Goodness of fit test for higher order binary Markov chain models

Figures & data

Table 1. Transition counts of Markov chain of depression data for the first- and second-order

	Transition time		t
First-order		t − 1	0	1
		0	3,951	1,455
		1	1,930	257
Second-order	t − 2	t − 1	t
	0	0	3,473	1,396
	1	0	680	221
	0	1	269	29
	1	1	568	151

Table 2. Estimates of parameters of covariate-dependent first- and second-order Markov models and testing the goodness of fit

First-order
Transition type	Covariates	Estimated value	s.e.	p-value
0→1	Constant	4.631	0.349	0.000
	Age	−0.096	0.005	0.000
	Sex	0.129	0.066	0.051
	BMI	0.011	0.006	0.078
	Drinking	−0.215	0.066	0.0011
1→1	Constant	8.378	0.825	0.000
	Age	−0.118	0.012	0.000
	Sex	0.019	0.148	0.896
	BMI	0.007	0.012	0.577
	Drinking	0.552	0.148	0.0002
Billingsley’s chi-square		3.94E-13	(p-value = 0.999)
Proposed test statistics		1.207	(p-value = 0.997)
Second-order
Transition type	Covariate	Estimated value	s.e.	p-value
0→0→1	Constant	4.360	0.385	0.000
	Age	−0.090	0.005	0.000
	Sex	0.224	0.068	0.0011
	BMI	0.009	0.006	0.153
	Drinking	−0.282	0.067	0.0003
1→0→1	Constant	−2.996	0.999	0.003
	Age	0.020	0.015	0.178
	Sex	0.335	0.174	0.053
	BMI	0.026	0.016	0.104
	Drinking	−0.278	0.160	0.082
0→1→1	Constant	2.261	2.250	0.315
	Age	0.006	0.034	0.870
	Sex	0.195	0.423	0.646
	BMI	−0.015	0.036	0.682
	Drinking	−0.101	0.410	0.805
1→1→1	Constant	9.295	1.217	0.000
	Age	−0.142	0.019	0.000
	Sex	0.026	0.220	0.907
	BMI	0.007	0.015	0.670
	Drinking	0.513	0.222	0.021
Billingsley’s chi-square		2.13E-08	(p-value = 0.999)
Proposed test statistics		2.057	(p-value = 0.999)

Table 3. Five hundred simulations for obtaining the estimates of associations based on the proposed first-order models

		Model 1—size 250		Model 2—size 500		Model 3—size 1000			Model 4—size 250		Model 5—size 500
Transition type
00		61		121		243			50		100
01		65		129		257			50		100
10		14		29		58			75		150
11		110		221		442			75		150
Correlation of response variables		0.4		0.4		0.4			0		0
Estimates of Parameters		Estimate	p-value	Estimate	p-value	Estimate	p-value		Estimate	p-value	Estimate		p-value
0 to 1	Constant	−0.868	0.115	−0.843	0.025	−0.830	0.002		−0.916	0.130	−0.877		0.040
	V1	1.422	0.011	1.386	0.000	1.360	0.000		0.972	0.143	0.942		0.043
	V2	0.301	0.438	0.290	0.383	0.287	0.304		0.341	0.396	0.357		0.313
	V3	0.093	0.482	0.109	0.469	0.101	0.464		0.517	0.321	0.469		0.237
	V4	0.305	0.423	0.262	0.380	0.259	0.293		0.759	0.263	0.751		0.126
1 to 1	Constant	−1.181	0.195	−1.114	0.084	−1.044	0.019		1.233	0.045	1.200		0.005
	V1	−1.776	0.059	−1.696	0.005	−1.685	0.000		−0.914	0.072	−0.907		0.012
	V2	−0.365	0.444	−0.257	0.453	−0.250	0.418		−0.364	0.352	−0.350		0.273
	V3	−0.013	0.483	−0.037	0.483	−0.061	0.468		−0.496	0.292	−0.491		0.161
	V4	−0.252	0.452	−0.253	0.448	−0.285	0.376		−0.693	0.157	−0.665		0.058
Billingsley		0.008	0.951	0.010	0.938	0.019	0.907		2.50E-05	0.999	1.06E-05		0.999
No. of tests accepting H0		500		500		500			500		500
Proposed test		8.646	0.462	8.267	0.481	8.172	0.481		8.159	0.486	8.403		0.470
No. of tests accepting H0		460		467		476			474		470
Proportion of rejection of H0		40/500		33/500		24/500			26/500		30/500
				Model 6—size 1000		Model 7—size 250			Model 8—size 500		Model 9—size 1000
Transition
00				201		26			51		102
01				200		99			200		399
10				300		75			148		298
11				300		50			101		201
Correlation of response variables				0		−0.4			−0.4		−0.4
Estimates of Parameters				Estimate	p-value	Estimate		p-value	Estimate	p-value	Estimate	p-value
0 to 1		Constant		−0.875	0.003	0.485		0.314	0.489	0.202	0.492	0.081
		V1		0.926	0.004	1.797		0.028	1.707	0.002	1.688	0.000
		V2		0.351	0.215	2.183		0.036	1.961	0.002	1.931	0.000
		V3		0.470	0.112	0.709		0.274	0.666	0.172	0.635	0.066
		V4		0.743	0.028	−0.715		0.269	−0.666	0.166	−0.653	0.063
1 to 1		Constant		1.190	0.000	2.463		0.003	2.315	0.000	2.291	0.000
		V1		−0.885	0.000	−1.731		0.013	−1.635	0.001	−1.613	0.000
		V2		−0.342	0.154	−1.621		0.011	−1.546	0.000	−1.529	0.000
		V3		−0.491	0.060	−0.645		0.242	−0.605	0.153	−0.611	0.042
		V4		−0.664	0.007	0.666		0.247	0.642	0.116	0.657	0.024
Billingsley				4.79E-06	0.999	0.011		0.935	0.011	0.934	0.011	0.923
No. of tests accepting H0				500		500			500		500
Proposed test				8.167	0.489	8.383		0.467	8.439	0.468	8.448	0.463
No. of tests accepting H0				472		471			475		473
Proportion of rejection of H0				28/500		29/500			25/500		27/500

Table 4. Five hundred simulations for obtaining the estimates of associations based on the proposed second-order models

			Model 1—size 250			Model 2—size 500			Model 3—size 1000
Transition
000			67			132			262
001			15			31			61
100			22			44			88
101			22			43			88
010			22			43			87
011			22			44			88
110			15			31			62
111			65			132			264
Correlation of response variables			0.4			0.4			0.4
Estimates of parameters			Estimate		p-value	Estimate	p-value		Estimate		p-value
0→0→1		Constant	−0.816		0.244	−0.724	0.136		−0.701		0.036
		V1	0.670		0.313	0.585	0.258		0.559		0.133
		V2	0.938		0.273	0.835	0.168		0.831		0.059
		V3	0.449		0.386	0.401	0.347		0.397		0.247
		V4	0.347		0.446	0.312	0.435		0.337		0.339
1→0→1		Constant	−1.025		0.232	−0.989	0.101		−0.951		0.031
		V1	0.605		0.334	0.601	0.242		0.582		0.116
		V2	0.861		0.272	0.855	0.117		0.796		0.036
		V3	0.425		0.404	0.397	0.369		0.419		0.235
		V4	0.335		0.438	0.295	0.411		0.268		0.353
0→1→1		Constant	1.035		0.235	0.990	0.101		0.938		0.031
		V1	−0.610		0.353	−0.595	0.240		−0.566		0.132
		V2	−0.902		0.262	−0.840	0.151		−0.806		0.049
		V3	−0.467		0.413	−0.432	0.341		−0.391		0.272
		V4	−0.348		0.446	−0.302	0.417		−0.304		0.364
1→1→1		Constant	1.333		0.244	1.255	0.116		1.235		0.027
		V1	−0.621		0.374	−0.613	0.255		−0.601		0.131
		V2	−0.814		0.263	−0.813	0.129		−0.781		0.032
		V3	−0.483		0.398	−0.418	0.356		−0.421		0.269
		V4	−0.341		0.437	−0.291	0.417		−0.293		0.353
Billingsley			2.86E-04		1.000	5.01E-05	1.000		1.66E-05		1.000
No. of tests accepting H0			500			500			500
Proposed test			16.913		0.445	16.650	0.459		15.804		0.503
No. of tests accepting H0			475			472			482
Proportion of rejection of H0			25/500			28/500			18/500
			Model 4—size 250			Model 5—size 500			Model 6—size 1000
Transition
000			31			63			126
001			31			62			125
100			32			63			124
101			31			62			125
010			31			63			125
011			31			62			125
110			31			63			125
111			32			62			125
Correlation of response variables			0			0			0
Estimates of parameters			Estimate	p-value		Estimate		p-value	Estimate	p-value
0→0→1	Constant		−0.771	0.266		−0.704		0.144	−0.724	0.034
	V1		0.615	0.343		0.577		0.260	0.581	0.125
	V2		0.912	0.288		0.847		0.163	0.838	0.053
	V3		0.412	0.400		0.378		0.362	0.406	0.240
	V4		0.381	0.441		0.320		0.422	0.317	0.373
1→0→1	Constant		−1.083	0.229		−0.976		0.111	−0.964	0.029
	V1		0.644	0.353		0.547		0.266	0.572	0.123
	V2		0.879	0.259		0.806		0.144	0.788	0.045
	V3		0.474	0.409		0.436		0.327	0.429	0.228
	V4		0.326	0.448		0.321		0.389	0.296	0.337
0→1→1	Constant		1.013	0.239		1.004		0.112	0.931	0.033
	V1		−0.580	0.359		−0.639		0.222	−0.557	0.133
	V2		−0.942	0.251		−0.795		0.177	−0.794	0.049
	V3		−0.434	0.409		−0.427		0.339	−0.401	0.251
	V4		−0.310	0.454		−0.306		0.419	−0.278	0.370
1→1→1	Constant		1.291	0.254		1.270		0.101	1.241	0.025
	V1		−0.611	0.377		−0.605		0.264	−0.597	0.146
	V2		−0.849	0.253		−0.789		0.130	−0.778	0.038
	V3		−0.450	0.434		−0.440		0.360	−0.429	0.261
	V4		−0.330	0.437		−0.300		0.421	−0.301	0.337
Billingsley			3.38E-04	1.000		7.01E-05		1.000	2.14E-05	1.000
No. of tests accepting H0			500			500			500
Proposed test			17.477	0.425		16.562		0.464	16.517	0.473
No. of tests accepting H0			559			476			473
Proportion of rejection of H0			41/500			24/500			27/500
			Model 7—size 250			Model 8—size 500			Model 9—size 1000
Transition
000			16			31			62
001			59			119			237
100			28			56			112
101			22			44			88
010			22			44			88
011			28			56			112
110			59			118			238
111			16			32			63
Correlation of response variables			−0.4			−0.4			−0.4
Estimates of parameters			Estimate		p-value	Estimate		p-value	Estimate	p-value
0→0→1	Constant		−0.068		0.479	−0.060		0.498	−0.063	0.468
	V1		−0.051		0.487	−0.053		0.485	−0.066	0.454
	V2		−0.053		0.466	−0.072		0.500	−0.061	0.472
	V3		−0.024		0.500	−0.046		0.479	−0.012	0.501
	V4		−0.091		0.498	−0.085		0.474	−0.092	0.479
1→0→1	Constant		−0.164		0.479	−0.173		0.456	−0.179	0.367
	V1		−0.090		0.473	−0.045		0.505	−0.042	0.489
	V2		−0.078		0.503	−0.061		0.494	−0.060	0.502
	V3		−0.056		0.493	0.034		0.514	−0.011	0.494
	V4		−0.096		0.477	−0.102		0.488	−0.085	0.481
0→1→1	Constant		0.041		0.489	0.082		0.492	0.092	0.501
	V1		0.090		0.455	0.032		0.489	0.052	0.495
	V2		0.139		0.489	0.096		0.476	0.053	0.511
	V3		0.351		0.409	0.152		0.465	0.014	0.491
	V4		0.418		0.459	0.219		0.489	0.133	0.496
1→1→1	Constant		0.326		0.484	0.183		0.504	0.185	0.460
	V1		−0.004		0.403	0.346		0.449	0.179	0.500
	V2		0.133		0.456	0.027		0.474	0.059	0.485
	V3		−0.005		0.240	0.114		0.361	0.249	0.450
	V4		0.289		0.361	0.212		0.452	0.272	0.477
Billingsley			9.51E-03		1.000	3.86E-03		1.000	1.12E-03	1.000
No. of tests accepting H0			500			500			500
Proposed test			15.810		0.508	17.039		0.442	16.388	0.478
No. of tests accepting H0			479			469			473
Proportion of rejection of H0			21/500			31/500			27/500