Author: Ioannis Kosmidis
Date: 2023-02-03

# Estimating the exponential of regression parameters using brglm2

#### Ioannis Kosmidis

#### 03 February 2023

# 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 (2002, sec. 5.4.2) on the effects of AZT in slowing the development of AIDS symptoms.

# AIDS and AZT use

The data analyzed in Agresti (2002, sec. 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
    #>   symptomatic asymptomatic  race AZT
    #> 1          14           93 White Yes
    #> 2          32           81 White  No
    #> 3          11           52 Black Yes
    #> 4          12           43 Black  No

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 (2002, Table 5.6).

    aids_mod <- glm(cbind(symptomatic, asymptomatic) ~ AZT + race,
                      family = binomial(), data = aids)
    summary(aids_mod)
    #>
    #> Call:
    #> glm(formula = cbind(symptomatic, asymptomatic) ~ AZT + race,
    #>     family = binomial(), data = aids)
    #>
    #> Coefficients:
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept) -1.07357    0.26294  -4.083 4.45e-05 ***
    #> AZTYes      -0.71946    0.27898  -2.579  0.00991 **
    #> raceWhite    0.05548    0.28861   0.192  0.84755
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for binomial family taken to be 1)
    #>
    #>     Null deviance: 8.3499  on 3  degrees of freedom
    #> Residual deviance: 1.3835  on 1  degrees of freedom
    #> AIC: 24.86
    #>
    #> Number of Fisher Scoring iterations: 4

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 0.01, 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")
    #>
    #> Call:
    #> expo.glm(object = aids_mod, type = "ML")
    #>
    #> Odds ratios
    #>             Estimate Std. Error   2.5 % 97.5 %
    #> (Intercept)  0.34178    0.08987 0.20414  0.572
    #> AZTYes       0.48702    0.13587 0.28189  0.841
    #> raceWhite    1.05705    0.30508 0.60038  1.861
    #>
    #>
    #> Type of estimator: ML (maximum likelihood)

As noted in Agresti (2002, sec. 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*")
    #>
    #> Call:
    #> expo.glm(object = aids_mod, type = "correction*")
    #>
    #> Odds ratios
    #>             Estimate Std. Error   2.5 % 97.5 %
    #> (Intercept)  0.33611    0.08915 0.19986  0.565
    #> AZTYes       0.47424    0.13509 0.27136  0.829
    #> raceWhite    1.00726    0.29467 0.56771  1.787
    #>
    #>
    #> Type of estimator: correction* (explicit mean bias correction with a multiplicative adjustment)
    expo(aids_mod, type = "Lylesetal2012")
    #>
    #> Call:
    #> expo.glm(object = aids_mod, type = "Lylesetal2012")
    #>
    #> Odds ratios
    #>             Estimate Std. Error   2.5 % 97.5 %
    #> (Intercept)  0.33592    0.08912 0.19972  0.565
    #> AZTYes       0.47390    0.13506 0.27108  0.828
    #> raceWhite    1.00643    0.29453 0.56713  1.786
    #>
    #>
    #> Type of estimator: Lylesetal2012 (Lyles et al., 2012; doi: 10.1016/j.jspi.2012.05.005)
    expo(aids_mod, type = "correction+")
    #>
    #> Call:
    #> expo.glm(object = aids_mod, type = "correction+")
    #>
    #> Odds ratios
    #>             Estimate Std. Error   2.5 % 97.5 %
    #> (Intercept)  0.33572    0.08909 0.19957  0.565
    #> AZTYes       0.47354    0.13503 0.27080  0.828
    #> raceWhite    1.00556    0.29439 0.56651  1.785
    #>
    #>
    #> Type of estimator: correction+ (explicit mean bias correction with an additive adjustment)

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")
    #>
    #> Call:
    #> expo.glm(object = aids_mod, type = "AS_median")
    #>
    #> Odds ratios
    #>             Estimate Std. Error   2.5 % 97.5 %
    #> (Intercept)  0.34454    0.09036 0.20606  0.576
    #> AZTYes       0.49023    0.13632 0.28426  0.845
    #> raceWhite    1.05401    0.30329 0.59967  1.853
    #>
    #>
    #> Type of estimator: AS_median (median bias-reducing adjusted score equations)

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, Kenne Pagui, and Sartori (2020) and Kosmidis and Firth (2020) 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
    #>
    #> Call:  glm(formula = HG ~ NV + PI + EH, family = binomial(), data = endometrial)
    #>
    #> Coefficients:
    #> (Intercept)           NV           PI           EH
    #>     4.30452     18.18556     -0.04218     -2.90261
    #>
    #> Degrees of Freedom: 78 Total (i.e. Null);  75 Residual
    #> Null Deviance:       104.9
    #> Residual Deviance: 55.39     AIC: 63.39

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

    library("detectseparation")
    #>
    #> Attaching package: 'detectseparation'
    #> The following objects are masked from 'package:brglm2':
    #>
    #>     check_infinite_estimates, detect_separation
    update(endometrialML, method = detect_separation)
    #> Implementation: ROI | Solver: lpsolve
    #> Separation: TRUE
    #> Existence of maximum likelihood estimates
    #> (Intercept)          NV          PI          EH
    #>           0         Inf           0           0
    #> 0: finite value, Inf: infinity, -Inf: -infinity

and a naive estimate of the associated odds ratio while controlling for `PI` and `EH` is 7.9047207^{7}, which is in reality infinite.

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

    expo(endometrialML, type = "correction*")
    #>
    #> Call:
    #> expo.glm(object = endometrialML, type = "correction*")
    #>
    #> Odds ratios
    #>              Estimate Std. Error     2.5 %  97.5 %
    #> (Intercept) 20.671820  33.136501  0.893141 478.451
    #> NV           8.496974   7.825239  1.397511  51.662
    #> PI           0.965089   0.036795  0.895602   1.040
    #> EH           0.056848   0.056344  0.008148   0.397
    #>
    #>
    #> Type of estimator: correction* (explicit mean bias correction with a multiplicative adjustment)
    expo(endometrialML, type = "correction+")
    #> Warning in log(trans_coefs): NaNs produced
    #>
    #> Call:
    #> expo.glm(object = endometrialML, type = "correction+")
    #>
    #> Odds ratios
    #>             Estimate Std. Error 2.5 % 97.5 %
    #> (Intercept) -4.71087        NaN   NaN    NaN
    #> NV          -3.78835        NaN   NaN    NaN
    #> PI           0.96509        NaN   NaN    NaN
    #> EH           0.05169        NaN   NaN    NaN
    #>
    #>
    #> Type of estimator: correction+ (explicit mean bias correction with an additive adjustment)
    expo(endometrialML, type = "Lylesetal2012")
    #>
    #> Call:
    #> expo.glm(object = endometrialML, type = "Lylesetal2012")
    #>
    #> Odds ratios
    #>              Estimate Std. Error     2.5 %  97.5 %
    #> (Intercept) 14.388911  23.599810  0.578015 358.193
    #> NV           5.622853   4.766859  1.067426  29.619
    #> PI           0.965089   0.035021  0.898834   1.036
    #> EH           0.054734   0.058473  0.006744   0.444
    #>
    #>
    #> Type of estimator: Lylesetal2012 (Lyles et al., 2012; doi: 10.1016/j.jspi.2012.05.005)

# `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*")
    #>
    #> Call:
    #> expo.brglmFit(object = aids_mod_br, type = "correction*")
    #>
    #> Odds ratios
    #>             Estimate Std. Error   2.5 % 97.5 %
    #> (Intercept)  0.33611    0.08915 0.19986  0.565
    #> AZTYes       0.47424    0.13509 0.27136  0.829
    #> raceWhite    1.00726    0.29467 0.56771  1.787
    #>
    #>
    #> Type of estimator: correction* (explicit mean bias correction with a multiplicative adjustment)

# References

Agresti, A. 2002. Categorical Data Analysis. Wiley.
Kosmidis, Ioannis, and David Firth. 2020. “Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models.” Biometrika 108 (1): 71–82. https://doi.org/10.1093/biomet/asaa052.
Kosmidis, Ioannis, Euloge Clovis Kenne Pagui, and Nicola Sartori. 2020. “Mean and Median Bias Reduction in Generalized Linear Models.” Statistics and Computing 30: 43–59. https://doi.org/10.1007/s11222-019-09860-6.