Figures and data in Evaluating distributional regression strategies for modelling self-reported sexual age-mixing

Figures
Tables
Additional files

18 figures, 15 tables and 1 additional file

Figures

Figure 1

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The sinh-arcsinh density with $μ = 0$ , $σ = 1$ , and a variety of assumptions about $ϵ$ and δ.

Figure 2

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Observed partner age distributions among women aged 34 years in all three data sets.

Figure 3

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Observed means, variances, skewnesses, and kurtoses of partner age by 5-year age bin and sex in all three data sets.

Figure 4

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Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among women aged 35–39 in the AHRI data set.

Posterior predictive distributions come from fitting each age bin/sex combination independently.

Figure 5

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Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for conventional regression and the most complex distributional model among men aged 16, 24, and 37 years in the AHRI data set.

Posterior predictive distributions come from regression models fit to the entire AHRI data set.

Figure 6

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Estimated sinh-arcsinh distributional parameters from the conventional regression model, and distributional models 1 and 4 fit to the AHRI data.

‘Conventional’ assumes no variation across age and sex, ‘Distributional 1’ allows for independent age and sex effects, and ‘Distributional 4’ includes sex-specific splines with respect to age.

Figure 7

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Estimated sinh-arcsinh distributional parameters for Distributional Model 4 fit to the three main data sets.

Appendix 1—figure 1

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Illustration of the effect of the deheaping algorithm on women aged exactly 24 years in the AHRI data.

Dark grey bars correspond to ages identified as potentially heaped (multiples of five away from 24). The red line is the expected count of observations estimated by excluding any potentially heaped ages.

Appendix 1—figure 2

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Observed sexual partner age distributions among women in the AHRI data.

The left panel is original data, and the right panel is the same data set after deheaping age differences from multiples of five.

Appendix 1—figure 3

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Overlaid quantile-quantile (QQ) plots for each probability distribution’s best fit to data in all three main data sets.

Presented quantiles range from 10th to 90th in increments of 10. Lines closer to the line of equality indicate better fit to empirical quantiles.

Appendix 1—figure 4

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Appendix 1—figure 5

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Appendix 1—figure 6

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Appendix 1—figure 7

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Appendix 1—figure 8

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Appendix 1—figure 9

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Appendix 1—figure 10

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Appendix 1—figure 11

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Tables

Table 1

Details of the five distributions tested in this analysis.

We define $x_{z} = (x - μ) / σ$ , $p (x)$ to be the standard normal PDF, $Φ (x)$ to be the standard normal cumulative density function, $S_{ϵ, δ} (x) = \sinh (ϵ + δ asinh (x))$ , and $C_{ϵ, δ} (x) = \cosh (ϵ + δ asinh (x))$ .

Distribution	Parameters	Domain	PDF
Normal	$\begin{matrix} μ (location) \\ σ > 0 (scale) \end{matrix}$	$ℝ$	$\frac{1}{σ \sqrt{2 π}} \exp [\frac{- x_{z}}{2}]$
Skew normal	$\begin{matrix} μ (location) \\ σ > 0 (scale) \\ ϵ (skewness) \end{matrix}$	$ℝ$	$\frac{2}{σ} p (x_{z}) Φ (ϵ x_{z})$
Gamma	$\begin{matrix} k > 0 (shape) \\ θ > 0 (scale) \end{matrix}$	$ℝ^{+}$	$\frac{1}{Γ (k) θ^{k}} x^{k - 1} \exp [\frac{- x}{θ}]$
Beta	$\begin{matrix} α > 0 (left) \\ β > 0 (right) \end{matrix}$	$ℝ^{(0, 1)}$	$\frac{x^{α - 1} {(1 - x)}^{β - 1}}{B (α, β)}$
Sinh-arcinh	$\begin{matrix} μ (location) \\ σ > 0 (scale) \\ ϵ (skewness) \\ δ > 0 (tail weight) \end{matrix}$	$ℝ$	$\frac{1}{σ \sqrt{2 π}} \frac{δ C_{ϵ, δ} (x_{z})}{\sqrt{1 + x_{z}^{2}}} \exp [- \frac{S_{ϵ, δ} {(x_{z})}^{2}}{2}]$

Table 2

Summary of five models fit in this analysis.

Model	Distributional?	Location	Other parameters
Conventional	No	Age-sex interaction	Constant
Distributional 1	Yes	Age-sex interaction	Age and sex effects
Distributional 2	Yes	Age-sex interaction	Age-sex interaction
Distributional 3	Yes	Sex-specific splines	Age-sex interaction
Distributional 4	Yes	Sex-specific splines	Sex-specific splines

Table 3

Share of subsets in which each dependent variable yields the highest ELPD given each probability distribution (excluding deheaped AHRI data).

Variable	Normal	Skew normal	Sinh-arcsinh
Age difference	22.2%	25.0%	16.7%
Linear age	8.3%	5.6%	16.7%
Log-age	19.4%	41.7%	30.6%
Log-ratio	50.0%	27.8%	36.1%

Table 4

Model comparison metrics averaged across all data subsets for all three data sets.

Higher ELPD values indicate better fit. Lower QQ RMSE values indicate more accurate prediction of empirical quantiles. Bolded rows are best across all three data sets.

Distribution	AHRI	Haiti 2016–17 DHS	Manicaland
ELPD
Gamma	−14847.2	−2917.9	−13152.8
Beta	−14748.0	−2896.5	−13003.5
Normal	−14593.7	−2868.4	−12856.8
Skew normal	−14505.1	−2854.0	−12778.5
Sinh-arcsinh	−14312.5	−2839.5	−12625.8
QQ RMSE
Gamma	0.83	0.82	0.95
Beta	0.99	0.82	1.11
Normal	0.82	0.68	0.97
Skew normal	0.77	0.65	0.85
Sinh-arcsinh	0.36	0.37	0.44

Table 5

ELPD and QQ RMSE values for all five distributional regression models fit to each data set.

The models increase in complexity from Conventional Regression to Distributional Model 4. Bolded ELPD values are more than two standard errors higher than the next best value in the column. Bolded QQ RMSE values are lowest in their column.

Model	AHRI	Haiti 2016–17 DHS	Manicaland
ELPD
Conventional	52689.2	4777.8	21011.3
Distributional 1	54335.2	5140.8	23192.5
Distributional 2	54794.8	5138.7	23472.1
Distributional 3	55534.2	5196.7	24313.7
Distributional 4	55841.9	5207.6	24516.1
QQ RMSE
Conventional	1.30	1.33	2.05
Distributional 1	1.15	0.98	1.89
Distributional 2	1.21	0.99	1.80
Distributional 3	0.93	0.91	1.34
Distributional 4	0.66	0.84	1.04

Appendix 1—table 1

ELPD and QQ RMSE values for all five models fit to deheaped AHRI data The models increase in complexity from Conventional Regression to Distributional Model 4.

Bolded ELPD values are more than two standard errors higher than the next best value in the column. Bolded QQ RMSE values are lowest in their column.

Model	AHRI deheaped
ELPD
Conventional	55296.2
Distributional 1	57097.4
Distributional 2	57503.7
Distributional 3	58219.2
Distributional 4	58504.0
QQ RMSE
Conventional	1.26
Distributional 1	1.06
Distributional 2	1.14
Distributional 3	0.92
Distributional 4	0.62

Appendix 1—table 2

Full ELPD and QQ RMSE table for women in the AHRI data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
AHRI Female 20-24
1	Sinh-arcsinh	−31750.94	0.00	0.00	0.32
2	Skew normal	−32056.39	−305.46	48.63	0.47
3	Normal	−32414.54	−663.61	60.54	0.62
4	Beta	−32953.92	−1202.98	112.08	0.77
5	Gamma	−33461.85	−1710.92	148.15	0.80
AHRI Female 25-29
1	Sinh-arcsinh	−24647.65	0.00	0.00	0.28
2	Skew normal	−24906.22	−258.57	43.27	0.52
3	Normal	−25238.71	−591.06	54.82	0.68
4	Beta	−25701.13	−1053.48	114.84	0.89
5	Gamma	−25995.81	−1348.16	132.15	0.90
AHRI Female 30-34
1	Sinh-arcsinh	−19831.53	0.00	0.00	0.44
2	Skew normal	−20200.44	−368.91	69.40	0.51
3	Normal	−20314.79	−483.26	52.24	0.80
4	Beta	−20575.61	−744.08	67.46	0.93
5	Gamma	−20708.35	−876.82	73.89	0.91
AHRI Female 35-39
1	Sinh-arcsinh	−15469.18	0.00	0.00	0.31
2	Skew normal	−15749.79	−280.61	53.04	0.77
3	Normal	−15834.32	−365.14	41.23	0.80
4	Beta	−16026.51	−557.33	53.99	1.18
5	Gamma	−16087.40	−618.22	57.06	1.04
AHRI Female 40-44
1	Sinh-arcsinh	−12556.61	0.00	0.00	0.45
2	Skew normal	−12876.71	−320.10	45.85	1.27
3	Normal	−12935.34	−378.73	52.38	0.92
4	Beta	−13137.69	−581.08	69.18	1.38
5	Gamma	−13150.66	−594.05	62.73	1.19
AHRI Female 45-49
1	Sinh-arcsinh	−10059.21	0.00	0.00	0.59
2	Skew normal	−10391.95	−332.74	42.75	1.36
3	Normal	−10433.64	−374.43	48.91	1.53
4	Gamma	−10527.00	−467.79	50.72	1.35
5	Beta	−10545.33	−486.12	56.02	1.58

Appendix 1—table 3

Full ELPD and QQ RMSE table for men in the AHRI data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
AHRI Male 20-24
1	Sinh-arcsinh	−20428.11	0.00	0.00	0.23
2	Skew normal	−20499.86	−71.75	17.12	0.25
3	Normal	−20503.89	−75.79	16.85	0.22
4	Beta	−20545.59	−117.49	23.21	0.22
5	Gamma	−20700.24	−272.13	43.53	0.29
AHRI Male 25-29
1	Sinh-arcsinh	−12664.21	0.00	0.00	0.26
2	Skew normal	−12727.03	−62.82	17.86	0.28
3	Beta	−12739.03	−74.82	18.65	0.31
4	Normal	−12753.25	−89.04	19.35	0.29
5	Gamma	−12788.26	−124.05	35.07	0.38
AHRI Male 30-34
1	Sinh-arcsinh	−9301.03	0.00	0.00	0.29
2	Skew normal	−9357.18	−56.15	14.08	0.43
3	Beta	−9371.86	−70.83	16.48	0.37
4	Normal	−9385.63	−84.60	14.67	0.46
5	Gamma	−9419.34	−118.31	35.11	0.27
AHRI Male 35-39
1	Sinh-arcsinh	−6746.89	0.00	0.00	0.30
2	Skew normal	−6812.77	−65.88	17.73	0.64
3	Normal	−6817.86	−70.97	23.24	0.70
4	Beta	−6830.95	−84.06	17.95	0.71
5	Gamma	−6832.47	−85.58	32.03	0.44
AHRI Male 40-44
1	Sinh-arcsinh	−4610.95	0.00	0.00	0.35
2	Skew normal	−4711.78	−100.84	18.66	0.92
3	Normal	−4713.78	−102.83	18.82	0.78
4	Gamma	−4718.28	−107.33	24.83	0.63
5	Beta	−4742.70	−131.75	17.28	1.07
AHRI Male 45-49
1	Sinh-arcsinh	−3683.47	0.00	0.00	0.34
2	Skew normal	−3770.59	−87.12	16.56	0.81
3	Gamma	−3776.33	−92.86	15.56	0.87
4	Normal	−3778.84	−95.37	14.50	1.17
5	Beta	−3805.78	−122.31	17.40	1.36

Appendix 1—table 4

Full ELPD and QQ RMSE table for women in the AHRI Deheaped data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
AHRI Deheaped Female 20-24
1	Sinh-arcsinh	−31411.24	0.00	0.00	0.26
2	Skew normal	−31797.37	−386.13	53.15	0.59
3	Normal	−32179.29	−768.05	65.50	0.56
4	Beta	−32737.57	−1326.32	118.47	0.76
5	Gamma	−33254.17	−1842.92	155.14	0.78
AHRI Deheaped Female 25-29
1	Sinh-arcsinh	−24439.47	0.00	0.00	0.27
2	Skew normal	−24768.06	−328.59	46.71	0.65
3	Normal	−25104.46	−664.99	58.32	0.82
4	Beta	−25574.33	−1134.86	119.65	1.03
5	Gamma	−25870.30	−1430.83	137.51	1.05
AHRI Deheaped Female 30-34
1	Sinh-arcsinh	−19680.77	0.00	0.00	0.41
2	Skew normal	−20112.70	−431.94	72.95	0.55
3	Normal	−20228.52	−547.76	56.19	0.81
4	Beta	−20492.23	−811.46	70.53	0.92
5	Gamma	−20624.82	−944.06	76.98	0.80
AHRI Deheaped Female 35-39
1	Sinh-arcsinh	−15381.68	0.00	0.00	0.26
2	Skew normal	−15703.77	−322.09	55.30	0.68
3	Normal	−15788.73	−407.05	43.67	0.82
4	Beta	−15983.57	−601.90	56.31	1.13
5	Gamma	−16044.22	−662.54	59.17	1.04
AHRI Deheaped Female 40-44
1	Sinh-arcsinh	−12491.91	0.00	0.00	0.25
2	Skew normal	−12846.63	−354.72	47.38	0.99
3	Normal	−12905.04	−413.12	54.14	0.89
4	Beta	−13109.82	−617.91	70.96	1.31
5	Gamma	−13121.45	−629.53	64.12	1.13
AHRI Deheaped Female 45-49
1	Sinh-arcsinh	−9981.83	0.00	0.00	0.53
2	Skew normal	−10357.85	−376.01	45.08	1.43
3	Normal	−10401.64	−419.80	51.57	1.46
4	Gamma	−10493.73	−511.90	52.90	1.37
5	Beta	−10513.46	−531.63	58.21	1.61

Appendix 1—table 5

Full ELPD and QQ RMSE table for men in the AHRI Deheaped data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
AHRI Deheaped Male 20-24
1	Sinh-arcsinh	−20310.35	0.00	0.00	0.27
2	Skew normal	−20429.90	−119.55	27.09	0.22
3	Normal	−20459.73	−149.38	35.78	0.29
4	Beta	−20574.15	−263.80	75.13	0.22
5	Gamma	−20899.52	−589.17	175.99	0.27
AHRI Deheaped Male 25-29
1	Sinh-arcsinh	−12585.54	0.00	0.00	0.28
2	Skew normal	−12680.59	−95.05	21.53	0.44
3	Beta	−12697.00	−111.46	23.31	0.37
4	Normal	−12701.76	−116.23	22.96	0.41
5	Gamma	−12763.81	−178.27	41.24	0.39
AHRI Deheaped Male 30-34
1	Sinh-arcsinh	−9227.26	0.00	0.00	0.37
2	Skew normal	−9302.42	−75.16	16.15	0.41
3	Beta	−9318.24	−90.97	19.07	0.39
4	Normal	−9327.58	−100.31	16.18	0.41
5	Gamma	−9372.32	−145.06	38.27	0.27
AHRI Deheaped Male 35-39
1	Sinh-arcsinh	−6694.86	0.00	0.00	0.30
2	Skew normal	−6774.11	−79.26	19.32	0.61
3	Normal	−6780.69	−85.84	25.42	0.44
4	Beta	−6791.95	−97.10	19.81	0.69
5	Gamma	−6796.41	−101.55	34.45	0.40
AHRI Deheaped Male 40-44
1	Sinh-arcsinh	−4591.04	0.00	0.00	0.49
2	Skew normal	−4700.54	−109.51	19.38	1.16
3	Normal	−4703.52	−112.49	19.93	1.00
4	Gamma	−4708.43	−117.40	25.94	0.89
5	Beta	−4731.41	−140.37	17.84	1.30
AHRI Deheaped Male 45-49
1	Sinh-arcsinh	−3680.18	0.00	0.00	0.30
2	Normal	−3796.06	−115.88	19.24	1.15
3	Skew normal	−3797.14	−116.95	23.48	1.02
4	Gamma	−3801.02	−120.83	24.51	0.98
5	Beta	−3817.97	−137.79	19.37	1.39

Appendix 1—table 6

Full ELPD and QQ RMSE table for women in the Haiti 2016–17 DHS data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
Haiti 2016-17 DHS Female 20-24
1	Sinh-arcsinh	−3259.31	0.00	0.00	0.49
2	Skew normal	−3263.46	−4.15	4.95	0.53
3	Normal	−3338.23	−78.92	19.54	0.91
4	Beta	−3441.91	−182.60	45.77	1.24
5	Gamma	−3504.85	−245.54	53.90	1.29
Haiti 2016-17 DHS Female 25-29
1	Sinh-arcsinh	−4447.43	0.00	0.00	0.26
2	Skew normal	−4471.22	−23.78	8.41	0.57
3	Normal	−4527.25	−79.82	18.72	0.86
4	Beta	−4625.97	−178.54	40.88	1.23
5	Gamma	−4678.20	−230.77	45.81	1.22
Haiti 2016-17 DHS Female 30-34
1	Sinh-arcsinh	−4720.12	0.00	0.00	0.44
2	Skew normal	−4749.57	−29.45	9.06	0.68
3	Normal	−4763.78	−43.66	10.51	0.62
4	Beta	−4809.11	−88.99	17.19	0.85
5	Gamma	−4836.82	−116.70	20.32	0.83
Haiti 2016-17 DHS Female 35-39
1	Sinh-arcsinh	−4490.82	0.00	0.00	0.33
2	Skew normal	−4518.58	−27.75	8.14	0.57
3	Normal	−4526.55	−35.73	8.59	0.73
4	Beta	−4561.27	−70.45	13.60	0.94
5	Gamma	−4577.84	−87.01	15.27	0.86
Haiti 2016-17 DHS Female 40-44
1	Sinh-arcsinh	−3601.02	0.00	0.00	0.35
2	Skew normal	−3629.45	−28.43	7.51	0.83
3	Normal	−3633.14	−32.11	7.96	0.71
4	Beta	−3641.61	−40.59	9.76	0.73
5	Gamma	−3644.89	−43.86	10.47	0.64
Haiti 2016-17 DHS Female 45-49
1	Sinh-arcsinh	−3106.27	0.00	0.00	0.39
2	Skew normal	−3133.10	−26.82	7.68	0.88
3	Gamma	−3133.61	−27.33	7.50	0.68
4	Normal	−3134.62	−28.35	7.46	0.81
5	Beta	−3136.89	−30.62	8.61	0.88

Appendix 1—table 7

Full ELPD and QQ RMSE table for men in the Haiti 2016–17 DHS data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
Haiti 2016-17 DHS Male 20-24
1	Skew normal	−468.98	0.00	0.00	0.43
2	Sinh-arcsinh	−469.60	−0.62	1.12	0.41
3	Normal	−475.31	−6.33	4.28	0.67
4	Beta	−483.53	−14.55	7.33	0.65
5	Gamma	−500.53	−31.55	13.35	0.94
Haiti 2016-17 DHS Male 25-29
1	Sinh-arcsinh	−1386.13	0.00	0.00	0.38
2	Skew normal	−1390.54	−4.41	3.19	0.49
3	Normal	−1395.47	−9.34	4.79	0.60
4	Beta	−1407.46	−21.32	7.31	0.62
5	Gamma	−1434.18	−48.04	11.75	0.79
Haiti 2016-17 DHS Male 30-34
1	Sinh-arcsinh	−2217.20	0.00	0.00	0.44
2	Skew normal	−2222.10	−4.89	3.42	0.69
3	Normal	−2223.97	−6.76	4.48	0.45
4	Beta	−2240.58	−23.37	9.32	0.52
5	Gamma	−2281.18	−63.98	17.91	0.73
Haiti 2016-17 DHS Male 35-39
1	Sinh-arcsinh	−2185.96	0.00	0.00	0.28
2	Skew normal	−2189.87	−3.91	2.67	0.69
3	Beta	−2191.05	−5.10	3.68	0.48
4	Normal	−2191.11	−5.16	3.55	0.49
5	Gamma	−2205.69	−19.73	9.57	0.52
Haiti 2016-17 DHS Male 40-44
1	Sinh-arcsinh	−2051.62	0.00	0.00	0.39
2	Skew normal	−2060.16	−8.54	4.21	0.72
3	Normal	−2060.38	−8.75	4.57	0.69
4	Beta	−2062.00	−10.37	4.87	0.70
5	Gamma	−2063.79	−12.17	5.73	0.47
Haiti 2016-17 DHS Male 45-49
1	Sinh-arcsinh	−2138.34	0.00	0.00	0.23
2	Normal	−2150.53	−12.19	6.38	0.35
3	Skew normal	−2151.51	−13.17	5.97	0.56
4	Gamma	−2152.88	−14.54	8.93	0.25
5	Beta	−2156.13	−17.79	6.14	0.48

Appendix 1—table 8

Full ELPD and QQ RMSE table for women in the Manicaland data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
Manicaland Female 20-24
1	Sinh-arcsinh	−16390.77	0.00	0.00	0.31
2	Skew normal	−16502.01	−111.25	21.22	0.44
3	Normal	−16779.93	−389.16	37.05	0.67
4	Beta	−17111.57	−720.80	62.02	0.86
5	Gamma	−17387.38	−996.61	76.80	1.02
Manicaland Female 25-29
1	Sinh-arcsinh	−18702.50	0.00	0.00	0.53
2	Skew normal	−18923.04	−220.53	25.27	0.94
3	Normal	−19080.66	−378.16	36.05	0.83
4	Beta	−19405.80	−703.30	64.97	1.05
5	Gamma	−19615.53	−913.03	76.38	1.09
Manicaland Female 30-34
1	Sinh-arcsinh	−16523.81	0.00	0.00	0.48
2	Skew normal	−16877.96	−354.15	40.36	0.87
3	Normal	−16886.62	−362.80	36.41	0.99
4	Beta	−17021.26	−497.44	43.60	1.12
5	Gamma	−17094.58	−570.76	49.53	0.93
Manicaland Female 35-39
1	Sinh-arcsinh	−14397.76	0.00	0.00	0.48
2	Skew normal	−14736.64	−338.88	28.35	1.25
3	Normal	−14798.55	−400.79	36.87	1.39
4	Beta	−14824.80	−427.04	33.02	1.47
5	Gamma	−14835.11	−437.35	34.49	1.14
Manicaland Female 40-44
1	Sinh-arcsinh	−12293.13	0.00	0.00	0.68
2	Skew normal	−12488.28	−195.15	21.36	1.03
3	Gamma	−12500.93	−207.80	22.18	1.03
4	Normal	−12508.91	−215.78	23.29	1.28
5	Beta	−12537.14	−244.01	25.41	1.22
Manicaland Female 45-49
1	Sinh-arcsinh	−9183.03	0.00	0.00	0.56
2	Skew normal	−9455.87	−272.83	23.57	1.68
3	Normal	−9477.33	−294.30	23.55	1.62
4	Gamma	−9497.31	−314.27	25.08	1.44
5	Beta	−9576.44	−393.40	32.07	1.94

Appendix 1—table 9

Full ELPD and QQ RMSE table for men in the Manicaland data set.

Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
Manicaland Male 20-24
1	Sinh-arcsinh	−9770.00	0.00	0.00	0.30
2	Skew normal	−9895.82	−125.83	33.35	0.40
3	Normal	−10139.11	−369.11	79.13	0.49
4	Beta	−10587.64	−817.64	181.23	0.56
5	Gamma	−11594.58	−1824.59	388.26	1.15
Manicaland Male 25-29
1	Sinh-arcsinh	−13978.59	0.00	0.00	0.40
2	Skew normal	−13990.39	−11.80	8.51	0.48
3	Normal	−14018.60	−40.00	17.48	0.45
4	Beta	−14152.35	−173.76	48.77	0.40
5	Gamma	−14500.47	−521.87	117.58	0.55
Manicaland Male 30-34
1	Sinh-arcsinh	−12949.24	0.00	0.00	0.31
2	Skew normal	−13016.44	−67.21	25.01	0.37
3	Normal	−13037.46	−88.22	16.57	0.49
4	Beta	−13070.31	−121.07	54.74	0.41
5	Gamma	−13285.47	−336.23	171.92	0.42
Manicaland Male 35-39
1	Sinh-arcsinh	−11496.14	0.00	0.00	0.27
2	Skew normal	−11528.36	−32.22	9.83	0.39
3	Normal	−11530.43	−34.29	9.72	0.26
4	Gamma	−11531.75	−35.61	12.47	0.24
5	Beta	−11582.63	−86.49	12.97	0.48
Manicaland Male 40-44
1	Sinh-arcsinh	−8714.06	0.00	0.00	0.35
2	Skew normal	−8749.78	−35.72	10.11	0.51
3	Gamma	−8777.08	−63.02	10.38	0.55
4	Normal	−8791.45	−77.38	12.23	0.76
5	Beta	−8860.22	−146.16	18.15	0.93
Manicaland Male 45-49
1	Sinh-arcsinh	−7110.27	0.00	0.00	0.42
2	Skew normal	−7177.03	−66.75	25.08	0.76
3	Gamma	−7213.99	−103.72	13.02	1.07
4	Normal	−7232.04	−121.77	13.09	1.28
5	Beta	−7312.35	−202.08	18.61	1.61

Appendix 1—table 10

LOO-CV estimated ELPD values, differences, and standard errors of differences, as well as QQ RMSE values, for all five regression models fit to all four data sets.

The ‘difference’ value of a row is the difference between that row’s ELPD value and data set-specific best ELPD value. Higher ELPD values and lower QQ RMSE values are better.

Rank	Model	ELPD	ELPD Diff	SE of Diff	QQ RMSE
AHRI
1	Distributional 4	55841.91	0.00	0.00	0.66
2	Distributional 3	55534.16	−307.75	32.36	0.93
3	Distributional 2	54794.79	−1047.12	51.69	1.21
4	Distributional 1	54335.19	−1506.72	72.32	1.15
5	Conventional	52689.21	−3152.70	100.59	1.30
AHRI Deaheaped
1	Distributional 4	58503.98	0.00	0.00	0.62
2	Distributional 3	58219.23	−284.75	28.64	0.92
3	Distributional 2	57503.68	−1000.30	47.14	1.14
4	Distributional 1	57097.39	−1406.59	64.48	1.06
5	Conventional	55296.25	−3207.73	99.42	1.26
Haiti 2016-17 DHS
1	Distributional 4	5207.57	0.00	0.00	0.84
2	Distributional 3	5196.69	−10.89	6.54	0.91
3	Distributional 1	5140.77	−66.80	12.27	0.98
4	Distributional 2	5138.75	−68.83	12.24	0.99
5	Conventional	4777.78	−429.80	30.54	1.33
Manicaland
1	Distributional 4	24516.15	0.00	0.00	1.04
2	Distributional 3	24313.74	−202.40	20.52	1.34
3	Distributional 2	23472.07	−1044.08	47.77	1.80
4	Distributional 1	23192.49	−1323.66	54.97	1.89
5	Conventional	21011.29	−3504.86	89.01	2.05

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Timothy M Wolock
Seth Flaxman
Kathryn A Risher
Tawanda Dadirai
Simon Gregson
Jeffrey W Eaton

(2021)

Evaluating distributional regression strategies for modelling self-reported sexual age-mixing

eLife 10:e68318.

https://doi.org/10.7554/eLife.68318

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The sinh-arcsinh density with μ=0, σ=1, and a variety of assumptions about ϵ and δ.

Observed partner age distributions among women aged 34 years in all three data sets.

Observed means, variances, skewnesses, and kurtoses of partner age by 5-year age bin and sex in all three data sets.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among women aged 35–39 in the AHRI data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for conventional regression and the most complex distributional model among men aged 16, 24, and 37 years in the AHRI data set.

Estimated sinh-arcsinh distributional parameters from the conventional regression model, and distributional models 1 and 4 fit to the AHRI data.

Estimated sinh-arcsinh distributional parameters for Distributional Model 4 fit to the three main data sets.

Illustration of the effect of the deheaping algorithm on women aged exactly 24 years in the AHRI data.

Observed sexual partner age distributions among women in the AHRI data.

Overlaid quantile-quantile (QQ) plots for each probability distribution’s best fit to data in all three main data sets.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among women in the AHRI data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among men in the AHRI data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among women in the AHRI Deheaped data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among men in the AHRI Deheaped data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among women in the Haiti 2016–17 DHS data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among men in the Haiti 2016–17 DHS data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among women in the Manicaland data set.

Observed partner age distributions (grey bars) and posterior predictive partner age distributions (lines) for each probability distribution among men in the Manicaland data set.

Details of the five distributions tested in this analysis.

Summary of five models fit in this analysis.

Share of subsets in which each dependent variable yields the highest ELPD given each probability distribution (excluding deheaped AHRI data).

Model comparison metrics averaged across all data subsets for all three data sets.

ELPD and QQ RMSE values for all five distributional regression models fit to each data set.

ELPD and QQ RMSE values for all five models fit to deheaped AHRI data The models increase in complexity from Conventional Regression to Distributional Model 4.

Full ELPD and QQ RMSE table for women in the AHRI data set.

Full ELPD and QQ RMSE table for men in the AHRI data set.

Full ELPD and QQ RMSE table for women in the AHRI Deheaped data set.

Full ELPD and QQ RMSE table for men in the AHRI Deheaped data set.

Full ELPD and QQ RMSE table for women in the Haiti 2016–17 DHS data set.

Full ELPD and QQ RMSE table for men in the Haiti 2016–17 DHS data set.

Full ELPD and QQ RMSE table for women in the Manicaland data set.

Full ELPD and QQ RMSE table for men in the Manicaland data set.

LOO-CV estimated ELPD values, differences, and standard errors of differences, as well as QQ RMSE values, for all five regression models fit to all four data sets.

Transparent reporting form

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

The sinh-arcsinh density with $μ = 0$ , $σ = 1$ , and a variety of assumptions about $ϵ$ and δ.