Package 'india'

Cook's distances

Description

Cook's distance is a measure to assess the influence of the ith observation on the model parameter estimates. This function computes the Cook's distance based on leave-one-out cases deletion for ordinary least squares, lad and ridge regression.

Usage

## S3 method for class 'lad'
cooks.distance(model, ...)
  ## S3 method for class 'ols'
cooks.distance(model, ...)
  ## S3 method for class 'ridge'
cooks.distance(model, type = "cov", ...)

Arguments

model	an R object, returned by ols, lad or ridge.
type	only required for 'ridge' objects, options available are "1st", "cov" and "both" to obtain the Cook's distance based on Equation (2.5), (2.6) or both by Walker and Birch (1988), respectively.
...	further arguments passed to or from other methods.

Value

A vector whose ith element contains the Cook's distance,

$D_i(\bold{M},c) = \frac{(\hat{\bold{\beta}}_{(i)} - \hat{\bold{\beta}})^T\bold{M} (\hat{\bold{\beta}}_{(i)} - \hat{\bold{\beta}})}{c},$

for $i = 1,\dots,n$, with $\bold{M}$ a positive definite matrix and $c > 0$. Specific choices of $\bold{M}$ and $c$ are done for objects of class ols, lad and ridge.

References

Cook, R.D., Weisberg, S. (1980). Characterizations of an empirical influence function for detecting influential cases in regression. Technometrics 22, 495-508. doi:10.1080/00401706.1980.10486199

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

Sun, R.B., Wei, B.C. (2004). On influence assessment for LAD regression. Statistics & Probability Letters 67, 97-110. doi:10.1016/j.spl.2003.08.018.

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

Examples

# Cook's distances for linear regression
fm <- ols(stack.loss ~ ., data = stackloss)
CD <- cooks.distance(fm)
plot(CD, ylab = "Cook's distances", ylim = c(0,0.8))
text(21, CD[21], label = as.character(21), pos = 3)

# Cook's distances for LAD regression
fm <- lad(stack.loss ~ ., data = stackloss)
CD <- cooks.distance(fm)
plot(CD, ylab = "Cook's distances", ylim = c(0,0.4))
text(17, CD[17], label = as.character(17), pos = 3)

# Cook's distances for ridge regression
data(portland)
fm <- ridge(y ~ ., data = portland)
CD <- cooks.distance(fm)
plot(CD, ylab = "Cook's distances", ylim = c(0,0.5))
text(8, CD[8], label = as.character(8), pos = 3)

---

Leverages

Description

Computes leverage measures from a fitted model object.

Usage

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

  ## S3 method for class 'ols'
hatvalues(model, ...)
  ## S3 method for class 'ridge'
hatvalues(model, ...)

Arguments

model	an R object, returned by lm, ols or ridge.
infl	influence structure as returned by lm.influence.
...	further arguments passed to or from other methods.

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 $h_{ii}(\lambda) = \bold{x}_i^T(\bold{X}^T\bold{X} + \lambda\bm{I})^{-1}\bold{x}_i$, $i=1,\dots,n$.

Note

This function never creates the prediction matrix and only obtains its diagonal elements from the singular value decomposition of $\bold{X}$.

Function hatvalues only is a wrapper for function leverages.

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.

Walker, E., Birch, J.B. (1988). Influence measures in ridge regression. Technometrics 30, 221-227. 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)

---

Likelihood Displacement

Description

Compute the likelihood displacement influence measure based on leave-one-out cases deletion for linear models, lad and ridge regression.

Usage

logLik.displacement(model, ...)
  ## S3 method for class 'lm'
logLik.displacement(model, pars = "full", ...)
  ## S3 method for class 'ols'
logLik.displacement(model, pars = "full", ...)
  ## S3 method for class 'lad'
logLik.displacement(model, method = "quasi", pars = "full", ...)
  ## S3 method for class 'ridge'
logLik.displacement(model, pars = "full", ...)

Arguments

model	an R object, returned by lm, ols, lad or ridge.
pars	should be considered the whole vector of parameters (pars = "full"), or only the vector of coefficients (pars = "coef").
method	only required for 'lad' objects, options available are "quasi" and "BR" to obtain the likelihood displacement based on Sun and Wei (2004) and Elian et al. (2000) approaches, respectively.
...	further arguments passed to or from other methods.

Value

A vector whose ith element contains the distance between the likelihood functions,

$LD_i(\bold{\beta},\sigma^2) = 2\{l(\hat{\bold{\beta}},\hat{\sigma}^2) - l(\hat{\bold{\beta}}_{(i)},\hat{\sigma}^2_{(i)})\},$

for pars = "full", where $\hat{\bold{\beta}}_{(i)}$ and $\hat{\sigma}^2_{(i)}$ denote the estimates of $\bold{\beta}$ and $\sigma^2$ when the ith observation is removed from the dataset. If we are interested only in $\bold{\beta}$ (i.e. pars = "coef") the likelihood displacement becomes

$LD_i(\bold{\beta}|\sigma^2) = 2\{l(\hat{\bold{\beta}},\hat{\sigma}^2) - \max_{\sigma^2} l(\hat{\bold{\beta}}_{(i)},\hat{\sigma}^2)\}.$

References

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

Cook, R.D., Pena, D., Weisberg, S. (1988). The likelihood displacement: A unifying principle for influence measures. Communications in Statistics - Theory and Methods 17, 623-640. doi:10.1080/03610928808829645.

Elian, S.N., Andre, C.D.S., Narula, S.C. (2000). Influence measure for the L1 regression. Communications in Statistics - Theory and Methods 29, 837-849. doi:10.1080/03610920008832518.

Sun, R.B., Wei, B.C. (2004). On influence assessment for LAD regression. Statistics & Probability Letters 67, 97-110. doi:10.1016/j.spl.2003.08.018.

Examples

# Likelihood displacement for linear regression
fm <- ols(stack.loss ~ ., data = stackloss)
LD <- logLik.displacement(fm)
plot(LD, ylab = "Likelihood displacement", ylim = c(0,9))
text(21, LD[21], label = as.character(21), pos = 3)

# Likelihood displacement for LAD regression
fm <- lad(stack.loss ~ ., data = stackloss)
LD <- logLik.displacement(fm)
plot(LD, ylab = "Likelihood displacement", ylim = c(0,1.5))
text(17, LD[17], label = as.character(17), pos = 3)

# Likelihood displacement for ridge regression
data(portland)
fm <- ridge(y ~ ., data = portland)
LD <- logLik.displacement(fm)
plot(LD, ylab = "Likelihood displacement", ylim = c(0,4))
text(8, LD[8], label = as.character(8), pos = 3)

---

Portland cement dataset

Description

This dataset comes from an experimental investigation of the heat evolved during the setting and hardening of Portland cements of varied composition and the dependence of this heat on the percentages of four compounds in the clinkers from which the cement was produced.

Usage

data(portland)

Format

A data frame with 13 observations on the following 5 variables.

y

The heat evolved after 180 days of curing, measured in calories per gram of cement.

x1

Tricalcium aluminate.

x2

Tricalcium silicate.

x3

Tetracalcium aluminoferrite.

x4

$\beta$-dicalcium silicate.

Source

Kaciranlar, S., Sakallioglu, S., Akdeniz, F., Styan, G.P.H., Werner, H.J. (1999). A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland cement. Sankhya, Series B 61, 443-459.

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Relative change in the condition number

Description

Compute the relative condition index to identify collinearity-influential points in linear models.

Usage

relative.condition(x)

Arguments

x	the model matrix $\bold{X}$.

Value

To assess the influence of the ith row of $\bold{X}$ on the condition index of $\bold{X}$, Hadi (1988) proposed the relative change,

$\delta_i = \frac{\kappa_{(i)} - \kappa}{\kappa},$

for $i=1,\dots,n$, where $\kappa = \kappa(\bold{X})$ and $\kappa_{(i)} = \kappa(\bold{X}_{(i)})$ denote the (scaled) condition index for $\bold{X}$ and $\bold{X}_{(i)}$, respectively.

References

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

Hadi, A.S. (1988). Diagnosing collinerity-influential observations. Computational Statistics & Data Analysis 7, 143-159. doi:10.1016/0167-9473(88)90089-8.

Examples

data(portland)
fm <- ridge(y ~ ., data = portland, x = TRUE)
x <- fm$x
rel <- relative.condition(x)
plot(rel, ylab = "Relative condition number", ylim = c(-0.1,0.4))
abline(h = 0, lty = 2, lwd = 2, col = "red")
text(3, rel[3], label = as.character(3), pos = 3)

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