. Author manuscript; available in PMC: 2024 Oct 9.

Published in final edited form as: Demography. 2024 Jun 1;61(3):643–664. doi: 10.1215/00703370-11330227

Table 2.

Bias, precision, and the root mean square error (RMSE) of predicting the average age of infant deaths, ${{}_{1}a}_{0}$ , and fitting different methods to the Under-5 Mortality Database

	Female			Male			Both Sexes Combined
	$μ$ (bias)	$φ$ (precision)	RMSE	$μ$ (bias)	$φ$ (precision)	RMSE	$μ$ (bias)	$φ$ (precision)	RMSE
A. Log-Quadratic Model, ${{}_{1}m}_{0}$ and $z (3 m)$	−0.0019 [−0.0046, 0.0009]	432.72 [399.18, 468.14]	0.0481 [0.0463, 0.0501]	−0.0023 [−0.0048, 0.0004]	486.64 [448.92, 526.48]	0.0454 [0.0437, 0.0473]	−0.0016 [−0.0037, 0.0005]	743.57 [685.95, 804.45]	0.0367 [0.0353, 0.0382]
B. Log-Quadratic Model, ${{}_{1}m}_{0}$ and $z (28 d)$	0.0006 [−0.0037, 0.0050]	163.43 [150.60, 176.79]	0.0783 [0.0752, 0.0815]	−0.0011 [−0.0054, 0.0032]	169.74 [156.42, 183.63]	0.0768 [0.0738, 0.0800]	−0.0010 [−0.0048, 0.0028]	215.24 [198.35, 232.84]	0.0682 [0.0656, 0.0710]
C. Log-Quadratic Model, ${{}_{1}q}_{0}$ and ${{}_{5}q}_{0}$	−0.0111 [−0.0248, 0.0025]	17.00 [15.68, 18.38]	0.2429 [0.2336, 0.2529]	−0.0160 [−0.0311, −0.0012]	14.16 [13.06, 15.31]	0.2664 [0.2562, 0.2774]	−0.0119 [−0.0250, 0.0010]	18.74 [17.29, 20.27]	0.2314 [0.2225, 0.2409]
D. Log-Quadratic Model, ${{}_{1}m}_{0}$ and ${{}_{4}m}_{1}$	−0.0111 [−0.0246, 0.0028]	17.07 [15.74, 18.44]	0.2424 [0.2331, 0.2523]	−0.0161 [−0.0309, −0.0008]	14.22 [13.12, 15.37]	0.2657 [0.2556, 0.2766]	−0.0119 [−0.0248, 0.0013]	18.84 [17.38, 20.36]	0.2308 [0.2220, 0.2402]
E. Alexander and Root (2022), ${{}_{1}q}_{0}$ and, ${{}_{1}q}_{0} / {{}_{5}q}_{0}$	−0.0570 [−0.0683, −0.0460]	26.21 [24.18, 28.35]	0.2036 [0.1957, 0.2118]	−0.0492 [−0.0612, −0.0375]	23.03 [21.25, 24.92]	0.2141 [0.2060, 0.2230]	−0.0507 [−0.0617, −0.0400]	27.63 [25.49, 29.89]	0.1969 [0.1894, 0.2050]
F. Linear Regression, $z (3 m)$	0.0007 [−0.0021, 0.0035]	399.25 [368.19, 431.93]	0.0501 [0.0481, 0.0521]	0.0005 [−0.0022, 0.0032]	428.12 [394.82, 463.16]	0.0484 [0.0465, 0.0503]	0.0004 [−0.0019, 0.0026]	622.90 [574.45, 673.89]	0.0401 [0.0385, 0.0417]
G. Linear Regression, $z (28 d)$	0.0020 [−0.0028, 0.0067]	141.63 [130.57, 153.14]	0.0841 [0.0809, 0.0876]	0.0015 [−0.0032, 0.0061]	148.98 [137.34, 161.09]	0.0820 [0.0788, 0.0854]	0.0011 [−0.0032, 0.0052]	178.78 [164.81, 193.31]	0.0748 [0.0720, 0.0780]
H. Log-Quadratic Model, ${{}_{1}q}_{0}$ and $k = 0$	0.0278 [0.0161, 0.0395]	22.78 [21.05, 24.64]	0.2115 [0.2034, 0.2200]	0.0307 [0.0185, 0.0428]	21.19 [19.59, 22.92]	0.2195 [0.2111, 0.2283]	0.0271 [0.0157, 0.0385]	23.95 [22.14, 25.91]	0.2062 [0.1983, 0.2145]
I. Log-Quadratic Model, ${{}_{1}m}_{0}$ and $k = 0$	0.0278 [0.0161, 0.0394]	22.74 [20.96, 24.56]	0.2116 [0.2036, 0.2203]	0.0306 [0.0186, 0.0427]	21.15 [19.50, 22.85]	0.2197 [0.2113, 0.2287]	0.0271 [0.0157, 0.0384]	23.91 [22.04, 25.83]	0.2064 [0.1986, 0.2149]
J. Log-Quadratic Model, ${{}_{5}m}_{0}$ and $k = 0$	0.0274 [0.0158, 0.0393]	22.67 [20.91, 24.50]	0.2119 [0.2040, 0.2207]	0.0302 [0.0182, 0.0426]	21.09 [19.44, 22.78]	0.2200 [0.2118, 0.2291]	0.0267 [0.0154, 0.0383]	23.82 [21.97, 25.74]	0.2067 [0.1991, 0.2153]
K. Andreev and Kingkade (2015), ${{}_{1}m}_{0}$	0.0032 [−0.0085, 0.0150]	22.60 [20.87, 24.39]	0.2105 [0.2026, 0.2190]	0.0155 [0.0035, 0.0275]	21.84 [20.18, 23.58]	0.2146 [0.2065, 0.2233]	0.0125 [0.0012, 0.0239]	24.33 [22.48, 26.26]	0.2032 [0.1956, 0.2114]
L. Preston et al. (2001), ${{}_{1}m}_{0}$	−0.4828 [−0.5012, −0.4638]	9.30 [8.59, 10.07]	0.5837 [0.5664, 0.6009]	−0.4526 [−0.4757, −0.4287]	5.86 [5.41, 6.35]	0.6128 [0.5919, 0.6340]	−0.4646 [−0.4852, −0.4433]	7.39 [6.83, 8.00]	0.5927 [0.5736, 0.6117]
M. Keyfitz (1970), ${{}_{1}m}_{0}$	−0.4368 [−0.4499, −0.4234]	18.03 [16.62, 19.45]	0.4963 [0.4840, 0.5090]	−0.3283 [−0.3439, −0.3126]	12.94 [11.93, 13.96]	0.4303 [0.4170, 0.4445]	−0.3798 [−0.3938, −0.3657]	15.95 [14.71, 17.21]	0.4550 [0.4424, 0.4683]
N. Coale and Demeny (1966), ${{}_{1}q}_{0}$	−0.5000 [−0.5188, −0.4811]	8.61 [7.95, 9.33]	0.6052 [0.5880, 0.6230]	−0.4634 [−0.4869, −0.4399]	5.53 [5.11, 5.99]	0.6290 [0.6082, 0.6506]	−0.4784 [−0.4994, −0.4573]	6.90 [6.36, 7.47]	0.6115 [0.5927, 0.6313]
O. Coale and Demeny (1966)—East, ${{}_{1}q}_{0}$	−1.2131 [−1.2475, −1.1791]	2.68 [2.47, 2.89]	1.3586 [1.3267, 1.3910]	−1.2474 [−1.2928, −1.2026]	1.54 [1.42, 1.66]	1.4855 [1.4443, 1.5271]	−1.2220 [−1.2617, −1.1828]	2.01 [1.85, 2.17]	1.4110 [1.3746, 1.4478]

Notes: Values are the median of the posterior probability distribution and the 2.5th and 97.5th percentiles. Andreev and Kingkade (2015), Preston et al. (2001), and Coale and Demeny (1966) did not report equations for both sexes combined. For these entries, average coefficients were calculated for comparability purposes, assuming a sex ratio at birth of 1.05 males per female. Keyfitz (1970) proposed only one general formula for both sexes combined. Resulting values of these approaches are shown in italics.