R: Jackknife Variance Estimates of Regression Coefficients

var.jack {pls} | R Documentation

## Jackknife Variance Estimates of Regression Coefficients

### Description

Calculates jackknife variance or covariance estimates of regression coefficients.

### Usage

    var.jack(object, ncomp = object$ncomp, covariance = FALSE, use.mean = TRUE)

### Arguments

`object` | an `mvr` object. A cross-validated model fitted with `jackknife = 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 is `FALSE`.
`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.

### Details

The original (Tukey) jackknife variance estimator is defined as (g-1)/g ∑_{i=1}^g(\tildeβ_{-i} - \barβ)^2, where g is the number of segments, \tildeβ_{-i} is the estimated coefficient when segment i is left out (called the jackknife replicates), and \barβ is the mean of the \tildeβ_{-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 ∑_{i=1}^g(\tildeβ_{-i} - \hatβ)^2, where \hatβ 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.)

### 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 ±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.

### Author(s)

Bjørn-Helge Mevik

### 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.

### See Also

`mvrCv`, `jack.test`

### Examples

    data(oliveoil)
    mod <- pcr(sensory ~ chemical, data = oliveoil, validation = "LOO",
               jackknife = TRUE)
    var.jack(mod, ncomp = 2)

[Package pls version 2.7-0 Index]