−0.021	0.203		$T_5$	276.61	13.271
	$T_{10}$	−0.019	0.194		$T_{10}$	276.63	13.217

Note: Logistic and Cauchy are Fractional Imputation using Logistic and Cauchy distributions for the response mechanism. $T_v$: t distribution function with degrees of freedom $v(=2,5,10)$.