var.jack.RdCalculates jackknife variance or covariance estimates of regression coefficients.
var.jack(object, ncomp = object$ncomp, covariance = FALSE, use.mean = TRUE)
| object | an |
|---|---|
| ncomp | the number of components to use for estimating the (co)variances |
| covariance | logical. If |
| use.mean | logical. If |
The original (Tukey) jackknife variance estimator is defined as \((g-1)/g \sum_{i=1}^g(\tilde\beta_{-i} - \bar\beta)^2\), where \(g\) is the number of segments, \(\tilde\beta_{-i}\) is the estimated coefficient when segment \(i\) is left out (called the jackknife replicates), and \(\bar\beta\) is the mean of the \(\tilde\beta_{-i}\). The most common case is delete-one jackknife, with \(g = n\), the number of observations.
This is the definition var.jack uses by default.
However, Martens and Martens (2000) defined the estimator as
\((g-1)/g \sum_{i=1}^g(\tilde\beta_{-i} - \hat\beta)^2\), where
\(\hat\beta\) is the coefficient estimate using the entire data set.
I.e., they use the original fitted coefficients instead of the
mean of the jackknife replicates. Most (all?) other jackknife
implementations for PLSR use this estimator. var.jack can be
made to use this definition with use.mean = FALSE. In
practice, the difference should be small if the number of
observations is sufficiently large. Note, however, that all
theoretical results about the jackknife refer to the `proper'
definition. (Also note that this option might disappear in a future
version.)
If covariance is FALSE, an \(p\times q \times c\)
array of variance estimates, where \(p\) is the number of
predictors, \(q\) is the number of responses, and \(c\) is the
number of components.
If covariance id TRUE, an \(pq\times pq \times c\)
array of variance-covariance estimates.
Note that the Tukey jackknife variance estimator is not unbiased for the variance of regression coefficients (Hinkley 1977). The bias depends on the \(X\) matrix. For ordinary least squares regression (OLSR), the bias can be calculated, and depends on the number of observations \(n\) and the number of parameters \(k\) in the mode. For the common case of an orthogonal design matrix with \(\pm 1\) levels, the delete-one jackknife estimate equals \((n-1)/(n-k)\) times the classical variance estimate for the regression coefficients in OLSR. Similar expressions hold for delete-d estimates. Modifications have been proposed to reduce or eliminate the bias for the OLSR case, however, they depend on the number of parameters used in the model. See e.g. Hinkley (1977) or Wu (1986).
Thus, the results of var.jack should be used with caution.
Tukey J.W. (1958) Bias and Confidence in Not-quite Large Samples. (Abstract of Preliminary Report). Annals of Mathematical Statistics, 29(2), 614.
Martens H. and Martens M. (2000) Modified Jack-knife Estimation of Parameter Uncertainty in Bilinear Modelling by Partial Least Squares Regression (PLSR). Food Quality and Preference, 11, 5--16.
Hinkley D.V. (1977), Jackknifing in Unbalanced Situations. Technometrics, 19(3), 285--292.
Wu C.F.J. (1986) Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis. Te Annals of Statistics, 14(4), 1261--1295.
data(oliveoil) mod <- pcr(sensory ~ chemical, data = oliveoil, validation = "LOO", jackknife = TRUE) var.jack(mod, ncomp = 2)#> , , 2 comps #> #> yellow green brown glossy transp #> Acidity 1024.4116919 1589.2686000 1.750141e+01 42.522264128 73.50823993 #> Peroxide 3.4451819 5.8716926 3.227187e-01 0.273051034 0.52181445 #> K232 583.6428901 961.3757680 2.190286e+01 22.819112503 69.31594523 #> K270 9.4454718 14.8484347 3.551073e-02 0.218596282 0.48383108 #> DK 0.1163998 0.1884952 9.368976e-04 0.005818676 0.01191753 #> syrup #> Acidity 8.6885127205 #> Peroxide 0.0171447602 #> K232 0.7877230726 #> K270 0.0352534553 #> DK 0.0004922534 #>