Computes p-values and confidence intervals for forward stepwise regression
fsInf(obj, sigma=NULL, alpha=0.1, k=NULL, type=c("active","all","aic"),
gridrange=c(-100,100), bits=NULL, mult=2, ntimes=2, verbose=FALSE)Object returned by fs function
Estimate of error standard deviation. If NULL (default), this is estimated
using the mean squared residual of the full least squares fit when n >= 2p, and
using the standard deviation of y when n < 2p. In the latter case, the user
should use estimateSigma function for a more accurate estimate
Significance level for confidence intervals (target is miscoverage alpha/2 in each tail)
See "type" argument below. Default is NULL, in which case k is taken to be the the number of steps computed in the forward stepwise path
Type of analysis desired: with "active" (default), p-values and confidence intervals are computed for each predictor as it is entered into the active step, all the way through k steps; with "all", p-values and confidence intervals are computed for all variables in the active model after k steps; with "aic", the number of steps k is first estimated using a modified AIC criterion, and then the same type of analysis as in "all" is carried out for this particular value of k.
Note that the AIC scheme is defined to choose a number of steps k after which the AIC criterion
increases ntimes in a row, where ntimes can be specified by the user (see below).
Under this definition, the AIC selection event is characterizable as a polyhedral set, and hence
the extra conditioning can be taken into account exactly. Also note that an analogous BIC scheme
can be specified through the mult argument (see below)
Grid range for constructing confidence intervals, on the standardized scale
Number of bits to be used for p-value and confidence interval calculations. Default is
NULL, in which case standard floating point calculations are performed. When not NULL,
multiple precision floating point calculations are performed with the specified number
of bits, using the R package Rmpfr (if this package is not installed, then a
warning is thrown, and standard floating point calculations are pursued).
Note: standard double precision uses 53 bits
so, e.g., a choice of 200 bits uses about 4 times double precision. The confidence
interval computation is sometimes numerically challenging, and the extra precision can be
helpful (though computationally more costly). In particular, extra precision might be tried
if the values in the output columns of tailarea differ noticeably from alpha/2.
Multiplier for the AIC-style penalty. Hence a value of 2 (default) gives AIC, whereas a value of log(n) would give BIC
Number of steps for which AIC-style criterion has to increase before minimizing point is declared
Print out progress along the way? Default is FALSE
Type of analysis (active, all, or aic)
Value of k specified in call
When type is "active", this is an estimated stopping point
declared by forwardStop; when type is "aic", this is the
value chosen by the modified AIC scheme
One sided P-values for active variables, uses the sign that a variable entered the model with.
Confidence intervals
Realized tail areas (lower and upper) for each confidence interval
Lower truncation limits for statistics
Upper truncation limits for statistics
Linear contrasts that define the observed statistics
Vector of outcomes
Variables in active set
Signs of active coefficients
Desired coverage (alpha/2 in each tail)
Value of error standard deviation (sigma) used
The call to fsInf
This function computes selective p-values and confidence intervals (selection intervals)
for forward stepwise regression. The default is to report the results for
each predictor after its entry into the model. See the "type" argument for other options.
The confidence interval construction involves numerical search and can be fragile:
if the observed statistic is too close to either end of the truncation interval
(vlo and vup, see references), then one or possibly both endpoints of the interval of
desired coverage cannot be computed, and default to +/- Inf. The output tailarea
gives the achieved Gaussian tail areas for the reported intervals---these should be close
to alpha/2, and can be used for error-checking purposes.
Ryan Tibshirani, Jonathan Taylor, Richard Lockhart, and Rob Tibshirani (2014). Exact post-selection inference for sequential regression procedures. arXiv:1401.3889.
Joshua Loftus and Jonathan Taylor (2014). A significance test for forward stepwise model selection. arXiv:1405.3920.
# NOT RUN {
set.seed(33)
n = 50
p = 10
sigma = 1
x = matrix(rnorm(n*p),n,p)
beta = c(3,2,rep(0,p-2))
y = x%*%beta + sigma*rnorm(n)
# run forward stepwise
fsfit = fs(x,y)
# compute sequential p-values and confidence intervals
# (sigma estimated from full model)
out.seq = fsInf(fsfit)
out.seq
# compute p-values and confidence intervals after AIC stopping
out.aic = fsInf(fsfit,type="aic")
out.aic
# compute p-values and confidence intervals after 5 fixed steps
out.fix = fsInf(fsfit,type="all",k=5)
out.fix
# }
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