Bias of using odds ratio estimates in multinomial logistic
          regressions to estimate relative risk or prevalence ratio and alternatives

Camey, Suzi Alves; Torman, Vanessa Bielefeldt Leotti; Hirakata, Vania Naomi; Cortes, Renan Xavier; Vigo, Alvaro

doi:10.1590/0102-311X00077313

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METHODOLOGICAL ISSUES • Cad. Saúde Pública 30 (1) • Jan 2014 • https://doi.org/10.1590/0102-311X00077313 copy

Bias of using odds ratio estimates in multinomial logistic regressions to estimate relative risk or prevalence ratio and alternatives

Viés da razão de chances estimada pela regressão logística multinomial para estimar o risco relativo ou a razão de prevalência e alternativas

El sesgo del odds ratio estimado por la regresión logística multinomial para estimar el riesgo relativo o la razón de prevalencia y alternativas

Authorship SCIMAGO INSTITUTIONS RANKINGS

Recent studies have emphasized that there is no justification for using the odds ratio (OR) as an approximation of the relative risk (RR) or prevalence ratio (PR). Erroneous interpretations of the OR as RR or PR must be avoided, as several studies have shown that the OR is not a good approximation for these measures when the outcome is common (> 10%). For multinomial outcomes it is usual to use the multinomial logistic regression. In this context, there are no studies showing the impact of the approximation of the OR in the estimates of RR or PR. This study aimed to present and discuss alternative methods to multinomial logistic regression based upon robust Poisson regression and the log-binomial model. The approaches were compared by simulating various possible scenarios. The results showed that the proposed models have more precise and accurate estimates for the RR or PR than the multinomial logistic regression, as in the case of the binary outcome. Thus also for multinomial outcomes the OR must not be used as an approximation of the RR or PR, since this may lead to incorrect conclusions.

Odds Ratio; Prevalence Ratio; Logistic Models; Relative Risk

Scenario	Marginal probabilities			Conditional probabilities
Scenario	P(X=+)	P(Y=B)	P(Y=C)	P(Y=A\|X=+)	P(Y=B\|X=+)	P(Y=C\|X=+)	P(Y=A\|X=-)	P(Y=B\|X=-)	P(Y=C\|X=-)
1	0.1	0.05	0.05	0.8377	0.0714	0.0909	0.9069	0.0476	0.0455
2	0.5	0.05	0.05	0.8733	0.0600	0.0667	0.9267	0.0400	0.0333
3	0.9	0.05	0.05	0.8957	0.0517	0.0526	0.9392	0.0345	0.0263
4	0.1	0.35	0.05	0.4091	0.5000	0.0909	0.6212	0.3333	0.0455
5	0.5	0.35	0.05	0.5133	0.4200	0.0667	0.6867	0.2800	0.0333
6	0.9	0.35	0.05	0.5853	0.3621	0.0526	0.7323	0.2414	0.0263
7	0.1	0.3	0.3	0.0260	0.4286	0.5454	0.4416	0.2857	0.2727
8	0.5	0.3	0.3	0.2400	0.3600	0.4000	0.5600	0.2400	0.2000
9	0.9	0.3	0.3	0.3739	0.3103	0.3158	0.6352	0.2069	0.1579

Scenario/n	Log-binomial model			Multinomial logistic regression			Ratio between biases
Scenario/n	Average of estimates	Average absolute bias	Coverage %	Average of estimates	Average absolute bias	Coverage %	Ratio between biases
1
100	3.592	2.092	95.26	4.797	3.297	93.99	1.58
500	1.610	0.110	96.74	1.792	0.292	95.75	2.65
1,000	1.535	0.034	96.19	1.684	0.184	94.92	5.41
2
100	1.808	0.308	99.70	1.962	0.462	99.67	1.50
500	1.692	0.192	95.73	1.813	0.313	95.67	1.63
1,000	1.580	0.080	95.62	1.684	0.184	95.06	2.30
3
100	0.485	1.015	88.68	0.428	1.072	89.81	1.06
500	1.626	0.126	95.22	1.697	0.197	96.36	1.56
1,000	1.948	0.448	96.77	2.058	0.558	97.41	1.25
4
100	1.433	0.067	97.51	3.22	1.720	95.13	25.67
500	1.507	0.007	95.37	2.418	0.918	74.00	131.14
1,000	1.503	0.003	94.59	2.342	0.842	53.49	280.67
5
100	1.597	0.097	96.09	2.259	0.759	90.23	7.82
500	1.515	0.015	94.96	2.05	0.550	67.77	36.67
1,000	1.510	0.010	95.29	2.033	0.533	42.61	53.30
6
100	1.952	0.452	95.97	2.386	0.886	98.72	1.96
500	1.621	0.121	95.48	2.059	0.559	92.20	4.62
1,000	1.555	0.055	95.24	1.966	0.466	86.71	8.47
7
100	1.416	0.084	97.76	5.987	4.487	97.84	53.42
500	1.495	0.005	95.67	23.678	22.178	0.19	4,435.60
1,000	1.504	0.004	95.27	32.44	30.94	0.00	7,735.00
8
100	1.614	0.114	95.62	4.238	2.738	60.84	24.02
500	1.521	0.021	95.02	3.623	2.123	3.12	101.10
1,000	1.511	0.011	94.89	3.561	2.061	0.00	187.36
9
100	1.789	0.289	95.61	3.105	1.605	99.55	5.55
500	1.643	0.143	95.23	2.874	1.374	72.67	9.61
1,000	1.567	0.067	95.06	2.700	1.200	47.24	17.91

Scenario/n	Log-binomial model			Multinomial logistic regression			Ratio between biases
Scenario/n	Average of estimates	Average absolute bias	Coverage %	Average of estimates	Average absolute bias	Coverage %	Ratio between biases
1
100	4.087	2.087	96.33	5.508	3.508	95.63	1.68
500	2.121	0.121	96.47	2.363	0.363	95.57	3.00
1,000	2.045	0.045	95.69	2.243	0.243	94.71	5.40
2
100	2.179	0.179	98.64	2.364	0.364	99.13	2.03
500	2.313	0.313	95.87	2.478	0.478	95.91	1.53
1,000	2.121	0.121	95.76	2.260	0.260	95.78	2.15
3
100	0.501	1.499	85.95	0.441	1.559	86.46	1.04
500	1.851	0.149	94.59	1.927	0.073	95.65	0.49
1,000	2.534	0.534	96.20	2.672	0.672	97.16	1.26
4
100	4.254	2.254	95.49	8.885	6.885	91.28	3.05
500	2.129	0.129	96.53	3.419	1.419	87.14	1.00
1,000	2.051	0.051	95.44	3.197	1.197	80.77	2.47
5
100	2.166	0.166	98.81	3.023	1.023	99.74	6.16
500	2.323	0.323	96.05	3.145	1.145	91.88	3.54
1,000	2.131	0.131	95.75	2.870	0.870	85.71	6.64
6
100	0.499	1.501	85.35	0.568	1.432	90.24	0.95
500	1.862	0.138	95.08	2.330	0.330	97.90	2.39
1,000	2.498	0.498	95.72	3.147	1.147	98.54	2.30
7
100	1.874	0.126	98.19	7.948	5.948	98.57	47.21
500	1.988	0.012	95.44	31.545	2.545	0.06	2,462.08
1,000	2.000	<0.001	95.53	43.148	41.148	0.00	200,853.14
8
100	2.204	0.204	95.97	5.784	3.784	62.45	18.55
500	2.033	0.033	95.06	4.845	2.845	3.52	86.21
1,000	2.016	0.016	94.91	4.751	2.751	0.02	171.94
9
100	2.095	0.095	94.19	3.569	1.569	99.49	16.52
500	2.296	0.296	95.51	4.022	2.022	78.29	6.83
1,000	2.121	0.121	95.32	3.655	1.655	55.74	13.68

Pre-pregnancy BMI (kg/m ² )	Total	Pregnancy weight gain
	Total	Insufficient	Adequate	Excessive
	n (%)	n (%)	n (%)	n (%)
< 18.5	26 (3.9)	12 (46.2)	11 (42.3)	3 (11.5)
18.5 ├ 25.0	414 (62.0)	123 (29.7)	138 (33.3)	153 (37.0)
25.0 ├ 30.0	148 (22.2)	18 (12.2)	32 (21.6)	98 (66.2)
≥ 30.0	79 (11.8)	19 (24.1)	15 (19.0)	45 (57.0)

Pre-pregnancy BMI (kg/m ² )	Log-binomial						Multinomial logistic regression
	Insufficient			Excessive			Insufficient			Excessive
	RR	95%CI	p-value	RR	95%CI	p-value	RR	95%CI	p-value	RR	95%CI	p-value
< 18.5	1.55	1.00, 2.41	0.05	0.31	0.11, 0.91	0.03	1.22	0.52, 2.87	0.64	0.25	0.07, 0.90	0.03
18.5 ├ 25.0	1.00			1.00			1.00			1.00
25.0 ├ 30.0	0.41	0.26, 0.64	< 0.01	1.79	1.51, 2.12	< 0.01	0.63	0.34, 1.18	0.15	2.76	1.74, 4.38	< 0.01
≥ 30.0	0.81	0.53, 1.23	0.32	1.54	1.23, 1.94	< 0.01	1.42	0.69, 2.92	0.34	2.71	1.44, 5.07	< 0.01

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[1] Correspondence S. A. Camey Departamento de Estatística, Universidade Federal do Rio Grande do Sul. Av. Bento Gonçalves 9500, prédio 43111, Porto Alegre, RS. 91509-900, Brasil. camey@mat.ufrgs.br