Background
Methods
Overview of count data regression models
Poisson model
Negative binomial model
Zero-inflated count models
Generalized linear models
Simulation study
Dataset generation
Distribution | Sample size, n | Proportion of Zeros | Level of Dispersion |
---|---|---|---|
Poisson | 50 | 0.1 | 0.01 |
NB | 200 | 0.3 | 1 |
ZIP | 600 | 0.5 | 5 |
ZINB | 1000 | 0.70 | 10 |
Models evaluation
Empirical study
Data description
Statistical analysis
Results
Simulation study
Data generated with poisson regression model
Sample Size | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|
50 | 100.0% | 53.5% | 100.0% | 90.4% |
200 | 100.0% | 53.6% | 100.0% | 93.4% |
600 | 100.0% | 54.9% | 100.0% | 93.5% |
1000 | 100.0% | 58.7% | 100.0% | 94.2% |
AIC | BIC | |||||||
---|---|---|---|---|---|---|---|---|
Sample Size | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
50 | 260.88 | 266.12 | 266.88 | 267.92 | 266.6152 | 273.3131 | 278.3513 | 280.1757 |
200 | 1032.82 | 1042.64 | 1038.82 | 1040.18 | 1042.7200 | 1054.9800 | 1058.6100 | 1061.4600 |
600 | 3092.07 | 3107.19 | 3098.07 | 3100.35 | 3105.2600 | 3123.6300 | 3124.4500 | 3128.8100 |
1000 | 5151.60 | 5173.91 | 5157.60 | 5160.71 | 5166.3200 | 5192.7200 | 5187.0400 | 5192.5700 |
Sample Size | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|
50 | 0.6201 | 0.6158 | 0.6202 | 0.6202 |
200 | 0.3039 | 0.3055 | 0.3055 | 0.3045 |
600 | 0.1727 | 0.1764 | 0.1736 | 0.1736 |
1000 | 0.1401 | 0.1420 | 0.1402 | 0.1402 |
Data generated with NB regression model
Sample Size | Dispersion | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
50 | 0.01 | 100.0% | 64.8% | 100.0% | 91.0% |
1 | 100.0% | 100.0% | 100.0% | 100.0% | |
5 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 100.0% | 100.0% | 100.0% | 100.0% | |
200 | 0.01 | 100.0% | 81.2% | 100.0% | 97.3% |
1 | 100.0% | 100.0% | 100.0% | 100.0% | |
5 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 100.0% | 100.0% | 100.0% | 100.0% | |
600 | 0.01 | 100.0% | 90.5% | 100.0% | 99.5% |
1 | 100.0% | 100.0% | 100.0% | 100.0% | |
5 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 100.0% | 100.0% | 100.0% | 100.0% | |
1000 | 0.01 | 100.0% | 94.0% | 100.0% | 99.6% |
1 | 100.0% | 100.0% | 100.0% | 100.0% | |
5 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 100.0% | 100.0% | 100.0% | 100.0% |
AIC | BIC | ||||||||
---|---|---|---|---|---|---|---|---|---|
Sample Size | Dispersion | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
50 | 0.01 | 265.51 | 270.12 | 271.51 | 272.78 | 271.24 | 277.42 | 282.98 | 285.40 |
1 | 635.25 | 341.14 | 634.90 | 346.91 | 640.98 | 348.79 | 646.37 | 360.29 | |
5 | 1346.69 | 276.80 | 858.14 | 282.22 | 1352.43 | 284.45 | 869.61 | 295.60 | |
10 | 1766.50 | 216.93 | 956.54 | 221.37 | 1772.24 | 224.58 | 968.01 | 234.75 | |
200 | 0.01 | 1051.89 | 1057.24 | 1057.89 | 1058.58 | 1061.79 | 1069.85 | 1077.68 | 1081.01 |
1 | 2618.54 | 1352.80 | 2612.87 | 1358.58 | 2628.44 | 1365.99 | 2632.66 | 1381.66 | |
5 | 5908.13 | 1095.27 | 3712.59 | 1100.91 | 5918.03 | 1108.46 | 3732.38 | 1124.01 | |
10 | 7900.24 | 862.34 | 4125.06 | 866.21 | 7910.14 | 875.53 | 4144.85 | 889.29 | |
600 | 0.01 | 3150.60 | 3153.55 | 3156.60 | 3155.11 | 3163.79 | 3170.49 | 3182.98 | 3185.66 |
1 | 7943.23 | 4057.43 | 7671.57 | 4063.25 | 7956.42 | 4075.02 | 7697.96 | 4094.03 | |
5 | 17,923.15 | 3274.81 | 11,317.07 | 3280.08 | 17,936.34 | 3292.41 | 11,343.45 | 3310.85 | |
10 | 24,241.73 | 2571.64 | 12,792.10 | 2575.34 | 24,254.92 | 2589.23 | 12,818.48 | 2606.12 | |
1000 | 0.01 | 5250.77 | 5251.31 | 5256.77 | 5253.24 | 5265.49 | 5270.52 | 5286.21 | 5287.51 |
1 | 13,213.43 | 6753.31 | 12,743.07 | 6758.97 | 13,228.15 | 6772.94 | 12,772.52 | 6793.33 | |
5 | 29,964.16 | 5455.19 | 18,966.36 | 5460.53 | 29,978.88 | 5474.82 | 18,995.81 | 5494.89 | |
10 | 40,391.38 | 4273.81 | 21,385.72 | 4277.43 | 40,406.10 | 4293.44 | 21,415.17 | 4311.78 |
Sample Size | Dispersion | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
50 | 0.01 | 0.6457 | 0.6501 | 0.6501 | 0.6477 |
1 | 2.0766 | 2.0981 | 2.0808 | 2.1070 | |
5 | 4.3471 | 4.7621 | 4.4190 | 4.7984 | |
10 | 6.0011 | 7.4481 | 6.2916 | 7.5888 | |
200 | 0.01 | 0.3263 | 0.3297 | 0.3263 | 0.3267 |
1 | 1.0397 | 1.0384 | 1.0412 | 1.0402 | |
5 | 2.2255 | 2.2529 | 2.2250 | 2.2651 | |
10 | 3.2408 | 3.0910 | 3.1349 | 3.3494 | |
600 | 0.01 | 0.1836 | 0.1828 | 0.1836 | 0.1839 |
1 | 0.6050 | 0.6020 | 0.6057 | 0.6024 | |
5 | 1.2813 | 1.2810 | 1.2853 | 1.2935 | |
10 | 1.8427 | 1.8309 | 1.8340 | 1.8844 | |
1000 | 0.01 | 0.1430 | 0.1427 | 0.1430 | 0.1429 |
1 | 0.4701 | 0.4677 | 0.4696 | 0.4683 | |
5 | 0.9983 | 0.9959 | 0.9972 | 1.0025 | |
10 | 1.4665 | 1.4614 | 1.4629 | 1.4951 |
Data generated with ZIP regression model
Sample Size | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
50 | 0.1 | 100.0% | 98.4% | 100.0% | 96.8% |
0.3 | 100.0% | 100.0% | 100.0% | 97.9% | |
0.5 | 100.0% | 100.0% | 100.0% | 96.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 95.0% | |
200 | 0.1 | 100.0% | 99.9% | 100.0% | 95.6% |
0.3 | 100.0% | 100.0% | 100.0% | 98.3% | |
0.5 | 100.0% | 100.0% | 100.0% | 96.7% | |
0.7 | 100.0% | 100.0% | 100.0% | 94.4% | |
600 | 0.1 | 100.0% | 100.0% | 100.0% | 95.0% |
0.3 | 100.0% | 100.0% | 100.0% | 97.4% | |
0.5 | 100.0% | 100.0% | 100.0% | 97.9% | |
0.7 | 100.0% | 100.0% | 100.0% | 94.9% | |
1000 | 0.1 | 100.0% | 100.0% | 100.0% | 95.2% |
0.3 | 100.0% | 100.0% | 100.0% | 99.4% | |
0.5 | 100.0% | 100.0% | 100.0% | 96.2% | |
0.7 | 100.0% | 100.0% | 100.0% | 95.0% |
AIC | BIC | ||||||||
---|---|---|---|---|---|---|---|---|---|
Sample Size | Proportion of Zeros | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
50 | 0.1 | 330.55 | 309.147 | 325.14 | 294.95 | 334.992 | 315.629 | 335.50 | 306.43 |
0.3 | 433.015 | 307.129 | 261.57 | 298.20 | 437.754 | 315.322 | 273.83 | 311.98 | |
0.5 | 475.166 | 256.639 | 204.79 | 235.84 | 479.329 | 263.507 | 215.14 | 244.69 | |
0.7 | 431.434 | 179.094 | 144.05 | 156.10 | 434.473 | 186.103 | 154.85 | 166.43 | |
200 | 0.1 | 1316.84 | 1225.51 | 1314.41 | 1109.70 | 1329.63 | 1239.56 | 1338.93 | 1130.06 |
0.3 | 1749.24 | 1215.96 | 970.82 | 1138.99 | 1756.33 | 1229.85 | 990.56 | 1160.73 | |
0.5 | 1936.33 | 1009.57 | 798.26 | 880.09 | 1942.94 | 1020.93 | 815.54 | 902.58 | |
0.7 | 1782.48 | 698.492 | 559.00 | 568.73 | 1793.98 | 714.057 | 580.31 | 590.12 | |
600 | 0.1 | 3960.45 | 3674.16 | 3703.56 | 3219.10 | 3969.41 | 3690.25 | 3719.15 | 3256.71 |
0.3 | 5250.21 | 3641.94 | 2902.96 | 3292.35 | 5267.11 | 3657.86 | 2926.41 | 3308.48 | |
0.5 | 5834.42 | 3016.79 | 2381.02 | 2516.09 | 5845.18 | 3032.01 | 2404.95 | 2553.38 | |
0.7 | 5387.17 | 2084.75 | 1664.94 | 1666.35 | 5410.3 | 2107.83 | 1695.38 | 1698.18 | |
1000 | 0.1 | 6596.51 | 6120.87 | 5475.69 | 5298.91 | 6611.8 | 6140.44 | 5520.85 | 5324.98 |
0.3 | 8756.95 | 6066.9 | 4846.20 | 5902.66 | 8782 | 6084.58 | 4866.44 | 5907.23 | |
0.5 | 9735.86 | 5026.48 | 3966.62 | 3995.33 | 9745.98 | 5047.67 | 3996.49 | 4027.83 | |
0.7 | 8987.2 | 3469.05 | 2768.75 | 2770.02 | 9004.26 | 3487.81 | 2799.32 | 2801.99 |
Sample Size | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
50 | 0.1 | 1.2051 | 1.2175 | 1.2041 | 1.2056 |
0.3 | 3.1300 | 3.1366 | 3.1202 | 3.1296 | |
0.5 | 5.2026 | 5.2005 | 5.1954 | 5.1954 | |
0.7 | 7.2560 | 7.2425 | 7.2506 | 7.2345 | |
200 | 0.1 | 1.0618 | 1.0624 | 1.0616 | 1.0638 |
0.3 | 3.1064 | 3.1057 | 3.1061 | 3.1059 | |
0.5 | 5.1843 | 5.1827 | 5.1841 | 5.1791 | |
0.7 | 7.2264 | 7.2231 | 7.2261 | 7.2265 | |
600 | 0.1 | 1.0350 | 1.0350 | 1.0348 | 1.0347 |
0.3 | 3.1078 | 3.1078 | 3.1078 | 3.1104 | |
0.5 | 5.1784 | 5.1784 | 5.1786 | 5.1814 | |
0.7 | 7.2242 | 7.2238 | 7.2241 | 7.2256 | |
1000 | 0.1 | 1.0299 | 1.0298 | 1.0297 | 1.0301 |
0.3 | 3.0995 | 3.0994 | 3.0995 | 3.1000 | |
0.5 | 5.1630 | 5.1628 | 5.1630 | 5.1633 | |
0.7 | 7.2381 | 7.2378 | 7.2380 | 7.2381 |
Data generated with ZINB regression model
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 100.0% | 98.3% | 100.0% | 97.4% |
0.3 | 100.0% | 100.0% | 100.0% | 98.6% | |
0.5 | 100.0% | 100.0% | 100.0% | 97.3% | |
0.7 | 100.0% | 100.0% | 100.0% | 94.6% | |
1 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 99.9% | |
5 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 99.8% | 99.0% | |
10 | 0.1 | 100.0% | 100.0% | 99.9% | 100.0% |
0.3 | 100.0% | 100.0% | 99.9% | 99.8% | |
0.5 | 100.0% | 100.0% | 99.9% | 99.4% | |
0.7 | 100.0% | 100.0% | 98.9% | 95.0% |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 100.0% | 99.8% | 100.0% | 98.8% |
0.3 | 100.0% | 100.0% | 100.0% | 98.8% | |
0.5 | 100.0% | 100.0% | 100.0% | 98.6% | |
0.7 | 100.0% | 100.0% | 100.0% | 96.4% | |
1 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 99.9% | |
5 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 100.0% | 99.8% | 100.0% | 99.3% |
0.3 | 100.0% | 100.0% | 100.0% | 99.8% | |
0.5 | 100.0% | 100.0% | 100.0% | 98.9% | |
0.7 | 100.0% | 100.0% | 100.0% | 98.2% | |
1 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% | |
5 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 100.0% | 99.8% | 100.0% | 99.8% |
0.3 | 100.0% | 100.0% | 100.0% | 99.9% | |
0.5 | 100.0% | 100.0% | 100.0% | 99.6% | |
0.7 | 100.0% | 100.0% | 100.0% | 99.1% | |
1 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% | |
5 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% | |
10 | 0.1 | 100.0% | 100.0% | 100.0% | 100.0% |
0.3 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.5 | 100.0% | 100.0% | 100.0% | 100.0% | |
0.7 | 100.0% | 100.0% | 100.0% | 100.0% |
AIC | BIC | ||||||||
---|---|---|---|---|---|---|---|---|---|
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
0.01 | 0.1 | 334.99 | 310.68 | 328.98 | 314.61 | 340.73 | 318.27 | 340.46 | 311.05 |
0.3 | 436.35 | 307.02 | 264.63 | 283.63 | 442.09 | 314.67 | 276.10 | 312.97 | |
0.5 | 476.33 | 256.23 | 207.07 | 230.73 | 482.07 | 263.88 | 218.54 | 249.45 | |
0.7 | 432.29 | 178.85 | 145.33 | 161.01 | 438.02 | 186.50 | 156.80 | 168.90 | |
1 | 0.1 | 662.56 | 328.90 | 599.90 | 331.82 | 668.30 | 336.55 | 611.37 | 348.00 |
0.3 | 684.04 | 288.86 | 451.98 | 289.87 | 689.77 | 296.51 | 463.45 | 306.57 | |
0.5 | 648.66 | 231.54 | 332.12 | 231.72 | 654.39 | 239.19 | 343.59 | 246.58 | |
0.7 | 516.19 | 159.30 | 206.90 | 158.71 | 521.93 | 166.95 | 218.37 | 172.42 | |
5 | 0.1 | 1299.27 | 257.12 | 761.95 | 260.82 | 1305.00 | 264.77 | 773.42 | 275.50 |
0.3 | 1152.80 | 212.91 | 573.34 | 216.00 | 1158.54 | 220.56 | 584.81 | 230.27 | |
0.5 | 955.49 | 165.12 | 396.99 | 167.50 | 961.23 | 172.77 | 408.46 | 181.53 | |
0.7 | 651.92 | 109.17 | 207.29 | 110.38 | 657.66 | 116.81 | 218.76 | 124.04 | |
10 | 0.1 | 1642.30 | 199.36 | 822.90 | 202.74 | 1648.04 | 207.01 | 834.37 | 216.88 |
0.3 | 1401.83 | 163.20 | 580.56 | 166.20 | 1407.56 | 170.84 | 592.04 | 180.44 | |
0.5 | 1103.92 | 125.17 | 380.16 | 128.91 | 1109.65 | 132.82 | 391.63 | 145.11 | |
0.7 | 757.17 | 83.21 | 178.00 | 85.78 | 762.91 | 90.84 | 189.47 | 103.55 |
AIC | BIC | ||||||||
---|---|---|---|---|---|---|---|---|---|
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
0.01 | 0.1 | 1333.77 | 1229.97 | 1332.39 | 1227.46 | 1343.66 | 1243.15 | 1352.18 | 1148.42 |
0.3 | 1761.76 | 1215.33 | 984.06 | 1067.55 | 1771.66 | 1228.53 | 1003.85 | 1164.05 | |
0.5 | 1942.31 | 1008.44 | 806.91 | 852.92 | 1952.21 | 1021.64 | 826.70 | 910.07 | |
0.7 | 1786.95 | 697.77 | 564.36 | 645.10 | 1796.84 | 710.96 | 584.15 | 594.94 | |
1 | 0.1 | 2742.24 | 1303.87 | 2488.31 | 1303.70 | 2752.14 | 1317.07 | 2508.10 | 1332.46 |
0.3 | 2833.77 | 1140.65 | 1820.51 | 1133.77 | 2843.66 | 1153.84 | 1840.30 | 1166.58 | |
0.5 | 2705.47 | 914.06 | 1386.86 | 904.33 | 2715.37 | 927.26 | 1406.65 | 933.00 | |
0.7 | 2236.29 | 624.40 | 892.96 | 619.49 | 2246.18 | 637.60 | 912.75 | 636.68 | |
5 | 0.1 | 5706.66 | 1016.90 | 3372.84 | 1020.43 | 5716.55 | 1030.09 | 3392.63 | 1045.04 |
0.3 | 5072.56 | 839.10 | 2614.08 | 841.96 | 5082.46 | 852.29 | 2633.87 | 865.30 | |
0.5 | 4292.57 | 646.81 | 1890.03 | 649.02 | 4302.47 | 660.00 | 1909.82 | 672.33 | |
0.7 | 3147.23 | 429.58 | 1125.24 | 431.35 | 3157.12 | 442.78 | 1145.03 | 453.84 | |
10 | 0.1 | 7469.74 | 794.21 | 3702.48 | 797.88 | 7479.63 | 807.40 | 3722.27 | 820.93 |
0.3 | 6449.26 | 648.03 | 2850.89 | 651.08 | 6459.15 | 661.23 | 2870.68 | 674.50 | |
0.5 | 5226.10 | 490.72 | 1996.52 | 493.30 | 5235.99 | 503.91 | 2016.31 | 516.53 | |
0.7 | 3693.37 | 322.88 | 1158.69 | 324.77 | 3703.26 | 336.07 | 1178.48 | 347.56 |
AIC | BIC | ||||||||
---|---|---|---|---|---|---|---|---|---|
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
0.01 | 0.1 | 4012.44 | 3687.82 | 3762.22 | 3645.61 | 4025.63 | 3705.40 | 3788.60 | 3307.96 |
0.3 | 5291.04 | 3640.69 | 2944.29 | 3009.53 | 5304.23 | 3658.28 | 2970.67 | 3330.99 | |
0.5 | 5858.06 | 3013.56 | 2408.92 | 2441.63 | 5871.25 | 3031.15 | 2435.30 | 2565.70 | |
0.7 | 5404.04 | 2082.93 | 1681.98 | 2019.25 | 5417.23 | 2100.52 | 1708.36 | 1712.98 | |
1 | 0.1 | 8325.06 | 3907.52 | 7023.07 | 3898.82 | 8338.25 | 3925.11 | 7049.45 | 3942.95 |
0.3 | 8638.02 | 3424.71 | 5545.94 | 3388.55 | 8651.21 | 3442.29 | 5572.33 | 3458.18 | |
0.5 | 8249.33 | 2739.12 | 4230.17 | 2700.33 | 8262.52 | 2756.71 | 4256.56 | 2744.53 | |
0.7 | 6855.00 | 1865.19 | 2742.46 | 1851.60 | 6868.19 | 1882.78 | 2768.84 | 1851.36 | |
5 | 0.1 | 17,280.45 | 3039.27 | 10,264.64 | 3042.69 | 17,293.64 | 3056.86 | 10,291.02 | 3074.23 |
0.3 | 15,597.37 | 2517.53 | 8119.98 | 2520.66 | 15,610.56 | 2535.11 | 8146.36 | 2550.90 | |
0.5 | 13,165.60 | 1934.56 | 5896.48 | 1936.43 | 13,178.79 | 1952.15 | 5922.86 | 1966.15 | |
0.7 | 9776.28 | 1278.72 | 3616.01 | 1280.44 | 9789.47 | 1296.31 | 3642.39 | 1309.09 | |
10 | 0.1 | 22,968.65 | 2366.89 | 11,539.50 | 2370.34 | 22,981.84 | 2384.48 | 11,565.88 | 2401.25 |
0.3 | 19,981.48 | 1934.71 | 9021.47 | 1937.67 | 19,994.67 | 1952.29 | 9047.85 | 1968.43 | |
0.5 | 16,344.06 | 1466.35 | 6497.22 | 1473.33 | 16,357.25 | 1483.94 | 6523.60 | 1499.15 | |
0.7 | 11,566.87 | 955.61 | 3866.63 | 962.28 | 11,580.06 | 973.20 | 3893.01 | 987.91 |
AIC | BIC | ||||||||
---|---|---|---|---|---|---|---|---|---|
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB | Poisson | NB | ZIP | ZINB |
0.01 | 0.1 | 6684.63 | 6143.38 | 5565.15 | 6052.92 | 6699.35 | 6163.02 | 5594.60 | 5420.31 |
0.3 | 8824.90 | 6064.34 | 4915.65 | 4924.89 | 8839.63 | 6083.97 | 4945.10 | 5931.34 | |
0.5 | 9777.68 | 5020.98 | 4015.15 | 4043.07 | 9792.40 | 5040.61 | 4044.60 | 4073.87 | |
0.7 | 9023.63 | 3466.21 | 2799.39 | 3385.59 | 9038.36 | 3485.85 | 2828.84 | 2833.13 | |
1 | 0.1 | 13,858.97 | 6504.03 | 11,178.92 | 6486.14 | 13,873.70 | 6523.66 | 11,208.37 | 6544.02 |
0.3 | 14,382.64 | 5698.27 | 9230.42 | 5630.25 | 14,397.36 | 5717.90 | 9259.87 | 5724.47 | |
0.5 | 13,733.09 | 4553.36 | 7032.62 | 4490.56 | 13,747.81 | 4572.99 | 7062.07 | 4518.57 | |
0.7 | 11,386.64 | 3105.34 | 4560.08 | 3089.41 | 11,401.36 | 3124.97 | 4589.52 | 3062.81 | |
5 | 0.1 | 28,903.48 | 5055.91 | 17,208.02 | 5059.30 | 28,918.20 | 5075.55 | 17,237.47 | 5094.76 |
0.3 | 26,089.74 | 4186.58 | 13,612.61 | 4189.29 | 26,104.46 | 4206.21 | 13,642.06 | 4223.62 | |
0.5 | 21,943.54 | 3211.80 | 9862.55 | 3213.03 | 21,958.27 | 3231.43 | 9891.99 | 3244.95 | |
0.7 | 16,243.74 | 2112.63 | 6053.74 | 2114.34 | 16,258.46 | 2132.26 | 6083.19 | 2145.26 | |
10 | 0.1 | 38,248.13 | 3928.96 | 19,291.56 | 3932.52 | 38,262.85 | 3948.59 | 19,321.01 | 3966.42 |
0.3 | 33,318.61 | 3211.97 | 15,134.62 | 3215.36 | 33,333.33 | 3231.60 | 15,164.07 | 3248.79 | |
0.5 | 26,982.05 | 2431.72 | 10,754.37 | 2434.39 | 26,996.77 | 2451.35 | 10,783.82 | 2467.68 | |
0.7 | 19,259.10 | 1582.60 | 6502.62 | 1592.23 | 19,273.82 | 1602.23 | 6532.07 | 1618.24 |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 1.2283 | 1.2316 | 1.2274 | 1.2307 |
0.3 | 3.1310 | 3.1385 | 3.1209 | 3.1402 | |
0.5 | 5.2114 | 5.2070 | 5.2052 | 5.2102 | |
0.7 | 7.2429 | 7.2286 | 7.2366 | 7.2279 | |
1 | 0.1 | 2.2478 | 2.2833 | 2.2567 | 2.2882 |
0.3 | 3.4686 | 3.5042 | 3.4598 | 3.5167 | |
0.5 | 5.2457 | 5.2841 | 5.2352 | 5.3016 | |
0.7 | 7.2643 | 7.3470 | 7.2495 | 7.3034 | |
5 | 0.1 | 4.4559 | 4.8665 | 4.5330 | 4.9490 |
0.3 | 5.0453 | 5.8211 | 5.1587 | 5.9835 | |
0.5 | 6.2162 | 7.2657 | 6.3823 | 7.1108 | |
0.7 | 7.6836 | 19.0239 | 20.2636 | 22.7636 | |
10 | 0.1 | 6.0659 | 8.2687 | 6.6112 | 9.1112 |
0.3 | 6.4278 | 11.9930 | 7.0274 | 9.5274 | |
0.5 | 7.2752 | 14.5155 | 7.0274 | 9.5274 | |
0.7 | 8.3214 | 15.8751 | 14.0274 | 16.5274 |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 1.0586 | 1.0591 | 1.0585 | 1.0565 |
0.3 | 3.1008 | 3.1002 | 3.1007 | 3.1018 | |
0.5 | 5.1761 | 5.1744 | 5.1759 | 5.1707 | |
0.7 | 7.2169 | 7.2130 | 7.2165 | 7.2146 | |
1 | 0.1 | 1.3805 | 1.3837 | 1.3802 | 1.3876 |
0.3 | 3.1443 | 3.1463 | 3.1426 | 3.1504 | |
0.5 | 5.1854 | 5.1823 | 5.1841 | 5.1840 | |
0.7 | 7.2180 | 7.2091 | 7.2166 | 7.2132 | |
5 | 0.1 | 2.3519 | 2.3926 | 2.3592 | 2.4141 |
0.3 | 3.4632 | 3.5050 | 3.4640 | 3.5303 | |
0.5 | 5.2182 | 5.2268 | 5.2143 | 5.2361 | |
0.7 | 7.2641 | 7.2626 | 7.2581 | 7.2598 | |
10 | 0.1 | 3.1613 | 3.2489 | 3.1768 | 3.3383 |
0.3 | 3.9775 | 4.0793 | 3.9982 | 4.1287 | |
0.5 | 5.4267 | 5.5666 | 5.4403 | 5.5871 | |
0.7 | 7.3654 | 7.5561 | 7.3998 | 7.7825 |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 1.0242 | 1.0242 | 1.0239 | 1.0243 |
0.3 | 3.0994 | 3.0993 | 3.0994 | 3.1010 | |
0.5 | 5.1722 | 5.1722 | 5.1723 | 5.1716 | |
0.7 | 7.2237 | 7.2233 | 7.2236 | 7.2253 | |
1 | 0.1 | 1.1257 | 1.1240 | 1.1248 | 1.1245 |
0.3 | 3.1168 | 3.1164 | 3.1165 | 3.1168 | |
0.5 | 5.1761 | 5.1760 | 5.1761 | 5.1772 | |
0.7 | 7.2476 | 7.2467 | 7.2474 | 7.2470 | |
5 | 0.1 | 1.5891 | 1.5964 | 1.5911 | 1.6048 |
0.3 | 3.1700 | 3.1734 | 3.1694 | 3.1776 | |
0.5 | 5.2008 | 5.1966 | 5.2002 | 5.1975 | |
0.7 | 7.2388 | 7.2288 | 7.2372 | 7.2304 | |
10 | 0.1 | 1.9827 | 2.0020 | 1.9896 | 2.0355 |
0.3 | 3.2512 | 3.2606 | 3.2539 | 3.2740 | |
0.5 | 5.1940 | 5.1913 | 5.1934 | 5.1893 | |
0.7 | 7.2050 | 7.1947 | 7.2016 | 7.1881 |
Dispersion | Proportion of Zeros | Poisson | NB | ZIP | ZINB |
---|---|---|---|---|---|
0.01 | 0.1 | 1.0317 | 1.0316 | 1.0316 | 1.0317 |
0.3 | 3.1048 | 3.1048 | 3.1049 | 3.1056 | |
0.5 | 5.1641 | 5.1639 | 5.1641 | 5.1643 | |
0.7 | 7.2360 | 7.2357 | 7.2359 | 7.2351 | |
1 | 0.1 | 1.0590 | 1.0588 | 1.0583 | 1.0590 |
0.3 | 3.0980 | 3.0979 | 3.0980 | 3.0990 | |
0.5 | 5.1694 | 5.1688 | 5.1692 | 5.1694 | |
0.7 | 7.2343 | 7.2338 | 7.2343 | 7.2340 | |
5 | 0.1 | 1.2959 | 1.2958 | 1.2943 | 1.3004 |
0.3 | 3.1097 | 3.1079 | 3.1089 | 3.1095 | |
0.5 | 5.1660 | 5.1646 | 5.1658 | 5.1644 | |
0.7 | 7.2351 | 7.2316 | 7.2341 | 7.2313 | |
10 | 0.1 | 1.6692 | 1.6708 | 1.6680 | 1.6957 |
0.3 | 3.1550 | 3.1544 | 3.1542 | 3.1567 | |
0.5 | 5.1485 | 5.1454 | 5.1479 | 5.1441 | |
0.7 | 7.2436 | 7.2350 | 7.2432 | 7.2339 |
Empirical study
Description of the study population
Characteristic | |
---|---|
Age (years), mean (SD) | 62.3 (40.66) |
Sex, n (%) | |
Male | 864 (39.87) |
Female | 1303 (60.13) |
Hospital admission type, n (%) | |
Elective | 378 (17.44) |
Emergency | 1748 (80.66) |
Urgent | 41 (1.89) |
Health insurance type, n (%) | |
Government | 96 (4.43) |
Medicaid | 304 (14.03) |
Medicare | 961 (44.35) |
Private | 789 (36.41) |
Self-Pay | 17 (0.78) |
LOS, mean (SD) | 8.0 (6.56) |
Comparison of fitted poisson, NB, ZIP, and ZINB regression models
Parameter | Poisson | NB | ZIP | ZINB | ||||
---|---|---|---|---|---|---|---|---|
Estimate (SE) | P-Value | Estimate (SE) | P-Value | Estimate (SE) | P-Value | Estimate (SE) | P-Value | |
Age | -0.001 (0.0002) | < .0001 | -0.001 (0.0004) | 0.0546 | -0.001 (0.0002) | 0.0002 | -0.001 (0.0004) | 0.0561 |
Sex | ||||||||
Female | 0.03 (0.02) | 0.1086 | 0.03 (0.03) | 0.4651 | 0.02 (0.02) | 0.1247 | 0.03 (0.03) | 0.4414 |
Male | reference | reference | reference | reference | ||||
Health insurance | ||||||||
Government | 0.32 (0.11) | 0.0045 | 0.32 (0.21) | 0.1228 | 0.26 (0.11) | 0.0215 | 0.26 (0.21) | 0.2129 |
Medicaid | 0.49 (0.11) | < .0001 | 0.49 (0.20) | 0.0131 | 0.44 (0.11) | < .0001 | 0.43 (0.20) | 0.0319 |
Medicare | 0.46 (0.11) | < .0001 | 0.46 (0.19) | 0.0189 | 0.41 (0.11) | 0.0001 | 0.40 (0.20) | 0.0445 |
Private | 0.41 (0.11) | 0.0001 | 0.40 (0.19) | 0.0372 | 0.35 (0.11) | 0.0009 | 0.35 (0.20) | 0.0796 |
Self-pay | reference | reference | reference | reference | ||||
Admission type | ||||||||
Elective | -0.30 (0.05) | < .0001 | -0.29 (0.12) | 0.0159 | -0.29 (0.05) | < .0001 | -0.29 (0.12) | 0.0160 |
Emergency | -0.17 (0.05) | 0.0011 | -0.16 (0.12) | 0.1649 | -0.16 (0.05) | 0.0025 | -0.16 (0.12) | 0.1667 |
Urgent | reference | reference | reference | reference | ||||
Zero Model | Estimate (SE) | P-Value | Estimate (SE) | P-Value | Estimate (SE) | P-Value | Estimate (SE) | P-Value |
Age | 0.003 (0.004) | 0.5015 | 0.07 (0.13) | 0.5722 | ||||
Sex | ||||||||
Female | -0.09 (0.43) | 0.8437 | 18.10 (4082.56) | 0.9965 | ||||
Male | reference | reference | ||||||
Health insurance | ||||||||
Government | -14.35 (528.33) | 0.9787 | -22.23 (16,904.72) | 0.9990 | ||||
Medicaid | -1.55 (1.23) | 0.2057 | -34.36 (4533.48) | 0.9940 | ||||
Medicare | -1.56 (1.17) | 0.1800 | -37.75 (4039.53) | 0.9925 | ||||
Private | -2.22 (1.19) | 0.0662 | -22.36 (5537.88) | 0.9968 | ||||
Self-pay | reference | reference | ||||||
Admission type | ||||||||
Elective | 12.99 (810.38) | 0.9871 | 1.81 (20,295.23) | 0.9999 | ||||
Emergency | 12.72 (810.38) | 0.9873 | 17.23 (19,158.48) | 0.9993 | ||||
Urgent | reference | reference | ||||||
Pearson Chi-Square (value/degrees of freedom) | 5.3016 | 1.1815 | 4.9001 | 1.1868 | ||||
AIC | 17,675.4989 | 13,134.7955 | 17,560.4274 | 13,150.2305 | ||||
BIC | 17,726.6288 | 13,191.6065 | 17,662.6872 | 13,258.1713 |