Inverse Probability Weights for the Analysis of Polytomous Outcomes

Polytomous outcomes are common in epidemiologic studies. Analyses based on multinomial models employ a likelihood that utilizes the data observed in all outcome categories simultaneously and permits inferences regarding associations across outcome categories. However, the potentially large number of estimated parameters produced by multinomial model fitting can lead to problems of estimation and inference (1). We have proposed an inverse-probability-of-exposure weighted multinomial model for analysis of polytomous outcomes, described its implementation, and illustrated it. The approach yields marginal estimates of associations, which are sometimes desirable as summary measures of association (2). This approach allows for confounding control and tends to be less susceptible to problems of estimation that arise when at least one outcome category is rare.

METHODS

Consider a study in which D denotes an outcome variable that has G + 1 mutually exclusive categories (0,1,2, . . ., G). Let E denote an exposure variable of primary interest, and Z= {Z₁, . . ., Z_k} denote the k covariates that are potential confounders of associations between E and D. We suppose for simplicity that E is binary while noting that the proposed method extends to nominal and continuous exposure variables (3).

If the investigator wants to compare the distribution of D among the exposed and unexposed, adjusting for confounding by Z, a multinomial regression model for the outcome might be fitted with the form:

(1)

where the first outcome category, D = 0, is the referent. The number of parameters in equation (1) increases with G and k. The multinomial model includes G intercepts, G coefficients describing associations between E and each index category of D, and a vector of coefficients of length G × k with coefficients describing associations between each covariate in Z and each index category of D.

Numerical problems may arise in estimation of the model in equation (1), in part because it includes a potentially large vector of coefficients to adjust for confounding by covariates Z. Inverse probability weighting provides an alternative approach to confounding control (4, 5). We propose an inverse-probability-of-exposure weighted (henceforth, “weighted”) multinomial logistic regression model to control confounding and allow estimation of marginal associations between exposure and outcome categories.

Standardized odds ratios can be estimated by fitting a weighted multinomial regression model for D with E as the only explanatory variable. Estimates of parameters β^*_g, are marginal estimates of association, equal to β_g integrated over the empirical distribution of Z. Robust and nonparametric bootstrap standard errors for the estimated coefficients can be obtained (Web Appendix 1, available at https://academic.oup.com/aje). Weighted multinomial models can be fitted that assume homogeneity of β^*_g for all g > 0 or some subset of these coefficients (Web Appendix 2). Simple weighted tabular analysis yields the standardized risk or prevalence of D at each level of E, and bootstrap estimation can be used to derive standard errors (Web Appendix 3).

For an example, we used data from the National Cancer Institute Black/White Cancer Survival Study, a cross-sectional study of 288 women with endometrial cancer (6, 7). We estimated associations between age and histological subtype among women with endometrial cancer. The explanatory variable of interest was coded AGE = 0 (50–64 years) or AGE = 1 (65–79). The polytomous outcome variable was histological cancer subtype, coded SUBTYPE = 0 (adenocarcinoma), SUBTYPE = 1 (adenosquamous carcinoma), or SUBTYPE = 2 (other). We assessed potential confounding by binary covariates for current smoking (SMOKING, yes/no), estrogen use (ESTROGEN, ever/never), and racial group (RACE, black/white). We estimated predicted probabilities of each individual’s observed level of AGE conditional on RACE, ESTROGEN, SMOKING, and product term ESTROGEN × SMOKING. To stabilize weights, we set the numerator of each weight equal to the marginal probability of the individual’s observed level of AGE. We estimated standardized odds ratios, prevalence ratios and differences, and associated 95% confidence intervals. Odds ratios for associations between AGE and SUBTYPE were estimated with a common referent (adenocarcinoma), and prevalence ratios and differences were calculated for each subtype.

We also used simulations to illustrate our approach and demonstrate that the proposed approach converges in many cases when multivariable multinomial models do not (Web Appendix 4). Web Figure 1 illustrates relationships in the simulation setup and Web Table 1 and Web Table 2 report the results of these simulations.

RESULTS

In the observed data, 19.8% of cases were adenocarcinoma, 15.6% were adenosquamous carcinoma, and 64.6% were other subtypes. Over half (63.2%) of the women were aged 65–79 years (AGE = 1). The mean stabilized inverse-probability-of-exposure weight was 1.00, with minimum and maximum of 0.46 and 2.87, respectively; the 5th, 25th, 50th, 75th, and 95th percentiles were 0.50, 0.89, 0.96, 1.08, and 1.33, respectively. Distributions of RACE and ESTROGEN were largely similar across AGE groups, but SMOKING and AGE were strongly associated in the observed data (Table 1). In the weighted data, covariate differences across AGE groups were diminished, and covariate distributions in each AGE group corresponded to their respective distributions in the total population. Crude odds ratios for histological subtype by AGE were 2.18 (95% confidence interval (CI): 1.04, 4.58) for adenosquamous carcinoma and 1.53 (95% CI: 0.82, 2.87) for “other” subtypes, while weighted estimates were 2.64 (95% CI: 1.15, 6.07) and 1.35 (95% CI: 0.69, 2.64), respectively. For adenocarcinoma, the standardized prevalence ratio for older age relative to younger age was 0.83 (95% CI: 0.69, 0.98), and the standardized prevalence difference was −12.6% (95% CI: −24.2, −0.9). For adenosquamous carcinoma, the standardized prevalence ratio for older age relative to younger age was 2.15 (95% CI: 1.09, 4.23), and the standardized prevalence difference was 10.7% (95% CI: 2.2, 19.2). For “other” histological subtypes, the standardized prevalence ratio for older versus younger age was 1.10 (95% CI: 0.66, 1.83), and the standardized prevalence difference was 1.9% (95% CI: −7.9, 11.6).

Table 1.

Cross-Sectional Data for Distributions of Covariates According to Age Group in Observed Data and Inverse-Probability-of-Exposure Weighted Data From a Study of 288 Women With Endometrial Cancer, National Cancer Institute Black/White Cancer Survival Study, United States, 1985–1987

Characteristic	Observed (Crude) Data							Inverse-Probability-of-Exposure Weighted Data
	Age 50–64 Years (n = 106)		Age 65–79 Years (n = 182)		Total (n = 288)		SAMDa	Age 50–64 Years (∑sw = 105.6)		Age 65–79 Years (∑sw = 180.2)		Total (∑sw = 285.9)		SAMD
	No.	%	No.	%	No.	%		∑sw	%	∑sw	%	∑sw	%
Race							0.07							<0.001
White	74	69.8	133	73.1	207	71.9		77.1	73.0	131.5	72.9	208.6	73.0
Black	32	30.2	49	26.9	81	28.1		28.5	27.0	48.8	27.1	77.3	27.0
Estrogen use							0.09							0.01
No	47	44.3	88	48.4	135	46.9		50.5	47.9	85.5	47.4	136.0	47.6
Yes	59	55.7	92	50.5	151	52.4		55.1	52.1	94.8	52.6	149.9	52.4
Missing	0	0.0	2	1.1	2	0.7
Ever smoker							0.51							0.01
No	85	80.2	175	96.2	260	90.3		95.3	90.3	163.0	90.4	258.4	90.4
Yes	21	19.8	7	3.8	28	9.7		10.3	9.7	17.2	9.6	27.5	9.6
Histological subtype
Adenocarcinoma	77	72.6	109	59. 9	186	64.6		76.1	72.1	107.3	59.5	183.4	64.2
Adenosquamous	11	10.4	34	18.7	45	15.6		9.8	9.3	36.1	20.0	45.9	16.1
Other subtype	18	17.0	39	21.4	57	19.8		19.6	18.6	36.9	20.5	56.5	19.8

Abbreviations: SAMD, standardized absolute mean difference; ∑_sw, sum of stabilized inverse-probability-of-exposure weights.

^a To assess balance of the distributions of covariates across age groups (AGE) in the crude and weighted data, we calculated the SAMD for each covariate, which is the absolute value of the difference in the mean of the covariate across levels of AGE, divided by the pooled standard error of the covariate (9).

DISCUSSION

We have described and illustrated a multinomial regression modeling approach using standardization by inverse-probability-of-exposure weights for some or all measured confounding variables. There are other approaches to handling multinomial model fitting with many parameters (such as shrinkage estimation approaches). However, the proposed form of weighting may be attractive to some investigators because it is simpler to implement in many settings than shrinkage estimation. It should be noted that the proposed approach is less efficient than maximum likelihood estimation (8). The exposure prediction model used to obtain the weights applied to the observed data is related to the exposure propensity score (Web Appendix 1) (2); correct specification of this exposure prediction model is required to obtain consistent estimates from a weighted multinomial model.

Inverse probability weighting for analysis of polytomous outcomes may help to address some routinely encountered difficulties in epidemiologic data analyses.