# SelectionBias

    library(SelectionBias)

# Introduction

Selecting a study population from a larger source population, based on the research question, is a common procedure, for example in an observational study with data from a population register. Subjects who fulfill all the selection criteria are included in the study population, and subjects who do not fulfill at least one selection criterion are excluded from the study population. These selections might introduce a systematic error when estimating a causal effect, commonly referred to as selection bias. Selection bias can also arise if the selections are involuntary, for example, if there are dropouts or other missing values for some individuals in the study. In an applied study, it is often of interest to assess the magnitude of potential biases using a sensitivity analysis, such as bounding the bias. Different bounds, the SV (Smith and VanderWeele), sharp, AF (assumption-free), GAF (generalized assumption-free), and CAF (counterfactual assumption-free), for the causal estimand under selection bias can be calculated in this R package `SelectionBias`. The content is:

- `zika_learner`: a simulated dataset of zika virus and microcephaly inspired both by data and a previous example (Araújo et al. 2018; Smith and VanderWeele 2019).
- `sensitivityparametersM()`: a function that calculates the sensitivity parameters for the SV and GAF bounds for an assumed model following the M-structure in Figure 1.
- `SVbound()`: a function that calculates the SV lower and upper bound for the risk ratio or risk difference in either the total or subpopulation for sensitivity parameters given by the user, or calculated from `sensitivityparametersM()`.
- `sharpbound()`: a function that calculates the sharp lower and upper bound for the risk ratio or risk difference in either the total or subpopulation for sensitivity parameters given by the user, or calculated from `sensitivityparametersM()`.
- `AFbound()`: a function that calculates the AF lower and upper bounds for the risk ratio or risk difference in either the total or subpopulation for a dataset or probabilities from the data.
- `GAFbound()`: a function that calculates the GAF lower and upper bounds for the risk ratio or risk difference in either the total or subpopulation for a dataset or probabilities from the data and sensitivity parameters either given by the user, or calculated from `sensitivityparametersM()`.
- `CAFbound()`: a function that calculates the CAF lower and upper bounds for the risk ratio or risk difference in either the total or subpopulation for a dataset or probabilities from the data and sensitivity parameters.
- `checksharpSVbound()`: a function that evaluates if the SV bound is sharp.

For the formulas of the bounds as well as the theory behind them, we refer to the original papers [Smith and VanderWeele (2019);Zetterstrom and Waernbaum (2022);Zetterstrom and Waernbaum (2023);Zetterstrom (2024);zetterstrom2025investigations].

Figure 1. The generalized M-structure.

# R package

## Simulated zika dataset

To illustrate the bounds, a simulated dataset, `zika_learner`, is constructed. It is inspired by a numerical zika example used in Smith and VanderWeele (2019) together with a case-control study that investigates the effect of zika virus on microcephaly (Araújo et al. 2018). The variables included are:

- Living area \((V)\)
- Socioeconomic status, SES \((U)\)
- Zika \((T)\)
- Microcephaly \((Y)\)
- Birth \((S_1)\)
- Public hospital \((S_2)\)

The relationships between the variables are illustrated in Figure 2 and Table 1. The prevalences of the variables, and strengths of dependencies between them, are chosen to mimic real data and the assumed values for the sensitivity parameters in Smith and VanderWeele (2019). The simulated data mimics a cohort with 5000 observations, even though the original study is a case-control study. For more details of the variables and the models, see Zetterstrom and Waernbaum (2023).

Figure 2. Causal model for the `zika_learner` dataset.

The causal dependencies are generated by the logit models described in Table 1.

Table 1. Data generating process for the dataset `zika_learner`. Models generating causal dependencies are logistic, \(g(X'\theta)\), for predictor variables \(X\) (including a constant 1 for the intercept) and model parameter \(\theta\) (including the intercept).
Model | Coefficients (\(\theta\))/Proportions | Function argument
--- | --- | ---
\(P(V=1)\) | \(0.85\) | `Vval`
\(P(U=1)\) | \(0.50\) | `Uval`
\(P(T=1|V)=g(V'\theta_T)\) | \((-6.20,1.75)\) | `Tcoef`
\(P(Y=1|T,U)=g[(T,U)'\theta_{Y}]\) | \((-5.20,5.00,-1.00)\) | `Ycoef`
\(P(S_1=1|V,U,T)=g[(V,U,T)'\theta_{S1}]\) | \((1.20,0.00,2.00,-4.00)\) | `Scoef`
\(P(S_2=1|V,U,T)=g[(V,U,T)'\theta_{S2}]\) | \((2.20,0.50,-2.75,0.00)\) | `Scoef`

The data was generated in `R`, version 4.2.0, using the package `arm`, version 1.13-1, with the following code:

    # Seed.
    set.seed(158118)

    # Number of observations.
    nObs = 5000

    # The unmeasured variable, living area (V).
    urban = rbinom(nObs, 1, 0.85)

    # The treatment variable, zika.
    zika_prob = arm::invlogit(-6.2 + 1.75 * urban)
    zika = rbinom(nObs, 1, zika_prob)

    # The unmeasured variable, SES (U).
    SES = rbinom(nObs, 1, 0.5)

    # The outcome variable, microcephaly.
    mic_ceph_prob = arm::invlogit(-5.2 + 5 * zika - 1 * SES)
    mic_ceph = rbinom(nObs, 1, mic_ceph_prob)

    # The first selection variable, birth.
    birth_prob = arm::invlogit(1.2 - 4 * zika + 2 * SES)
    birth = rbinom(nObs, 1, birth_prob)

    # The second selection variable, hospital.
    hospital_prob = arm::invlogit(2.2 + 0.5 * urban - 2.75 * SES)
    hospital = rbinom(nObs, 1, hospital_prob)

    # The selection indicator.
    sel_ind = birth * hospital

The resulting proportions of the `zika_learner` data, for the total dataset, the subset with \(S_1=1\) and the subset with \(S_1=S_2=1\) are seen in Tables 2-4.

Table 2. Proportions for the simulated dataset, by treatment status and overall.
| Not zika infected (N=4939) | Zika infected (N=61) | Overall (N=5000)
--- | --- | --- | ---
Microcephaly |  |  |
Mean | 0.003 | 0.361 | 0.008
Living area |  |  |
Mean | 0.849 | 0.951 | 0.850
SES |  |  |
Mean | 0.499 | 0.426 | 0.498

Table 3. Proportions for the simulated dataset, by treatment status and overall, after the first selection.
| Not zika infected (N=4268) | Zika infected (N=11) | Overall (N=4279)
--- | --- | --- | ---
Microcephaly |  |  |
Mean | 0.003 | 0.273 | 0.004
Living area |  |  |
Mean | 0.845 | 1.000 | 0.846
SES |  |  |
Mean | 0.556 | 0.818 | 0.557

Table 4. Proportions for the simulated dataset, by treatment status and overall, after both selections.
| Not zika infected (N=2869) | Zika infected (N=7) | Overall (N=2876)
--- | --- | --- | ---
Microcephaly |  |  |
Mean | 0.004 | 0.286 | 0.005
Living area |  |  |
Mean | 0.858 | 1.000 | 0.858
SES |  |  |
Mean | 0.382 | 0.714 | 0.382

The dataset and data generating process (DGP) can be used to test the functions in `SelectionBias`.

## `sensitivityparametersM()`

The sensitivity parameters for the SV, sharp, and GAF bounds are calculated for the generalized M-structure, illustrated in Figure 1. The sensitivity parameters are only calculated for an assumed model structure, since they depend on the unobserved variable, U. However, the observed probabilities of the outcome, \(P(Y=1|T=t,I_S=1)\), \(t=0,1\), are inputs. The code and the output are:

    # SV bound
    sensparSV = sensitivityparametersM(whichEst = "RR_tot",
                       whichBound = "SV",
                       Vval = matrix(c(1, 0, 0.85, 0.15), ncol = 2),
                       Uval = matrix(c(1, 0, 0.5, 0.5), ncol = 2),
                       Tcoef = c(-6.2, 1.75),
                       Ycoef = c(-5.2, 5.0, -1.0),
                       Scoef = matrix(c(1.2, 2.2, 0.0, 0.5,
                                        2.0, -2.75, -4.0, 0.0),
                                      ncol = 4),
                       Mmodel = "L",
                       pY1_T1_S1 = 0.286,
                       pY1_T0_S1 = 0.004)
    print(sensparSV)
    #> $parameter
    #>  [1] "BF_11"     "BF_00"     "BF_10"     "BF_01"     "RR_UY|T=1" "RR_UY|T=0"
    #>  [7] "RR_SU|11"  "RR_SU|00"  "RR_SU|10"  "RR_SU|01"
    #>
    #> $value
    #>  [1] 1.2102 1.3532 1.3208 1.3895 1.9448 2.7089 1.5566 1.7058 1.9998 1.7998
    #>
    #> attr(,"class")
    #> [1] "sensparams"
    #> attr(,"row.names")
    #>  [1]  1  2  3  4  5  6  7  8  9 10

    # Sharp bound
    sensparSharp = sensitivityparametersM(whichEst = "RR_tot",
                       whichBound = "sharp",
                       Vval = matrix(c(1, 0, 0.85, 0.15), ncol = 2),
                       Uval = matrix(c(1, 0, 0.5, 0.5), ncol = 2),
                       Tcoef = c(-6.2, 1.75),
                       Ycoef = c(-5.2, 5.0, -1.0),
                       Scoef = matrix(c(1.2, 2.2, 0.0, 0.5,
                                        2.0, -2.75, -4.0, 0.0),
                                      ncol = 4),
                       Mmodel = "L",
                       pY1_T1_S1 = 0.286,
                       pY1_T0_S1 = 0.004)

    print(sensparSharp)
    #> $parameter
    #>  [1] "BF_11"     "BF_00"     "BF_10"     "BF_01"     "RR_UY|T=1" "RR_UY|T=0"
    #>  [7] "RR_SU|11"  "RR_SU|00"  "RR_SU|10"  "RR_SU|01"
    #>
    #> $value
    #>  [1] 1.2102 1.3532 1.3208 1.3895 1.9448 2.7089 1.5566 1.7058 1.9998 1.7998
    #>
    #> attr(,"class")
    #> [1] "sensparams"
    #> attr(,"row.names")
    #>  [1]  1  2  3  4  5  6  7  8  9 10

    # GAF bound
    sensparGAF = sensitivityparametersM(whichEst = "RR_tot",
                       whichBound = "GAF",
                       Vval = matrix(c(1, 0, 0.85, 0.15), ncol = 2),
                       Uval = matrix(c(1, 0, 0.5, 0.5), ncol = 2),
                       Tcoef = c(-6.2, 1.75),
                       Ycoef = c(-5.2, 5.0, -1.0),
                       Scoef = matrix(c(1.2, 2.2, 0.0, 0.5,
                                        2.0, -2.75, -4.0, 0.0),
                                      ncol = 4),
                       Mmodel = "L",
                       pY1_T1_S1 = 0.286,
                       pY1_T0_S1 = 0.004)

    print(sensparGAF)
    #> $parameter
    #> [1] "m_T" "M_T"
    #>
    #> $value
    #> [1] 0.0020 0.4502
    #>
    #> attr(,"class")
    #> [1] "sensparams"
    #> attr(,"row.names")
    #> [1] 1 2

The first argument is `whichEst`, where the user inputs the causal estimand of interest. It must be one of the four `"RR_tot"`, `"RD_tot"`, `"RR_sub"` or `"RD_sub"`. The second argument is `whichBound`, where the user inputs the bound they want to use, either `"SV"`, `"sharp"`, or `"GAF"`. Third, the argument `Vval` takes the matrix for V as input. The first column contains the values that V can take, and the second column contains the corresponding probabilities. In this example, V is binary, so the first two elements in the matrix are 1 and 0. However, any discrete V can be used. An approximation of a continuous V can be used, if it is discretized. The fourth argument is `Uval`, which takes the matrix for U as input. The matrix U has a similar structure as V. The fifth argument is `Tcoef`, containing the coefficients used in the model for T. The first entry in `Tcoef` is the intercept of the model, and the second the slope for V. The sixth argument is `Ycoef`, containing the coefficient vector for the outcome model, where the first entry is the intercept, the second the slope coefficient for T and third is the slope coefficient for U. The seventh argument is `Scoef`, the coefficient matrix for the selection variables. The number of rows is equal to the number of selection variables, and the number of columns is equal to four. The columns represent the intercept, and slope coefficients for V, U and T, respectively. A summary of the code notation is seen in the last column of Table 3. The eighth argument is `Mmodel`, which indicates whether the models in the M-structure are probit (`Mmodel = "P"`) or logit (`Mmodel = "L"`). The ninth and tenth arguments are `pY1_T1_S1` and `pY1_T0_S1`. They are the observed probabilities \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\). The output is the sensitivity parameters for the chosen bound.

In the zika example, the estimand of interest is the risk ratio in the total population, `whichEst = "RR_tot"`, the DGP is found in Table 1, logistic models are used in the DGP, and the probabilities are found in Table 4. For the SV and sharp bounds, the output is \(RR_{UY|T=1}=1.9448\), \(RR_{UY|T=0}=2.7089\), \(RR_{SU|11}=1.5566\), \(RR_{SU|00}=1.7058\),\(RR_{SU|10}=1.9998\), and \(RR_{SU|01}=1.7998\), which gives \(BF_{11}=1.2102\) and \(BF_{00}=1.3532\), \(BF_{10}=1.3208\), and \(BF_{01}=1.3895\). For the GAF bound, the output is \(M_T=0.4502\) and \(m_T=0.002\).

## `SVbound()`

The SV bound can be calculated using the function `SVbound()`. The first argument is `whichEst`, indicating the causal estimand of interest (`"RR_tot"`, `"RD_tot"`, `"RR_sub"` or `"RD_sub"`). The second argument is `sens`, which is either output from `sensitivityparametersM()`, a named list or a data frame with column names parameter and value, which includes the sensitivity parameters. This input is optional as the sensitivity parameters can also be specified directly in the function with the below arguments. The third and fourth arguments are the observed conditional probabilities \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\), which are needed to calculate bounds for the causal estimands and not the selection bias itself. The optional fifth and sixth arguments are the treatment probabilities \(P(T=1|I_S=1)\) and \(P(T=0|I_S=1)\). These are only needed for the alternative SV bound for the causal risk difference in the subpopulation. If they are not included, the original SV bound in the risk difference is calculated. The subsequent arguments are the sensitivity parameters provided by the user. The default value for all sensitivity parameters are `NULL`, and the user must then specify numeric values on the sensitivity parameters that are necessary for the bound for the chosen estimand. These arguments are also optional in case the argument `sens` is used. The sensitivity parameters can either be calculated using the function `sensitivityparametersM()`, or found elsewhere. For sensitivity parameters found elsewhere, `SVbound()` is not restricted to the generalized M-structure. However, the necessary assumptions for the SV bound must still be fulfilled (Smith and VanderWeele 2019). The output is the SV bound. The code and output are:

    SVbound(whichEst = "RR_tot",
            pY1_T1_S1 = 0.286,
            pY1_T0_S1 = 0.004,
            RR_UY_T1 = 1.9448,
            RR_UY_T0 = 2.7089,
            RR_SU_11 = 1.5566,
            RR_SU_00 = 1.7058,
            RR_SU_10 = 1.9998,
            RR_SU_01 = 1.7998)
    #>      [,1]             [,2]
    #> [1,] "SV lower bound" 43.66
    #> [2,] "SV upper bound" 131.22

As before in the zika example, the causal estimand is the risk ratio in the total population, `whichEst = "RR_tot"`. The sensitivity parameters are \(RR_{UY|T=1}=1.9448\), \(RR_{UY|T=0}=2.7089\), \(RR_{SU|11}=1.5566\), \(RR_{SU|00}=1.7058\),\(RR_{SU|10}=1.9998\), and \(RR_{SU|01}=1.7998\), calculated above in `sensitivityparametersM()`, which gives a lower SV bound equal to 43.66 and a higher SV bound equal to 131.22. Alternatively, the output from the function `sensitivityparametersM()` can be used directly as input which gives the code:

    SVbound(whichEst = "RR_tot",
            sens = sensparSV,
            pY1_T1_S1 = 0.286,
            pY1_T0_S1 = 0.004)
    #>      [,1]             [,2]
    #> [1,] "SV lower bound" 43.66
    #> [2,] "SV upper bound" 131.22

This approach gives the same bounds as specifying the sensitivity parameters manually.

## `sharpbound()`

The sharp bound can be calculated using the function `sharpbound()`. The input arguments are very similar to that of `SVbound()`. The first argument is `whichEst`, indicating the causal estimand of interest (`"RR_tot"`, `"RD_tot"`, `"RR_sub"` or `"RD_sub"`). The second argument is `sens`, which is either output from `sensitivityparametersM()`, a named list or a data frame with column names parameter and value, which includes the sensitivity parameters. This input is optional as the sensitivity parameters can also be specified directly in the function with the below arguments. The third and fourth arguments are the observed conditional probabilities \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\), which are needed to calculate bounds for the causal estimands and not the selection bias itself. The fifth and sixth arguments are the treatment probabilities \(P(T=1|I_S=1)\) and \(P(T=0|I_S=1)\). These are only needed for the bounds for the causal estimands in the subpopulation. The seventh and eighth arguments are the probabilities \(P(I_S=1|T=1)\) and \(P(I_S=1|T=0)\) which are needed for the bounds for the causal estimands in the total population. If they are unknown, they can be set to 0 for more conservative bounds. The subsequent arguments are the sensitivity parameters provided by the user. The default value for all sensitivity parameters are `NULL`, and the user must then specify numeric values on the sensitivity parameters that are necessary for the bound for the chosen estimand. These arguments are also optional in case the argument `sens` is used. The sensitivity parameters can either be calculated using the function `sensitivityparametersM()`, or found elsewhere. For sensitivity parameters found elsewhere, `sharpbound()` is not restricted to the generalized M-structure. However, the necessary assumptions for the sharp bound must still be fulfilled (Zetterstrom, Sjölander, and Waernbaum 2025). The output is the sharp bound. The code and output with the two options to specify the sensitivity parameters are:

    sharpbound(whichEst = "RR_tot",
            pY1_T1_S1 = 0.286,
            pY1_T0_S1 = 0.004,
            pS1_T1 = 0.11,
            pS1_T0 = 0.58,
            RR_UY_T1 = 1.9448,
            RR_UY_T0 = 2.7089,
            RR_SU_11 = 1.5566,
            RR_SU_00 = 1.7058,
            RR_SU_10 = 1.9998,
            RR_SU_01 = 1.7998)
    #>      [,1]                [,2]
    #> [1,] "Sharp lower bound" 73.91
    #> [2,] "Sharp upper bound" 78.99

    sharpbound(whichEst = "RR_tot",
            sens = sensparSharp,
            pY1_T1_S1 = 0.286,
            pY1_T0_S1 = 0.004,
            pS1_T1 = 0.11,
            pS1_T0 = 0.58,)
    #>      [,1]                [,2]
    #> [1,] "Sharp lower bound" 73.91
    #> [2,] "Sharp upper bound" 78.99

As before in the zika example, the causal estimand is the risk ratio in the total population, `whichEst = "RR_tot"`. The observed probabilities come from the dataset, and the sensitivity parameters are \(RR_{UY|T=1}=1.9448\), \(RR_{UY|T=0}=2.7089\), \(RR_{SU|11}=1.5566\), \(RR_{SU|00}=1.7058\), \(RR_{SU|10}=1.9998\), and \(RR_{SU|01}=1.7998\), calculated above in `sensitivityparametersM()`, which gives a lower sharp bound equal to 73.91 and a higher sharp bound equal to 78.99.

## `checksharpSVbound()`

The sharpness of the SV bound can be evaluated using `checksharpSVbound()` (Zetterstrom, Sjölander, and Waernbaum 2025). The first argument is `whichEst`, indicating the causal estimand of interest (`"RR_tot"`, `"RR_sub"` or `"RD_sub"`). Note that the SV bound for the risk difference in the total population is not sharp. The second argument is `sens`, which is either output from `sensitivityparametersM()`, a named list or a data frame with column names parameter and value, which includes the bounding factors (\(BF\)). This input is optional as the bounding factors can also be specified directly in the function with the argument below. The third argument is `BF`, the bounding factors \(BF_{00}\) and \(BF{10}\) for the total population, and \(BF_0\) and \(BF_1\) for the subpopulation. The fourth argument is `pY1`, the probabilities \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\). Note that the order of the bounding factors and probabilities matter. The output is two strings stating whether the lower and upper SV bound are sharp or not. Note that the SV bound for the risk ratio in the total population and risk difference in the subpopulation can only be arbitrarily sharp, see (Zetterstrom, Sjölander, and Waernbaum 2025) for more details. The code and output are:

    checksharpSVbound(whichEst = "RR_tot",
            sens = sensparSV,
            pY1 = c(0.286, 0.004))
    #> [[1]]
    #> [1] "The lower SV bound for the risk ratio in the total population is arbitrarily sharp. See vignette for details."
    #>
    #> [[2]]
    #> [1] "The upper SV bound for the risk ratio in the total population is arbitrarily sharp. See vignette for details."

Here, both the lower and upper SV bounds are arbitrarily sharp.

## `AFbound()`

The AF bound is calculated using the function `AFbound()`. The first argument is the causal estimand of interest (`"RR_tot"`, `"RD_tot"`, `"RR_sub"` or `"RD_sub"`). The second argument is `outcome`, where the user inputs either the observed numeric vector with the outcome variable or a vector with the conditional outcome probabilities, \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\). The third argument is `treatment`, where the user inputs either the observed numeric vector with the treatment variable or a vector with the conditional treatment probabilities, \(P(T=1|I_S=1)\) and \(P(T=0|I_S=1)\). The fourth argument is `selection` where the user can either input the observed selection vector or the selection probability. Its default value is `NULL` since it is only required when the causal estimands in the total population are of interest. If the subpopulation is of interest and `selection = NULL`, the outcome and treatment vectors must only include the selected subjects. The output is the lower and upper AF bounds. The code and output are:

    attach(zika_learner)

    AFbound(whichEst = "RR_tot",
            outcome = mic_ceph[sel_ind == 1],
            treatment = zika[sel_ind == 1],
            selection = mean(sel_ind))
    #>      [,1]             [,2]
    #> [1,] "AF lower bound" 0
    #> [2,] "AF upper bound" 454.09

Similar to before, `whichEst = "RR_tot"`. Furthermore, the outcome and treatment variables are microcephaly and zika. The selection probability is specified since the other variables are restricted to those subjects with \(I_S=1\). The output is the lower and upper AF bounds, which are 0 and 454.09 in the zika example.

If the raw data is not available, one can input the conditional probabilities instead. In this example, these probabilities are:

    AFbound(whichEst = "RR_tot",
            outcome = c(0.286, 0.004),
            treatment = c(0.002, 0.998),
            selection = mean(sel_ind))
    #>      [,1]             [,2]
    #> [1,] "AF lower bound" 0
    #> [2,] "AF upper bound" 435.14

The difference in these two examples comes from rounding errors in the estimated probabilities.

## `GAFbound()`

The GAF bound is calculated using the function `GAFbound()`. The first argument is the causal estimand of interest (`"RR_tot"`, `"RD_tot"`, `"RR_sub"` or `"RD_sub"`). The second argument is `sens`, which is either output from `sensitivityparametersM()`, a named list or a data frame with column names “parameter” and “value”, which includes the sensitivity parameters. This input is optional as the sensitivity parameters can also be specified directly in the function with the below arguments. The third and fourth arguments are `M` and `m` which are the two sensitivity parameters for the GAF bound. The sensitivity parameters can either be calculated using `sensitivityparametersM()`, or found elsewhere. For sensitivity parameters found elsewhere, `GAFbound()` is not restricted to the generalized M-structure. However, the necessary assumptions for the GAF bound must still be fulfilled (Zetterstrom 2024). The fifth argument is `outcome`, where the user inputs either the observed numeric vector with the outcome variable or a vector with the conditional outcome probabilities, \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\). The fifth argument is `treatment`, where the user inputs either the observed numeric vector with the treatment variable or a vector with the conditional treatment probabilities, \(P(T=1|I_S=1)\) and \(P(T=0|I_S=1)\). The sixth argument is `selection` where the user can either input the observed selection vector or selection probability. Its default value is `NULL` since it is only required when the causal estimands in the total population are of interest. If the subpopulation is of interest and `selection = NULL`, the outcome and treatment vectors must only include the selected subjects. The output is the lower and upper GAF bounds. The code and output are:

    GAFbound(whichEst = "RR_tot",
             M = 0.4502,
             m = 0.002,
             outcome = mic_ceph[sel_ind == 1],
             treatment = zika[sel_ind == 1],
             selection = mean(sel_ind))
    #>      [,1]              [,2]
    #> [1,] "GAF lower bound" 0.01
    #> [2,] "GAF upper bound" 147.42

    GAFbound(whichEst = "RR_tot",
             sens = sensparGAF,
             outcome = mic_ceph[sel_ind == 1],
             treatment = zika[sel_ind == 1],
             selection = mean(sel_ind))
    #>      [,1]              [,2]
    #> [1,] "GAF lower bound" 0.01
    #> [2,] "GAF upper bound" 147.42

Similar to before, `whichEst = "RR_tot"`. The sensitivity parameters are the output from `sensitivityparametersM()`. Furthermore, the outcome and treatment variables are microcephaly and zika. The selection probability is specified since the other variables are restricted to those subjects with \(I_S=1\). The output is the lower and upper GAF bounds, which are 0.01 and 147.42 in the zika example. If the raw data is not available, one can input the conditional probabilities instead, similar to the AF bound.

## `CAFbound()`

The CAF bound is calculated using the function `CAFbound()`. The first argument is the causal estimand of interest (`"RR_tot"`, `"RD_tot"`, `"RR_sub"` or `"RD_sub"`). The second and third arguments are `M` and `m` which are the two sensitivity parameters for the CAF bound. The fourth argument is `outcome`, where the user inputs either the observed numeric vector with the outcome variable or a vector with the conditional outcome probabilities, \(P(Y=1|T=1,I_S=1)\) and \(P(Y=1|T=0,I_S=1)\). The fifth argument is `treatment`, where the user inputs either the observed numeric vector with the treatment variable or a vector with the conditional treatment probabilities, \(P(T=1|I_S=1)\) and \(P(T=0|I_S=1)\). The sixth argument is `selection` where the user can either input the observed selection vector or selection probability. Its default value is `NULL` since it is only required when the causal estimands in the total population are of interest. If the subpopulation is of interest and `selection = NULL`, the outcome and treatment vectors must only include the selected subjects. The output is the lower and upper CAF bounds. The code and output are:

    CAFbound(whichEst = "RR_tot",
             M = 0.3,
             m = 0.005,
             outcome = c(0.286, 0.004),
             treatment = c(0.002, 0.998),
             selection = mean(sel_ind))
    #>      [,1]              [,2]
    #> [1,] "CAF lower bound" 0.04
    #> [2,] "CAF upper bound" 67.78

Similar to before, `whichEst = "RR_tot"`. The sensitivity parameters are chosen by the user. Furthermore, the outcome and treatment variables are microcephaly and zika. The selection probability is specified since other variables are restricted to those subjects with \(I_S=1\). The output is the lower and upper CAF bounds, which are 0.04 and 67.78 in the zika example. If the raw data is not available, one can input the conditional probabilities instead, similar to the AF and GAF bounds.

# References

Araújo, Thalia Velho Barreto de, Ricardo Arraes de Alencar Ximenes, Demócrito de Barros Miranda-Filho, Wayner Vieira Souza, Ulisses Ramos Montarroyos, Ana Paula Lopes de Melo, Sandra Valongueiro, et al. 2018. “Association Between Microcephaly, Zika Virus Infection, and Other Risk Factors in Brazil: Final Report of a Case-Control Study.” The Lancet Infectious Diseases 18 (3): 328–36.
Smith, Louisa H, and Tyler J VanderWeele. 2019. “Bounding Bias Due to Selection.” Epidemiology 30 (4): 509–16.
Zetterstrom, Stina. 2024. “Bounds for Selection Bias Using Outcome Probabilities.” Epidemiologic Methods 13 (1): 20230033.
Zetterstrom, Stina, Arvid Sjölander, and Ingeborg Waernbaum. 2025. “Investigations of Sharp Bounds for Causal Effects Under Selection Bias.” Statistical Methods in Medical Research, 09622802251374168.
Zetterstrom, Stina, and Ingeborg Waernbaum. 2022. “Selection Bias and Multiple Inclusion Criteria in Observational Studies.” Epidemiologic Methods 11 (1): 1–21.
———. 2023. “SelectionBias: An R Package for Bounding Selection Bias.” arXiv. https://doi.org/10.48550/ARXIV.2302.06518.