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 4 Performance measures for the estimation of the regression coefficient with GHQ scores on raw scale

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

Scenario	Validation statistics
Likert	$\hat{Q} = 0.03227$		U = 0.02143
MCAR	$E [{\bar{Q}}_{m}]$	bias	$E [{\bar{U}}_{m}]$	$Var ({\bar{Q}}_{m})$	(1 + m ^- 1)E[B _m]	coverage of $\hat{Q}$
Regression, non-rounded	0.03181	-0.00046	0.02211	0.00027	0.00022	0.922
Post-imputation rounding	0.03192	-0.00035	0.02219	0.00027	0.00022	0.920
Truncated regression	0.03247	0.00020	0.02217	0.00028	0.00022	0.914
Predictive mean matching	0.02551	-0.00676	0.02223	0.00019	0.00016	0.918
MAR
Regression, non-rounded	0.02888	-0.00340	0.02270	0.00080	0.00067	0.927
Post-imputation rounding	0.02926	-0.00301	0.02279	0.00079	0.00066	0.924
Truncated regression	0.03010	-0.00217	0.02278	0.00084	0.00066	0.926
Predictive mean matching	0.01727	-0.01500	0.02298	0.00036	0.00036	0.911
C-GHQ	$\hat{Q} = 0.04794$		U = 0.03967
MCAR	$E [{\bar{Q}}_{m}]$	bias	$E [{\bar{U}}_{m}]$	$Var ({\bar{Q}}_{m})$	(1 + m ^- 1)E[B _m]	coverage of $\hat{Q}$
Regression, non-rounded	0.04694	-0.00100	0.04014	0.00077	0.00076	0.946
Post-imputation rounding	0.04805	0.00012	0.04123	0.00080	0.00069	0.932
Truncated regression	0.04360	-0.00433	0.04130	0.00086	0.00073	0.925
Predictive mean matching	0.03738	-0.01055	0.04033	0.00053	0.00056	0.939
MAR
Regression, non-rounded	0.04283	-0.00511	0.04056	0.00232	0.00219	0.939
Post-imputation rounding	0.04932	0.00138	0.04158	0.00224	0.00195	0.928
Truncated regression	0.06470	0.01676	0.04133	0.00243	0.00210	0.906
Predictive mean matching	0.02442	-0.02352	0.04087	0.00109	0.00121	0.929
Standard	$\hat{Q} = 0.05236$		U = 0.04202
MCAR	$E [{\bar{Q}}_{m}]$	bias	$E [{\bar{U}}_{m}]$	$Var ({\bar{Q}}_{m})$	(1 + m ^- 1)E[B _m]	coverage of $\hat{Q}$
Regression, non-rounded	0.05066	-0.00170	0.04336	0.00101	0.00085	0.938
Post-imputation rounding	0.05252	0.00016	0.04550	0.00106	0.00067	0.889
Truncated regression	0.04613	-0.00623	0.04218	0.00101	0.00096	0.930
Predictive mean matching	0.04069	-0.01167	0.04368	0.00065	0.00061	0.929
MAR
Regression, non-rounded	0.04557	-0.00679	0.04441	0.00278	0.00261	0.942
Post-imputation rounding	0.06226	0.00990	0.04590	0.00238	0.00187	0.911
Truncated regression	0.09485	0.04249	0.04092	0.00233	0.00259	0.857
Predictive mean matching	0.02669	-0.02567	0.04517	0.00122	0.00139	0.939

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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