R: Functions for order restricted linear regression estimation...

orlm {ic.infer}

R Documentation

Functions for order restricted linear regression estimation and testing

Description

Function orlm calculates order-restricted linear models (linear equality and inequality constraints). It uses the internal function boot.orlm for bootstrapping, which in turn uses the internal functions orlm_forboot... . The remaining functions extract coefficients, provide a residual plot, give a short printout or a more extensive summary.

Usage

orlm(model, ui, ci, ...)
## S3 method for class 'lm'
orlm(model, ui, ci, index = 2:length(coef(model)), meq = 0, 
    orig.out = FALSE, boot = FALSE, B = 1000, fixed = FALSE, 
    tol = sqrt(.Machine$double.eps), ...)
## Default S3 method:
orlm(model, ui, ci, index = NULL, meq = 0, 
    tol = sqrt(.Machine$double.eps), df.error = NULL, ...)
boot.orlm(model, B = 1000, fixed = FALSE, ui, ci, index, meq)
orlm_forboot.fixed(data, indices, ...)
orlm_forboot(data, indices, index = index, ...)
## S3 method for class 'orlm'
coef(object, ...)
## S3 method for class 'orlm'
plot(x, caption = "Residuals vs Fitted", 
        panel = if (add.smooth) panel.smooth else points, sub.caption = NULL, 
        main = "", ..., id.n = 3, labels.id = names(x$residuals), cex.id = 0.75, 
        add.smooth = getOption("add.smooth"), label.pos = c(4, 2), 
        cex.caption = 1)
## S3 method for class 'orlm'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'orlm'
summary(object, display.unrestr = FALSE, brief = FALSE, 
        digits = max(3, getOption("digits") - 3), 
        scientific = FALSE, overall.tests = TRUE, 
        bootCIs = TRUE, bty = "perc", level = 0.95, ...)

Arguments

`model`	a linear model object (class `lm`) with data included OR a covariance matrix of Y and all regressors (in this order)
`ui`	matrix (or vector in case of one single restriction only) defining the left-hand side of the restriction `ui%*%beta >= ci`, where beta is the parameter vector; the first few of these restrictions can be declared equality- instead of inequality restrictions (cf. argument `meq`); if only part of the elements of beta are subject to restrictions, the columns of `ui` can be restricted to these elements, if their index numbers are provided in `index`; by default, `index` excludes the intercept, i.e. the columns of `ui` refer to the non-intercept elements of `coef(model)` Rows of `ui` must be linearly independent; in case of linearly dependent rows the function gives an error message with a hint which subset of rows is independent. Note that the restrictions must define a (possibly translated) cone, i.e. e.g. interval restrictions on a parameter are not permitted. See `contr.diff` for examples of how to comfortably define various types of restriction.
`ci`	vector on the right-hand side of the restriction (cf. `ui`)
`index`	index numbers of the components of beta, which are subject to the specified constraints as `ui%*%beta[index] >= ci`, default is `index = 2:length(coef(model))`, i.e. `ui` is supposed to have columns for all coefficients except the intercept; CAUTIONs: - `index` refers to the position of the coefficient in the model. The first coefficient is usually the intercept (which is therefore per default excluded from restrictions). - If the intercept is included into restrictions (model with intercept, index containing the element `1`, intercept-related column of ui not consisting of zeroes only), R-squared values may become unreasonable, if the restriction on the intercept is active.
`meq`	integer number (default 0) giving the number of rows of `ui` that are used for equality restrictions instead of inequality restrictions.
`orig.out`	should the original model be included in the output list ? (default: `FALSE`)
`boot`	should bootstrapping be conducted ? (default: `FALSE`)
`B`	number of bootstrap samples (default: `1000`)
`fixed`	should bootstrapping consider the sample as fixed and bootstrap residuals ? (default: `FALSE`)
`data`	data handed to bootstrap sampling routine
`indices`	indices for sampling
`tol`	numerical tolerance value; estimates closer to 0 than `tol` are set to exactly 0
`df.error`	error degrees of freedom (number of observations minus number of colummns of covariance matrix) for `orlm.default`; required in order to calculate adequate covariance matrix and tests; valid coefficient estimates can also be obtained for arbitrary values of `df.error`
`...`	Further options
`object`	object of class `orlm` (created by function `orlm`)
`x`	object of class `orlm` (created by function `orlm`)
`caption`	like in function `plot.lm`
`panel`	like in function `plot.lm`
`sub.caption`	like in function `plot.lm`
`main`	like in function `plot.lm`
`id.n`	like in function `plot.lm`
`labels.id`	like in function `plot.lm`
`cex.id`	like in function `plot.lm`
`add.smooth`	like in function `plot.lm`
`label.pos`	like in function `plot.lm`
`cex.caption`	like in function `plot.lm`
`digits`	number of digits to display
`display.unrestr`	if `TRUE`, also display unrestricted model; default: `FALSE`
`brief`	if `TRUE`, suppress printing of restrictions; default: `FALSE`
`scientific`	if `FALSE`, suppresses scientific format; default: `FALSE`
`overall.tests`	if `FALSE`, suppresses output of overall model tests; default: `TRUE`; for models with large sets of restrictions, tests can take up substantial time because of weight calculation
`bootCIs`	if `FALSE`, suppresses bootstrap confidence intervals, even though the `obj` contains a `bootout` element; default: `TRUE`
`bty`	type of bootstrap confidence interval; any of `"perc"`, `"bca"`, `"norm"` or `"basic"`, cf. function `boot.ci` from package `boot`, default: `"perc"`
`level`	confidence level for bootstrap confidence intervals, default: `0.95`

Details

Function orlm performs order restricted linear model analysis. Functions coef.orlm, plot.orlm, print.orlm, and summary.orlm provide methods for reporting the results on an object of S3 class orlm. The functions directly referring to bootstrapping are internal and should not be called by the user but are called from within function orlm if option boot is set to TRUE.

Of course, bootstrapping is not possible, if function orlm is applied to a covariance matrix, since the raw data are not available in this case. Also note that the intercept is not estimated in this case but can easily be estimated from the resulting estimate if the variable means are known (cf. example).

The output from summary.orlm provides information about the restrictions, a comparison of $R^2$-values for unrestricted and restricted model, restricted estimates, and

- if requested (option boot set to TRUE in function orlm and option bootCIs set to TRUE in the summary function) with bootstrap confidence intervals,

- if requested (option overall.tests set to TRUE) several restriction-related tests (implemented by calls to ic.test): The analogue to the overall F-Test in the ordinary linear model is the test of all coefficients but intercept equal to 0 within the restricted parameter space. In addition, three tests related to the restriction are reported:

Test 1: H0: Restriction valid with equality vs. H1: at least one inequality

Test 2: H0: Restriction valid vs. H1: restriction violated

Test 3: H0: Restriction violated or valid with equality vs. H1: all restrictions valid with inequality

Test 3 is conducted in case of no equality-restrictions only.

Value

The output of function orlm belongs to S3 classes orlm and orest. It is a list with the following items:

`b.restr`	restricted estimate
`b.unrestr`	unrestricted estimate
`R2`	R-squared
`residuals`	residuals of restricted model
`fitted.values`	fitted values of restricted model
`weights`	observation weights
`orig.R2`	R-squared of unrestricted model
`df.error`	error degrees of freedom of unrestricted model
`s2`	MSE of unrestricted model
`Sigma`	variance covariance matrix of beta-hat in unrestricted model
`origmodel`	unrestricted model itself (`NULL`, if `orig.out=FALSE`)
`ui`	as input
`ci`	as input
`restr.index`	the input vector index
`meq`	as input
`iact`	active restrictions, i.e. restrictions that are satisfied with equality in the solution, as output by `solve.QP`
`bootout`	object of class boot obtained by bootstrapping, will be used by summary.orlm for calculating bootstrap confidence intervals; `NULL` if `boot=FALSE`

Note

Package versions up to 1.1-4 had a bug in function ic.test that caused the p-value of the overall model test to be too large.

Author(s)

Ulrike Groemping, BHT Berlin

References

Shapiro, A. (1988) Towards a unified theory of inequality-constrained testing in multivariate analysis. International Statistical Review 56, 49–62

Examples

limo <- lm(swiss)
## restricted linear model with restrictions that
## - Education and Examination have same coefficient
## - Catholic and Infant.Mortality have non-negative coefficients
orlimo <- orlm(limo, ui=rbind(c(0,1,-1,0,0),c(0,0,0,1,0),c(0,0,0,0,1)), meq=1)
orlimo
plot(orlimo)
summary(orlimo)
## same model using index vector
orlimo <- orlm(limo, ui=rbind(c(1,-1,0,0),c(0,0,1,0),c(0,0,0,1)), index=3:6, meq=1)

## reduced number of bootstrap samples below reasonable size for example run time
orlimo <- orlm(limo, ui=rbind(c(1,-1,0,0),c(0,0,1,0),c(0,0,0,1)), 
    index=3:6, meq=1, boot=TRUE, B=100)
summary(orlimo)

## bootstrap considering data as fixed 
orlimof <- orlm(limo, ui=rbind(c(1,-1,0,0),c(0,0,1,0),c(0,0,0,1)), 
    index=3:6, meq=1, boot=TRUE, B=100, fixed=TRUE)
summary(orlimof, brief=TRUE)

[Package ic.infer version 1.1-7 Index]