Package 'budgetIVr'

Partially identify causal effects with invalid instruments

Description

Computes the set of possible values of a causal parameter consistent with observational data and given budget constraints. See Penn et al. (2025) for technical definitions.

Usage

budgetIV(
  beta_y,
  beta_phi,
  phi_basis = NULL,
  tau_vec = NULL,
  b_vec = NULL,
  ATE_search_domain = NULL,
  X_baseline = NULL,
  delta_beta_y = NULL
)

Arguments

beta_y	Either $1 \times d_{Z}$ matrix or a $d_{Z}$-dimensional vector representing the (estimated) cross covariance $\mathrm{Cov}(Y, Z)$.
beta_phi	A $d_{\Phi} \times d_{Z}$ matrix representing the (estimated) cross covariance $\mathrm{Cov}(\Phi (X), Z)$.
phi_basis	A $d_{\Phi}$-dimensional expression (separated by commas) with each term representing a component of $\Phi (X)$. The expression consists of $d_{X}$ unique vars. The default value NULL can be used for a $d_{X} = d_{\Phi}$-dimensional linear model.
tau_vec	A $K$-dimensional vector of increasing, positive thresholds representing degrees of IV invalidity. The default value NULL can be used for a single threshold at $0$.
b_vec	A $K$-dimensional vector of increasing positive integers representing the maximum number of IVs that can surpass each threshold. The default value NULL can be used for a single threshold at $0$, with at least $50\%$ of IVs assumed to be valid.
ATE_search_domain	A $d_{X}$-column data.frame with column names equal to the vars in phi_basis. Rows correspond to values of the treatment $X$. The default value NULL can be used to generate a small $d_{X}$-dimensional grid.
X_baseline	Either a data.frame or list representing a baseline treatment $x_0$, with names equal to the vars in phi_basis. The default value NULL can be used for the baseline treatment $0$ for each of of the $d_{X}$ vars.
delta_beta_y	A $d_{Z}$-dimensional vector of positive half-widths for box-shaped confidence bounds on beta_y. The default value NULL can be used to not include finite sample uncertainty.

Details

Instrumental variables are defined by three structural assumptions: (A1) they are associated with the treatment; (A2) they are unconfounded with the outcome; and (A3) exclusively effect the outcome through the treatment. Of these, only (A1) can be tested without further assumptions. The budgetIV function allows for valid causal inference when some proportion (possibly a small minority) of candidate instruments satisfy both (A2) and (A3). Tuneable thresholds decided by the user also allow for bounds on the degree of invalidity for each instrument (i.e., bounds on the proportion of $\mathrm{Cov}(Y, Z)$ not explained by the causal effect of $X$ on $Z$). Full technical details are included in Penn et al. (2025).

budgetIV assumes that treatment effects are homogeneous, which implies a structural equation of the form $Y = \theta \cdot \Phi(X) + g_y(Z, \epsilon_x)$, where $\theta$ and $\Phi(X)$ are a $d_{\Phi}$-dimensional vector and vector-valued function respectively. A valid basis expansion $\Phi (X)$ is assumed (e.g., linear, logistic, polynomial, RBF, neural embedding, PCA, UMAP etc.). It is also assumed that $d_{\Phi} <= d_{Z}$, which allows us to treat the basis functions as a complete linear model (see Theil (1953), or Sanderson et al. (2019) for a modern MR focused discussion). The parameters $\theta$ capture the unknown treatment effect. Violation of (A2) and/or (A3) will bias classical IV approaches through the statistical dependence between $Z$ and $g_y(Z, \epsilon_x)$, summarized by the covariance parameter $\gamma := \mathrm{Cov} (g_y(Z, \epsilon_x), Z)$.

budgetIV constrains $\gamma$ through a series of positive thresholds $0 \leq \tau_1 < \tau_2 < \ldots < \tau_K$ and corresponding integer budgets $0 < b_1 < b_2 < \ldots < b_K \leq d_Z$. It is assumed for each $i \in \{ 1, \ldots, K\}$ that no more than $b_i$ components of $\gamma$ are greater in magnitude than $\tau_i$. For instance, taking $d_Z = 100$, $K = 1$, $b_1 = 5$ and $\tau_1 = 0$ means assuming $5$ of the $100$ candidates are valid instrumental variables (in the sense that their ratio estimates $\theta_j := \mathrm{Cov}(Y, Z_j)/\mathrm{Cov}(\Phi(X), Z_j)$ are unbiased).

With delta_beta_y = NULL, budgetIV returns the identified set of causal effects that agree with both the budget constraints described above and the values of $\mathrm{Cov}(Y, Z)$ and $\mathrm{Cov}(Y, Z)$, assumed to be exactly precise. Unlike classical partial identification methods (see Manski (1990) for a canonical example), the non-convex mixed-integer budget constraints yield a possibly disconnected solution set. Each connected subset has a different interpretation as to which of the candidate instruments $Z$ are valid up to each threshold.

delta_beta_y represents box-constraints to quantify uncertainty in beta_y. In the examples, delta_beta_y is calculated through a Bonferroni correction and gives an (asymptotically) valid confidence set over beta_y. Under the so-called "no measurement error" assumption (see Bowden et al. (2016)), which is commonly applied in Mendelian randomization, it is assumed that the estimate of beta_y is the dominant source of finite-sample uncertainty, with uncertainty in beta_x considered negligible. With an (asymptotically) valid confidence set for delta_beta_y, and under the "no measurement error" assumption, budgetIV returns an (asymptotically) valid confidence set for $\theta$ when using just a single exposure.

Value

A data.table with each row corresponding to a set of bounds on the ATE at a given point in ATE_search_domain. Columns include: a non-unique identifier curve_index with a one-to-one mapping with U; lower_ATE_bound and upper_ATE_bound for the corresponding bounds on the ATE; a list U for the corresponding budget assignment; and a column for each unique variable in ATE_search_domain to indicate the treatment value at which the bounds are being calculated.

References

Jordan Penn, Lee Gunderson, Gecia Bravo-Hermsdorff, Ricardo Silva, and David Watson. (2024). BudgetIV: Optimal Partial Identification of Causal Effects with Mostly Invalid Instruments. AISTATS 2025.

Jack Bowden, Fabiola Del Greco M, Cosetta Minelli, George Davey Smith, Nuala A Sheehan, and John R Thompson. (2016). Assessing the suitability of summary data for two-sample Mendelian randomization analyses using MR-Egger regression: the role of the I^2 statistic. Int. J. Epidemiol. 46.6, pp. 1985–1998.

Charles F Manski. (1990). Nonparametric bounds on treatment effects. Am. Econ. Rev. 80.2, pp. 219–323.

Henri Theil. (1953). Repeated least-squares applied to complete equation systems. Centraal Planbureau Memorandum.

Eleanor Sanderson, George Davey Smith, Frank Windmeijer and Jack Bowden. (2019). An examination of multivariable Mendelian randomization in the single-sample and two-sample summary data settings. Int. J. Epidemiol. 48.3, pp. 713–727.

Examples

data(simulated_data_budgetIV)

beta_y <- simulated_data_budgetIV$beta_y

beta_phi_1 <- simulated_data_budgetIV$beta_phi_1
beta_phi_2 <- simulated_data_budgetIV$beta_phi_2

beta_phi <- matrix(c(beta_phi_1, beta_phi_2), nrow = 2, byrow = TRUE)

delta_beta_y <- simulated_data_budgetIV$delta_beta_y

tau_vec = c(0)
b_vec = c(3)

x_vals <- seq(from = 0, to = 1, length.out = 500)

ATE_search_domain <- expand.grid("x" = x_vals)

phi_basis <- expression(x, x^2)

X_baseline <- list("x" = c(0))

solution_set <- budgetIV(beta_y = beta_y, 
                         beta_phi = beta_phi, 
                         phi_basis = phi_basis, 
                         tau_vec = tau_vec, 
                         b_vec = b_vec, 
                         ATE_search_domain = ATE_search_domain, 
                         X_baseline = X_baseline,
                         delta_beta_y = delta_beta_y)

---

Efficient partial identification of a scalar causal effect parameter with invalid instruments

Description

Partial identification and coverage of a causal effect parameter using summary statistics and budget constraint assumptions. See Penn et al. (2025) for technical definitions.

Usage

budgetIV_scalar(
  beta_y,
  beta_phi,
  tau_vec = NULL,
  b_vec = NULL,
  delta_beta_y = NULL,
  bounds_only = TRUE
)

Arguments

beta_y	A $d_{Z}$-dimensional vector representing the (estimated) cross covariance $\mathrm{Cov}(Y, Z)$.
beta_phi	A $d_{Z}$-dimensional vector representing the (estimated) cross covariance $\mathrm{Cov}(\Phi (X), Z)$.
tau_vec	A $K$-dimensional vector of increasing, positive thresholds representing degrees of IV invalidity. The default value NULL can be used for a single threshold at $0$.
b_vec	A $K$-dimensional vector of increasing positive integers representing the maximum number of IVs that can surpass each threshold. The default value NULL can be used for a single threshold at $0$, with at least $50\%$ of IVs assumed to be valid.
delta_beta_y	A $d_{Z}$-dimensional vector of positive half-widths for box-shaped confidence bounds on beta_y. The default value NULL can be used to not include finite sample uncertainty.
bounds_only	A boolean TRUE or FALSE. TRUE will store overlapping intervals in the confidence set as a single interval, while FALSE will store different intervals for different values of budget_assignment (see return value of Penn et al. (2025) for further details). The default is TRUE. If TRUE (default), the output consists only of disjoint bounds. Otherwise, if FALSE, the output consists of bounds for possibly touching intervals (but never overlapping), as well as the budget assignment corresponding to each bound.

Details

Instrumental variables are defined by three structural assumptions: (A1) they are associated with the treatment; (A2) they are unconfounded with the outcome; and (A3) they exclusively effect the outcome through the treatment. Assumption (A1) has a simple statistical test, whereas for many data generating processes (A2) and (A3) are unprovably false. The budgetIV and budgetIV_scalar algorithms allow for valid causal inference when some proportion, possibly a small minority, of candidate instruments satisfy both (A2) and (A3).

budgetIV & budgetIV_scalar assume a homogeneous treatment effect, which implies the separable structural equation $Y = \theta \Phi(X) + g_y(Z, \epsilon_x)$. The difference between the algorithms is that budgetIV_scalar assumes $\Phi(X)$ and $\theta$ take scalar values, which is exploited for super-exponential computational speedup and allows for causal inference with thousands of candidate instruments $Z$. Both methods assume ground truth knowledge of the functional form of $\Phi (X)$, e.g., a linear, logistic, Cox hazard, principal component based or other model. The parameter $\theta$ captures the unknown treatment effect. Violation of (A2) and/or (A3) will bias classical IV approaches through the statistical dependence between $Z$ and $g_y(Z, \epsilon_x)$, summarized by the covariance parameter $\gamma := \mathrm{Cov} (g_y(Z, \epsilon_x), Z)$.

budgetIV & budgetIV_scalar constrain $\gamma$ through a series of positive thresholds $0 \leq \tau_1 < \tau_2 < \ldots < \tau_K$ and corresponding integer budgets $0 < b_1 < b_2 < \ldots < b_K \leq d_Z$. It is assumed for each $i \in \{ 1, \ldots, K\}$ that no more than $b_i$ components of $\gamma$ are greater in magnitude than $\tau_i$. For instance, taking $d_Z = 100$, $K = 1$, $b_1 = 5$ and $\tau_1 = 0$ means assuming $5$ of the $100$ candidates are valid instrumental variables (in the sense that their ratio estimates $\theta_j := \mathrm{Cov}(Y, Z_j)/\mathrm{Cov}(\Phi(X), Z_j)$ are unbiased).

With delta_beta_y = NA, budgetIV & budgetIV_scalar return the identified set of causal effects that agree with both the budget constraints described above and the values of $\mathrm{Cov}(Y, Z)$ and $\mathrm{Cov}(Y, Z)$, assumed to be exactly precise. Unlike classical partial identification methods (see Manski (1990) ofr a canonical example), the non-convex mixed-integer budget constraints yield a possibly disconnected identified set. Each connected subset has a different interpretation as to which of the candidate instruments $Z$ are valid up to each threshold. budgetIV_scalar returns these interpretations alongside the corresponding bounds on $\theta$.

When delta_beta_y is not null, it is used as box-constraints to quantify uncertainty in beta_y. In the examples, delta_beta_y is calculated through a Bonferroni correction and gives an (asymptotically) valid confidence set over beta_y. Under the so-called "no measurement error" (NOME) assumption (see Bowden et al. (2016)) which is commonly applied in Mendelian randomisation, it is assumed that the estimate of beta_y is the dominant source of finite-sample uncertainty, with uncertainty in beta_x entirely negligible. With an (asymptotically) valid confidence set over delta_beta_y and under the "no measurement error" assumption, budgetIV_scalar returns an (asymptotically) valid confidence set for $\theta$.

Value

A data.table with each row corresponding to bounds on the scalar causal effect parameter $\theta$ corresponding to a particular budget assignment $U$ (see Penn et al. (2025)). The return table has the following rows: a logical is_point determining whether the upper and lower bounds are equivalent; numerical lower_bound and upper_bound giving the lower and upper bounds; and a list budget_assignment giving the value of $U$ for each candidate instrument. budget_assignment will only be returned if bounds_only == FALSE as input by the user.

A list of two entries: intervals, which is a two-column matrix with rows corresponding to disjoint bounds containing plausible values of $\theta$; and points, which is a one-column matrix consisting of lone plausible values of $\theta$—relevant when using $\tau_1 = 0$.

References

Jordan Penn, Lee Gunderson, Gecia Bravo-Hermsdorff, Ricardo Silva, and David Watson. (2024). BudgetIV: Optimal Partial Identification of Causal Effects with Mostly Invalid Instruments. arXiv preprint, 2411.06913.

Jack Bowden, Fabiola Del Greco M, Cosetta Minelli, George Davey Smith, Nuala A Sheehan, and John R Thompson. (2016). Assessing the suitability of summary data for two-sample Mendelian randomization analyses using MR-Egger regression: the role of the I^2 statistic. Int. J. Epidemiol. 46.6, pp. 1985–1998.

Charles F Manski. (1990). Nonparametric bounds on treatment effects. Am. Econ. Rev. 80.2, pp. 219–323.

Examples

data(Do_et_al_summary_statistics)

candidatesHDL = Do_et_al_summary_statistics[Do_et_al_summary_statistics$pHDL <= 1e-8, ]

candidate_labels <- candidatesHDL$rsID
d_Z <- length(candidate_labels)

beta_x <- candidatesHDL$betaHDL

beta_y <- candidatesHDL$betaCAD

SE_beta_y <- abs(beta_y) / qnorm(1-candidatesHDL$pCAD/2)

alpha = 0.05
delta_beta_y <- qnorm(1 - alpha/(2*d_Z))*SE_beta_y

feasible_region <- budgetIV_scalar(
                                   beta_y = candidatesHDL$betaCAD,
                                   beta_phi = beta_x,
                                   tau_vec = c(0),
                                   b_vec = c(30),
                                   delta_beta_y = delta_beta_y,
                                   bounds_only = FALSE
                                   )

---

Summary statistics from Do et al. (2013)

Description

Common variants associated with plasma triglycerides and risk for coronary artery disease. Preprocessed and harmonized summary statistics from a Mendelian randomization analysis, including summary statistics for variants' association with plasma triglyceride levels, serum HDL levels, serum LDL levels and risk of coronary artery disease (CAD). Dataset previously applied in the mode-based estimate approach of Hartwig et al. (2017). Each row of the dataset corresponds to a single genetic variant (single nucleotide polymorphism) found to be associated with either the HDL, LDL, or triglyceride biomarkers across a population of 180,000 (HDL, LDL) or 86,000 (triglyceride) individuals. Got further biological and statistical details, see Do et al. (2013).

Usage

data(Do_et_al_summary_statistics)

Format

A data frame with 185 rows and 14 variables:

Details

X

A unique identifier from 1 to 185.

rsID

A unique string specifying each SNP using the rsID format.

chr

String specifying the chromosomal position of each SNP.

a1

Character specifying one allele of the SNP (all 185 SNPs are assumed to be biallelic).

a2

Character specifying the other allele of the SNP.

betaLDL

Effect size (linear regression) for association between SNP allele and LDL.

pLDL

p-value for testing association between SNP allele and LDL.

betaHDL

Effect size (linear regression) for association between SNP allele and HDL.

pHDL

p-value for testing association between SNP allele and HDL.

betaTri

Effect size (linear regression) for association between SNP allele and triglyceride.

pTri

p-value for testing association between SNP allele and triglyceride.

betaCAD

Effect size (logistic regression) for association between SNP allele and CAD.

pCAD

p-value for testing association between SNP allele and CAD.

References

Ron Do et al. (2013). Common variants associated with plasma triglycerides and risk for coronary artery disease. Nat Genet. 45.11, pp. 1345–52.

Fernando Pires Hartwig, George Davey Smith, and Jack Bowden. (2017). Robust inference in summary data Mendelian randomization via the zero modal pleiotropy assumption. Int. J. Epidemiol. 46.6, pp. 1985–1998.

Examples

# Extracting relevant summary statistics to investigate the causal effect of HDL on CAD risk.

data(Do_et_al_summary_statistics)

candidatesHDL = Do_et_al_summary_statistics[Do_et_al_summary_statistics$pHDL <= 1e-8, ]

candidate_labels <- candidatesHDL$rsID
d_Z <- length(candidate_labels)

beta_x <- candidatesHDL$betaHDL

beta_y <- candidatesHDL$betaCAD

SE_beta_y <- abs(beta_y) / qnorm(1-candidatesHDL$pCAD/2)

# For confidence set in budgetIV/budgetIV_scalar.
alpha = 0.05
delta_beta_y <- qnorm(1 - alpha/(2*d_Z))*SE_beta_y

---

Simulated summary statistics with invalid instruments and nonlinear treatment effect

Description

Example dataset from the nonlinear simulation study using 6 candidate instruments, 3 of which are invalid with violation of IV assumptions (A2) and (A3). See Appx. C.2 of Penn et al. (2025) for technical details or visit the source code for reproducibility, both referenced below. The ground truth causal effect is $\Phi^* (X) = (X - 0.25)^2 - 0.25^2$. $\beta_{\Phi}$ is taken with respect to the basis functions $\Phi = (X, X^2)$.

Usage

data(simulated_data_budgetIV)

Format

A data frame with 6 rows and 4 columns.

Details

beta_y

Components of the estimator $\mathrm{Cov} (Y, Z)$.

beta_phi_1

Components of the estimator $\mathrm{Cov} ( \Phi_1 (X), Z )$.

beta_phi_2

Components of the estimator $\mathrm{Cov} ( \Phi_2 (X), Z )$.

delta_beta_y

Components of the standard error $\mathrm{Se} (\mathrm{Cov} (Y, Z))$.

Source

The code that generated this dataset was written by the authors and can be found in https://github.com/jpenn2023/budgetIVr/blob/main/paper/simulate_nonlinear_data.R. The dataset is saved as "my_dat R = 0.5 SNR_y = 1.csv".

References

Jordan Penn, Lee Gunderson, Gecia Bravo-Hermsdorff, Ricardo Silva, and David Watson. (2024). BudgetIV: Optimal Partial Identification of Causal Effects with Mostly Invalid Instruments. arXiv preprint, 2411.06913.

Examples

data(simulated_data_budgetIV)

beta_y <- simulated_data_budgetIV$beta_y

beta_phi_1 <- simulated_data_budgetIV$beta_phi_1
beta_phi_2 <- simulated_data_budgetIV$beta_phi_2

d_Z <- length(beta_phi_1)

beta_phi <- matrix(c(beta_phi_1, beta_phi_2), nrow = 2, byrow = TRUE)

delta_beta_y <- simulated_data_budgetIV$delta_beta_y

---