Calculates jackknife variance or covariance estimates of regression coefficients.
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.)
Arguments
- object
an
mvrobject. A cross-validated model fitted withjackknife = TRUE.- ncomp
the number of components to use for estimating the (co)variances
- covariance
logical. If
TRUE, covariances are calculated; otherwise only variances. The default isFALSE.- use.mean
logical. If
TRUE(default), the mean coefficients are used when estimating the (co)variances; otherwise the coefficients from a model fitted to the entire data set. See Details.
Value
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.
Warning
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.
References
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.
Examples
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
#>