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. 2017 May 22;36(21):3361–3379. doi: 10.1002/sim.7345

Table 2.

Results of fitting the state‐space model by using the maximum likelihood method.

Subject
1 2 3 4 5 6 7 8 9 10
Best model M10 M8 M9 M6 M11 M8 M11 M12 M10 M5
α 0.210 0.953 0.344 1.520 0.318 1.971 0.630 0.172 0.150 0.201
[0.151,0.292] [0.628,1.439] [0.253,0.467] [1.015,2.266] [0.233,0.435] [1.329,2.905] [0.449,0.882] [0.128,0.231] [0.106,0.211] [0.140,0.288]
β 8.915 32.131 10.916 45.146 8.536 59.929 19.510 5.886 5.284 6.544
[6.303,12.610] [21.157,48.798] [7.883,15.115] [30.420,67.002] [6.094,11.955] [40.536,88.601] [13.806,27.572] [4.236,8.178] [3.595,7.766] [4.429,9.669]
σ 0.112 0.161 0.119 0.121 0.093 0.152 0.108 0.101 0.116 0.209
[0.103,0.121] [0.150,0.173] [0.109,0.130] [0.113,0.129] [0.085,0.102] [0.142,0.163] [0.098,0.120] [0.094,0.109] [0.109,0.123] [0.195,0.224]
a 36.299 36.203 36.485 36.384 36.632 36.522 36.518 36.519 36.645 36.123
[36.247,36.351] [36.187,36.219] [36.466,36.504] [36.370,36.397] [36.614,36.650] [36.509,36.535] [36.501,36.536] [36.492,36.546] [36.620,36.670] [36.087,36.160]
b 1 0.193 0.197 0.174 0.098 −0.034 0.107 0.130 0.141 0.129 0.138
[0.153,0.232] [0.179,0.216] [0.147,0.200] [0.074,0.122] [−0.072,0.003] [0.088,0.126] [0.101,0.159] [0.114,0.169] [0.099,0.159] [0.082,0.193]
b 2 0.015 0.024 −0.028 −0.079 0.115 −0.063 −0.001 −0.370 −0.012 0.002
[−0.021,0.050] [−0.010,0.058] [−0.061,0.006] [−0.096,−0.061] [0.063,0.167] [−0.081,−0.046] [−0.024,0.022] [−0.416,−0.323] [−0.067,0.043] [−0.082,0.085]
b 3 0.009 −0.057 0.027 0.032 −0.098 0.018 0.025 −0.089 −0.087 −0.126
[−0.112,0.130] [−0.085,−0.029] [−0.001,0.056] [0.012,0.051] [−0.148,−0.048] [−0.003,0.039] [0.001,0.049] [−0.132,−0.045] [−0.136,−0.038] [−0.192,−0.060]
b 4 −0.039 0.021 −0.023 0.007 0.124 −0.028 −0.032 0.167 −0.078 −0.153
[−0.075,−0.003] [−0.010,0.053] [−0.051,0.005] [−0.017,0.030] [0.094,0.153] [−0.054,−0.001] [−0.061,−0.003] [0.105,0.229] [−0.125,−0.030] [−0.250,−0.056]
b 5 0.058 −0.046 0.013 −0.020 −0.129 −0.030 0.029 0.174 0.049 −0.003
[−0.006,0.121] [−0.097,0.005] [−0.028,0.054] [−0.039,−0.002] [−0.165,−0.092] [−0.061,0.000] [0.001,0.057] [0.137,0.212] [−0.040,0.139] [−0.153,0.148]
b 6 −0.014 0.046 −0.065 −0.022 0.097 0.028 −0.001 −0.038 0.100 NA
[−0.230,0.202] [−0.005,0.098] [−0.106,−0.024] [−0.051,0.006] [0.045,0.149] [−0.001,0.058] [−0.030,0.029] [−0.122,0.047] [0.040,0.160]
b 7 −0.024 0.019 0.006 NA −0.078 −0.026 0.024 −0.112 −0.003 NA
[−0.485,0.437] [−0.030,0.068] [−0.034,0.046] [−0.124,−0.032] [−0.068,0.016] [−0.015,0.062] [−0.149,−0.075] [−0.104,0.099]
b 8 0.037 0.034 −0.063 NA 0.088 0.037 0.060 −0.098 −0.058 NA
[−0.256,0.331] [−0.037,0.105] [−0.145,0.020] [0.047,0.129] [−0.011,0.085] [0.011,0.109] [−0.180,−0.017] [−0.171,0.056]
b 9 −0.086 NA −0.045 NA −0.060 NA 0.007 0.010 −0.036 NA
[−0.231,0.060] [−0.142,0.051] [−0.090,−0.029] [−0.019,0.033] [−0.033,0.053] [−0.136,0.064]
b 10 0.061 NA NA NA 0.101 NA 0.050 0.120 0.053 NA
[−0.170,0.293] [0.062,0.140] [0.002,0.097] [0.062,0.178] [−0.036,0.141]
b 11 NA NA NA NA −0.058 NA 0.031 0.023 NA NA
[−0.100,−0.016] [−0.032,0.094] [−0.023,0.069]
b 12 NA NA NA NA NA NA NA −0.114 NA NA
[−0.145,−0.082]
c 1 −0.193 −0.108 −0.182 −0.270 −0.184 −0.170 −0.254 −0.417 −0.172 −0.276
[−0.304,−0.081] [−0.140,−0.075] [−0.218,−0.146] [−0.289,−0.252] [−0.204,−0.164] [−0.190,−0.150] [−0.274,−0.233] [−0.463,−0.371] [−0.218,−0.126] [−0.334,−0.217]
c 2 −0.051 −0.131 −0.130 −0.034 −0.107 −0.058 −0.038 −0.216 −0.035 −0.057
[−0.085,−0.018] [−0.154,−0.108] [−0.159,−0.101] [−0.058,−0.010] [−0.133,−0.080] [−0.082,−0.035] [−0.059,−0.017] [−0.263,−0.169] [−0.069,−0.001] [−0.107,−0.006]
c 3 −0.089 −0.029 −0.022 −0.056 0.006 −0.023 −0.021 0.062 −0.070 −0.019
[−0.120,−0.059] [−0.065,0.008] [−0.051,0.006] [−0.076,−0.037] [−0.041,0.052] [−0.046,−0.000] [−0.044,0.002] [0.018,0.106] [−0.126,−0.014] [−0.096,0.059]
c 4 −0.030 −0.001 −0.035 0.005 −0.016 −0.012 0.016 0.248 −0.047 0.042
[−0.174,0.114] [−0.042,0.041] [−0.063,−0.007] [−0.014,0.025] [−0.077,0.046] [−0.038,0.014] [−0.013,0.044] [0.202,0.295] [−0.118,0.024] [−0.065,0.148]
c 5 −0.003 −0.058 0.027 −0.017 0.010 −0.037 −0.016 −0.040 −0.082 −0.126
[−0.256,0.250] [−0.117,0.001] [−0.007,0.062] [−0.040,0.007] [−0.047,0.067] [−0.067,−0.007] [−0.044,0.011] [−0.099,0.019] [−0.134,−0.029] [−0.186,−0.066]
c 6 −0.053 −0.003 0.023 −0.034 −0.007 −0.025 −0.010 −0.277 0.004 NA
[−0.143,0.037] [−0.055,0.050] [−0.030,0.076] [−0.059,−0.008] [−0.052,0.038] [−0.062,0.013] [−0.036,0.015] [−0.312,−0.243] [−0.107,0.114]
c 7 0.068 −0.030 −0.031 NA 0.010 −0.040 −0.052 −0.075 0.059 NA
[−0.040,0.176] [−0.080,0.020] [−0.073,0.012] [−0.038,0.057] [−0.086,0.006] [−0.081,−0.023] [−0.136,−0.015] [0.014,0.104]
c 8 −0.050 0.044 0.067 NA −0.044 0.027 0.027 0.214 0.074 NA
[−0.324,0.223] [−0.002,0.089] [0.006,0.127] [−0.102,0.013] [−0.028,0.082] [−0.014,0.068] [0.171,0.257] [−0.009,0.157]
c 9 −0.027 NA −0.085 NA 0.006 NA 0.007 0.079 −0.067 NA
[−0.675,0.620] [−0.134,−0.036] [−0.059,0.071] [−0.049,0.063] [0.036,0.122] [−0.169,0.034]
c 10 0.025 NA NA NA −0.019 NA 0.013 −0.107 −0.057 NA
[−0.494,0.543] [−0.088,0.050] [−0.032,0.058] [−0.172,−0.041] [−0.156,0.042]
c 11 NA NA NA NA 0.041 NA 0.067 −0.022 NA NA
[−0.011,0.093] [0.025,0.108] [−0.057,0.014]
c 12 NA NA NA NA NA NA NA 0.012 NA NA
[−0.053,0.077]
Log‐ 468.905 164.278 206.631 415.909 343.887 234.46 354.47 408.464 330.137 −178.022
likelihood
Subject
11 12 13 14 15 16 17 18 19 20
Best model M12 M12 M5 M7 M6 M9 M5 M2 M10 M10
α 0.452 0.439 3.626 3.246 12.593 1.000 9.367 2.084 0.456 0.360
[0.329,0.620] [0.331,0.580] [2.249,5.744] [1.984,5.224] [6.756,21.022] † [5.696,14.622] [1.258,3.414] [0.326,0.638] [0.265,0.487]
β 13.791 12.092 96.044 100.347 345.852 25.939 267.147 55.485 14.223 9.981
[9.918,19.178] [8.985,16.274] [59.924,153.936] [61.670,163.280] [194.670,614.443] † [166.127,429.594] [33.748,91.222] [10.062,20.106] [7.205,13.827]
σ 0.098 0.090 0.169 0.185 0.109 0.088 0.127 0.144 0.126 0.092
[0.090,0.108] [0.081,0.100] [0.159,0.180] [0.174,0.197] [0.103,0.116] † [0.120,0.134] [0.135,0.153] [0.115,0.137] [0.082,0.104]
a 36.536 36.741 36.715 36.393 36.687 36.407 36.512 36.280 36.529 36.439
[36.520,36.553] [36.722,36.760] [36.699,36.730] [36.378,36.407] [36.678,36.696] [36.393,36.421] [36.501,36.522] [36.266,36.294] [36.512,36.547] [36.423,36.455]
b 1 0.014 0.113 −0.000 0.142 0.011 0.091 0.028 0.100 0.085 0.121
[−0.014,0.042] [0.090,0.136] [−0.025,0.025] [0.121,0.164] [−0.004,0.025] [0.073,0.110] [0.012,0.045] [0.077,0.122] [0.058,0.112] [0.088,0.153]
b 2 0.055 −0.150 −0.026 −0.059 −0.019 −0.058 −0.041 −0.024 −0.103 −0.021
[0.029,0.082] [−0.188,−0.112] [−0.051,−0.001] [−0.081,−0.037] [−0.032,−0.007] [−0.079,−0.038] [−0.057,−0.026] [−0.043,−0.004] [−0.129,−0.077] [−0.052,0.011]
b 3 −0.042 0.148 −0.006 0.009 −0.013 −0.012 0.021 NA 0.005 −0.023
[−0.088,0.004] [0.109,0.187] [−0.031,0.019] [−0.016,0.034] [−0.026,−0.000] [−0.036,0.012] [0.004,0.038] [−0.026,0.036] [−0.052,0.006]
b 4 0.001 −0.011 −0.011 −0.011 −0.009 −0.027 −0.005 NA −0.001 −0.059
[−0.080,0.083] [−0.063,0.041] [−0.040,0.018] [−0.038,0.015] [−0.023,0.006] [−0.053,−0.000] [−0.021,0.011] [−0.031,0.029] [−0.092,−0.027]
b 5 0.004 −0.081 −0.017 −0.012 −0.031 0.031 0.011 NA 0.019 0.008
[−0.083,0.091] [−0.108,−0.054] [−0.049,0.015] [−0.048,0.024] [−0.045,−0.016] [0.000,0.061] [−0.007,0.029] [−0.026,0.064] [−0.033,0.049]
b 6 −0.015 0.058 NA 0.010 −0.027 0.021 NA NA −0.041 −0.039
[−0.074,0.044] [0.013,0.104] [−0.041,0.062] [−0.042,−0.011] [−0.030,0.071] [−0.085,0.002] [−0.077,−0.002]
b 7 0.074 0.051 NA −0.043 NA 0.001 NA NA 0.002 −0.056
[0.036,0.112] [−0.027,0.129] [−0.081,−0.005] [−0.043,0.046] [−0.032,0.036] [−0.102,−0.010]
b 8 −0.097 −0.108 NA NA NA −0.015 NA NA 0.016 0.001
[−0.121,−0.072] [−0.170,−0.045] [−0.097,0.066] [−0.017,0.049] [−0.047,0.049]
b 9 0.043 −0.025 NA NA NA −0.057 NA NA 0.030 −0.030
[−0.005,0.090] [−0.081,0.032] [−0.109,−0.004] [−0.038,0.098] [−0.064,0.003]
b 10 −0.014 0.031 NA NA NA NA NA NA −0.075 0.012
[−0.063,0.036] [−0.019,0.082] [−0.114,−0.036] [−0.122,0.145]
b 11 −0.012 −0.061 NA NA NA NA NA NA NA NA
[−0.153,0.129] [−0.180,0.058]
b 12 −0.035 −0.047 NA NA NA NA NA NA NA NA
[−0.187,0.116] [−0.194,0.100]
c 1 −0.238 −0.071 −0.235 −0.164 −0.225 −0.181 −0.228 −0.198 −0.171 −0.200
[−0.260,−0.216] [−0.105,−0.036] [−0.255,−0.214] [−0.187,−0.140] [−0.236,−0.213] [−0.202,−0.160] [−0.242,−0.214] [−0.219,−0.178] [−0.200,−0.143] [−0.228,−0.172]
c 2 0.000 −0.173 0.004 −0.076 −0.009 −0.097 −0.012 −0.036 −0.059 −0.017
[−0.028,0.029] [−0.215,−0.131] [−0.018,0.025] [−0.101,−0.052] [−0.021,0.003] [−0.120,−0.074] [−0.028,0.003] [−0.057,−0.015] [−0.097,−0.022] [−0.039,0.004]
c 3 −0.116 −0.099 −0.053 −0.010 −0.031 −0.047 −0.062 NA −0.061 −0.043
[−0.147,−0.086] [−0.150,−0.048] [−0.078,−0.029] [−0.033,0.013] [−0.043,−0.018] [−0.066,−0.028] [−0.077,−0.047] [−0.091,−0.031] [−0.078,−0.007]
c 4 0.121 0.088 −0.029 −0.024 0.029 −0.046 −0.003 NA −0.025 0.022
[0.095,0.147] [0.059,0.118] [−0.055,−0.003] [−0.051,0.002] [0.015,0.043] [−0.066,−0.026] [−0.019,0.013] [−0.054,0.004] [−0.030,0.075]
c 5 −0.130 −0.033 −0.040 −0.046 0.007 −0.031 −0.030 NA 0.035 0.022
[−0.166,−0.093] [−0.090,0.024] [−0.069,−0.012] [−0.075,−0.016] [−0.008,0.022] [−0.067,0.005] [−0.045,−0.011] [−0.006,0.076] [−0.007,0.051]
c 6 0.051 −0.078 NA 0.061 −0.009 0.039 NA NA −0.033 0.014
[0.020,0.083] [−0.124,−0.031] [0.028,0.094] [−0.025,0.007] [0.004,0.074] [−0.083,0.017] [−0.024,0.052]
c 7 −0.017 0.088 NA −0.020 NA −0.031 NA NA −0.009 0.010
[−0.104,0.070] [0.036,0.139] [−0.065,0.024] [−0.051,−0.012] [−0.048,0.030] [−0.040,0.061]
c 8 −0.003 0.060 NA NA NA 0.052 NA NA 0.004 0.017
[−0.117,0.112] [−0.015,0.135] [0.023,0.082] [−0.051,0.059] [−0.025,0.060]
c 9 0.021 −0.057 NA NA NA −0.034 NA NA 0.036 −0.017
[−0.050,0.091] [−0.092,−0.022] [−0.101,0.034] [−0.031,0.103] [−0.064,0.029]
c 10 −0.023 −0.032 NA NA NA NA NA NA 0.007 −0.093
[−0.061,0.014] [−0.100,0.036] [−0.123,0.136] [−0.132,−0.053]
c 11 0.087 0.101 NA NA NA NA NA NA NA NA
[0.046,0.128] [0.049,0.153]
c 12 −0.092 −0.111 NA NA NA NA NA NA NA NA
[−0.143,−0.040] [−0.156,−0.067]
Log‐ 343.440 287.380 111.443 88.146 479.948 267.320 365.655 217.357 266.617 253.890
likelihood

Parameter estimates and their 95% confidence intervals (in brackets) of the best Akaike information criterion model are shown for each subject. Log‐likelihood was approximated by using the non‐Gaussian filter with the state space discretized with 512 intervals.

†

Confidence intervals were not calculated because the Hessian of log‐likelihood was singular.