Comparison of methods for imputing limited-range variables: a simulation study

Rodwell, Laura; Lee, Katherine J; Romaniuk, Helena; Carlin, John B

doi:10.1186/1471-2288-14-57

BMC Medical Research Methodology

Table 3 Performance measures for the estimation of the marginal mean with transformed GHQ scores

From: Comparison of methods for imputing limited-range variables: a simulation study

Scenario	Validation statistics
Likert	$\hat{Q} = 10.2311$		U = 0.1855
MCAR	$E [{\bar{Q}}_{m}]$	bias	$E [{\bar{U}}_{m}]$	$Var ({\bar{Q}}_{m})$	(1 + m ^- 1)E[B _m]	coverage for $\hat{Q}$
Regression, non-rounded	10.2366	0.0055	0.1861	0.0181	0.0174	0.947
Post-imputation rounding	10.1446	-0.0865	0.1857	0.0416	0.0170	0.820
Truncated regression	10.2227	-0.0084	0.1840	0.0181	0.0163	0.935
Predictive mean matching	10.1926	-0.0385	0.1837	0.0174	0.0148	0.916
MAR
Regression, non-rounded	10.2119	-0.0192	0.1846	0.0223	0.0197	0.928
Post-imputation rounding	10.0985	-0.1326	0.1842	0.0628	0.0193	0.758
Truncated regression	10.2010	-0.0301	0.1825	0.0217	0.0180	0.915
Predictive mean matching	10.1401	-0.0910	0.1819	0.0216	0.0164	0.858
C-GHQ	$\hat{Q} = 3.3179$		U = 0.1055
MCAR	$E [{\bar{Q}}_{m}]$	bias	$E [{\bar{U}}_{m}]$	$Var ({\bar{Q}}_{m})$	(1 + m ^- 1)E[B _m]	coverage for $\hat{Q}$
Regression, non-rounded	3.3268	0.0088	0.1087	0.0057	0.0063	0.960
Post-imputation rounding	3.3231	0.0051	0.1053	0.0058	0.0052	0.931
Truncated regression	3.5563	0.2384	0.1028	0.0053	0.0048	0.096
Predictive mean matching	3.3009	-0.0170	0.1049	0.0057	0.0053	0.941
MAR
Regression, non-rounded	3.3265	0.0086	0.1094	0.0065	0.0077	0.969
Post-imputation rounding	3.3185	0.0005	0.1055	0.0065	0.0063	0.954
Truncated regression	3.5452	0.2273	0.1027	0.0060	0.0055	0.160
Predictive mean matching	3.2722	-0.0457	0.1045	0.0069	0.0059	0.886
Standard	$\hat{Q} = 1.8081$		U = 0.0947
MCAR	$E [{\bar{Q}}_{m}]$	bias	$E [{\bar{U}}_{m}]$	$Var ({\bar{Q}}_{m})$	(1 + m ^- 1)E[B _m]	coverage for $\hat{Q}$
Regression, non-rounded	263.63	261.82	30917	53900000	998000000	0.992
Post-imputation rounding	1.7996	-0.0086	0.1055	0.0037	0.0074	0.992
Truncated regression	2.2661	0.4580	0.0953	0.0056	0.0047	0.000
Predictive mean matching	1.8052	-0.0029	0.0943	0.0043	0.0041	0.940
MAR
Regression, non-rounded^Ɨ
Post-imputation rounding	1.8035	-0.0046	0.1074	0.0043	0.0095	0.989
Truncated regression	2.2603	0.4522	0.0951	0.0061	0.0054	0.000
Predictive mean matching	1.7829	-0.0252	0.0937	0.0053	0.0045	0.909

^ƗValues larger than those obtained under the MCAR condition Key: $\hat{Q}$ = complete data estimate; U = estimated variance of $\hat{Q}$ from complete data; $E [{\bar{Q}}_{m}]$ = average of MI-based point estimates across 1000 simulated datasets; bias = difference between $E [{\bar{Q}}_{m}]$ and $\hat{Q}$ ; $E [{\bar{U}}_{m}]$ = average of estimated within-imputation variance across simulated datasets; $Var ({\bar{Q}}_{m})$ = variance of the MI point estimates across simulated datasets; (1 + m^- 1)E[B_m] = average of estimated between-imputation variance (with adjustment for number of imputations) across simulated datasets; coverage = proportion of (nominally) 95% confidence intervals that contain the complete data estimate.

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ISSN: 1471-2288

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