Nonignorable item nonresponse in panel data

Figures & data

Table 1. Example of $D_1,D_2,D_3$ when k = 3 and n = 30.

Entire data set				$D_1$				$D_2$				$D_3$
Unit	$y_1$	$y_2$	$y_3$	Unit	$y_1$	$y_2$	$y_3$	Unit	$y_1$	$y_2$	$y_3$	Unit	$y_1$	$y_2$	$y_3$
1	?	?	?	2	$✓$	$✓$	$✓$	2	$✓$	$✓$	$✓$	2	$✓$	$✓$	$✓$
2	$✓$	$✓$	$✓$	3	?	$✓$	$✓$	5	$✓$	$✓$	$✓$	5	$✓$	$✓$	$✓$
3	?	$✓$	$✓$	5	$✓$	$✓$	$✓$	8	$✓$	$✓$	$✓$	7	$✓$	$✓$	?
4	$✓$	?	?	8	$✓$	$✓$	$✓$	11	✓	?	$✓$	8	$✓$	$✓$	$✓$
5	$✓$	$✓$	$✓$	12	✓	$✓$	$✓$	12	✓	$✓$	$✓$	10	$✓$	$✓$	?
6	?	?	$✓$	15	$✓$	$✓$	$✓$	14	$✓$	?	$✓$	12	✓	$✓$	$✓$
7	$✓$	$✓$	?	16	?	$✓$	$✓$	15	$✓$	$✓$	$✓$	15	✓	$✓$	$✓$
8	✓	$✓$	$✓$	17	$✓$	$✓$	$✓$	17	$✓$	$✓$	$✓$	17	✓	$✓$	$✓$
9	?	?	$✓$	21	$✓$	$✓$	$✓$	21	$✓$	$✓$	$✓$	18	✓	$✓$	?
10	$✓$	$✓$	?	23	$✓$	$✓$	$✓$	22	$✓$	?	$✓$	20	✓	$✓$	?
11	$✓$	?	$✓$	24	?	$✓$	$✓$	23	$✓$	$✓$	$✓$	21	$✓$	$✓$	$✓$
12	✓	$✓$	$✓$	28	$✓$	$✓$	$✓$	27	$✓$	?	$✓$	23	$✓$	$✓$	$✓$
13	?	$✓$	?					28	$✓$	$✓$	$✓$	28	$✓$	$✓$	$✓$
14	$✓$	?	$✓$					30	$✓$	?	$✓$	29	$✓$	$✓$	?
15	$✓$	$✓$	$✓$
16	?	$✓$	$✓$
17	$✓$	$✓$	$✓$
18	$✓$	$✓$	?
19	?	?	?
20	$✓$	$✓$	?
21	$✓$	$✓$	$✓$
22	$✓$	?	$✓$
23	$✓$	$✓$	$✓$
24	?	$✓$	$✓$
25	?	?	?
26	?	$✓$	?
27	$✓$	?	$✓$
28	$✓$	$✓$	$✓$
29	$✓$	$✓$	?
30	$✓$	?	$✓$

Table 2. Simulation results for a single discrete covariate $\mathit{x}=z$ and $\mathit{y}$ with identically distributed components (n = 2000 with 1000 simulations).

		Estimation of mean			Estimation in propensity
Case		$\hat{\mu }$	${\hat{\mu }}^N$	${\hat{\mu }}^F$	$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$
I	Estimate	36.0097	37.3052	36.0029	2.4204	−0.0291	−0.0294	−0.0300
	Bias(%)	0.0269	3.6255	0.0080	−3.1830	−2.9003	−2.0861	0.0493
	SD	0.5927	0.1660	0.1477	0.6110	0.0099	0.0100	0.0099
	SE	0.6197	0.1639	0.1504
	CP	0.9580	0.0000	0.9520
II	Estimate	36.0342	35.1975	36.0041	−3.0067	0.0199	0.0202	0.0196
	Bias(%)	0.0951	−2.2291	0.0114	0.2243	−0.2509	0.9542	−1.9291
	SD	0.4321	0.1760	0.1548	0.4128	0.0086	0.0086	0.0087
	SE	0.4675	0.1698	0.1502
	CP	0.9610	0.0050	0.9470
III	Estimate	40.9571	43.3364	40.9965	2.7493	−0.0298	−0.0297	−0.0300
	Bias(%)	−0.1046	5.6987	−0.0085	−1.8100	−0.7331	−1.1194	−0.1108
	SD	0.6585	0.2138	0.2068	0.4435	0.0084	0.0088	0.0080
	SE	0.6987	0.2155	0.2113
	CP	0.9520	0.0000	0.9520
IV	Estimate	41.0381	39.4074	41.0018	−3.3208	0.0196	0.0202	0.0202
	Bias(%)	0.0930	−3.8843	0.0044	0.6295	−1.9536	1.0658	1.0930
	SD	0.6682	0.2376	0.2125	0.3398	0.0088	0.0080	0.0081
	SE	0.7262	0.2316	0.2119
	CP	0.9680	0.0000	0.9370

Table 3. Simulation results for a single continuous covariate $\mathit{x}=z$ and $\mathit{y}$ with identically distributed components (n = 2000 with 1000 simulations).

		Estimation of mean			Estimation in propensity
Case		$\hat{\mu }$	${\hat{\mu }}^N$	${\hat{\mu }}^F$	$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$
V	Estimate	30.0268	31.4417	29.9946	1.7769	−0.0304	−0.0302	−0.0298
	Bias(%)	0.0893	4.8056	−0.0180	−1.2853	1.4452	0.5892	−0.6491
	SD	0.3932	0.1884	0.1767	0.4663	0.0114	0.0111	0.0115
	SE	0.4333	0.1814	0.1687
	CP	0.9770	0.0000	0.9390
VI	Estimate	30.0194	31.4549	30.0098	1.7658	−0.0300	−0.0298	−0.0301
	Bias(%)	0.0647	4.8497	0.0327	−1.8974	0.0565	−0.7405	0.3017
	SD	0.3697	0.1805	0.1659	0.4278	0.0102	0.0093	0.0102
	SE	0.3918	0.1810	0.1686
	CP	0.9620	0.0000	0.9470
VII	Estimate	30.0441	31.4541	30.0040	1.7513	−0.0292	−0.0274	−0.0373
	Bias(%)	0.1471	4.8470	0.0135	−2.7030	−2.7809	−8.6993	24.4380
	SD	0.3511	0.1828	0.1676	2.3737	0.0270	0.1044	0.2069
	SE	0.3926	0.1810	0.1686
	CP	0.9720	0.0000	0.9490

Table 4. Simulation results for $\mathit{x}=(u,z)$ with a categorical z and a continuous u (n = 2000 with 1000 simulations).

		Estimation of mean
Case		$\hat{\mu }$	${\hat{\mu }}_N$	${\hat{\mu }}_F$
VIII	Estimate	24.0085	26.2166	23.9988
	Bias(%)	0.0354	9.2358	−0.0052
	SD	0.3526	0.2484	0.2627
	SE	0.3763	0.2406	0.2543
	CP	0.9620	0.0000	0.9450
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\eta }$
	Estimate	0.5253	−0.0290	−0.0297	−0.0292	0.0384
	Bias(%)	−12.4446	−3.4645	−1.0577	−2.7879	−3.9445
	SD	0.2599	0.0122	0.0122	0.0124	0.0152
		Estimation of mean
		$\hat{\mu }$	${\hat{\mu }}_N$	${\hat{\mu }}_F$
VIV	Estimate	24.0020	27.8904	24.0086
	Bias(%)	0.0083	16.2098	0.0358
	SD	0.4641	0.2085	0.2479
	SE	0.4623	0.2118	0.2541
	CP	0.9780	0.0000	0.9500
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\eta }$
	Estimate	1.2077	−0.0326	−0.0270	−0.0290	−0.0468
	Bias(%)	−28.9598	8.7831	−9.9284	−3.4798	17.0849
	SD	2.4568	0.4101	0.0505	0.2869	0.1186
		Estimation of mean
Case		$\hat{\mu }$	${\hat{\mu }}_N$	${\hat{\mu }}_F$
X	Estimate	32.5066	35.3083	32.5009
	Bias(%)	0.0203	8.6409	0.0028
	SD	0.3980	0.2608	0.2785
	SE	0.4354	0.2596	0.2828
	CP	0.9690	0.0000	0.9570
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\eta }$
	Estimate	0.9710	−0.0300	−0.0295	−0.0306	0.0499
	Bias(%)	−2.8962	−0.0200	−1.6826	2.0488	−0.2297
	SD	0.2568	0.0273	0.0257	0.0268	0.0119
		Estimation of mean
		$\hat{\mu }$	${\hat{\mu }}_N$	${\hat{\mu }}_F$
XI	Estimate	32.3482	37.5787	32.5173
	Bias(%)	−0.4670	15.6268	0.0532
	SD	1.0728	0.2053	0.2869
	SE	1.7310	0.1811	0.2661
	CP	0.9400	0.0000	0.9500
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\eta }$
	Estimate	2.6765	−0.0276	−0.0283	−0.0332	−0.0533
	Bias(%)	−4.4104	−8.1382	−5.5310	10.6065	6.6456
	SD	0.6086	0.0808	0.0672	0.0743	0.0224

Table 5. Simulation results for $\mathit{x}=(u,z)$ and multivariate $\mathit{y}$ (n = 2000 with 1000 simulations); SDimp(%) $=(1-$ SD of ${\hat{\mu }}_j^{GR}/$ SD of ${\hat{\mu }}_j)\times 100\mathrm{\%}$.

			Estimation of mean
Case			${\hat{\mu }}_j$	${\hat{\mu }}_j^{GR}$	${\hat{\mu }}_j^N$	${\hat{\mu }}_j^F$
XII	j = 1	Estimate	9.0131	9.0137	9.6294	9.0064
		Bias(%)	0.1456	0.1522	6.9935	0.0715
		SD	0.2116	0.1843	0.1856	0.1469
		SE	0.2179	0.1897	0.1851	0.1463
		CP	0.9550	0.9470	0.0920	0.9540
		SDimp(%)		12.8762
	j = 2	Estimate	11.0105	11.0204	11.7195	11.0076
		Bias(%)	0.0959	0.1852	6.5413	0.0687
		SD	0.2327	0.2043	0.1971	0.1590
		SE	0.2377	0.2064	0.2001	0.1580
		CP	0.9590	0.9580	0.0650	0.9370
		SDimp(%)		12.1987
	j = 3	Estimate	11.0105	16.0257	17.1169	16.0077
		Bias(%)	0.2687	0.1608	6.9809	0.0484
		SD	0.3810	0.3146	0.3308	0.2624
		SE	0.3819	0.3188	0.3272	0.2580
		CP	0.9510	0.9570	0.0900	0.9400
		SDimp(%)		17.4088
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\gamma }$
	Estimate	0.1036	−0.0201	−0.0206	−0.0208	0.0516
	Bias(%)	3.5808	0.7052	3.0718	4.2204	3.1186
	SD	0.1584	0.0251	0.0210	0.0186	0.0263
			Estimation of mean
Case			${\hat{\mu }}_j$	${\hat{\mu }}_j^{GR}$	${\hat{\mu }}_j^N$	${\hat{\mu }}_j^F$
XIII	j = 1	Estimate	9.0092	9.0094	8.8764	9.0034
		Bias(%)	0.1020	0.1040	−1.3734	0.0381
		SD	0.2116	0.1830	0.1784	0.1470
		SE	0.2124	0.1865	0.1769	0.1459
		CP	0.9560	0.9530	0.9050	0.9450
		SDimp(%)		13.5115
	j = 2	Estimate	11.0071	11.0110	10.7956	11.0005
		Bias(%)	0.0642	0.1004	−1.8585	0.0046
		SD	0.2283	0.2002	0.1836	0.1532
		SE	0.2328	0.2044	0.1904	0.1567
		CP	0.9430	0.9460	0.8150	0.9430
		SDimp(%)		12.3331
	j = 3	Estimate	11.0071	16.0159	15.8899	16.0062
		Bias(%)	0.1387	0.0994	−0.6879	0.0390
		SD	0.3563	0.3071	0.3090	0.2535
		SE	0.3663	0.3084	0.3118	0.2570
		CP	0.9400	0.9630	0.9230	0.9410
		SDimp(%)		13.8014
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\gamma }$
	Estimate	−1.2133	0.0186	0.0204	0.0209	−0.1014
	Bias(%)	1.1097	−7.0930	1.8714	4.4117	1.4445
	SD	0.1326	0.0255	0.0197	0.0166	0.0247
			Estimation of mean
Case			${\hat{\mu }}_j$	${\hat{\mu }}_j^{GR}$	${\hat{\mu }}_j^N$	${\hat{\mu }}_j^F$
XIV	j = 1	Estimate	10.2059	10.2035	12.2317	10.1935
		Bias(%)	0.0579	0.0345	19.9190	−0.0638
		SD	0.3286	0.2885	0.2803	0.2353
		SE	0.3416	0.2953	0.2880	0.2354
		CP	0.9560	0.9550	0.0000	0.9520
		SDimp(%)		12.1951
	j = 2	Estimate	14.4110	14.4100	16.5011	14.3915
		Bias(%)	0.0764	0.0698	14.5906	−0.0588
		SD	0.3461	0.2952	0.2964	0.2436
		SE	0.3459	0.2966	0.2986	0.2439
		CP	0.9550	0.9500	0.0000	0.9520
		SDimp(%)		14.6967
	j = 3	Estimate	14.4110	16.2097	20.1238	16.1886
		Bias(%)	0.0064	0.0601	24.2210	−0.0705
		SD	0.6786	0.5654	0.5603	0.4542
		SE	0.6633	0.5645	0.5535	0.4517
		CP	0.9490	0.9530	0.0000	0.9410
		SDimp(%)		16.6760
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\gamma }$
	Estimate	0.0853	−0.0209	−0.0206	−0.0189	0.0466
	Bias(%)	−14.7282	4.4709	2.8384	−5.4701	−6.8876
	SD	0.1920	0.0305	0.0186	0.0303	0.1200
			Estimation of mean
Case			${\hat{\mu }}_j$	${\hat{\mu }}_j^{GR}$	${\hat{\mu }}_j^N$	${\hat{\mu }}_j^F$
XV	j = 1	Estimate	10.2144	10.2160	9.0712	10.2000
		Bias(%)	0.1410	0.1566	−11.0664	0.0004
		SD	0.3727	0.3264	0.2830	0.2346
		SE	0.3870	0.3436	0.2867	0.2355
		CP	0.9650	0.9620	0.0260	0.9450
		SDimp(%)		12.4127
	j = 2	Estimate	14.4282	14.4221	13.2057
		Bias(%)	0.1961	0.1535	−8.2938	0.0095
		SD	0.4093	0.3620	0.3017	0.2433
		SE	0.4235	0.3799	0.2965	0.2436
		CP	0.9660	0.9610	0.0230	0.9400
		SDimp(%)		11.5561
	j = 3	Estimate	14.4282	16.2286	14.0698	16.1996
		Bias(%)	0.1344	0.1768	−13.1493	−0.0026
		SD	0.6812	0.5890	0.5366	0.4531
		SE	0.7135	0.6175	0.5504	0.4518
		CP	0.9600	0.9600	0.0400	0.9470
		SDimp(%)		13.5300
		Estimation in propensity
		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\gamma }$
	Estimate	−1.5441	0.0105	0.0267	0.0240	−0.1351
	Bias(%)	28.6770	−47.6082	33.4260	19.9365	35.0864
	SD	9.8790	0.2753	0.1834	0.1707	1.3114

Table 6. The number of observed values in each nonresponse pattern

	Nonresponse pattern
Group	Weight in 1997	Weight in 1999	Weight in 2001	Number of observations
$\mathrm{\Delta }=0$	?	?	?	504
$\mathrm{\Delta }=1$	$✓$	?	?	119
	?	$✓$	?	67
	?	?	$✓$	90
$\mathrm{\Delta }=2$	$✓$	$✓$	?	158
	$✓$	?	$✓$	134
	?	$✓$	$✓$	143
$\mathrm{\Delta }=3$	$✓$	$✓$	$✓$	1799
Total				3014

Table 7. Estimation based on NLSW data.

		Estimation of marginal population mean
ERNYR-P		$\hat{\mu }$	${\hat{\mu }}^N$
	Estimate	132.2044	158.1434
	SE	1.2182	0.6510
	Estimation of parameters in propensity
Parameter		$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$
	Estimate	−0.8844	−0.0245	0.0489	−0.0374

Table 8. The number of observed values in each of nonresponse pattern.

	Missing pattern
Group	LBXSCH	BPXSY1	BMDAVSAD	Number of observations
$\mathrm{\Delta }=0$	?	?	?	2007
$\mathrm{\Delta }=1$	$✓$	?	?	94
	?	$✓$	?	91
	?	?	$✓$	90
$\mathrm{\Delta }=2$	$✓$	$✓$	?	374
	$✓$	?	$✓$	422
	?	$✓$	$✓$	444
$\mathrm{\Delta }=3$	$✓$	$✓$	$✓$	5578
Total				9100

Table 9. Estimation based on NHANES data.

Results for $z=$ DMDHRAGE
		Estimation of mean
Component		${\hat{\mu }}_j$	${\hat{\mu }}_j^{GR}$	${\hat{\mu }}_j^N$
LBXSCH	Estimate	144.0550	144.0248	184.5257
	SE	0.8946	0.9050	0.4634
BPXSY1	Estimate	89.1456	89.2445	119.2604
	SE	16.6209	16.6619	21.5088
BMDAVSAD	Estimate	0.5489	0.5500	0.2094
	SE	0.1059	0.1066	0.0560
	Estimation in propensity
	$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\gamma }$
Estimate	0.1740	0.0066	−0.0159	−0.1152	−0.0183
SE	2.4623	0.0105	0.0134	0.1161	0.0113
Results for $z=$ INDHHIN2
		Estimation of mean
Component		${\hat{\mu }}_j$	${\hat{\mu }}_j^{GR}$	${\hat{\mu }}_j^N$
LBXSCH	Estimate	144.0251	143.9984	184.5257
	SE	0.8394	0.8407	0.5822
BPXSY1	Estimate	89.0743	89.1804	119.2758
	SE	0.5200	0.5169	0.2308
BMDAVSAD	Estimate	16.6357	16.6790	21.5052
	SE	0.1020	0.1038	0.0658
	Estimation in propensity
	$\hat{\alpha }$	${\widehat{\text{β}}}_1$	${\widehat{\text{β}}}_2$	${\widehat{\text{β}}}_3$	$\hat{\gamma }$
Estimate	−1.5423	−0.0035	0.0055	−0.0978	0.0090
SE	6.2862	0.0346	0.0205	0.2393	0.0105