boot.lasso.proj: P-values based on the bootstrapped lasso projection method
In hdi: High-Dimensional Inference

Description Usage Arguments Value Author(s) References Examples

Compute p-values based on the lasso projection method, also known as the de-sparsified Lasso, using the bootstrap to approximate the distribution of the estimator.

boot.lasso.proj(x, y, family = "gaussian", standardize = TRUE,
                multiplecorr.method = "WY",
                parallel = FALSE, ncores = getOption("mc.cores", 2L),
                betainit = "cv lasso", sigma = NULL, Z = NULL, verbose = FALSE,
                return.Z = FALSE, robust= FALSE,
                B = 1000, boot.shortcut = FALSE,
                return.bootdist = FALSE, wild = FALSE,
                gaussian.stub = FALSE)

`x`	Design matrix (without intercept).
`y`	Response vector.
`family`	family
`standardize`	Should design matrix be standardized to unit column standard deviation.
`multiplecorr.method`	Either "WY" or any of `p.adjust.methods`.
`parallel`	Should parallelization be used? (logical)
`ncores`	Number of cores used for parallelization.
`betainit`	Either a numeric vector, corresponding to a sparse estimate of the coefficient vector, or the method to be used for the initial estimation, "scaled lasso" or "cv lasso".
`sigma`	Estimate of the standard deviation of the error term. This estimate needs to be compatible with the initial estimate (see betainit) provided or calculated. Otherwise, results will not be correct.
`Z`	user input, also see `return.Z` below
`verbose`	A boolean to enable reporting on the progress of the computations. (Only prints out information when Z is not provided by the user)
`return.Z`	An option to return the intermediate result which only depends on the design matrix x. This intermediate results can be used when calling the function again and the design matrix is the same as before.

`robust`	Uses a robust variance estimation procedure to be able to deal with model misspecification.
`B`	The number of bootstrap samples to be used.
`boot.shortcut`	A boolean to enable the computational shortcut for the bootstrap. If set to true, the lasso is not re-tuned for each bootstrap iteration, but it uses the tuning parameter computed on the original data instead.
`return.bootdist`	A boolean specifying if one is to return the computed bootstrap distributions to the estimator. (Matrix size: ncol(x)*B) If the multiple testing method was chosen to be WY, the bootstrap distribution computer under the complete null hypothesis is returned as well. This option is required if one wants to compute confidence intervals afterwards.
`wild`	Perform the wild bootstrap based on N(0,1) distributed random variables
`gaussian.stub`	DEVELOPER OPTION. Only enable if you know what you are doing. A boolean to run stub code instead of actually bootstrapping the estimator. It generates a finite sample distribution for each estimate by sampling B samples from N(0,\hat{s.e.}_j^2). (Note: we do not sample from the multivariate gaussian with the covariance matrix. Therefore, no dependencies are modelled at all.) Useful for debugging and for checking if the bootstrap is way off for some reason.

`pval`	Individual p-values for each parameter.
`pval.corr`	Multiple testing corrected p-values for each parameter.

`sigmahat`	\widehat{σ} coming from the scaled lasso.
`Z`	Only different from NULL if the option return.Z is on. This is an intermediate result from the computation which only depends on the design matrix x. These are the residuals of the nodewise regressions.
`B`	The number of bootstrap samples used.
`boot.shortcut`	If the bootstrap shortcut has been used or not.
`lambda`	What tuning parameter was used for the bootstrap shortcut. NULL if no shortcut was used or if no valid lambda was available to use for the shortcut.
`cboot.dist`	Only different from NULL if the option return.bootdist is on. This is a ncol(x)*B matrix where each row contains the computed centered bootstrap distribution for that estimate.
`cboot.dist.underH0`	Only different from NULL if the option return.bootdist is on and if the multiple testing method is WY. This is a ncol(x)*B matrix where each row contains the computed centered bootstrap distribution for that estimate. These bootstrap distributions were computed under the complete null hypothesis (b_1 = ... = b_p = 0).

Ruben Dezeure

van de Geer, S., Bühlmann, P., Ritov, Y. and Dezeure, R. (2014) On asymptotically optimal confidence regions and tests for high-dimensional models. Annals of Statistics 42, 1166–1202._

Zhang, C., Zhang, S. (2014) Confidence intervals for low dimensional parameters in high dimensional linear models. Journal of the Royal Statistical Society: Series B 76, 217–242.

Bühlmann, P. and van de Geer, S. (2015) High-dimensional inference in misspecified linear models. Electronic Journal of Statistics 9, 1449–1473.

Dezeure, R., Bühlmann, P. and Zhang, C. (2016) High-dimensional simultaneous inference with the bootstrap http://arxiv.org/abs/1606.03940

x <- matrix(rnorm(100 * 10), nrow = 100, ncol = 10)
y <- x[,1] + x[,2] + rnorm(100)


fit.lasso <- boot.lasso.proj(x, y)
which(fit.lasso$pval.corr < 0.05) # typically: '1' and '2' and no other



## Use the computational shortcut for the bootstrap to speed up
## computations
fit.lasso.shortcut <- boot.lasso.proj(x, y, boot.shortcut = TRUE)
which(fit.lasso.shortcut$pval.corr < 0.05) # typically: '1' and '2' and no other



## Return the bootstrap distribution as well and compute confidence intervals based on it
fit.lasso.allinfo <- boot.lasso.proj(x, y, return.bootdist = TRUE)
confint(fit.lasso.allinfo, level = 0.95)
confint(fit.lasso.allinfo, parm = 1:3)

## Use the scaled lasso for the initial estimate
fit.lasso.scaled <- boot.lasso.proj(x, y, betainit = "scaled lasso")
which(fit.lasso.scaled$pval.corr < 0.05)

## Use a robust estimate for the standard error
fit.lasso.robust <- boot.lasso.proj(x, y, robust = TRUE)
which(fit.lasso.robust$pval.corr < 0.05)