Estimating the exponential of regression parameters using **brglm2**

In brglm2: Bias Reduction in Generalized Linear Models

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The expo() method

The brglm2 R package provides the expo() method for estimating exponentiated parameters of generalized linear models using various methods.

The expo() method uses a supplied "brglmFit" or "glm" object to estimate exponentiated parameters of generalized linear models with maximum likelihood or various mean and median bias reduction methods. expo() is useful for computing (corrected) estimates of the multiplicative impact of a unit increase on a covariate on the mean of a Poisson log-linear model (family = poisson("log") in glm()) while adjusting for other covariates, the odds ratio associated with a unit increase on a covariate in a logistic regression model (family = binomial("logit") in glm()) while adjusting for other covariates, the relative risk associated with a unit increase on a covariate in a relative risk regression model (family = binomial("log") in glm()) while adjusting for other covariates, among others.

The vignette demonstrates the use of expo() and the associated methods by reproducing part of the analyses in @agresti:02[, Section 5.4.2] on the effects of AZT in slowing the development of AIDS symptoms.

AIDS and AZT use

The data analyzed in @agresti:02[, Section 5.4.2] is from a 3-year study on the effects of AZT in slowing the development of AIDS symptoms. 338 veterans whose immune systems were beginning to falter after infection with the AIDS virus were randomly assigned either to receive AZT immediately or to wait until their T cells showed severe immune weakness. See ?aids for more details.

The aids data set cross-classifies the veterans' race (race), whether they received AZT immediately (AZT), and whether they developed AIDS symptoms during the 3-year study (symptomatic and asymptomatic).

library("brglm2")
data("aids", package = "brglm2")
aids

We now use a logistic regression model to model the probability of developing symptoms in terms of AZT and race, and reproduce part of the compute output in @agresti:02[, Table 5.6].

aids_mod <- glm(cbind(symptomatic, asymptomatic) ~ AZT + race, 
                  family = binomial(), data = aids)
summary(aids_mod)

The Wald test for the hypothesis of conditional independence of AZT treatment and development of AIDS symptoms, controlling for race, returns a p-value of r round(coef(summary(aids_mod))["AZTYes", "Pr(>|z|)"], 3), showing evidence of association.

The predicted probabilities for each combination of levels

The maximum likelihood estimates of the odds ratio between immediate AZT use and development of AIDS symptoms can be inferred from aids_mod through the expo() method, which also estimates standard errors using the delta method, and returns approximate 95% confidence intervals (see ?expo for details).

expo(aids_mod, type = "ML")

As noted in @agresti:02[, Section 5.4.2], for each race, the estimated odds of symptoms are half as high for those who took AZT immediately, with value $0.49$ and a nominally 95\% Wald confidence interval $(0.28, 0.84)$.

The expo() method can be used to estimate the odds ratios using three methods that return estimates of the odds ratios with asymptotically smaller mean bias than the maximum likelihood estimator

expo(aids_mod, type = "correction*")
expo(aids_mod, type = "Lylesetal2012")
expo(aids_mod, type = "correction+")

and one method that returns estimates of the odds ratios with asymptotically smaller median bias than the maximum likelihood estimator

expo(aids_mod, type = "AS_median")

The estimated odds ratios and associated inferences from the methods that correct for mean and median bias are similar to those from maximum likelihood.

Infinite odds ratio estimates

When expo() is called with type = correction*, type = correction+, type = Lylesetal2012, and type = AS_median, then the estimates of the odds ratios can be shown to be always finite and greater than zero. The reason is that the corresponding odds-ratio estimators depend on regression parameter estimates that are finite even if the maximum likelihood estimates are infinite. See, @kosmidis:2019 and @kosmidis+firth:21 for details.

As an example, consider the estimated odds ratios from a logistic regression model fitted on the endometrial data set using maximum likelihood.

data("endometrial", package = "brglm2")
endometrialML <- glm(HG ~ NV + PI + EH, data = endometrial, family = binomial())
endometrialML

The estimate of the coefficient for NV is in reality infinite as it can be verified using the detectseparation R package

library("detectseparation")
update(endometrialML, method = detect_separation)

and a naive estimate of the associated odds ratio while controlling for PI and EH is r exp(coef(endometrialML)["NV"]), which is in reality infinite.

In contrast, expo() returns finite reduced-mean-bias estimates of the odds ratios

expo(endometrialML, type = "correction*")
expo(endometrialML, type = "correction+")
expo(endometrialML, type = "Lylesetal2012")

brglmFit objects

The expo() method also works seamlessly with brglmFit objects, returning the same results as above. For example,

aids_mod_br <- update(aids_mod, method = "brglmFit")
expo(aids_mod_br, type = "correction*")

References

brglm2 documentation built on Oct. 12, 2023, 1:07 a.m.