R: Two Stage Curvature Identification with Boosting

tsci_boosting {TSCI}

R Documentation

Two Stage Curvature Identification with Boosting

Description

tsci_boosting implements Two Stage Curvature Identification (Guo and Buehlmann 2022) with boosting. Through a data-dependent way, it tests for the smallest sufficiently large violation space among a pre-specified sequence of nested violation space candidates. Point and uncertainty estimates of the treatment effect for all violation space candidates including the selected violation space will be returned amongst other relevant statistics.

Usage

tsci_boosting(
  Y,
  D,
  Z,
  X = NULL,
  W = X,
  vio_space,
  create_nested_sequence = TRUE,
  sel_method = c("comparison", "conservative"),
  split_prop = 2/3,
  nrounds = 50,
  eta = 0.3,
  max_depth = c(1:6),
  subsample = 1,
  colsample_bytree = 1,
  early_stopping = TRUE,
  nfolds = 5,
  self_predict = FALSE,
  sd_boot = TRUE,
  iv_threshold = 10,
  threshold_boot = TRUE,
  alpha = 0.05,
  intercept = TRUE,
  parallel = c("no", "multicore", "snow"),
  nsplits = 10,
  mult_split_method = c("FWER", "DML"),
  ncores = 1,
  cl = NULL,
  raw_output = NULL,
  B = 300
)

Arguments

`Y`	observations of the outcome variable. Either a numeric vector of length n or a numeric matrix with dimension n by 1. If outcome variable is binary use dummy encoding.
`D`	observations of the treatment variable. Either a numeric vector of length n or a numeric matrix with dimension n by 1. If treatment variable is binary use dummy encoding.
`Z`	observations of the instrumental variable(s). Either a vector of length n or a matrix with dimension n by s. If observations are not numeric dummy encoding will be applied.
`X`	observations of baseline covariate(s). Either a vector of length n or a matrix with dimension n by p or `NULL` (if no covariates should be included). If observations are not numeric dummy encoding will be applied.
`W`	(transformed) observations of baseline covariate(s) used to fit the outcome model. Either a vector of length n or a matrix with dimension n by p_w or `NULL` (if no covariates should be included). If observations are not numeric dummy encoding will be applied.
`vio_space`	list with vectors of length n and/or matrices with n rows as elements to specify the violation space candidates. If observations are not numeric dummy encoding will be applied. See Details for more information.
`create_nested_sequence`	logical. If `TRUE`, the violation space candidates (in form of matrices) are defined sequentially starting with an empty violation matrix and subsequently adding the next element of `vio_space` to the current violation matrix. If `FALSE,` the violation space candidates (in form of matrices) are defined as the empty space and the elements of `vio_space`. See Details for more information.
`sel_method`	The selection method used to estimate the treatment effect. Either "comparison" or "conservative". See Details.
`split_prop`	proportion of observations used to fit the outcome model. Has to be a value in (0, 1).
`nrounds`	number of boosting iterations. Can either be a single integer value or a vector of integer values to try.
`eta`	learning rate of the boosting algorithm. Can either be a single numeric value or a vector of numeric values to try.
`max_depth`	maximal tree depth. Can either be a single integer value or a vector of integer values to try.
`subsample`	subsample ratio of the training instance. Can either be a single numeric value or a vector of numeric values to try. Has to be a numeric value in (0, 1].
`colsample_bytree`	subsample ratio of columns when constructing each tree. Can either be a single numeric value or a vector of numeric values to try. Has to be a numeric value in (0, 1].
`early_stopping`	logical. If `TRUE`, early stopping will be applied to choose the optimal number of boosting iteration using the cross-validation mean squared error.
`nfolds`	a positive integer value specifying the number of folds used for cross-validation to choose best parameter combination.
`self_predict`	logical. If `FALSE`, it sets the diagonal of the hat matrix of each tree to zero to avoid self-prediction and rescales the off-diagonal elements accordingly.
`sd_boot`	logical. if `TRUE`, it determines the standard error using a bootstrap approach.
`iv_threshold`	a numeric value specifying the minimum of the threshold of IV strength test.
`threshold_boot`	logical. if `TRUE`, it determines the threshold of the IV strength using a bootstrap approach. If `FALSE`, it does not perform a bootstrap. See Details.
`alpha`	the significance level. Has to be a numeric value between 0 and 1.
`intercept`	logical. If `TRUE`, an intercept is included in the outcome model.
`parallel`	one out of `"no"`, `"multicore"`, or `"snow"` specifying the parallelization method used.
`nsplits`	number of times the data will be split. Has to be an integer larger or equal 1. See Details.
`mult_split_method`	method to calculate the standard errors, p-values and to construct the confidence intervals if multi-splitting is performed. Default is "DML" if `nsplits` == 1 and "FWER" otherwise. See Details.
`ncores`	the number of cores to use. Has to be an integer value larger or equal 1.
`cl`	either a parallel or snow cluster or `NULL`.
`raw_output`	logical. If `TRUE`, the coefficient and standard error estimates of each split will be returned. This is only needed for the use of the function `confint` if `mult_split_method` equals "FWER". Default is `TRUE` if `mult_split_method` is `TRUE` and `FALSE` otherwise.
`B`	number of bootstrap samples. Has to be a positive integer value. Bootstrap methods are used to calculate the IV strength threshold if `threshold_boot` is `TRUE` and for the violation space selection.

Details

The treatment and outcome models are assumed to be of the following forms:

D_i = f(Z_i, X_i) + \delta_i

Y_i = \beta \cdot D_i + h(Z_i, X_i) + \phi(X_i) + \epsilon_i

where f(Z_i, X_i) is estimated using L2 boosting with regression trees as base learners, h(Z_i X_i) is approximated using the violation space candidates and \phi(X_i) is approximated by a linear combination of the columns in W. The errors are allowed to be heteroscedastic. To avoid overfitting bias the data is randomly split into two subsets A1 and A2 where the proportion of observations in the two sets is specified by split_prop. A2 is used to train the random forest and A1 is used to perform violation space selection and to estimate the treatment effect.

The package xgboost is used for boosting. If any of nrounds, eta, max_depth, subsample or colsample_bytree has more than one value, the best parameter combination is chosen by minimizing the cross-validation mean squared error.

The violation space candidates should be in a nested sequence as the violation space selection is performed by comparing the treatment estimate obtained by each violation space candidate with the estimates of all violation space candidates further down the list vio_space that provide enough IV strength. Only if no significant difference was found in all of those comparisons, the violation space candidate will be selected. If sel_method is 'comparison', the treatment effect estimate of this violation space candidate will be returned. If sel_method is 'conservative', the treatment effect estimate of the successive violation space candidate will be returned provided that the IV strength is large enough. The specification of suitable violation space candidates is a crucial step because a poor approximation of g(Z_i, X_i) might not address the bias caused by the violation of the IV assumption sufficiently well. The function create_monomials can be used to create a predefined sequence of violation space candidates consisting of monomials. The function create_interactions can be used to create a predefined sequence of violation space candidates consisting of two-way interactions between the instrumens themselves and between the instruments and the instruments and baseline covariates.

W should be chosen to be flexible enough to approximate the functional form of \phi well as otherwise the treatment estimator might be biased.

The instrumental variable(s) are considered strong enough for violation space candidate V_q if the estimated IV strength using this violation space candidate is larger than the obtained value of the threshold of the IV strength. The formula of the threshold of the IV strength has the form \min \{\max \{ 2 \cdot \mathrm{Trace} [ \mathrm{M} (V_q) ], \mathrm{iv{\_}threshold} \} + S (V_q), 40 \} if threshold_boot is TRUE, and \min \{\max \{ 2 \cdot \mathrm{Trace} [ \mathrm{M} (V_q) ], \mathrm{iv{\_}threshold} \}, 40 \} if threshold_boot is FALSE. The matrix \mathrm{M} (V_q) depends on the hat matrix obtained from estimating f(Z_i, X_i), the violation space candidate V_q and the variables to include in the outcome model W. S (V_q) is obtained using a bootstrap and aims to adjust for the estimation error of the IV strength. Usually, the value of the threshold of the IV strength obtained using the bootstrap approach is larger. Thus, using threshold_boot equals TRUE leads to a more conservative IV strength test. For more information see subsection 3.3 in Guo and Buehlmann (2022).

nsplits specifies the number of data splits that should be performed. For each data split the output statistics such as the point estimates of the treatment effect are calculated. Those statistics will then be aggregated over the different data splits. It is recommended to perform multiple data splits as data splitting introduces additional randomness. By aggregating the results of multiple data splits, the effects of this randomness can be decreased. If nsplits is larger than 1, point estimates are aggregated by medians. Standard errors, p-values and confidence intervals are obtained by the method specified by the parameter mult_split_method. 'DML' uses the approach by Chernozhukov et al. (2018). 'FWER' uses the approach by Meinshausen et al. (2009) and controls for the family-wise error rate. 'FWER' does not provide standard errors. For large sample sizes, a large values for nsplits can lead to a high running time as for each split a new hat matrix must be calculated.

There are three possibilities to set the argument parallel, namely "no" for serial evaluation (default), "multicore" for parallel evaluation using forking, and "snow" for parallel evaluation using a parallel socket cluster. It is recommended to select RNGkind ("L'Ecuyer-CMRG") and to set a seed to ensure that the parallel computing of the package TSCI is reproducible. This ensures that each processor receives a different substream of the pseudo random number generator stream. Thus, the results are reproducible if the arguments (including ncores) remain unchanged. There is an optional argument cl to specify a custom cluster if parallel = "snow".

See also Carl et al. (2023) for more details.

Value

A list containing the following elements:

Coef_all: a series of point estimates of the treatment effect obtained by the different violation space candidates.
sd_all: standard errors of the estimates of the treatmnet effect obtained by the different violation space candidates.
pval_all: p-values of the treatment effect estimates obtained by the different violation space candidates.
CI_all: confidence intervals for the treatment effect obtained by the different violation space candidates.
Coef_sel: the point estimator of the treatment effect obtained by the selected violation space candidate(s).
sd_sel: the standard error of Coef_sel.
pval_sel: p-value of the treatment effect estimate obtained by the selected violation space candidate(s).
CI_sel: confidence interval for the treatment effect obtained by the selected violation space candidate(s).
iv_str: IV strength using the different violation space candidates.
iv_thol: the threshold for the IV strength using the different violation space candidates.
Qmax: the frequency each violation space candidate was the largest violation space candidate for which the IV strength was considered large enough determined by the IV strength test over the multiple data splits. If 0, the IV Strength test failed for the first violation space candidate. Otherwise, violation space selection was performed.
q_comp: the frequency each violation space candidate was selected by the comparison method over the multiple data splits.
q_cons: the frequency each violation space candidate was selected by the conservative method over the multiple data splits.
invalidity: the frequency the instrumental variable(s) were considered valid, invalid or too weak to test for violations. The instrumental variables are considered too weak to test for violations if the IV strength is already too weak using the first violation space candidate (besides the empty violation space). Testing for violations is always performed by using the comparison method.
mse: the out-of-sample mean squared error of the fitted treatment model.
FirstStage_model: the method used to fit the treatment model.
n_A1: number of observations in A1.
n_A2: number of observations in A2.
nsplits: number of data splits performed.
mult_split_method: the method used to calculate the standard errors and p-values.
alpha: the significance level used.

References

Zijian Guo, and Peter Buehlmann. Two Stage Curvature Identification with Machine Learning: Causal Inference with Possibly Invalid Instrumental Variables. arXiv:2203.12808, 2022
Nicolai Meinshausen, Lukas Meier, and Peter Buehlmann. P-values for high-dimensional regression. Journal of the American Statistical Association, 104(488):1671-1681, 2009. 16, 18
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural parameters: Double/debiased machine learning. The Econometrics Journal, 21(1), 2018. 4, 16, 18
David Carl, Corinne Emmenegger, Peter Buehlmann, and Zijian Guo. TSCI: two stage curvature identification for causal inference with invalid instruments. arXiv:2304.00513, 2023

Examples

### a small example without baseline covariates
if (require("MASS")) {
  # sample size
  n <- 100
  # the IV strength
  a <- 1
  # the violation strength
  tau <- 1
  # true effect
  beta <- 1
  # treatment model
  f <- function(x) {1 + a * (x + x^2)}
  # outcome model
  g <- function(x) {1 + tau * x}

  # generate data
  mu_error <- rep(0, 2)
  Cov_error <- matrix(c(1, 0.5, 0.5, 1), 2, 2)
  Error <- MASS::mvrnorm(n, mu_error, Cov_error)
  # instrumental variable
  Z <- rnorm(n)
  # treatment variable
  D <- f(Z) + Error[, 1]
  # outcome variable
  Y <- beta * D + g(Z) + Error[, 2]

  # Two Stage L2 Boosting
  # create violation space candidates
  vio_space <- create_monomials(Z, 2, "monomials_main")
  # perform two stage curvature identification
  output_BO <- tsci_boosting(Y, D, Z, vio_space = vio_space, nsplits = 1,
                             max_depth = 2, nrounds = 10, B = 100)
  summary(output_BO)
}

[Package TSCI version 3.0.4 Index]