india (version 0.1-2)

leverages: Leverages

Description

Computes leverage measures from a fitted model object.

Usage

leverages(model, ...)
# S3 method for lm
leverages(model, infl = lm.influence(model, do.coef = FALSE), ...)
# S3 method for nls
leverages(model, ...)
# S3 method for ols
leverages(model, ...)
# S3 method for ridge
leverages(model, ...)
# S3 method for nls
hatvalues(model, ...)
# S3 method for ols
hatvalues(model, ...)
# S3 method for ridge
hatvalues(model, ...)

Value

A vector containing the diagonal of the prediction (or ‘hat’) matrix.

For linear regression (i.e., for "lm" or "ols" objects) the prediction matrix assumes the form

$$\bold{H} = \bold{X}(\bold{X}^T\bold{X})^{-1}\bold{X}^T,$$

in which case, \(h_{ii} = \bold{x}_i^T(\bold{X}^T\bold{X})^{-1}\bold{x}_i\) for \(i=1,\dots,n\). Whereas for ridge regression, the prediction matrix is given by

$$\bold{H}(\lambda) = \bold{X}(\bold{X}^T\bold{X} + \lambda\bold{I})^{-1}\bold{X}^T,$$

where \(\lambda\) represents the ridge parameter. Thus, the diagonal elements of \(\bold{H}(\lambda)\), are \(h_{ii}(\lambda) = \bold{x}_i^T(\bold{X}^T\bold{X} + \lambda\bm{I})^{-1}\bold{x}_i\), \(i=1,\dots,n\).

In nonlinear regression, the tangent plane leverage matrix is given by

$$\hat{\bold{H}} = \hat{\bold{F}}(\hat{\bold{F}}^T\hat{\bold{F}})^{-1}\hat{\bold{F}}^T,$$

where \(\bold{F} = \bold{F}(\bold{\beta})\) is the \(n\times p\) local model matrix with ith row \(\partial f_i(\bold{\beta})/\partial\bold{\beta}\) and \(\hat{\bold{F}} = \bold{F}(\hat{\bold{\beta}})\).

Arguments

model

an R object, returned by lm, nls, ols or ridge.

infl

influence structure as returned by lm.influence.

...

further arguments passed to or from other methods.

References

Chatterjee, S., Hadi, A.S. (1988). Sensivity Analysis in Linear Regression. Wiley, New York.

Cook, R.D., Weisberg, S. (1982). Residuals and Influence in Regression. Chapman and Hall, London.

Ross, W.H. (1987). The geometry of case deletion and the assessment of influence in nonlinear regression. The Canadian Journal of Statistics 15, 91-103. tools:::Rd_expr_doi("10.2307/3315198")

St. Laurent, R.T., Cook, R.D. (1992). Leverage and superleverage in nonlinear regression. Journal of the Amercian Statistical Association 87, 985-990. tools:::Rd_expr_doi("10.1080/01621459.1992.10476253")

Walker, E., Birch, J.B. (1988). Influence measures in ridge regression. Technometrics 30, 221-227. tools:::Rd_expr_doi("10.1080/00401706.1988.10488370")

Examples

# Leverages for linear regression
fm <- ols(stack.loss ~ ., data = stackloss)
lev <- leverages(fm)
cutoff <- 2 * mean(lev)
plot(lev, ylab = "Leverages", ylim = c(0,0.45))
abline(h = cutoff, lty = 2, lwd = 2, col = "red")
text(17, lev[17], label = as.character(17), pos = 3)

# Leverages for ridge regression
data(portland)
fm <- ridge(y ~ ., data = portland)
lev <- leverages(fm)
cutoff <- 2 * mean(lev)
plot(lev, ylab = "Leverages", ylim = c(0,0.7))
abline(h = cutoff, lty = 2, lwd = 2, col = "red")
text(10, lev[10], label = as.character(10), pos = 3)

# Leverages for nonlinear regression
data(skeena)
model <- recruits ~ b1 * spawners * exp(-b2 * spawners)
fm <- nls(model, data = skeena, start = list(b1 = 3, b2 = 0))
lev <- leverages(fm)
plot(lev, ylab = "Leverages", ylim = c(0,0.25))
obs <- c(1,9)
text(obs, lev[obs], label = as.character(obs), pos = 3)