Jackknife approximate t tests of regression coefficients

Performes approximate t tests of regression coefficients based on jackknife variance estimates.

Usage

jack.test(object, ncomp = object$ncomp, use.mean = TRUE)

# S3 method for class 'jacktest'
print(x, P.values = TRUE, ...)

Arguments

object

an mvr object. A cross-validated model fitted with jackknife = TRUE.

ncomp

the number of components to use for estimating the variances

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 var.jack for details.

x

an jacktest object, the result of jack.test.

P.values

logical. Whether to print \(p\) values (default).

...

Further arguments sent to the underlying print function printCoefmat.

Value

jack.test returns an object of class "jacktest", with components

coefficients

The estimated regression coefficients

sd

The square root of the jackknife variance estimates

tvalues

The \(t\) statistics

df

The `degrees of freedom' used for calculating \(p\) values

pvalues

The calculated \(p\) values

print.jacktest returns the "jacktest" object (invisibly).

Details

jack.test uses the variance estimates from var.jack to perform \(t\) tests of the regression coefficients. The resulting object has a print method, print.jacktest, which uses printCoefmat for the actual printing.

Warning

The jackknife variance estimates are known to be biased (see var.jack). Also, the distribution of the regression coefficient estimates and the jackknife variance estimates are unknown (at least in PLSR/PCR). Consequently, the distribution (and in particular, the degrees of freedom) of the resulting \(t\) statistics is unknown. The present code simply assumes a \(t\) distribution with \(m - 1\) degrees of freedom, where \(m\) is the number of cross-validation segments.

Therefore, the resulting \(p\) values should not be used uncritically, and should perhaps be regarded as mere indicator of (non-)significance.

Finally, also keep in mind that as the number of predictor variables increase, the problem of multiple tests increases correspondingly.

References

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.

See also

var.jack, mvrCv

Author

Bjørn-Helge Mevik

Examples

data(oliveoil)
mod <- pcr(sensory ~ chemical, data = oliveoil, validation = "LOO", jackknife = TRUE)
jack.test(mod, ncomp = 2)
#> Response yellow (2 comps):
#>           Estimate Std. Error Df t value Pr(>|t|)
#> Acidity  -53.88406   32.00643 15 -1.6835   0.1130
#> Peroxide  -1.64614    1.85612 15 -0.8869   0.3891
#> K232      -9.49515   24.15870 15 -0.3930   0.6998
#> K270      -3.66858    3.07335 15 -1.1937   0.2511
#> DK        -0.35815    0.34117 15 -1.0498   0.3105
#> 
#> Response green (2 comps):
#>          Estimate Std. Error Df t value Pr(>|t|)
#> Acidity  68.69511   39.86563 15  1.7232   0.1054
#> Peroxide  1.41003    2.42316 15  0.5819   0.5693
#> K232     12.06057   31.00606 15  0.3890   0.7028
#> K270      4.67437    3.85337 15  1.2131   0.2439
#> DK        0.45638    0.43416 15  1.0512   0.3098
#> 
#> Response brown (2 comps):
#>           Estimate Std. Error Df t value Pr(>|t|)  
#> Acidity  -5.974767   4.183469 15 -1.4282  0.17373  
#> Peroxide  1.271210   0.568083 15  2.2377  0.04084 *
#> K232     -0.958874   4.680049 15 -0.2049  0.84042  
#> K270     -0.401311   0.188443 15 -2.1296  0.05017 .
#> DK       -0.039263   0.030609 15 -1.2827  0.21905  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Response glossy (2 comps):
#>           Estimate Std. Error Df t value Pr(>|t|)  
#> Acidity  -7.693126   6.520910 15 -1.1798  0.25647  
#> Peroxide -1.126323   0.522543 15 -2.1555  0.04778 *
#> K232     -1.413252   4.776935 15 -0.2958  0.77140  
#> K270     -0.527123   0.467543 15 -1.1274  0.27727  
#> DK       -0.051409   0.076280 15 -0.6739  0.51060  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Response transp (2 comps):
#>            Estimate Std. Error Df t value Pr(>|t|)  
#> Acidity  -13.630902   8.573695 15 -1.5899  0.13272  
#> Peroxide  -1.286214   0.722367 15 -1.7806  0.09524 .
#> K232      -2.458184   8.325620 15 -0.2953  0.77185  
#> K270      -0.931303   0.695580 15 -1.3389  0.20055  
#> DK        -0.090869   0.109167 15 -0.8324  0.41825  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Response syrup (2 comps):
#>          Estimate Std. Error Df t value  Pr(>|t|)    
#> Acidity  1.848304   2.947628 15  0.6270 0.5400554    
#> Peroxide 0.666474   0.130938 15  5.0900 0.0001331 ***
#> K232     0.365127   0.887538 15  0.4114 0.6866016    
#> K270     0.128133   0.187759 15  0.6824 0.5053697    
#> DK       0.012474   0.022187 15  0.5622 0.5822789    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1