# The Art of Influence

### A Practical Guide to Working with Influence Functions

#### Klaus Kähler Holst

# Influence functions

Estimators that have parametric convergence rates can often be fully characterized by their influence function (IF), also referred to as an influence curve or canonical gradient (Bickel et al. 1998; Vaart 1998). The IF allows for the direct estimation of properties of the estimator, including its asymptotic variance. Moreover, estimates of the IF enable the simple combination and transformation of estimators into new ones. This vignette describes how to estimate and manipulate IFs using the R-package `lava` (K. K. Holst and Budtz-Jørgensen 2013).

Formally, let \(Z_1,\ldots,Z_n\) be iid \(k\)-dimensional stochastic variables, \(Z_i=(Y_{i},A_{i},W_{i})\sim P_{0}\), and \(\widehat{\theta}\) a consistent estimator for the parameter \(\theta\in\mathbb{R}^p\). When \(\widehat{\theta}\) is a regular and asymptotic linear (RAL) estimator, it has a unique iid decomposition \[\begin{align*} \sqrt{n}(\widehat{\theta}-\theta) = \frac{1}{\sqrt{n}}\sum_{i=1}^n \operatorname{IC}(Z_i; P_{0}) + o_{P}(1), \end{align*}\] where the function \(\operatorname{IC}\) is the unique Influence Function s.t. \(\mathbb{E}\{\operatorname{IC}(Z_{i}; P_{0})\}=0\) and \(\mathbb{V}\!\text{ar}\{\operatorname{IC}(Z_{i}; P_{0})^{2}\}<\infty\) (Tsiatis 2006; Vaart 1998). The influence function thus fully characterizes the asymptotic behaviour of the estimator and by the central limit theorem it follows that the estimator converges weakly to a Gaussian distribution \[ \sqrt{n}(\widehat{\theta}-\theta) \overset{\mathcal{D}}{\longrightarrow} \mathcal{N}(0, \mathbb{V}\!\text{ar}\{\operatorname{IC}(Z; P_{0})\}), \] where the empirical variance of the plugin estimator, \(\mathbb{P}_{n}\operatorname{IC}(Z; \widehat{P})^{\otimes 2} = \frac{1}{n}\sum_{i=1}^n \operatorname{IC}(Z_{i}; \widehat{P})\operatorname{IC}(Z_{i}; \widehat{P})^{\top}\) can be used to obtain a consistent estimate of the asymptotic variance. Note, in practice the estimate \(\widehat{P}\) used in the plugin-estimate, needs only to capture the parts of the distribution of \(Z\) that is necessary to evaluate the IF. In some cases this nuisance parameter can be estimated using flexible machine learning components and in other (parametric) cases be derived directly from \(\widehat{\theta}\).

The IFs are easily derived for the parameters of many parametric statistical models as illustrated in the next example sections. More generally, the IF can also be derived for a smooth target parameter \(\Psi: \mathcal{P}\to\mathbb{R}\) where \(\mathcal{P}\) is a family of probability distributions forming the statistical model, which often can be left completely non-parametric. Formally, the parameter must be pathwise differentiable see (Vaart 1998) in the sense that there exists linear bounded function \(\dot\Psi\colon L_{2}(P_{0})\to\mathbb{R}\) such that \([\Psi(P_{t}) - \Psi(P_{0}))]t^{-1} \to \dot\Psi(P_{0})(g)\) as \(t\to 0\) for any parametric submodel \(P_t\) with score model \(g(z)= \partial/(\partial t) \log (p_t)(z)|_{t=0}\). Riesz’s representation theorem then tells us that the directional derivative has a unique representer, \(\phi_{P_{0}}\) lying in the closure of the submodel score space (the tangent space), s.t. \[\begin{align*} \dot\Psi(P_0)(g) = \langle\phi_{P_0}, g\rangle = \int \phi_{P_0}(Z)g(X)\,dP_0 \end{align*}\] The unique representer is exactly the IF which can be found by solving the above integral equation. For more details on how to derive influence functions, we refer to (Laan and Rose 2011; Hines et al. 2022).

As an example we might be interested in the target parameter \(\Psi(P) = \mathbb{E}_P(Z)\) which can be shown to have the unique (and thereby efficient) influence function \(Z\mapsto Z-\mathbb{E}_P(Z)\) under the non-parametric model. Another target parameter could be \(\Psi_{a}(P) = \mathbb{E}_{P}[\mathbb{E}_{P}(Y\mid A=a, W)]\) which is often a key interest in causal inference and which has the IF \[\begin{align*} \operatorname{IC}(Y,A,W; P) = \frac{\mathbf{1}(A=a)}{\mathbb{P}(A=a\mid W)}(Y-\mathbb{E}_{P}[Y\mid A=a,W]) + \mathbb{E}_{P}[Y\mid A=a,W] - \Psi_{a}(P) \end{align*}\] See section on average treatment effects.

# Working with influence functions with `lava::estimate`

The main functions for working with influence functions are

- `estimate` which prepares a model object and estimates the IF and corresponding robust standard errors. Can also be used to transform model parameters by application of the Delta Theorem.
- `merge` method for combining estimates via their estimated IFs
- `IC`, `coef`, `vcov` methods to extract the estimated IF, coefficients, and asymptotic covariance matrix
- `subset` for extracting subset of parameters and IFs
- `transform` method for transforming parameter estimates
- `labels` for renaming parameters

The `estimate` function is the primary tool for obtaining parameter estimates and related information. It returns an object of the class type `estimate`, which is a general container for holding information about estimated parameters. The estimate function takes as input either a model object (the first argument `x`), or a parameter vector and corresponding influence function (IF) matrix specified using the `coef` and `IC` arguments. If the primary goal is to apply the delta method or test linear hypotheses, it is also possible to provide the asymptotic variance estimate via the `vcov` argument, without specifying the IC matrix.

    estimate(x=, ...)
    estimate(coef=, IC=, ...)
    estimate(coef=, vcov=, ...)

A typical call could look like

    merge(subset(estimate(x), 1), estimate(coef=p, IC=ic, id=id)) |>
      transform(function(x) c(exp(x), exp(x[1]))) |> # parameter transformation
      labels(c("a", "b")) # rename parameters

Direct calculations is also possible, if `a` and `b` are estimate objects we can obtain new transformed parameter estimates with the syntax

    c(exp(b)^0.5, exp(b * a) / (1 + exp(-b)))

with the necessary derivatives exactly calculated automatically. We refer to section IF building blocks for more details.

# Examples

To illustrate the methods we consider data arising from the model \(Y_{ij} \sim Bernoulli\{\operatorname{expit}(X_{ij} + A_{i} + W_{i})\}, A_{i} \sim Bernoulli\{\operatorname{expit}(W_{i})\}\) with independent covariates \(X_{ij}\sim\mathcal{N}(0,1), W_{i}\sim\mathcal{N}(0,1)\).

    m <- lvm() |>
      regression(y1 ~ x1 + a + w) |>
      regression(y2 ~ x2 + a + w) |>
      regression(y3 ~ x3 + a + w) |>
      regression(y4 ~ x4 + a + w) |>
      regression(a ~ w) |>
      distribution(~ y1 + y2 + y3 + y4 + a, value = binomial.lvm()) |>
      distribution(~id, value = Sequence.lvm(integer = TRUE))

We simulate from the model where \(Y_3\) is only observed for half of the subjects

    n <- 4e2
    dw <- sim(m, n, seed = 1) |>
      transform(y3 = y3 * ifelse(id > n / 2, NA, 1))
    Print(dw)
    #>     y1 x1       a w        y2 x2       y3 x3       y4 x4        id
    #> 1   1  -0.6265  1  1.0744  0  -1.08691 1  -1.5570  1   0.34419  1
    #> 2   1   0.1836  1  1.8957  1  -1.82608 1   1.9232  1   0.01272  2
    #> 3   0  -0.8356  0 -0.6030  0   0.99528 0  -1.8568  0  -0.87345  3
    #> 4   0   1.5953  0 -0.3909  1  -0.01186 0  -2.1061  1   0.34280  4
    #> 5   0   0.3295  1 -0.4162  0  -0.59963 1   0.6976  1  -0.17739  5
    #> ---
    #> 396 0  -0.92431 0 -1.02939 0  -0.30825 NA -1.05037 0  -0.008056 396
    #> 397 1   1.59291 1 -0.01093 1   0.01552 NA  1.63787 1   1.033784 397
    #> 398 0   0.04501 0 -1.22499 0  -0.44232 NA -1.20733 0  -0.799127 398
    #> 399 0  -0.71513 0 -2.59611 0  -1.63801 NA -2.62616 0   1.004233 399
    #> 400 1   0.86522 1  1.16912 0  -0.64140 NA  0.01746 1  -0.311973 400
    ## Data in long format
    dl <- reshape(dw,
            varying = list(paste0("y",1:4),
                           paste0("x",1:4)),
            v.names=c("y", "x"), direction="long") |>
      na.omit()
    dl <- dl[order(dl$id), ]
    ## dl <- mets::fast.reshape(dw, varying = c("y", "x")) |> na.omit()
    Print(dl)
    #>       a w      id  time y x
    #> 1.1   1 1.074  1   1    1 -0.6265
    #> 1.2   1 1.074  1   2    0 -1.0869
    #> 1.3   1 1.074  1   3    1 -1.5570
    #> 1.4   1 1.074  1   4    1  0.3442
    #> 2.1   1 1.896  2   1    1  0.1836
    #> ---
    #> 399.2 0 -2.596 399 2    0 -1.6380
    #> 399.4 0 -2.596 399 4    0  1.0042
    #> 400.1 1  1.169 400 1    1  0.8652
    #> 400.2 1  1.169 400 2    0 -0.6414
    #> 400.4 1  1.169 400 4    1 -0.3120

## Example: population mean

Here we first consider the problem of estimating the IF of the mean. For a general transformation \(f: \mathbb{R}^k\to\mathbb{R}^p\) we have that \[ \sqrt{n}\{\mathbb{P}_{n}f(X) - \mathbb{E}[f(X)]\} = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} f(X_{i}) - \mathbb{E}[f(X)] \] and hence for the problem of estimating the proportion of the binary outcome \(Y_1\), the IF is given by \(\mathbf{1}(Y_{1}=1) - \mathbb{P}(Y_{1}=1)\).

To estimate this parameter and its IF we will use the `estimate` function

    inp <- as.matrix(dw[, c("y1", "y2")])
    e <- estimate(inp[, 1, drop = FALSE], type="mean")
    class(e)
    #> [1] "estimate"
    e
    #>    Estimate Std.Err   2.5%  97.5%    P-value
    #> y1     0.61 0.02439 0.5622 0.6578 4.435e-138

The reported standard errors from the `estimate` method are the robust standard errors obtained from the IF. The variance estimate and the parameters can be extracted with the `vcov` and `coef` methods. The IF itself can be extracted with the `IC` (or `influence`) method:

    IC(e) |> Print()
    #>     y1
    #> 1    0.39
    #> 2    0.39
    #> 3   -0.61
    #> 4   -0.61
    #> 5   -0.61
    #> ---
    #> 396 -0.61
    #> 397  0.39
    #> 398 -0.61
    #> 399 -0.61
    #> 400  0.39

It is also possible to simultaneously estimate the proportions of each of the two binary outcomes

    estimate(inp)
    #>    Estimate Std.Err   2.5%  97.5%    P-value
    #> y1    0.610 0.02439 0.5622 0.6578 4.435e-138
    #> y2    0.535 0.02494 0.4861 0.5839 4.316e-102

or alternatively the input can be a model object, here a `mlm` object:

    e <- lm(cbind(y1, y2) ~ 1, data = dw) |>
      estimate()
    IC(e) |> head()
    #>   y1:(Intercept) y2:(Intercept)
    #> 1           0.39         -0.535
    #> 2           0.39          0.465
    #> 3          -0.61         -0.535
    #> 4          -0.61          0.465
    #> 5          -0.61         -0.535
    #> 6          -0.61          0.465

Different methods are available for inspecting an `estimate` object

    summary(e)
    #> Call: estimate.default(f = FALSE, contrast = contrast, vcov = vcov(object),
    #>     coef = p)
    #> ────────────────────────────────────────────────────────────
    #>                Estimate Std.Err   2.5%  97.5%    P-value
    #> y1:(Intercept)    0.610 0.02439 0.5622 0.6578 4.435e-138
    #> y2:(Intercept)    0.535 0.02494 0.4861 0.5839 4.316e-102
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [y1:(Intercept)] = 0
    #>   [y2:(Intercept)] = 0
    #>
    #> chisq = 955.6986, df = 2, p-value < 2.2e-16
    ## extract parameter coefficients
    coef(e)
    #> y1:(Intercept) y2:(Intercept)
    #>          0.610          0.535
    ## ## Asymptotic (robust) variance estimate
    vcov(e)
    #>                y1:(Intercept) y2:(Intercept)
    #> y1:(Intercept)     5.9475e-04   0.0000841250
    #> y2:(Intercept)     8.4125e-05   0.0006219375
    ## Matrix with estimates and confidence limits
    estimate(e, null=0, level = 0.99) |> parameter()
    #>                Estimate    Std.Err      0.5%     99.5%       P-value
    #> y1:(Intercept)    0.610 0.02438750 0.5471820 0.6728180 4.434692e-138
    #> y2:(Intercept)    0.535 0.02493867 0.4707622 0.5992378 4.316104e-102
    ## Influence curve
    IC(e) |> head()
    #>   y1:(Intercept) y2:(Intercept)
    #> 1           0.39         -0.535
    #> 2           0.39          0.465
    #> 3          -0.61         -0.535
    #> 4          -0.61          0.465
    #> 5          -0.61         -0.535
    #> 6          -0.61          0.465
    ## Join estimates
    ee <- merge(e, e)
    ee
    #>                  Estimate Std.Err   2.5%  97.5%    P-value
    #> y1:(Intercept)      0.610 0.02439 0.5622 0.6578 4.435e-138
    #> y2:(Intercept)      0.535 0.02494 0.4861 0.5839 4.316e-102
    #> y1:(Intercept).1    0.610 0.02439 0.5622 0.6578 4.435e-138
    #> y2:(Intercept).1    0.535 0.02494 0.4861 0.5839 4.316e-102
    ## Forest plots
    plot(ee, null=0.5, digits=2)

## Example: generalized linear model

For a \(Z\)-estimator defined by the score equation \(E[U(Z; \theta)] = 0\), the IF is given by \[\begin{align*} IC(Z; \theta) = -\mathbb{E}\Big\{\frac{\partial}{\partial\theta^\top}U(\theta; Z)\Big\}^{-1}U(Z; \theta) \end{align*}\] In particular, for a maximum likelihood estimator the score, \(U\), is given by the partial derivative of the log-likelihood function.

As an example, we can obtain the estimates with robust standard errors for a logistic regression model:

    g <- glm(y1 ~ a + x1, data = dw, family = binomial)
    estimate(g)
    #>             Estimate Std.Err    2.5%   97.5%   P-value
    #> (Intercept)  -0.2687  0.1622 -0.5867 0.04931 9.772e-02
    #> a             1.5595  0.2428  1.0835 2.03545 1.348e-10
    #> x1            0.9728  0.1435  0.6916 1.25397 1.198e-11

We can compare that to the usual (non-robust) standard errors by supplying the model-based covariance matrix explicitly:

    estimate(g, vcov = vcov(g))
    #>             Estimate Std.Err    2.5%   97.5%   P-value
    #> (Intercept)  -0.2687  0.1589 -0.5802 0.04281 9.091e-02
    #> a             1.5595  0.2423  1.0846 2.03433 1.220e-10
    #> x1            0.9728  0.1396  0.6992 1.24634 3.177e-12
    # estimate(g, vcov = TRUE) alternative syntax to obtain model-based SEs

The IF can be extracted from the `estimate` object or directly from the model object

    IC(g) |> head()
    #>   (Intercept)          a         x1
    #> 1  0.09816353   3.715892 -0.8478763
    #> 2 -0.08203584   2.562573  0.7085752
    #> 3 -2.74896196   3.316937  1.8772112
    #> 4 -6.78328520   4.052090 -9.0268560
    #> 5  0.47533946 -11.818085 -4.1056905
    #> 6 -2.77584564   3.340948  1.8677174

The same estimates can be obtained with a cumulative link regression model which also generalizes to ordinal outcomes. Here we consider the proportional odds model given by \[\begin{align*} \log\left(\frac{\mathbb{P}(Y\leq j\mid x)}{1-\mathbb{P}(Y\leq j\mid x)}\right) = \alpha_{j} + \beta^{t}x, \quad j=1,\ldots,J \end{align*}\]

    ordreg(y1 ~ a + x1, dw, family=binomial(logit)) |> estimate()
    #>     Estimate Std.Err     2.5%  97.5%   P-value
    #> 0|1   0.2687  0.1622 -0.04932 0.5867 9.772e-02
    #> a     1.5595  0.2429  1.08349 2.0355 1.350e-10
    #> x1    0.9728  0.1435  0.69157 1.2540 1.200e-11

Note that the `sandwich::estfun` function from the `sandwich` library (Zeileis, Köll, and Graham 2020) can also estimate the IF for different parametric models, but does not provide the tools for combining and transforming these.

## Example: right-censored outcomess

To illustrate the methods on survival data we will use the Mayo Clinic Primary Biliary Cholangitis Data (Therneau and Grambsch 2000)

    library("survival")
    data(pbc, package="survival")

The Cox proportional hazards model can be fitted with the `mets::phreg` method which can estimate the IF for both the partial likelihood parameters and the baseline hazard. Here we fit a survival model with right-censored event times

    fit.phreg <- mets::phreg(Surv(time, status > 0) ~ age + sex, data = pbc)
    fit.phreg
    #> Call:
    #> mets::phreg(formula = Surv(time, status > 0) ~ age + sex, data = pbc)
    #>
    #>    n events
    #>  418    186
    #> coeffients:
    #>        Estimate       S.E.    dU^-1/2 P-value
    #> age   0.0220977  0.0070372  0.0072712  0.0017
    #> sexf -0.2999507  0.2022144  0.2097533  0.1380
    #>
    #> exp(coeffients):
    #>      Estimate    2.5%  97.5%
    #> age   1.02234 1.00834 1.0365
    #> sexf  0.74085 0.49843 1.1012
    IC(fit.phreg) |> head()
    #>              age       sexf
    #> [1,]  0.12691175  2.9551968
    #> [2,] -0.16011629 -4.3755455
    #> [3,]  0.19322595 -8.1786480
    #> [4,]  0.04668109  0.8548021
    #> [5,] -0.22936186  0.9721761
    #> [6,]  0.07015171  0.5644414

The IF for the baseline cumulative hazard at a specific time point \[\begin{align*} \Lambda_0(t) = \int_0^t \lambda_0(u)\,du, \end{align*}\] where \(\lambda_0(t)\) is the baseline hazard, can be estimated in similar way:

    baseline <- function(object, time, ...) {
      ic <- mets::IC(object, baseline = TRUE, time = time, ...)
      est <- mets::predictCumhaz(object$cumhaz, new.time = time)[1, 2]
      estimate(NULL, coef = est, IC = ic, labels = paste0("chaz:", time))
    }
    tt <- 2000
    baseline(fit.phreg, tt)
    #>           Estimate Std.Err    2.5%  97.5% P-value
    #> chaz:2000    0.178 0.07597 0.02913 0.3269 0.01911

The `estimate` and `IF` methods are also available for parametric survival models via `survival::survreg`, here a Weibull model:

    survival::survreg(Surv(time, status > 0) ~ age + sex, data = pbc, dist="weibull") |>
      estimate()
    #>             Estimate  Std.Err     2.5%     97.5%    P-value
    #> (Intercept)  9.02697 0.382437  8.27741  9.776530 3.521e-123
    #> age         -0.01919 0.006362 -0.03166 -0.006723  2.554e-03
    #> sexf         0.28170 0.174338 -0.06000  0.623392  1.061e-01
    #> scale        0.87751 0.070693  0.73896  1.016067  2.220e-35

## Example: random effects model / structural equation model

General structural equation models (SEMs) can be estimated with `lava::lvm`. Here we fit a random effects probit model \[ \mathbb{P}(Y_{ij} = 1 \mid U_{i}, W_{ij})=\Phi(\mu_{j} + \beta_{j} W_{ij} + U_{i}), \quad U_{i}\sim\mathcal{N}(0,\sigma_{u}^{2}),\quad j=1,2 \] to the simulated dataset

    sem <- lvm(y1 + y2 ~ 1 * u + w) |>
      latent(~ u) |>
      ordinal(K=2, ~ y1 + y2)
    semfit <- estimate(sem, data = dw)

    ## Robust standard errors
    estimate(semfit)
    #>      Estimate Std.Err    2.5%    97.5%   P-value
    #> y2   -0.21037 0.09391 -0.3944 -0.02630 2.509e-02
    #> u     0.36025 0.06659  0.2297  0.49075 6.295e-08
    #> y1~w  0.55425 0.06930  0.4184  0.69008 1.272e-15
    #> y2~w  0.59388 0.07510  0.4467  0.74108 2.623e-15
    #> u~~u -0.09496 0.07360 -0.2392  0.04929 1.970e-01

## Example: quantile

Let \(\beta\) denote the \(\tau\)th quantile of \(X\), with IF \[\begin{align*} \operatorname{IC}(x; P_{0}) = \tau - \mathbf{1}(x\leq \beta)f_{0}(\beta)^{-1} \end{align*}\]

where \(f_{0}\) is the density function of \(X\).

To calculate the variance estimate, an estimate of the density is thus needed which can be obtained by a kernel estimate. Alternatively, the resampling method of (Zeng and Lin 2008) can be applied. Here we use a kernel smoother (additional arguments to the `estimate` function are parsed on to `stats::density.default`) to estimate the quantiles and IF for the 25%, 50%, and 75% quantiles of \(W\) and \(X_1\)

    eq <- estimate(dw[, c("w", "x1")], type = "quantile", probs = c(0.25, 0.5, 0.75))
    eq
    #>        Estimate Std.Err    2.5%    97.5%   P-value
    #> w.25%  -0.81214 0.07277 -0.9548 -0.66951 6.390e-29
    #> w.50%  -0.11062 0.07201 -0.2518  0.03052 1.245e-01
    #> w.75%   0.67716 0.07784  0.5246  0.82973 3.353e-18
    #> x1.25% -0.57510 0.06340 -0.6994 -0.45084 1.177e-19
    #> x1.50% -0.02664 0.06078 -0.1458  0.09249 6.611e-01
    #> x1.75%  0.69590 0.06696  0.5647  0.82715 2.683e-25
    IC(eq) |> head()
    #>          w.25%     w.50%      w.75%    x1.25%    x1.50%     x1.75%
    #> [1,] 0.8402973  1.440254  2.6966215 -2.196201 -1.215663 -0.7732142
    #> [2,] 0.8402973  1.440254  2.6966215  0.732067  1.215663 -0.7732142
    #> [3,] 0.8402973 -1.440254 -0.8988738 -2.196201 -1.215663 -0.7732142
    #> [4,] 0.8402973 -1.440254 -0.8988738  0.732067  1.215663  2.3196427
    #> [5,] 0.8402973 -1.440254 -0.8988738  0.732067  1.215663 -0.7732142
    #> [6,] 0.8402973 -1.440254 -0.8988738 -2.196201 -1.215663 -0.7732142

# Combining influence functions

A key benefit of working with the IFs of estimators is that this allows for transforming or combining different estimates while easily deriving the resulting IF and thereby asymptotic distribution of the new estimator.

Let \(\widehat{\theta}_{1}, \ldots, \widehat{\theta}_{M}\) be \(M\) different estimators with decompositions \[\begin{align*} \sqrt{n}(\widehat{\theta}_{m}-\theta_{m}) = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \operatorname{IC}_m(Z_i; P_{0}) + o_{P}(1) \end{align*}\] based on iid data \(Z_1,\ldots,Z_n\). It then follows immediately (Vaart 1998 Theorem 18.10[vi]) that the joint distribution of \(\widehat{\theta} - {\theta} = (\widehat{\theta}_{1}^{\top},\ldots,\widehat{\theta}_{M}^{\top})^\top - ({\theta}_{1}^{\top},\ldots,{\theta}_{M}^{\top})^\top\) is given by \[\begin{align*} \sqrt{n}(\widehat{\theta}-\theta) &= \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \underbrace{[\operatorname{IC}_{1}(Z_i; P_{0})^\top,\ldots,\operatorname{IC}_{M}(Z_i; P_{0})^\top]^{\top}}_{\overline{\operatorname{IC}}(Z_i; P_{0})} + o_{P}(1) \\ &\overset{\mathcal{D}}{\longrightarrow}\mathcal{N}(0,\Sigma) \end{align*}\] by the CLT, and under regulatory conditions \(\mathbb{P}_{n}\overline{\operatorname{IC}}(Z_i; \widehat{P})^{\otimes 2} \overset{P}{\longrightarrow}\Sigma\) as \(n\to\infty\).

To illustrate this we consider two marginal logistic regression models fitted separately for \(Y_1\) and \(Y_2\) and combine the estimates and IFs using the `merge` method

    g1 <- glm(y1 ~ a, family=binomial, data=dw)
    g2 <- glm(y2 ~ a, family=binomial, data=dw)
    e <- merge(g1, g2)
    summary(e)
    #> Call: estimate.default(f = FALSE, contrast = contrast, vcov = vcov(object),
    #>     coef = p)
    #> ────────────────────────────────────────────────────────────
    #>               Estimate Std.Err    2.5%    97.5%   P-value
    #> (Intercept)    -0.1861  0.1442 -0.4688  0.09655 1.969e-01
    #> a               1.3239  0.2173  0.8981  1.74978 1.105e-09
    #> (Intercept).1  -0.6168  0.1505 -0.9117 -0.32185 4.152e-05
    #> a.1             1.5060  0.2148  1.0849  1.92712 2.385e-12
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [(Intercept)] = 0
    #>   [a] = 0
    #>   [(Intercept).1] = 0
    #>   [a.1] = 0
    #>
    #> chisq = 96.4362, df = 4, p-value < 2.2e-16

As we have access to the joint asymptotic distribution we can for example test for whether the odds-ratio is the same for the two responses:

    estimate(e, cbind(0,1,0,-1), null=0)
    #>             Estimate Std.Err    2.5%  97.5% P-value
    #> [a] - [a.1]  -0.1821  0.3003 -0.7707 0.4065  0.5443
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] - [a.1] = 0
    #>
    #> chisq = 0.3677, df = 1, p-value = 0.5443

More details an be found in the Section on hypothesis testing.

## Imbalanced data

Let \(O_{1} = (Z_{1}R_{1}, R_{1}), \ldots, O_{N}=(Z_{N}R_{N}, R_{N})\) be iid with \(R_{i}\!\perp\!\!\!\!\perp\!Z_i\) and let the full-data IF for some estimator of a parameter \(\theta\in\mathbb{R}^p\) be \(IC(\cdot; P_{0})\). For convenience let the data be ordered \(R_{i}=\mathbf{1}(i\leq n)\) where \(n\) is the number of observed data points, then the complete-case estimator is consistent and based on same IF

\[\begin{align*} \sqrt{n}(\widehat{\theta}-\theta) = \frac{1}{\sqrt{n}}\sum_{i=1}^n IC(Z_i; P_{0}) + o_{P}(1). \end{align*}\]

This estimator can also be decomposed in terms of the observed data \(O_1,\ldots,O_N\) noting that

\[\begin{align*} \sqrt{N}(\widehat{\theta}-\theta) = \frac{1}{\sqrt{N}}\sum_{i=1}^N IC(Z_i; P)\frac{R_i N}{n} + o_{P}(1). \end{align*}\]

where the term \(\frac{R_i N}{n}\) corresponds to an inverse probability weighting with the empirical plugin estimate of the proportion of observed data \(R=1\). Under a missing completely at random assumption we can therefore combine estimators that are estimated on different datasets. Let the observed data be \((Z_{11}R_{11}, R_{11}, Z_{21}R_{21}, R_{21}), \ldots, (Z_{1N}R_{1N}, R_{1N}, Z_{2N}R_{2N}, R_{2N}))\) with complete-case estimators \(\widehat{\theta}_1\) and \(\widehat{\theta}_2\) for parameters \(\theta_1\) and \(\theta_2\) based on \((Z_{11}R_{11}, \ldots, Z_{1N}R_{1N})\) and \((Z_{21}R_{21}, \ldots, Z_{2N}R_{2N})\), respectively, and let the corresponding IFs be \(IC_{1}(\cdot; P_{0})\) and \(IC_{2}(\cdot;\ P)\). It then follows that

\[\begin{align*} \sqrt{N}\left\{ \begin{pmatrix} \widehat{\theta}_1 \\ \widehat{\theta}_2 \end{pmatrix} - \begin{pmatrix} \vphantom{\widehat{\theta}_1}\theta_1 \\ \vphantom{\widehat{\theta}_1}\theta_2 \end{pmatrix} \right\} = \frac{1}{\sqrt{N}}\sum_{i=1}^N \begin{pmatrix} IC_1(Z_{1i}; P_{0})\frac{R_{1i}N}{R_{1\bullet}} \\ IC_2(Z_{2i}; P_{0})\frac{R_{2i}N}{R_{2\bullet}} \end{pmatrix} + o_{P}(1) \end{align*}\]

with \(R_{k\bullet} = \sum_{i=1}^{N}R_{ki}.\) Returning to the example, we can combine the marginal estimates of two model objects that have been estimated from different datasets (as the outcome \(Y_3\) is only available in half of the data) with the `merge` function

    g2 <- glm(y2 ~ 1, family = binomial, data = dw)
    summary(g2)
    #>
    #> Call:
    #> glm(formula = y2 ~ 1, family = binomial, data = dw)
    #>
    #> Coefficients:
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)   0.1402     0.1002   1.399    0.162
    #>
    #> (Dispersion parameter for binomial family taken to be 1)
    #>
    #>     Null deviance: 552.56  on 399  degrees of freedom
    #> Residual deviance: 552.56  on 399  degrees of freedom
    #> AIC: 554.56
    #>
    #> Number of Fisher Scoring iterations: 3
    dwc <- na.omit(dw)
    g3 <- glm(y3 ~ 1, family = binomial, data = dwc)
    summary(g3)
    #>
    #> Call:
    #> glm(formula = y3 ~ 1, family = binomial, data = dwc)
    #>
    #> Coefficients:
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)   0.2615     0.1426   1.833   0.0668 .
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for binomial family taken to be 1)
    #>
    #>     Null deviance: 273.87  on 199  degrees of freedom
    #> Residual deviance: 273.87  on 199  degrees of freedom
    #> AIC: 275.87
    #>
    #> Number of Fisher Scoring iterations: 3

    e2 <- estimate(g2, id = dw$id)
    e3 <- estimate(g3, id = "id", data=dwc)

    ecomb <- merge(e2, e3)
    IC(ecomb) |> Print()
    #>     (Intercept) (Intercept).1
    #> 1   -2.151       3.540
    #> 2    1.869       3.540
    #> 3   -2.151      -4.598
    #> 4    1.869      -4.598
    #> 5   -2.151       3.540
    #> ---
    #> 396 -2.151       0.000
    #> 397  1.869       0.000
    #> 398 -2.151       0.000
    #> 399 -2.151       0.000
    #> 400 -2.151       0.000
    vcov(ecomb)
    #>               (Intercept) (Intercept).1
    #> (Intercept)   0.010049191   0.002102598
    #> (Intercept).1 0.002102598   0.020342639
    ## Same marginals as
    list(vcov(e2), vcov(e3))
    #> [[1]]
    #>             (Intercept)
    #> (Intercept)  0.01004919
    #>
    #> [[2]]
    #>             (Intercept)
    #> (Intercept)  0.02034264

Note, it is also possible to directly specify the id-variables in the `merge` call:

    merge(e2, e3, id = list(dw$id, dwc$id))
    #>               Estimate Std.Err     2.5%  97.5% P-value
    #> (Intercept)     0.1402  0.1002 -0.05625 0.3367 0.16186
    #> (Intercept).1   0.2615  0.1426 -0.01807 0.5410 0.06676

In the above example the `id` argument defines the identifier that makes it possible to link the rows in the different IFs that should be glued together. If omitted then the `id` will automatically be extracted from the model-specific `IC` method (deriving it from the original data.frame used for estimating the model). This automatically works with all models and `IC` methods described in this document.

    estimate(g2) |>
      IC() |> head()
    #>   (Intercept)
    #> 1   -2.150532
    #> 2    1.869154
    #> 3   -2.150532
    #> 4    1.869154
    #> 5   -2.150532
    #> 6    1.869154
    vcov(merge(estimate(g2), estimate(g3)))
    #>               (Intercept) (Intercept).1
    #> (Intercept)   0.010049191   0.002102598
    #> (Intercept).1 0.002102598   0.020342639
    merge(estimate(g2), estimate(g3)) |>
      (rownames %++% head %++% IC)()
    #> [1] "1"   "10"  "100" "101" "102" "103"

To force that the id variables are not overlapping between the merged model objects, i.e., assuming that there is complete independence between the estimates, the argument `id=NULL` can be used

    merge(g1, g2, id = NULL) |> (Print %++% IC)()
    #>     (Intercept) a          (Intercept).1
    #> 1    8.882e-15   5.128e+00  0.000e+00
    #> 2    8.882e-15   5.128e+00  0.000e+00
    #> 3   -7.547e+00   7.547e+00  0.000e+00
    #> 4   -7.547e+00   7.547e+00  0.000e+00
    #> 5   -2.665e-14  -1.600e+01  0.000e+00
    #> ---
    #> 796  0.000       0.000     -4.301
    #> 797  0.000       0.000      3.738
    #> 798  0.000       0.000     -4.301
    #> 799  0.000       0.000     -4.301
    #> 800  0.000       0.000     -4.301
    merge(g1, g2, id = NULL) |> vcov()
    #>                 (Intercept)             a (Intercept).1
    #> (Intercept)    2.079760e-02 -2.079760e-02 -1.587849e-28
    #> a             -2.079760e-02  4.720777e-02 -1.556391e-25
    #> (Intercept).1 -1.587849e-28 -1.556391e-25  1.004919e-02

## Renaming and subsetting parameters

To only keep a subset of the parameters the `keep` argument can be used.

    merge(g1, g2, keep = c("(Intercept)", "(Intercept).1"))
    #>               Estimate Std.Err     2.5%   97.5% P-value
    #> (Intercept)    -0.1861  0.1442 -0.46876 0.09655  0.1969
    #> (Intercept).1   0.1402  0.1002 -0.05625 0.33671  0.1619

The argument can be given either as character vector or a vector of indices:

    merge(g1, g2, keep=c(1, 3))
    #>               Estimate Std.Err     2.5%   97.5% P-value
    #> (Intercept)    -0.1861  0.1442 -0.46876 0.09655  0.1969
    #> (Intercept).1   0.1402  0.1002 -0.05625 0.33671  0.1619

or as a vector of perl-style regular expressions

    merge(g1, g2, keep = "cept", regex = TRUE)
    #>               Estimate Std.Err     2.5%   97.5% P-value
    #> (Intercept)    -0.1861  0.1442 -0.46876 0.09655  0.1969
    #> (Intercept).1   0.1402  0.1002 -0.05625 0.33671  0.1619
    merge(g1, g2, keep = c("\\)$", "^a$"), regex = TRUE, ignore.case = TRUE)
    #>             Estimate Std.Err    2.5%   97.5%   P-value
    #> (Intercept)  -0.1861  0.1442 -0.4688 0.09655 1.969e-01
    #> a             1.3239  0.2173  0.8981 1.74978 1.105e-09

When merging estimates unique parameter names are created. It is also possible to rename the parameters with the `labels` argument

    merge(g1, g2, labels = c("a", "b", "c")) |> estimate(keep = c("a", "c"))
    #>   Estimate Std.Err     2.5%   97.5% P-value
    #> a  -0.1861  0.1442 -0.46876 0.09655  0.1969
    #> c   0.1402  0.1002 -0.05625 0.33671  0.1619
    merge(g1, g2,
          labels = c("a", "b", "c"),
          keep = c("a", "c")
    )
    #>   Estimate Std.Err     2.5%   97.5% P-value
    #> a  -0.1861  0.1442 -0.46876 0.09655  0.1969
    #> c   0.1402  0.1002 -0.05625 0.33671  0.1619
    estimate(g1, labels=c("a", "b"))
    #>   Estimate Std.Err    2.5%   97.5%   P-value
    #> a  -0.1861  0.1442 -0.4688 0.09655 1.969e-01
    #> b   1.3239  0.2173  0.8981 1.74978 1.105e-09

Finally, the `subset` argument can be used to subset the parameters and IFs before the actual merging is being done

    merge(g1, g2, subset="(Intercept)")
    #>               Estimate Std.Err     2.5%   97.5% P-value
    #> (Intercept)    -0.1861  0.1442 -0.46876 0.09655  0.1969
    #> (Intercept).1   0.1402  0.1002 -0.05625 0.33671  0.1619

## Clustered data (non-iid case)

Let \(Z_i = (Z_{i1},\ldots,Z_{iN_{i}})\) and assume that \((Z_{i}, N_{i}) \sim P\), \(i=1,\ldots,n\) are iid and \(N_i\!\perp\!\!\!\!\perp\!Z_{ij}\). The variables \(Z_{i1},\ldots,Z_{iN_{i}}\) we assume are exchangeable but not necessarily independent. Define \(N = \sum_{i=1}^{n} N_i\), and assume that a parameter estimate, \(\widehat{\theta}\in\mathbb{R}^p\) has the decomposition \[ \sqrt{N}(\widehat{\theta}-\theta) = \frac{1}{\sqrt{N}} \sum_{i=1}^{n} \sum_{k=1}^{N_{i}} IC(Z_{ik}; P_{0}) + o_{P}(1). \] It then follows that \[ \sqrt{n}(\widehat{\theta}-\theta) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \widetilde{\operatorname{IC}}(Z_{i}; P_{0}) + o_{P}(1) \] with \(\widetilde{\operatorname{IC}}(Z_{i}; P_{0}) = \sum_{k=1}^{N_{i}} \frac{n}{N}IC(Z_{ik}; P_{0})\), \(i=1,\ldots,n\) which are iid an therefore admits the usual CLT to derive the asymptotic variance of \(\widehat{\theta}\). Turning back to the example data, we can estimate the marginal model

    g0 <- glm(y ~ a + w + x, data = dl, family = binomial())

The asymptotic variance estimate ignoring that the observations are not independent is not consistent. Instead we can calculate the cluster robust standard errors from the above iid decomposition

    estimate(g0, id=dl$id)
    #>             Estimate Std.Err    2.5%   97.5%   P-value
    #> (Intercept)  -0.1147 0.09351 -0.2979 0.06862 2.201e-01
    #> a             1.0178 0.13016  0.7627 1.27288 5.303e-15
    #> w             0.9825 0.07421  0.8370 1.12791 5.278e-40
    #> x             0.9485 0.07835  0.7949 1.10203 9.976e-34

We can confirm that this situation is equivalent to the variance estimates we obtain from a GEE marginal model with working independence correlation structure (Halekoh, Højsgaard, and Yan 2006)

    gee0 <- geepack::geeglm(y ~ a + w + x, data = dl, id = dl$id, family=binomial)
    summary(gee0)
    #>
    #> Call:
    #> geepack::geeglm(formula = y ~ a + w + x, family = binomial, data = dl,
    #>     id = dl$id)
    #>
    #>  Coefficients:
    #>             Estimate  Std.err    Wald Pr(>|W|)
    #> (Intercept) -0.11466  0.09351   1.504     0.22
    #> a            1.01777  0.13016  61.145 5.33e-15 ***
    #> w            0.98246  0.07421 175.250  < 2e-16 ***
    #> x            0.94845  0.07835 146.523  < 2e-16 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Correlation structure = independence
    #> Estimated Scale Parameters:
    #>
    #>             Estimate Std.err
    #> (Intercept)   0.9538   0.098
    #> Number of clusters:   400  Maximum cluster size: 4

## Computational aspects

Working with large and potentially multiple different IFs can be memory-intensive. A remedy is to use the idea of aggregating the IFs by introducing a random coarser grouping variable. Following the same arguments as in the previous section, the aggregated IF will still be iid and allows us to estimate the asymptotic variance. Obviously, the same grouping must be used across estimates when combining IFs.

    set.seed(1)
    y <- cbind(rnorm(1e5))
    N <- 2e2 ## Number of aggregated groups, the number of observations in the new IF
    id <- foldr(nrow(y), N, list=FALSE)
    Print(cbind(table(id)))
    #>     [,1]
    #> 1   500
    #> 2   500
    #> 3   500
    #> 4   500
    #> 5   500
    #> ---
    #> 196 500
    #> 197 500
    #> 198 500
    #> 199 500
    #> 200 500

    ## Aggregated IF
    e <- estimate(cbind(y), id = id)
    object.size(e)
    #> 18992 bytes
    e
    #>     Estimate Std.Err      2.5%    97.5% P-value
    #> p1 -0.002244 0.00332 -0.008751 0.004263  0.4991

# IF building blocks: transformations and the delta theorem

Let \(\phi\colon \mathbb{R}^p\to\mathbb{R}^m\) be differentiable at \(\theta\) and assume that \(\widehat{\theta}_n\) is RAL estimator with IF given by \(\operatorname{IC}(\cdot; P_{0})\) such that \[\begin{align*} \sqrt{n}(\widehat{\theta}_n - \theta) = \frac{1}{\sqrt{n}}\sum_{i=1}^n \operatorname{IC}(Z_i; P_{0}) + o_{P}(1), \end{align*}\] then by the delta method (Vaart 1998 Theorem 3.1) \[\begin{align*} \sqrt{n}\{\phi(\widehat{\theta}_n) - \phi(\theta)\} = \frac{1}{\sqrt{n}}\sum_{i=1}^n \nabla\phi(\theta)\operatorname{IC}(Z_i; P_{0}) + o_{P}(1), \end{align*}\]

where \(\phi\colon \theta\mapsto (\phi_{1}(\theta),\ldots,\phi_{m}(\theta))^\top\) and \(\nabla\) is the partial derivative operator \[\begin{align*} \nabla\phi(\theta) = \begin{pmatrix} \tfrac{\partial}{\partial\theta_1}\phi_1(\theta) & \cdots & \tfrac{\partial}{\partial\theta_p}\phi_1(\theta) \\ \vdots & \ddots & \vdots \\ \tfrac{\partial}{\partial\theta_1}\phi_m(\theta) & \cdots & \tfrac{\partial}{\partial\theta_p}\phi_m(\theta) \\ \end{pmatrix}. \end{align*}\]

Together with the ability to derive the joint IF from marginal IFs, this provides us with a powerful tool for constructing new estimates using the IFs as the fundamental building blocks.

To apply the delta method the transformation of the parameters function must be supplied to the `estimate` method (argument `f`)

    estimate(g1, sum)
    #>    Estimate Std.Err   2.5% 97.5%   P-value
    #> p1    1.138  0.1625 0.8193 1.456 2.532e-12
    estimate(g1, function(p) list(a = sum(p))) # named list
    #>   Estimate Std.Err   2.5% 97.5%   P-value
    #> a    1.138  0.1625 0.8193 1.456 2.532e-12
    ## Multiple parameters
    estimate(g1, function(x) c(x, x[1] + exp(x[2]), inv = 1 / x[2]))
    #>               Estimate Std.Err    2.5%   97.5%   P-value
    #> (Intercept)    -0.1861  0.1442 -0.4688 0.09655 1.969e-01
    #> a               1.3239  0.2173  0.8981 1.74978 1.105e-09
    #> (Intercept).1   3.5721  0.7289  2.1435 5.00062 9.539e-07
    #> inv.a           0.7553  0.1240  0.5124 0.99828 1.105e-09
    estimate(g1, exp)
    #>             Estimate Std.Err   2.5% 97.5%   P-value
    #> (Intercept)   0.8302  0.1197 0.5955 1.065 4.087e-12
    #> a             3.7582  0.8166 2.1578 5.359 4.175e-06

The gradient can be provided as the attribute `grad` and otherwise numerical differentiation is applied.

An often more convenient syntax is to directly transform the `estimate` objects via standard mathematical R operations. An advantage of this is also that the necessary derivatives are automatically calculated exactly. To illustrate this consider the following simple example where we consider two `estimate` objects each with a single parameter estimate

    a <- estimate(coef=c("a"=0.5), IC=scale(rnorm(10)), id=1:10)
    b <- estimate(coef=c("b"=0.8), IC=scale(rnorm(10)), id=1:10)

Parameter transformation can now be calculated directly as in the following examples

    a * b
    #>   Estimate Std.Err    2.5% 97.5% P-value
    #> a      0.4  0.3156 -0.2185 1.018  0.2049
    (3 * cos(a) / sqrt(b) + 1) / a
    #>   Estimate Std.Err   2.5% 97.5% P-value
    #> a    7.887   6.089 -4.047 19.82  0.1952
    c(sum=sum(e), sum2=a+b,
      prod=prod(e), prod2=a*b) # sum and prod function
    #>        Estimate Std.Err      2.5%    97.5% P-value
    #> sum   -0.002244 0.00332 -0.008751 0.004263 0.49911
    #> sum2   1.300000 0.47822  0.362698 2.237302 0.00656
    #> prod  -0.002244 0.00332 -0.008751 0.004263 0.49911
    #> prod2  0.400000 0.31556 -0.218490 1.018490 0.20495
    e <- c(a,b) # merge
    e %*% e # inner prod.
    #>    Estimate Std.Err   2.5% 97.5% P-value
    #> p1     0.89  0.6311 -0.347 2.127  0.1585
    c(1, 2) %*% e
    #>    Estimate Std.Err   2.5% 97.5%  P-value
    #> p1      2.1  0.7399 0.6499  3.55 0.004535
    c(pow = a^b) # power-function, rename parameter
    #>     Estimate Std.Err    2.5% 97.5% P-value
    #> pow   0.5743  0.2692 0.04679 1.102 0.03286
    a^c(0.5, 2)
    #>    Estimate Std.Err    2.5% 97.5%   P-value
    #> p1   0.7071  0.2121  0.2913 1.123 0.0008581
    #> p2   0.2500  0.3000 -0.3380 0.838 0.4046568
    c(e["a"] * e["b"] / a, e["b"])
    #>   Estimate Std.Err  2.5% 97.5%  P-value
    #> a      0.8     0.3 0.212 1.388 0.007661
    #> b      0.8     0.3 0.212 1.388 0.007661

For the `%*%` operator we can also use a general contrast matrix (see also Section on Linear contrasts

    B <- rbind(c(1,-1), c(1,0), c(0,1))
    B %*% e
    #>           Estimate Std.Err     2.5%  97.5%  P-value
    #> [a] - [b]     -0.3  0.3624 -1.01020 0.4102 0.407718
    #> a              0.5  0.3000 -0.08799 1.0880 0.095581
    #> b              0.8  0.3000  0.21201 1.3880 0.007661
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] - [b] = 0
    #>   [a] = 0
    #>   [b] = 0
    #>
    #> chisq = 8.075, df = 2, p-value = 0.01764
    B %*% e == c(1,1,0)
    #>           Estimate Std.Err     2.5%  97.5%   P-value
    #> [a] - [b]     -0.3  0.3624 -1.01020 0.4102 0.0003337
    #> a              0.5  0.3000 -0.08799 1.0880 0.0955807
    #> b              0.8  0.3000  0.21201 1.3880 0.0076608
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [[a] - [b]] = 1
    #>   [a] = 1
    #>   [b] = 0
    #>
    #> chisq = 13.26, df = 2, p-value = 0.001317

The following transformations are implemented

- trigonometric functions: `cos`, `sin`, `tan`
- inverse trigonometric functions: `acos`, `asin`, `atan`
- hyperbolic functions: `cosh`, `sinh`, `tanh`
- inverse hyperbolic functions: `acosh`, `asinh`, `atanh`
- other mathematical functions: `log`, `log1p`, `exp`, `expm1`, `sqrt`
- mathematical operators: `+`, `-`, `*`, `/`, `^`, `prod`, `sum`, `%*%`

and any compositions of these functions, which also means we can immediately apply most existing user-defined functions. To give an example consider the `logit` function

    lava::logit
    #> function (p)
    #> log(p/(1 - p))
    #> <bytecode: 0x55a6d1639558>
    #> <environment: namespace:lava>
    logit(b)
    #>   Estimate Std.Err   2.5% 97.5% P-value
    #> b    1.386   1.875 -2.289 5.061  0.4597
    expit(c(a,b))
    #>   Estimate Std.Err   2.5%  97.5%   P-value
    #> a   0.6225 0.07050 0.4843 0.7606 1.055e-18
    #> b   0.6900 0.06417 0.5642 0.8158 5.811e-27

In the above example we also demonstrated the user of the subset operator `[]` and the merge method `c()`. These are basically equivalent to the `subset` and `merge` methods. There is also a `transform` method for constructing parameter transformations above, and the `labels` function for renaming parameters. Together we can combine these methods via the pipe operator to create readable code:

    merge(a, b) |>  # merges the two `estimate` objects
      transform(prod) |> # calculates product of parameter estimates
      subset(1) |> # nothing happens here as the result was already 1-dim.
      labels("prod") # rename parameter
    #>      Estimate Std.Err    2.5% 97.5% P-value
    #> prod      0.4  0.3156 -0.2185 1.018  0.2049

Finally, the `with` function can be used to reference parameter names

    e <- c("e1"=a, "e2"=b)
    with(e, c(est = e1*e2))
    #>     Estimate Std.Err    2.5% 97.5% P-value
    #> est      0.4  0.3156 -0.2185 1.018  0.2049

## Example: Pearson correlation

As a simple toy example consider the problem of estimating the covariance of two variables \(X_1\) and \(X_2\) \[\begin{align*} \widehat{\mathbb{C}\!\text{ov}}(X_1,X_2) = \mathbb{P}_n(X_1-\mathbb{P}_n X_1)(Y_1-\mathbb{P}_n Y_1). \end{align*}\] It is easily verified that the IF of the sample estimate of \((\mathbb{E}X_{1}, \mathbb{E}X_{2}, \mathbb{E}\{X_{1}X_{2}\})^\top\) given by is \(\operatorname{IC}(X1,X2; P_{0}) = (X_{1}-\mathbb{E}X_{1}, X_{2}-\mathbb{E}X_{2}, X_{1}X_{2}-\mathbb{E}\{X_{1}X_{2}\})^\top\). By the delta theorem with \(\phi(x,y,z) = z-xy\) we have \(\nabla\phi(x,y,z) = (-y, -x, 1)^\top\) and thus the IF for the sample covariance estimate becomes \[\begin{align*} \operatorname{IC}_{x_1, x_2}(X_1, X_2; P_{0}) = (X_1 - \mathbb{E}X_1)(X_2 - \mathbb{E}X_2) - \mathbb{C}\!\text{ov}(X_1,X_2) \end{align*}\]

We can implement this directly using the `estimate` function via the `IC` argument which allows us to provide a user-specificed IF and with the point estimate given by the `coef` argument

    Cov <- function(x, y, ...) {
      est <- mean(x * y)-mean(x)*mean(y)
        estimate(
          coef = est,
          IC = (x - mean(x)) * (y - mean(y)) - est,
          ...
        )
    }
    with(dw, Cov(x1, x2))
    #>  Estimate Std.Err     2.5%  97.5% P-value
    #>  0.004043 0.04976 -0.09349 0.1016  0.9352

As an illustration we could also derive this estimate from simpler building blocks of \(\mathbb{E}X_{1}\), \(\mathbb{E}X_{2}\), and \(\mathbb{E}(X_{1}X_{2})\).

    est <- lm(cbind(x1, x2, x1 * x2) ~ 1, data = dw) |>
      estimate(labels = c("E1", "E2", "E12"))
    est
    #>      Estimate Std.Err     2.5%   97.5% P-value
    #> E1   0.038089 0.04842 -0.05681 0.13298  0.4315
    #> E2  -0.037026 0.05187 -0.13869 0.06464  0.4753
    #> E12  0.002633 0.05003 -0.09541 0.10068  0.9580

    est["E12"] - est["E2"]*est["E1"]
    #>     Estimate Std.Err     2.5%  97.5% P-value
    #> E12 0.004043 0.04976 -0.09349 0.1016  0.9352
    # transform(e1, function(x) c(x, cov=with(as.list(x), E12 - E2* E1))) # Same result

The variance estimates can be estimated in the same way and combined to estimate the correlation

    v12 <- with(dw, Cov(x1, x2, id = id))
    v1  <- with(dw, Cov(x1, x1, id = id))
    v2  <- with(dw, Cov(x2, x2, id = id))

    rho <- c(rho = v12 / sqrt(v1 * v2))
    rho
    #>     Estimate Std.Err     2.5%  97.5% P-value
    #> rho 0.004025 0.04953 -0.09306 0.1011  0.9352

by using a variance stabilizing transformation, Fishers \(z\)-transform (Lehmann and Romano 2023), \(z = \operatorname{arctanh}(\widehat{\rho}) = \frac{1}{2}\log\left(\frac{1+\widehat{\rho}}{1-\widehat{\rho}}\right)\), confidence limits with general better coverage can be obtained

    estimate(atanh(rho), back.transform = tanh)
    #>     Estimate Std.Err     2.5%  97.5% P-value
    #> rho 0.004025         -0.09279 0.1008  0.9352

The confidence limits are calculated on the \(\operatorname{arctanh}\)-scale and transformed back to the original correlation scale via the `back.transform` argument. In this case, where the estimates are far away from the boundary of the parameter space, the variance stabilizing transform does almost not have any impact, and the confidence limits agrees with the original symmetric confidence limits.

## Linear contrasts and hypothesis testing

An important special case of parameter transformations are linear transformations. A particular interest may be formulated around testing null-hypotheses of the form \[\begin{align*} H_0\colon\quad \mathbf{B}\theta = \mathbf{b}_0 \end{align*}\]

where \(\mathbf{B}\in\mathbb{R}^{m\times p}\) is a matrix of estimable contrasts and \(\mathbf{b}_0\in\mathbb{R}^{m}\).

As an example consider marginal models for the binary response variables \(Y_1, Y_2, Y_3, Y_4\)

    g <- lapply(
      list(y1 ~ a, y2 ~ a, y3 ~ a), #, y4 ~ a+x4),
      function(f) glm(f, family = binomial, data = dw)
    )
    gg <- Reduce(merge, g)
    gg
    #>               Estimate Std.Err    2.5%    97.5%   P-value
    #> (Intercept)    -0.1861  0.1442 -0.4688  0.09655 1.969e-01
    #> a               1.3239  0.2173  0.8981  1.74978 1.105e-09
    #> (Intercept).1  -0.6168  0.1505 -0.9117 -0.32185 4.152e-05
    #> a.1             1.5060  0.2148  1.0849  1.92712 2.385e-12
    #> (Intercept).2  -0.2938  0.2063 -0.6982  0.11064 1.545e-01
    #> a.2             1.1047  0.2962  0.5242  1.68515 1.914e-04

A linear transformation can be specified via the `f` as a matrix argument instead of function object

    B <- cbind(0,1, 0,-1, 0,0)
    estimate(gg, B)
    #>             Estimate Std.Err    2.5%  97.5% P-value
    #> [a] - [a.1]  -0.1821  0.3003 -0.7707 0.4065  0.5443
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] - [a.1] = 0
    #>
    #> chisq = 0.3677, df = 1, p-value = 0.5443

The \(\mathbf{b}_0\) vector (default assumed to be zero) can be specified via the `null` argument

    estimate(gg, B, null=1)
    #>             Estimate Std.Err    2.5%  97.5%   P-value
    #> [a] - [a.1]  -0.1821  0.3003 -0.7707 0.4065 8.281e-05
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] - [a.1] = 1
    #>
    #> chisq = 15.49, df = 1, p-value = 8.281e-05

For testing multiple hypotheses we use that \((\mathbf{B}\widehat{\theta}-\mathbf{b}_0)^{\top} (\mathbf{B}\widehat{\Sigma}\mathbf{B}^{\top})^{-1} (\mathbf{B}\widehat{\theta}-\mathbf{b}_0) \sim \chi^2_{\operatorname{rank}(B)}\) under the null hypothesis where \(\widehat{\Sigma}\) is the estimated variance of \(\theta\) (i.e., `vcov(gg)`)

    B <- rbind(cbind(0,1, 0,-1, 0,0),
               cbind(0,1, 0,0, 0,-1))
    estimate(gg, B)
    #>             Estimate Std.Err    2.5%  97.5% P-value
    #> [a] - [a.1]  -0.1821  0.3003 -0.7707 0.4065  0.5443
    #> [a] - [a.2]   0.2192  0.3637 -0.4936 0.9321  0.5466
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] - [a.1] = 0
    #>   [a] - [a.2] = 0
    #>
    #> chisq = 1.343, df = 2, p-value = 0.5109

Such linear statistics can also be specified directly as expressions of the parameter names

    estimate(gg, a + a.1, 2*a - a.2, a, null=c(2,1,1))
    #>              Estimate Std.Err   2.5% 97.5%  P-value
    #> [a] + [a.1]     2.830  0.3107 2.2210 3.439 0.007557
    #> 2[a] - [a.2]    1.543  0.5208 0.5224 2.564 0.296991
    #> a               1.324  0.2173 0.8981 1.750 0.135985
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] + [a.1] = 2
    #>   2[a] - [a.2] = 1
    #>   [a] = 1
    #>
    #> chisq = 7.557, df = 3, p-value = 0.05612

We refer to the function `lava::contr` and `lava::parsedesigns` for defining contrast matrices.

    contr(list(1, c(1, 2), c(1, 4)), n = 5)
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    1   -1    0    0    0
    #> [3,]    1    0    0   -1    0

A particular useful contrast is the following for considering all pairwise comparisons of different exposure estimates:

    pairwise.diff(3)
    #>      [,1] [,2] [,3]
    #> [1,]    1   -1    0
    #> [2,]    1    0   -1
    #> [3,]    0    1   -1
    estimate(gg, pairwise.diff(3), null=c(1,1,1), use=c(2,4,6))
    #>               Estimate Std.Err    2.5%  97.5%   P-value
    #> [a] - [a.1]    -0.1821  0.3003 -0.7707 0.4065 8.281e-05
    #> [a] - [a.2]     0.2192  0.3637 -0.4936 0.9321 3.182e-02
    #> [a.1] - [a.2]   0.4013  0.3506 -0.2858 1.0885 8.773e-02
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   [a] - [a.1] = 1
    #>   [a] - [a.2] = 1
    #>   [a.1] - [a.2] = 1
    #>
    #> chisq = 11.96, df = 2, p-value = 0.002523

When conducting multiple tests each at a nominal-level of \(\alpha\) the overall type I error is not controlled at \(\alpha\)-level. The influence function also allows for adjusting for multiple comparisons. Let \(Z_{1},\ldots,Z_{p}\) denote \(Z\)-statistics from \(p\) distinct two-sided hypothesis tests which we will assume is asymptotically distributed under the null hypothesis as a zero-mean Gaussian distribution with correlation matrix \(R.\) Let $ Z_{max} = _{i=1,,p} |Z_i|$ then the family-wise error rate (FWER) under the null can be approximated by \[ P(Z_{max} > z) = 1-\int_{-z}^{z} \cdots \int_{-z}^{z} \phi_{R}(x_{1},\ldots,x_{p}) \,dx_{1}\cdots\,dx_{p} \] where \(\phi_{R}\) is the multivariate normal density function with mean 0 and variance given by the correlation matrix \(R\). The adjusted \(p\)-values can then be calculated as \[ P(Z_{max} > \Phi^{-1}(1-p/2)) \] where \(\Phi\) is the standard Gaussian CDF. As described in in (Pipper, Ritz, and Bisgaard 2012) the joint distribution of \(Z_{1},\ldots,Z_{p}\) can be estimated from the IFs. This is implemented in the `alpha_zmax` method

    gg0 <- estimate(gg, use="^a", regex=TRUE, null=rep(.8, 3))
    alpha_zmax(gg0)
    #>     Estimate  P-value Adj.P-value
    #> a      1.324 0.015891    0.046873
    #> a.1    1.506 0.001015    0.003043
    #> a.2    1.105 0.303571    0.661496
    #> attr(,"adjusted.significance.level")
    #> [1] 0.01698

While this always yields a more powerful test compared to Bonferroni adjustments, a more powerful closed-testing procedure (Marcus, Eric, and Gabriel 1976), can be generally obtained by considering all intersection hypotheses.

Figure: closed testing via Wald tests of all intersections hypotheses.

To reject the \(H_{1}\) at an correct \(\alpha\)-level we should test all intersection hypotheses involving \(H_{1}\) and check if there all are rejected at the \(\alpha\)-level. The adjusted \(p\)-values can here be obtained as the maximum \(p\)-value across all the composite hypothesis tests. Unfortunately, this only works for relatively few comparisons as the number of tests grows exponentially.

    closed_testing(gg0, test = test_wald)
    #> Call: closed_testing(object = gg0, test = test_wald)
    #>
    #>     Estimate     adj.p
    #> a      1.324 1.105e-09
    #> a.1    1.506 2.385e-12
    #> a.2    1.105 1.914e-04

## Averaging

Some parameters of interest are expressed as averages over functions of the observed data and estimated parameters of a model. The asymptotic distribution can in some of these cases also be derived from the influence function. Let \(Z_{1},\ldots,Z_{n}\) be iid observations, \(Z_{1}\sim P_{0}\) and let \(X_{i}\subset Z_{i}\).

Assume that \(\widehat{\theta}\) is RAL estimator of \(\theta\in\Omega\subset \mathbb{R}^{p}\) \[\sqrt{n}(\widehat{\theta}-\theta) = \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\phi(Z_{i}; P_{0}) + o_{P}(1). \] Let \(f:\mathcal{X}\times\Omega\to\mathbb{R}\) be continuous differentiable in \(\theta\) \[\sqrt{n}\{f(X; \widehat{\theta})-f(X; \theta)\} = \frac{1}{\sqrt{n}}\nabla_{\theta}f(X;\theta)\sum_{i=1}^{n}\phi(Z_{i}; P) + o_{P}(1). \]

Let \(\Psi = P_{0}f(X;\theta)\) and \(\widehat{\Psi} = P_{n} f(X;\widehat{\theta})\). \(P_{0}\) and \(P_{n}\) are here everywhere the integrals wrt. \(X\). It is easily verified that \[\begin{align*} \widehat{\Psi}-\Psi &= (P_{n}-P_{0})(f(X; \theta)-\Psi) + P[f(X;\widehat{\theta})-f(X;\theta)] \\ &\quad + (P_{n}-P_{0})[f(X;\widehat{\theta})-f(X;\theta)] \end{align*}\]

From Lemma 19.24 (Vaart 1998) it follows that for the last term \[\begin{align*} \sqrt{n}(P_{n}-P_{0})[f(X;\widehat{\theta})-f(X;\theta)] = o_{P}(1) \end{align*}\] when \(f\) for example is Lipschitz and more generally when \(f(X;\theta)\) forms a \(P_{0}\)-Donsker class.

It therefore follows that \[\begin{align*} \sqrt{n}(\widehat{\Psi}-\Psi) &= \sqrt{n}P_{n}(f(X; \theta)-\Psi) + \frac{1}{\sqrt{n}}P\nabla_{\theta}f(X;\theta)\sum_{i=1}^{n}\phi(Z_{i}; P_{0}) + o_{P}(1) \\ &= \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\{f(X;\theta)-\Psi\} + \frac{1}{\sqrt{n}}P\nabla_{\theta}f(X;\theta)\sum_{i=1}^{n}\phi(Z_{i}, P_{0}) + o_{P}(1) \end{align*}\] Hence the IF for \(\widehat{\Psi}\) becomes \[ IC(Z; P_{0}) = f(X;\theta)-\Psi + [P_{0}\nabla_{\theta}f(X;\theta)]\phi(Z). \]

Turning back to the example we can estimate the logistic regression model \(\operatorname{logit}(E\{Y_1 | A,X_1,W\}) = \beta_0 + \beta_a A + \beta_{x_1} X_1 + \beta_w W\), and from this we want to estimate the target parameter \[ \theta(P) = \mathbb{E}_{P}[E(Y\mid A=1, X_{1}, W)]. \] To do this we need first to estimate the model and then define a function that gives the predicted probability \(\mathbb{P}(Y=1\mid,A=a,X_{1},W)\) for any observed values of \(X_1,W\) but with the treatment variable \(A\) kept fixed at the value \(1\)

    g <- glm(y1 ~ a + x1 + w, data=dw, family=binomial)
    pr <- function(p, data, ...)
      with(data, expit(p[1] + p["a"] + p["x1"]*x1 + p["w"]*w))
    pr(coef(g), dw) |> head()
    #> [1] 0.8307 0.9651 0.4234 0.9337 0.7669 0.4834

The target parameter can now be estimated with the syntax

    id <- foldr(NROW(dw), 100, list=FALSE)
    ea <- estimate(g, pr, average=TRUE, id=id)
    ea
    #>     Estimate Std.Err   2.5%  97.5%    P-value
    #> val   0.7006 0.03124 0.6394 0.7618 2.035e-111
    IC(ea) |> head()
    #>        val
    #> 1  0.05127
    #> 2 -0.69871
    #> 3  0.33584
    #> 4 -0.15036
    #> 5  0.29918
    #> 6 -0.08264

## Average Treatment Effects

Let the observed data be \(Z = (W,A,Y)\), where \(W\) are covariates, \(A\) a binary treatment variable, and \(Y\) the outcome. In the following we are interested in estimating the target parameter \(\psi_a(P) = E_P[Y(a)]\), where \(Y(a)\) is the potential outcome we would have observed if treatment \(a\) had been administered, possibly contrary to the actual treatment that was observed, i.e., \(Y = Y(A)\). To assess the treatment effect we can then the consider the average treatment effect (ATE) \[E_P[Y(1)]-E_P[Y(0)],\] or some other contrast of interest \(g(\psi_1(P), \psi_0(P))\). Under the following assumptions

1. Stable Unit Treatment Values Assumption (the treatment of a specific subject is not affecting the potential outcome of other subjects)
2. Positivity, \(P(A\mid W)>\epsilon\) for some \(\epsilon>0\) and baseline covariates \(W\)
3. No unmeasured confounders, \(Y(a)\perp \!\!\! \perp A|W\)

then the target parameter can be identified from the observed data distribution as \[E(E[Y|W,A=a]) = E(E[Y(a)|W]) = E[Y(a)]\] or \[E[Y I(A=a)/P(A=a|W)] = E[Y(a)].\]

This suggests estimators based on outcome regression (\(g\)-computation) or inverse probability weighting. More generally, under the above assumption we can constructor a one-step estimator from the Efficient Influence Function combining these two \[ E\left[\frac{I(A=a)}{\Pi_a(W)}(Y-Q(W,A)) + Q(W,a)\right].\] In practice, this requires plugin estimates of both the outcome model, \(Q(W,A) := E(Y\mid A, W)\), and of the treatment propensity model \(\Pi_a(W) := P(A=a\mid W)\). The corresponding estimator is consistent even if just one of the two nuisance models is correctly specified.

To illustrate this we manually construct the above estimating equations based on logistic regressions models for the two nuisance models

    qmod <- glm(y1 ~ a * w, family = binomial, data = dw) # E(Y|W,A) := Q(W,A)
    amod <- glm(a ~ w, family = binomial, data = dw) # P(A=1|W) = Pi_1(W)
    q0 <- predict(qmod, transform(dw, a=0), type="response") # Q(W,0)
    q1 <- predict(qmod, transform(dw, a=1), type="response") # Q(W,1)
    p1 <- predict(amod, dw, type="response") # P(A=1|W)
    e0 <- with(dw, (1-a) / (1-p1) * (y1 - q0) + q0)
    e1 <- with(dw, a / p1 * (y1 - q1) + q1)
    head(cbind(e0, e1))
    #>        e0      e1
    #> 1  0.7381  1.0447
    #> 2  0.8434  1.0119
    #> 3 -0.2967  0.6236
    #> 4 -0.3878  0.6656
    #> 5  0.4649 -0.8175
    #> 6 -0.3952  0.6685

We can then easily construct estimates of the two expected potential outcomes and estimate their joint asymptotic distribution

    est0 <- estimate(coef=mean(e0), IC=e0-mean(e0)) # E[Y(0)]
    est1 <- estimate(coef=mean(e1), IC=e1-mean(e1)) # E[Y(1)]

    potential_outcomes <- merge(est0, est1, paired=TRUE, labels=c("y(0)", "y(1)"))
    potential_outcomes
    #>      Estimate Std.Err   2.5%  97.5%   P-value
    #> y(0)   0.5359 0.03535 0.4667 0.6052 6.290e-52
    #> y(1)   0.7050 0.03721 0.6321 0.7779 4.934e-80
    head(IC(potential_outcomes))
    #>       y(0)     y(1)
    #> 1  0.20211  0.33973
    #> 2  0.30750  0.30690
    #> 3 -0.83262 -0.08136
    #> 4 -0.92373 -0.03944
    #> 5 -0.07105 -1.52251
    #> 6 -0.93115 -0.03652
    vcov(potential_outcomes)
    #>           y(0)      y(1)
    #> y(0) 1.249e-03 6.938e-05
    #> y(1) 6.938e-05 1.385e-03

Finally, we can obtain the Average Treatment Effect (ATE) (risk-difference) \[\begin{align*} E[Y(1)] - E[Y(0)] \end{align*}\]

    estimate(potential_outcomes, cbind(-1, 1), labels="ate")
    #>     Estimate Std.Err    2.5% 97.5%   P-value
    #> ate   0.1691 0.04996 0.07115 0.267 0.0007138
    #> ────────────────────────────────────────────────────────────
    #> Null Hypothesis:
    #>   -[y(0)] + [y(1)] = 0
    #>
    #> chisq = 11.45, df = 1, p-value = 0.0007138

Alternatively, we could get the estimate of the treatment effect on the log-odds scale:

    est <- with(potential_outcomes, logit(`y(1)`)-logit(`y(0)`))
    est
    #>      Estimate Std.Err  2.5% 97.5%  P-value
    #> y(1)   0.7272  0.2226 0.291 1.163 0.001085
    transform(est, labels="OR", back.transform=exp)
    #>    Estimate Std.Err  2.5% 97.5%  P-value
    #> OR    2.069         1.338 3.201 0.001085
    #logor <- function(p) logit(p[2]) - logit(p[1])
    #transform(potential_outcomes, logor, labels="logOR")
    #transform(potential_outcomes, logor, labels="OR", back.transform=exp)

We refer to the `targeted` package (Klaus K. Holst, Sommer, and Nordland 2026) for a general implementation where the nuisance models can be based on machine learning models using cross-fitting. The above estimation can be obtained with the `targeted::cate` function in the following way

    amod <- targeted::learner_glm(a ~ w, family=binomial)
    qmod <- targeted::learner_glm(y1 ~ a * w, family=binomial)
    est <- targeted::cate(qmod, amod, data=dw, second.order = FALSE)
    #>             Estimate Std.Err    2.5%  97.5%   P-value
    #> E[y1(1)]      0.7050 0.03721 0.63207 0.7779 4.934e-80
    #> E[y1(0)]      0.5359 0.03535 0.46666 0.6052 6.290e-52
    #> ───────────
    #> (Intercept)   0.1691 0.04996 0.07115 0.2670 7.138e-04

# SessionInfo

    sessionInfo()
    #> R version 4.5.3 (2026-03-11)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /home/kkzh/.asdf/installs/r/4.5.3/lib/R/lib/libRblas.so
    #> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0  LAPACK version 3.12.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> time zone: Europe/Copenhagen
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] survival_3.8-6 lava_1.9.1
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] tidyr_1.3.2            sass_0.4.10            future_1.70.0
    #>  [4] generics_0.1.4         lattice_0.22-9         listenv_0.10.1
    #>  [7] digest_0.6.39          magrittr_2.0.4         evaluate_1.0.5
    #> [10] grid_4.5.3             mvtnorm_1.3-7          fastmap_1.2.0
    #> [13] jsonlite_2.0.0         Matrix_1.7-4           backports_1.5.0
    #> [16] graph_1.88.1           purrr_1.2.1            Rgraphviz_2.54.0
    #> [19] codetools_0.2-20       numDeriv_2016.8-1.1    jquerylib_0.1.4
    #> [22] cli_3.6.6              rlang_1.2.0            mets_1.3.10
    #> [25] parallelly_1.47.0      future.apply_1.20.2    splines_4.5.3
    #> [28] RcppArmadillo_15.2.6-1 geepack_1.3.13         cachem_1.1.0
    #> [31] yaml_2.3.12            otel_0.2.0             tools_4.5.3
    #> [34] parallel_4.5.3         dplyr_1.2.0            globals_0.19.1
    #> [37] BiocGenerics_0.56.0    broom_1.0.12           vctrs_0.7.1
    #> [40] R6_2.6.1               stats4_4.5.3           lifecycle_1.0.5
    #> [43] MASS_7.3-65            pkgconfig_2.0.3        timereg_2.0.7
    #> [46] progressr_0.19.0       bslib_0.10.0           pillar_1.11.1
    #> [49] glue_1.8.0             Rcpp_1.1.1-1.1         tidyselect_1.2.1
    #> [52] xfun_0.56              tibble_3.3.1           knitr_1.51
    #> [55] htmltools_0.5.9        rmarkdown_2.31         compiler_4.5.3

# Bibliography

Bickel, Peter J., Chris A. J. Klaassen, Ya’Acov Ritov, and Jon A. Wellner. 1998. Efficient and Adaptive Estimation for Semiparametric Models. Springer.
Halekoh, Ulrich, Søren Højsgaard, and Jun Yan. 2006. “The R Package Geepack for Generalized Estimating Equations.” Journal of Statistical Software 15/2: 1–11. https://doi.org/10.18637/jss.v015.i02.
Hines, Oliver, Oliver Dukes, Karla Diaz-Ordaz, and Stijn Vansteelandt. 2022. “Demystifying Statistical Learning Based on Efficient Influence Functions.” The American Statistician, February, 1–13. https://doi.org/10.1080/00031305.2021.2021984.
Holst, K. K., and E. Budtz-Jørgensen. 2013. “Linear Latent Variable Models: The Lava-Package.” Computational Statistics 28 (4): 1385–1452. https://doi.org/10.1007/s00180-012-0344-y.
Holst, Klaus K., Benedikt Sommer, and Andreas Nordland. 2026. Targeted: Targeted Inference. https://kkholst.github.io/targeted/.
Laan, Mark J. van der, and Sherri Rose. 2011. Targeted Learning. Causal Inference for Observational and Experimental Data. Springer. https://doi.org/10.1007/978-1-4419-9782-1.
Lehmann, E. L., and Joseph P. Romano. 2023. Testing Statistical Hypotheses. Fourth. Springer Texts in Statistics. New York: Springer. https://doi.org/10.1007/978-3-030-70578-7.
Marcus, Ruth, Pertiz Eric, and K. R. Gabriel. 1976. “On Closed Testing Procedures with Special Reference to Ordered Analysis of Variance.” Biometrika 63 (3): 655–60. https://doi.org/10.1093/biomet/63.3.655.
Pipper, Christian B., Christian Ritz, and Hans Bisgaard. 2012. “A Versatile Method for Confirmatory Evaluation of the Effects of a Covariate in Multiple Models.” Journal of the Royal Statistical Society C, Appl. Statist. 61 (2): 315–26. https://doi.org/10.1111/j.1467-9876.2011.01005.x.
Therneau, T., and P. Grambsch. 2000. Modeling Survival Data: Extending the Cox Model. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4757-3294-8.
Tsiatis, A. 2006. Semiparametric Theory and Missing Data. Springer Series in Statistics. Springer New York. https://doi.org/10.1007/0-387-37345-4.
Vaart, A. W. van der. 1998. Asymptotic Statistics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511802256.
Zeileis, Achim, Susanne Köll, and Nathaniel Graham. 2020. “Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R.” Journal of Statistical Software 95 (1): 1–36. https://doi.org/10.18637/jss.v095.i01.
Zeng, Donglin, and D. Y. Lin. 2008. “Efficient Resampling Methods for Nonsmooth Estimating Functions.” Biostatistics 9 (2): 355–63. https://doi.org/10.1093/biostatistics/kxm034.