Validation of an Ordinary Linear Model

The validate function when used on an object created by ols does resampling validation of a multiple linear regression model, with or without backward step-down variable deletion. Uses resampling to estimate the optimism in various measures of predictive accuracy which include \(R^2\), \(MSE\) (mean squared error with a denominator of \(n\)), the \(g\)-index, and the intercept and slope of an overall calibration \(a + b\hat{y}\). The "corrected" slope can be thought of as shrinkage factor that takes into account overfitting. validate.ols can also be used when a model for a continuous response is going to be applied to a binary response. A Somers' \(D_{xy}\) for this case is computed for each resample by dichotomizing y. This can be used to obtain an ordinary receiver operating characteristic curve area using the formula \(0.5(D_{xy} + 1)\). The Nagelkerke-Maddala \(R^2\) index for the dichotomized y is also given. See predab.resample for information about confidence limits and for the list of resampling methods.

The LaTeX needspace package must be in effect to use the latex method.

Usage

# fit <- fitting.function(formula=response ~ terms, x=TRUE, y=TRUE)
# S3 method for class 'ols'
validate(fit, method="boot", B=40,
         bw=FALSE, rule="aic", type="residual", sls=0.05, aics=0, 
         force=NULL, estimates=TRUE, pr=FALSE, u=NULL, rel=">",
         tolerance=1e-7, ...)

Arguments

fit: a fit derived by ols. The options x=TRUE and y=TRUE must have been specified. See validate for a description of arguments method - pr.
method,B,bw,rule,type,sls,aics,force,estimates,pr: see validate and predab.resample and fastbw
u: If specifed, y is also dichotomized at the cutoff u for the purpose of getting a bias-corrected estimate of \(D_{xy}\).
rel: relationship for dichotomizing predicted y. Defaults to ">" to use y>u. rel can also be "<", ">=", and "<=".
tolerance: tolerance for singularity; passed to lm.fit.qr.
...: other arguments to pass to predab.resample, such as group, cluster, and subset

Value

matrix with rows corresponding to R-square, MSE, g, intercept, slope, and optionally \(D_{xy}\) and \(R^2\), and columns for the original index, resample estimates, indexes applied to whole or omitted sample using model derived from resample, average optimism, corrected index, and number of successful resamples.

Side Effects

prints a summary, and optionally statistics for each re-fit

Author

Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com

Examples

set.seed(1)
x1 <- runif(200)
x2 <- sample(0:3, 200, TRUE)
x3 <- rnorm(200)
distance <- (x1 + x2/3 + rnorm(200))^2

f <- ols(sqrt(distance) ~ rcs(x1,4) + scored(x2) + x3, x=TRUE, y=TRUE)

#Validate full model fit (from all observations) but for x1 < .75
validate(f, B=20, subset=x1 < .75)   # normally B=300
#>           index.orig training   test optimism index.corrected   Lower Upper  n
#> R-square      0.0939    0.122 0.0592   0.0625          0.0313 -0.1181 0.154 20
#> MSE           0.6047    0.593 0.6278  -0.0348          0.6395  0.4857 0.803 20
#> g             0.2896    0.327 0.2654   0.0616          0.2280  0.0849 0.405 20
#> Intercept     0.0863    0.065 0.2358  -0.1708          0.2571 -0.3944 0.698 20
#> Slope         0.9016    0.919 0.7651   0.1535          0.7481  0.3518 1.324 20

#Validate stepwise model with typical (not so good) stopping rule
validate(f, B=20, bw=TRUE, rule="p", sls=.1, type="individual")
#> 
#> 		Backwards Step-down - Original Model
#> 
#>  Deleted Chi-Sq d.f. P      Residual d.f. P      AIC   R2   
#>  x3      0.99   1    0.3204 0.99     1    0.3204 -1.01 0.128
#> 
#> Approximate Estimates after Deleting Factors
#> 
#>               Coef   S.E.  Wald Z         P
#> Intercept  0.94530 0.2548  3.7100 0.0002072
#> x1        -0.65558 1.0361 -0.6327 0.5269109
#> x1'        3.11974 2.9290  1.0651 0.2868256
#> x1''      -8.11867 9.5505 -0.8501 0.3952801
#> x2         0.05955 0.1540  0.3868 0.6989377
#> x2=2       0.27098 0.2766  0.9797 0.3272444
#> x2=3       0.36983 0.4068  0.9091 0.3632803
#> 
#> Factors in Final Model
#> 
#> [1] x1 x2
#>           index.orig training   test optimism index.corrected   Lower Upper  n
#> R-square       0.128    0.150 0.0818   0.0679          0.0602 -0.0532 0.178 20
#> MSE            0.668    0.631 0.7034  -0.0727          0.7407  0.5570 0.959 20
#> g              0.360    0.373 0.3094   0.0636          0.2961  0.1390 0.427 20
#> Intercept      0.000    0.000 0.1973  -0.1973          0.1973 -0.2860 0.647 20
#> Slope          1.000    1.000 0.8627   0.1373          0.8627  0.4063 1.290 20
#> 
#> Factors Retained in Backwards Elimination
#> 
#>  x1 x2 x3
#>     *    
#>  *  *    
#>  *     * 
#>     *    
#>  *  *    
#>  *  *    
#>  *  *  * 
#>  *  *  * 
#>  *  *    
#>  *  *    
#>  *  *  * 
#>  *  *    
#>     *    
#>  *  *    
#>  *  *    
#>  *  *  * 
#>  *  *    
#>  *  *  * 
#>  *  *  * 
#>  *  *  * 
#> 
#> Frequencies of Numbers of Factors Retained
#> 
#>  1  2  3 
#>  3 10  7