`expo()`

methodThe **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.

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.

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`

objectsThe `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)
```

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.