Suppose \(X\) denotes an exposure of interest that takes values 0 or 1. This could represent two different medical treatments, environmental exposures, economic policies, or genetic variants. We will most often use biomedical examples because we are biostatisticians.

We consider the setting where it is of interest to quantify the effect of intervening with \(X\) on outcome that we will denote \(Y\). The outcome could represent some numeric value, it could be the presence or absence of a condition, or it could be the time between two events, such as time from cancer diagnosis to death. Let \(Y(X = x)\) denote the potential outcome if all subjects in the population would hypothetically be exposed to \(X = x\).

To quantify the effect of \(X\), we must summarize the distribution of \(Y(X = x)\) with some statistic. If \(Y\) is dichotomous, it is natural to use \(p\{Y(X = x) = 1\}\), called the risk. If \(Y\) is continuous, the mean is a natural summary statistic \(E\{Y(X = x)\}\). If \(Y\) is a continuous time-to-event, the probability of exceeding a particular value \(t\) is a reasonable statistic: \(p\{Y(X = x) > t\}\). In general, denote the summary statistic of choice as \(T\{Y(X = x)\}\). The summary statistic can also be applied to conditional distributions, which we will denote, e.g., \(T\{Y | X = x\}\).

To quantify the effect of \(X\), we
must also decide on a *contrast* to measure the causal effect.
The point of the contrast is to compare the chosen summary statistic
between the \(X = 1\) and \(X = 0\) interventions. Typical choices
would be the difference \(T\{Y(X = 1)\} -
T\{Y(X = 0)\}\) or the ratio \(T\{Y(X =
1)\} / T\{Y(X = 0)\}\). It may also be of interest to quantify
and report the summary statistics within each group \((T\{Y(X = 1)\}, T\{Y(X = 0)\})\).

In observational studies, the relationship between \(X\) and \(Y\) is likely confounded by a set of other variables \(\boldsymbol{Z}\). This means that the values of the outcome \(Y\) are determined by at least a subset of \(\boldsymbol{Z}\) and the values of the exposure \(X\) are determined by a subset of \(\boldsymbol{Z}\). Naively estimating the contrast would lead to biased estimates of the causal effect.

See Sjölander (2016) and Sjölander (2018) for more details. Suppose that
the covariates \(\boldsymbol{Z}\) are
sufficient for confounding control. For more information on what
constitutes a sufficient adjustment set, see Witte and Didelez (2019). For a given summary
statistic \(T\), then \[
T\{Y(X = x)\} = E_{\boldsymbol{Z}}[T\{Y | X = x, \boldsymbol{Z}\}],
\] where the expectation is taken with respect to the population
distribution of \(\boldsymbol{Z}\).
This is also known as the *g-formula* or *adjustment
formula*.

In order to estimate this quantity based on an independent and identically distributed sample \((X_1, Y_1, \boldsymbol{Z}_1), \ldots, (X_n, Y_n, \boldsymbol{Z}_n)\), we proceed by

- Specifying and estimating a regression model for \(Y\) given \(X\) and \(\boldsymbol{Z}\).
- Use the fitted model to obtain estimates of \(T\{Y_i | X_i = x, \boldsymbol{Z}_i\}\) for \(i = 1, \ldots, n\). This is done by creating a copy of the observed dataset, replacing the observed \(X_i\) with \(x\) for each individual, and using the fitted model to get predicted values for the copy of the observed data. Denote these predicted values as \(\hat{T}\{Y_i | X_i = x, \boldsymbol{Z}_i\}\).
- Average over the empirical distribution of \(\boldsymbol{Z}\) to obtain the estimate \[ \hat{T}\{Y(X = x)\} = \frac{1}{n}\sum_{i = 1}^n \hat{T}\{Y_i | X_i = x, \boldsymbol{Z}_i\}. \]

One can do this for each level of \(X = 0, 1\) and compute the desired contrast. Under the assumptions that 1) \(\boldsymbol{Z}\) is sufficient for confounding control, and 2) the regression model in step 1 is correctly specified, then this estimator is consistent and asymptotically normal.

A doubly-robust estimator is an estimator that is consistent for a
given estimand when one or more of the models used in the forming of the
estimator is correctly specified for confounding. So far, we only have
one model used in our estimator: the *outcome model*. We will now
introduce a model for the exposure, and see how we can combine them.
First, some terminology:

*Misspecified model* - The true generating mechanism is not
contained in the possible mechanisms that are possible under the
selected model.

*Correctly Specified* - A model is correctly specified if it
is not misspecified.

*Correctly specified for confounding* - A correctly specified
model that contains a sufficient set of confounders.

If we can model \(P(X=1|\boldsymbol{Z})\), and it is correctly specified and contains all confounders, then we can use that to estimate the probability that each individual \(i\) received the treatment that they did \(W_i = \frac{X_i}{P(X_i=1|\boldsymbol{Z}_i)} + \frac{1-X_i}{1-P(X_i=1|\boldsymbol{Z}_i)}\). Let \(\hat{p}_i\) denote the estimated probability that subject \(i\) received treatment \(1\).

Any consistent estimation method can be used for the outcome and
exposure models. As long as *either* the outcome model
*or* the propensity score model is correctly specified for
confounding, then a doubly robust estimator is consistent for the ATE.
In this package, for generalized linear models, we use the estimator as
described by Gabriel et al. (2024).

Recently, there seems to be a general misconception that combining an
adjusted outcome model and a propensity score model always gives a
doubly robust estimator. **This is not true** – it matters
how you combine them!

Here we will use regression standardization to estimate the average
causal effect of the exposure (quitting smoking) in the variable
`qsmk`

on the weight gain outcome in the variable
`wt82_71`

in the `nhefs_complete`

dataset that
comes with the `causaldata`

package. The data was collected
as part of a project by the National Center for Health Statistics and
the National Institute on Aging in collaboration with other agencies of
the United States Public Health Service. It was designed to investigate
the relationships between clinical, nutritional, and behavioral factors
and subsequent morbidity, mortality, and hospital utilization and
changes in risk factors, functional limitation, and
institutionalization. The dataset includes 1566 individuals and contains
among others the following variables:

- seqn: person id}
- wt82_71: weight gain in kilograms between 1971 and 1982
- qsmk: quit smoking between between 1st questionnaire and 1982, 1 = yes, 0 = no
- sex: 0 = male, 1 = female
- race: 0 = white, 1 = black or other
- age: age in years at baseline
- education: level of education in 1971, 1 = 8th grade or less, 2 = high school dropout, 3 = high school, 4 - dropout, 5 = college or more
- smokeintensity: number of cigarettes smoked per day in 1971
- smokeyrs: number of years smoked
- exercise: level of physical activity in 1971, 0 = much exercise, 1 = moderate exercise, 2 = little or no exercise
- active: level of activity in 1971, 0 = very active, 1 = moderately active, 2 = inactive, 3 = missing
- wt71: weight in kilograms in 1971
- ht: height in centimeters in 1971

```
nhefs_dat <- causaldata::nhefs_complete
summary(nhefs_dat)
#> seqn qsmk death yrdth
#> Min. : 233 Min. :0.0000 Min. :0.0000 Min. :83.00
#> 1st Qu.:10625 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:85.00
#> Median :20702 Median :0.0000 Median :0.0000 Median :88.00
#> Mean :16639 Mean :0.2573 Mean :0.1858 Mean :87.64
#> 3rd Qu.:22771 3rd Qu.:1.0000 3rd Qu.:0.0000 3rd Qu.:90.00
#> Max. :25061 Max. :1.0000 Max. :1.0000 Max. :92.00
#> NA's :1275
#> modth dadth sbp dbp sex
#> Min. : 1.000 Min. : 1.0 Min. : 87.0 Min. : 47.00 0:762
#> 1st Qu.: 3.000 1st Qu.: 8.0 1st Qu.:115.0 1st Qu.: 70.00 1:804
#> Median : 6.000 Median :16.0 Median :126.0 Median : 77.00
#> Mean : 6.383 Mean :16.1 Mean :128.6 Mean : 77.74
#> 3rd Qu.:10.000 3rd Qu.:24.5 3rd Qu.:139.0 3rd Qu.: 85.00
#> Max. :12.000 Max. :31.0 Max. :229.0 Max. :130.00
#> NA's :1271 NA's :1271 NA's :29 NA's :33
#> age race income marital school
#> Min. :25.00 0:1360 Min. :11.00 Min. :2.000 Min. : 0.00
#> 1st Qu.:33.00 1: 206 1st Qu.:17.00 1st Qu.:2.000 1st Qu.:10.00
#> Median :43.00 Median :19.00 Median :2.000 Median :12.00
#> Mean :43.66 Mean :17.99 Mean :2.496 Mean :11.17
#> 3rd Qu.:53.00 3rd Qu.:20.00 3rd Qu.:2.000 3rd Qu.:12.00
#> Max. :74.00 Max. :22.00 Max. :8.000 Max. :17.00
#> NA's :59
#> education ht wt71 wt82 wt82_71
#> 1:291 Min. :142.9 Min. : 39.58 Min. : 35.38 Min. :-41.280
#> 2:340 1st Qu.:161.8 1st Qu.: 59.53 1st Qu.: 61.69 1st Qu.: -1.478
#> 3:637 Median :168.2 Median : 69.23 Median : 72.12 Median : 2.604
#> 4:121 Mean :168.7 Mean : 70.83 Mean : 73.47 Mean : 2.638
#> 5:177 3rd Qu.:175.3 3rd Qu.: 79.80 3rd Qu.: 83.46 3rd Qu.: 6.690
#> Max. :198.1 Max. :151.73 Max. :136.53 Max. : 48.538
#>
#> birthplace smokeintensity smkintensity82_71 smokeyrs
#> Min. : 1.00 Min. : 1.00 Min. :-80.000 Min. : 1.00
#> 1st Qu.:22.00 1st Qu.:10.00 1st Qu.:-10.000 1st Qu.:15.00
#> Median :34.00 Median :20.00 Median : -1.000 Median :24.00
#> Mean :31.67 Mean :20.53 Mean : -4.633 Mean :24.59
#> 3rd Qu.:42.00 3rd Qu.:30.00 3rd Qu.: 1.000 3rd Qu.:33.00
#> Max. :56.00 Max. :80.00 Max. : 50.000 Max. :64.00
#> NA's :90
#> asthma bronch tb hf
#> Min. :0.00000 Min. :0.00000 Min. :0.00000 Min. :0.000000
#> 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.000000
#> Median :0.00000 Median :0.00000 Median :0.00000 Median :0.000000
#> Mean :0.04853 Mean :0.08365 Mean :0.01341 Mean :0.005109
#> 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.000000
#> Max. :1.00000 Max. :1.00000 Max. :1.00000 Max. :1.000000
#>
#> hbp pepticulcer colitis hepatitis
#> Min. :0.000 Min. :0.0000 Min. :0.00000 Min. :0.00000
#> 1st Qu.:0.000 1st Qu.:0.0000 1st Qu.:0.00000 1st Qu.:0.00000
#> Median :1.000 Median :0.0000 Median :0.00000 Median :0.00000
#> Mean :1.059 Mean :0.1015 Mean :0.03448 Mean :0.01788
#> 3rd Qu.:2.000 3rd Qu.:0.0000 3rd Qu.:0.00000 3rd Qu.:0.00000
#> Max. :2.000 Max. :1.0000 Max. :1.00000 Max. :1.00000
#>
#> chroniccough hayfever diabetes polio
#> Min. :0.00000 Min. :0.00000 Min. :0.0000 Min. :0.00000
#> 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.0000 1st Qu.:0.00000
#> Median :0.00000 Median :0.00000 Median :0.0000 Median :0.00000
#> Mean :0.05109 Mean :0.08621 Mean :0.9898 Mean :0.01405
#> 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:2.0000 3rd Qu.:0.00000
#> Max. :1.00000 Max. :1.00000 Max. :2.0000 Max. :1.00000
#>
#> tumor nervousbreak alcoholpy alcoholfreq
#> Min. :0.00000 Min. :0.00000 Min. :0.0000 Min. :0.000
#> 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:1.0000 1st Qu.:1.000
#> Median :0.00000 Median :0.00000 Median :1.0000 Median :2.000
#> Mean :0.02363 Mean :0.02746 Mean :0.8787 Mean :1.913
#> 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:1.0000 3rd Qu.:3.000
#> Max. :1.00000 Max. :1.00000 Max. :2.0000 Max. :5.000
#>
#> alcoholtype alcoholhowmuch pica headache
#> Min. :1.000 Min. : 1.000 Min. :0.000 Min. :0.0000
#> 1st Qu.:1.000 1st Qu.: 2.000 1st Qu.:0.000 1st Qu.:0.0000
#> Median :3.000 Median : 2.000 Median :0.000 Median :1.0000
#> Mean :2.466 Mean : 3.293 Mean :0.986 Mean :0.6328
#> 3rd Qu.:4.000 3rd Qu.: 4.000 3rd Qu.:2.000 3rd Qu.:1.0000
#> Max. :4.000 Max. :48.000 Max. :2.000 Max. :1.0000
#> NA's :397
#> otherpain weakheart allergies nerves
#> Min. :0.0000 Min. :0.00000 Min. :0.00000 Min. :0.000
#> 1st Qu.:0.0000 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.000
#> Median :0.0000 Median :0.00000 Median :0.00000 Median :0.000
#> Mean :0.2433 Mean :0.02235 Mean :0.06322 Mean :0.145
#> 3rd Qu.:0.0000 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.000
#> Max. :1.0000 Max. :1.00000 Max. :1.00000 Max. :1.000
#>
#> lackpep hbpmed boweltrouble wtloss
#> Min. :0.00000 Min. :0.000 Min. :0.000 Min. :0.00000
#> 1st Qu.:0.00000 1st Qu.:0.000 1st Qu.:0.000 1st Qu.:0.00000
#> Median :0.00000 Median :1.000 Median :1.000 Median :0.00000
#> Mean :0.05045 Mean :1.015 Mean :1.046 Mean :0.02618
#> 3rd Qu.:0.00000 3rd Qu.:2.000 3rd Qu.:2.000 3rd Qu.:0.00000
#> Max. :1.00000 Max. :2.000 Max. :2.000 Max. :1.00000
#>
#> infection active exercise birthcontrol pregnancies
#> Min. :0.000 0:702 0:300 Min. :0.000 Min. : 1.000
#> 1st Qu.:0.000 1:715 1:661 1st Qu.:0.000 1st Qu.: 2.000
#> Median :0.000 2:149 2:605 Median :1.000 Median : 3.000
#> Mean :0.145 Mean :1.081 Mean : 3.694
#> 3rd Qu.:0.000 3rd Qu.:2.000 3rd Qu.: 5.000
#> Max. :1.000 Max. :2.000 Max. :15.000
#> NA's :866
#> cholesterol hightax82 price71 price82
#> Min. : 78.0 Min. :0.0000 Min. :1.507 Min. :1.452
#> 1st Qu.:189.0 1st Qu.:0.0000 1st Qu.:2.037 1st Qu.:1.740
#> Median :216.0 Median :0.0000 Median :2.168 Median :1.815
#> Mean :219.9 Mean :0.1653 Mean :2.138 Mean :1.806
#> 3rd Qu.:245.0 3rd Qu.:0.0000 3rd Qu.:2.242 3rd Qu.:1.868
#> Max. :416.0 Max. :1.0000 Max. :2.693 Max. :2.103
#> NA's :16 NA's :90 NA's :90 NA's :90
#> tax71 tax82 price71_82 tax71_82
#> Min. :0.5249 Min. :0.2200 Min. :-0.2027 Min. :0.0360
#> 1st Qu.:0.9449 1st Qu.:0.4399 1st Qu.: 0.2010 1st Qu.:0.4610
#> Median :1.0498 Median :0.5060 Median : 0.3360 Median :0.5440
#> Mean :1.0580 Mean :0.5058 Mean : 0.3324 Mean :0.5522
#> 3rd Qu.:1.1548 3rd Qu.:0.5719 3rd Qu.: 0.4438 3rd Qu.:0.6220
#> Max. :1.5225 Max. :0.7479 Max. : 0.6121 Max. :0.8844
#> NA's :90 NA's :90 NA's :90 NA's :90
#> id censored older
#> Min. : 1.0 Min. :0 Min. :0.0000
#> 1st Qu.: 414.2 1st Qu.:0 1st Qu.:0.0000
#> Median : 824.5 Median :0 Median :0.0000
#> Mean : 821.0 Mean :0 Mean :0.2989
#> 3rd Qu.:1228.8 3rd Qu.:0 3rd Qu.:1.0000
#> Max. :1629.0 Max. :0 Max. :1.0000
#>
```

We will assume that the set of confounders in \(\boldsymbol{Z}\) includes sex, race, age, education, number of cigarettes smoked per year, the number of years smoked, level of physical activity, and baseline weight in 1971. This equivalent to assuming that the counterfactual weight gain is independent of the exposure conditional on these variables. In other words, we are assuming the following directed acyclic graph:

For the specific forms of the conditional expectations required in the outcome we assume a linear regression model with both linear and quadratic forms of the continuous covariates. We can fit this as

```
m <- glm(wt82_71 ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat)
summary(m)
#>
#> Call:
#> glm(formula = wt82_71 ~ qsmk + sex + race + age + I(age^2) +
#> as.factor(education) + smokeintensity + I(smokeintensity^2) +
#> smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
#> wt71 + I(wt71^2), data = nhefs_dat)
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -1.6586176 4.3137734 -0.384 0.700666
#> qsmk 3.4626218 0.4384543 7.897 5.36e-15 ***
#> sex1 -1.4650496 0.4683410 -3.128 0.001792 **
#> race1 0.5864117 0.5816949 1.008 0.313560
#> age 0.3626624 0.1633431 2.220 0.026546 *
#> I(age^2) -0.0061377 0.0017263 -3.555 0.000389 ***
#> as.factor(education)2 0.8185263 0.6067815 1.349 0.177546
#> as.factor(education)3 0.5715004 0.5561211 1.028 0.304273
#> as.factor(education)4 1.5085173 0.8323778 1.812 0.070134 .
#> as.factor(education)5 -0.1708264 0.7413289 -0.230 0.817786
#> smokeintensity 0.0651533 0.0503115 1.295 0.195514
#> I(smokeintensity^2) -0.0010468 0.0009373 -1.117 0.264261
#> smokeyrs 0.1333931 0.0917319 1.454 0.146104
#> I(smokeyrs^2) -0.0018270 0.0015438 -1.183 0.236818
#> as.factor(exercise)1 0.3206824 0.5349616 0.599 0.548961
#> as.factor(exercise)2 0.3628786 0.5589557 0.649 0.516300
#> as.factor(active)1 -0.9429574 0.4100208 -2.300 0.021593 *
#> as.factor(active)2 -0.2580374 0.6847219 -0.377 0.706337
#> wt71 0.0373642 0.0831658 0.449 0.653297
#> I(wt71^2) -0.0009158 0.0005235 -1.749 0.080426 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for gaussian family taken to be 53.59474)
#>
#> Null deviance: 97176 on 1565 degrees of freedom
#> Residual deviance: 82857 on 1546 degrees of freedom
#> AIC: 10701
#>
#> Number of Fisher Scoring iterations: 2
```

To perform regression standardization to estimate the causal effect
we use `standardize_glm`

. We must specify the same outcome
regression model as a formula, provide the data, describe which values
of the exposure we wish to estimate the counterfactual means, specify
which contrasts we want, and specify the reference level for the
contrasts. The following command estimates that model and we obtain the
group-wise estimates, the difference, and the ratio.

```
m2 <- standardize_glm(wt82_71 ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat,
values = list(qsmk = c(0,1)),
contrasts = c("difference", "ratio"),
reference = 0)
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
m2
#> Outcome formula: wt82_71 ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: gaussian
#> Outcome link function: identity
#> Exposure: qsmk
#>
#> Tables:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.75 0.217 1.32 2.17
#> 2 1 5.21 0.420 4.39 6.03
#>
#> Reference level: qsmk = 0
#> Contrast: difference
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.00 0.000 0.00 0.00
#> 2 1 3.46 0.466 2.55 4.38
#>
#> Reference level: qsmk = 0
#> Contrast: ratio
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.00 0.000 1.00 1.00
#> 2 1 2.98 0.435 2.13 3.83
plot(m2)
```

The output from the model shows the estimated potential outcome means in each exposure level, the difference and ratio thereof, and the associated standard error estimates, confidence intervals, and p-values. Inference is done using the sandwich method for variance calculation. Under the assumptions that the outcome model is correctly specified and contains all confounders, these are consistent estimates of the causal effects of interest. We can also get plots of the effects by using the plot function.

To obtain the estimates and inference for the different contrasts as
a tidy data frame, suitable for saving as a data table or using in
downstream analyses or reports, we provide the `tidy`

function:

```
(tidy(m2) -> tidy_m2) |> print()
#> qsmk Estimate Std.Error lower.0.95 upper.0.95 contrast transform
#> 1 0 1.747216 0.2174047 1.321111 2.173322 none identity
#> 2 1 5.209838 0.4199393 4.386772 6.032904 none identity
#> 3 0 0.000000 0.0000000 0.000000 0.000000 difference identity
#> 4 1 3.462622 0.4660848 2.549112 4.376131 difference identity
#> 5 0 1.000000 0.0000000 1.000000 1.000000 ratio identity
#> 6 1 2.981793 0.4349643 2.129279 3.834308 ratio identity
```

To obtain doubly robust inference we use the following command. Note that we now specify a model for the exposure, the propensity score model.

```
m2_dr <- standardize_glm_dr(formula_outcome = wt82_71 ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
formula_exposure = qsmk ~ sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat,
values = list(qsmk = c(0,1)),
contrast = c("difference", "ratio"),
reference = 0)
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
m2_dr
#> Doubly robust estimator with:
#>
#> Exposure formula: qsmk ~ sex + race + age + I(age^2) + as.factor(education) + smokeintensity +
#> I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) + as.factor(exercise) +
#> as.factor(active) + wt71 + I(wt71^2)
#> Exposure link function: logit
#> Outcome formula: wt82_71 ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: gaussian
#> Outcome link function: identity
#> Exposure: qsmk
#>
#> Tables:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.76 0.218 1.34 2.19
#> 2 1 5.19 0.438 4.33 6.04
#>
#> Reference level: qsmk = 0
#> Contrast: difference
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.00 0.00 0.00 0.00
#> 2 1 3.42 0.48 2.48 4.37
#>
#> Reference level: qsmk = 0
#> Contrast: ratio
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.00 0.000 1.0 1.00
#> 2 1 2.94 0.431 2.1 3.79
(tidy(m2_dr) -> tidy_m2_dr) |> print()
#> qsmk Estimate Std.Error lower.0.95 upper.0.95 contrast transform
#> 1 0 1.762981 0.2176810 1.336334 2.189628 none identity
#> 2 1 5.186772 0.4378466 4.328609 6.044936 none identity
#> 3 0 0.000000 0.0000000 0.000000 0.000000 difference identity
#> 4 1 3.423792 0.4804026 2.482220 4.365363 difference identity
#> 5 0 1.000000 0.0000000 1.000000 1.000000 ratio identity
#> 6 1 2.942047 0.4310190 2.097265 3.786829 ratio identity
```

Based on these results, we can report that the estimated effect of smoking on average weight change as measured by the difference in potential outcome means is 3.46 (2.55 to 4.38) and as measured by the ratio of potential outcome means is 2.98 (2.13 to 3.83). Using the doubly-robust estimation method, we obtain similarly 3.42 (2.48 to 4.37) and as measured by the ratio of potential outcome means is 2.94 (2.10 to 3.79). In particular, if we believe that the necessary assumptions are valid, we can conclude that smoking causes an increase in weight which the numeric summaries indicate that smoking increases the weight change over 11 years by about 3.5 pounds (additive scale) or smoking increases the weight change over 11 years by a multiplicative factor of about 3.

The doubly-robust method requires specification of an exposure model
in addition to the specification of the outcome model. Both of these
estimated models can be accessed and inspected by looking at the
`fit_outcome`

and `fit_exposure`

elements of the
return object:

```
m2_dr$res$fit_outcome
#>
#> Call: fit_glm(formula = formula_outcome, family = family_outcome, data = data,
#> response = "outcome", weights = weights)
#>
#> Coefficients:
#> (Intercept) qsmk sex1
#> -1.931e+00 3.424e+00 -1.989e+00
#> race1 age I(age^2)
#> 4.707e-01 6.207e-01 -8.678e-03
#> as.factor(education)2 as.factor(education)3 as.factor(education)4
#> 1.935e+00 8.366e-01 2.564e+00
#> as.factor(education)5 smokeintensity I(smokeintensity^2)
#> 8.339e-01 1.130e-01 -2.240e-03
#> smokeyrs I(smokeyrs^2) as.factor(exercise)1
#> 1.034e-01 -1.473e-03 7.865e-01
#> as.factor(exercise)2 as.factor(active)1 as.factor(active)2
#> 1.231e+00 -8.251e-01 -4.576e-01
#> wt71 I(wt71^2)
#> -1.221e-01 8.616e-06
#>
#> Degrees of Freedom: 1565 Total (i.e. Null); 1546 Residual
#> Null Deviance: 212700
#> Residual Deviance: 175100 AIC: 11030
m2_dr$res$fit_exposure
#>
#> Call: fit_glm(formula = formula_exposure, family = family_exposure,
#> data = data, response = "exposure", weights = weights)
#>
#> Coefficients:
#> (Intercept) sex1 race1
#> -2.2425191 -0.5274782 -0.8392636
#> age I(age^2) as.factor(education)2
#> 0.1212052 -0.0008246 -0.0287755
#> as.factor(education)3 as.factor(education)4 as.factor(education)5
#> 0.0864318 0.0636010 0.4759606
#> smokeintensity I(smokeintensity^2) smokeyrs
#> -0.0772704 0.0010451 -0.0735966
#> I(smokeyrs^2) as.factor(exercise)1 as.factor(exercise)2
#> 0.0008441 0.3548405 0.3957040
#> as.factor(active)1 as.factor(active)2 wt71
#> 0.0319445 0.1767840 -0.0152357
#> I(wt71^2)
#> 0.0001352
#>
#> Degrees of Freedom: 1565 Total (i.e. Null); 1547 Residual
#> Null Deviance: 1786
#> Residual Deviance: 1677 AIC: 1715
```

This is useful when we want to inspect the distribution of the propensity scores by exposure status. This is an important thing to check in practice, we are looking for any practical violations of the positivity assumption.

```
hist(m2_dr$res$fit_exposure$fitted[nhefs_dat$qsmk == 0],
xlim = c(0, 1), main = "qsmk = 0", xlab = "estimated propensity")
```

```
hist(m2_dr$res$fit_exposure$fitted[nhefs_dat$qsmk == 1],
xlim = c(0, 1), main = "qsmk = 1", xlab = "estimated propensity")
```

The function `standardize_glm`

and its doubly-robust
counterpart `standardize_glm_dr`

support any outcome variable
type the can be used with `glm`

. This includes binary, count,
and more. Just like in `glm`

, we specify the model we would
like to use for the outcome by specifying the `family`

argument of `standardize_glm`

or the
`family_outcome`

argument of `standardize_glm_dr`

.
By default this is `"gaussian"`

which corresponds to linear
regression. For a binary outcome we might like to use
`"binomial"`

which corresponds to logistic regression. Let us
see how it works using the a binary outcome that we create by
dichotomizing the weight change at 0.

```
nhefs_dat$gained_weight <- 1.0 * (nhefs_dat$wt82_71 > 0)
m3 <- standardize_glm(gained_weight ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat,
family = "binomial",
values = list(qsmk = c(0,1)),
contrasts = c("difference", "ratio"),
reference = 0)
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
m3
#> Outcome formula: gained_weight ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: quasibinomial
#> Outcome link function: logit
#> Exposure: qsmk
#>
#> Tables:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.639 0.0140 0.611 0.666
#> 2 1 0.772 0.0195 0.733 0.810
#>
#> Reference level: qsmk = 0
#> Contrast: difference
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.000 0.0000 0.0000 0.000
#> 2 1 0.133 0.0236 0.0866 0.179
#>
#> Reference level: qsmk = 0
#> Contrast: ratio
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.00 0.0000 1.00 1.00
#> 2 1 1.21 0.0398 1.13 1.29
```

Here we interpret the estimates in terms of probabilities, or the risk of gaining weight over 11 years. In particular, smoking causes a 13% additive increase in the risk of gaining weight, or a 1.2 fold multiplicative increase in the risk of gaining weight. With binary outcomes, we can use transformations to get different contrasts, and also to improve the performance of the ratio contrast. By using the log transformation combined with the difference contrast, our estimates become the log risk ratios. We can exponential them to get the risk ratios, but often it is better to construct confidence intervals of ratios on the log scale and back transform.

```
m3_log <- standardize_glm(gained_weight ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat,
family = "binomial",
values = list(qsmk = c(0,1)),
contrasts = c("difference"),
transforms = c("log"),
reference = 0)
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
m3_log
#> Outcome formula: gained_weight ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: quasibinomial
#> Outcome link function: logit
#> Exposure: qsmk
#>
#> Tables:
#> Transform:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 -0.448 0.0220 -0.492 -0.405
#> 2 1 -0.259 0.0253 -0.309 -0.210
#>
#> Transform:
#> Reference level: qsmk = 0
#> Contrast: difference
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.000 0.0000 0.000 0.000
#> 2 1 0.189 0.0329 0.125 0.254
```

The estimates reported in the table are log risk ratios, which we now back transform by exponentiating and compare to the ratios estimated previously.

```
tm3_log <- tidy(m3_log)
tm3_log$rr <- exp(tm3_log$Estimate)
tm3_log$rr.lower <- exp(tm3_log$lower.0.95)
tm3_log$rr.upper <- exp(tm3_log$upper.0.95)
subset(tm3_log, contrast == "difference" & qsmk == 1)
#> qsmk Estimate Std.Error lower.0.95 upper.0.95 contrast transform rr
#> 4 1 0.1890909 0.03292713 0.1245549 0.2536269 difference log 1.208151
#> rr.lower rr.upper
#> 4 1.132644 1.288691
m3
#> Outcome formula: gained_weight ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: quasibinomial
#> Outcome link function: logit
#> Exposure: qsmk
#>
#> Tables:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.639 0.0140 0.611 0.666
#> 2 1 0.772 0.0195 0.733 0.810
#>
#> Reference level: qsmk = 0
#> Contrast: difference
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.000 0.0000 0.0000 0.000
#> 2 1 0.133 0.0236 0.0866 0.179
#>
#> Reference level: qsmk = 0
#> Contrast: ratio
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.00 0.0000 1.00 1.00
#> 2 1 1.21 0.0398 1.13 1.29
```

In this case, the estimates and inference using transformation are nearly identical to those without transformation.

Other available transformations include the odds, and logit (log odds). These can be used to estimate causal odds ratios, untransformed and transformed, respectively.

```
m3_odds <- standardize_glm(gained_weight ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat,
family = "binomial",
values = list(qsmk = c(0,1)),
contrasts = c("ratio"),
transforms = c("odds"),
reference = 0)
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
m3_odds
#> Outcome formula: gained_weight ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: quasibinomial
#> Outcome link function: logit
#> Exposure: qsmk
#>
#> Tables:
#> Transform:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.77 0.108 1.56 1.98
#> 2 1 3.38 0.374 2.64 4.11
#>
#> Transform:
#> Reference level: qsmk = 0
#> Contrast: ratio
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 1.00 0.000 1.00 1.00
#> 2 1 1.91 0.238 1.44 2.38
m3_logit <- standardize_glm(gained_weight ~ qsmk + sex + race + age + I(age^2) +
as.factor(education) + smokeintensity + I(smokeintensity^2) +
smokeyrs + I(smokeyrs^2) + as.factor(exercise) + as.factor(active) +
wt71 + I(wt71^2),
data = nhefs_dat,
family = "binomial",
values = list(qsmk = c(0,1)),
contrasts = c("difference"),
transforms = c("logit"),
reference = 0)
#> Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts argument
#> ignored
m3_logit
#> Outcome formula: gained_weight ~ qsmk + sex + race + age + I(age^2) + as.factor(education) +
#> smokeintensity + I(smokeintensity^2) + smokeyrs + I(smokeyrs^2) +
#> as.factor(exercise) + as.factor(active) + wt71 + I(wt71^2)
#> Outcome family: quasibinomial
#> Outcome link function: logit
#> Exposure: qsmk
#>
#> Tables:
#> Transform:
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.569 0.0609 0.45 0.689
#> 2 1 1.217 0.1108 1.00 1.434
#>
#> Transform:
#> Reference level: qsmk = 0
#> Contrast: difference
#> qsmk Estimate Std.Error lower.0.95 upper.0.95
#> 1 0 0.000 0.000 0.000 0.000
#> 2 1 0.648 0.124 0.404 0.892
m3_logOR <- tidy(m3_logit) |>
subset(contrast == "difference" & qsmk == 1)
sprintf("%.2f (%.2f to %.2f)", exp(m3_logOR$Estimate),
exp(m3_logOR$lower.0.95), exp(m3_logOR$upper.0.95))
#> [1] "1.91 (1.50 to 2.44)"
```

In this case the estimate is identical, but the confidence interval is slightly different because it is symmetric on the odds ratio scale in the first case, and symmetric on the log odds ratio scale in the transformed case.

Gabriel, Erin E, Michael C Sachs, Torben Martinussen, Ingeborg
Waernbaum, Els Goetghebeur, Stijn Vansteelandt, and Arvid Sjölander.
2024. “Inverse Probability of Treatment Weighting with Generalized
Linear Outcome Models for Doubly Robust Estimation.”
*Statistics in Medicine* 43 (3): 534–47.

Sjölander, Arvid. 2016. “Regression Standardization with the r
Package stdReg.” *European Journal of Epidemiology* 31:
563–74.

———. 2018. “Estimation of Causal Effect Measures with the
r-Package stdReg.” *European Journal of Epidemiology* 33:
847–58.

Witte, Janine, and Vanessa Didelez. 2019. “Covariate Selection
Strategies for Causal Inference: Classification and Comparison.”
*Biometrical Journal* 61 (5): 1270–89.