Improved methods for estimating fraction of missing information in multiple imputation¹

1. The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the National Center for Health Statistics (NCHS) or Centers for Disease Control (CDC), USA.

Figures & data

Table 1. Changes of E(γ_m), D_γ, and R_Dγ with the increase of m at different γ₀ levels, where D_γ = 100(E(γ_m)−γ₀)/γ₀ and R_Dγ = 100(γ_m−γ_m+1)/γ_m+1

m	E(γm)		RDγ		Dγ
	γ0 = 0.2	γ0 = 0.01	γ0 = 0.2	γ0 = 0.01	γ0 = 0.2	γ0 = 0.01
2	0.361	0.01536	23.33	14.12	80.59	53.64
5	0.250	0.01205	3.872	2.957	25.23	20.47
10	0.224	0.01102	1.0240	0.8502	11.84	10.16
20	0.212	0.01051	0.2664	0.2306	5.750	5.062
40	0.206	0.01025	0.0682	0.0602	2.837	2.528
60	0.204	0.01017	0.0306	0.0272	1.883	1.684
100	0.202	0.01010	0.0111	0.0099	1.126	1.010
200	0.201	0.01005	0.0028	0.0025	0.561	0.505

Figure 1. The m–E(γ_m) relationship curve at γ₀ = 0.2 and 0.15 as determined by Equation (14).

Figure 2. Effects of γ₀ levels on D_γ as defined by Equation (15) and R_Dγ as defined by Equation (16): a. D_γ at m = 5; b. D_γ at m = 2.

Figure 3. Effects of m on γ_m at δ = 29% for analytic model = Anal-2: a. MI model = MI-1; b. MI model = MI-2.

Table 2. Coefficient of variations (%) of B_m, U_m, and γ_m for SIZE100

m	MI-1, Anal-1			MI-2, Anal-2
	Bm	Um	γm	Bm	Um	γm
3	20.24	0.185	20.19	1.52	0.0553	1.50
5	11.12	0.179	11.10	1.00	0.0200	1.01
10	7.75	0.135	7.75	0.88	0.0353	0.91
20	5.91	0.098	5.89	0.49	0.0282	0.48
30	6.24	0.077	6.26	0.24	0.0150	0.24
40	3.60	0.048	3.59	0.61	0.0180	0.59
60	2.74	0.058	2.73	0.39	0.0091	0.39
80	2.48	0.041	2.47	0.17	0.0062	0.17
100	2.45	0.037	2.45	0.15	0.0081	0.16

Table 3. Comparison of different γ₀ estimation methods for SIZE20 with imputation model = MI-2 and analytic model = Anal-2 in the PWS12 MI trials. The best γ̂₀ was calculated by Equation (19) using (B̅₁₀₀)/(U̅₁₀₀) as the estimate of B₀/U₀, where B̅₁₀₀ and U̅₁₀₀ were the mean of the 30 replicates of B₁₀₀ and U₁₀₀, respectively

m	SIZE20, MI-2, Anal-2
	Control γ̂0 = γm	Improved
		γ̂0 = γm≥100	γ̂0 = cm/(1 + cm)	γ̂0 = γm(m/(1 + m)
3	0.00379		0.00283	0.00284
5	0.00318		0.00265	0.00265
10	0.00286		0.00260	0.00260
20	0.00293		0.00279	0.00279
30	0.00278		0.00269	0.00269
40	0.00273		0.00267	0.00267
60	0.00277		0.00273	0.00273
80	0.00272		0.00269	0.00269
100	0.00270	0.00270	0.00267	0.00267
(∞)	Best γ̂0: 0.00267