lad: Least absolute deviation
In Blossom: Statistical Comparisons with Distance-Function Based Permutation Tests

Description Usage Arguments Details Value Author(s) References See Also Examples

Least absolute deviation (LAD) regression is an alternative to ordinary least squares (OLS) regression that has greater power for thick-tailed symmetric and asymmetric error distributions (Cade and Richards 1996). LAD regression estimates the conditional median (a conditional 0.50 quantile) of a dependent variable given the independent variable(s) by minimizing sums of absolute deviations between observed and predicted values. LAD regression can be used anywhere OLS regression would be used but is often more desirable because it is less sensitive to outlying data points and is more efficient for skewed error distributions as well as some symmetric error distributions.

1 2	lad(formula, data, contrasts = NULL, number.perms = 5000, quant, test = FALSE, all.quants = FALSE, OLS,weights)

`formula`	an object of class `formula` or a statement that can be coerced to this class.
`data`	a data frame or object that can be coerced to one using `as.data.frame` containing the variables and response in the model. If missing the formula is evaluated in the environment from which the original function call was made.
`contrasts`	an optional list see `contrasts` for further details.
`number.perms`	the number of permutations used if a Monte Carlo resampling procedure is specified.
`quant`	a numeric value which specifies a quantile for which all subsequent testing is done.
`test`	a logical indicating whether to test if all slope parameters are equal to zero.
`all.quants`	a logical indicating whether all possible quantile regression estimates should be returned.
`weights`	a vector of weights to be applied to the response and model frame matrix.
`OLS`	a logical indicating whether ordinary least squares regression should be performed.

The lad command can be used to fit a variety of least absolute deviation regressions. The hypothesis.test command allows the specification of reduced parameter LAD regression model to compare with the full parameter regression model. The regressions are run using the lad command and the tests performed with the hypothesis.test command. If the quant = num option is specified, all subsequent testing is done on the specified conditional quantile. By default the model will include a constant term. To specify the exclusion of the constant term use -1 in the formula specification.

The number.perms = num option allows the user to specify more or fewer permutations than the default of 5,000 used in approximating probabilities. The save.test = TRUE option specifies that predicted values, residuals, and model variables are to be stored as part of the LAD object. The fitted values and residuals can be accessed using the commands predict(LadObj) and residuals(LadObj).

The quant = num option specifies a regression quantile, where the number specified must be greater than 0.0 and less than 1.0. Specifying all.quants = TRUE yields all quantile regression estimates and when combined with a save.test=TRUE, the parameter estimates by quantile are saved as part of the LAD object and can be retrieved using the command QuantValues.

The test=TRUE option in a lad command is used to test an intercept only null model against the specified model and the hypothesis.test command can be used to test a reduced parameter null model. The hypothesis.test command requires two LAD objects similar to the anova command commonly used for lm or glm objects (Note that it is not possible to test a hypothesis when all quantiles were selected with the all.quants = TRUE option. The dependent variable should be the same in both LAD objects and a reduced number of the same independent variables used in the first LAD object should be present in the second LAD object. The rank.score option bases hypothesis tests on a scoring function of the sign of the residuals for the reduced parameter model specified by hypothesis.test. Asymptotic Chi-square distributional and permutation approximations of P-values are both provided. The double.permutation option provides double permutation for null models that are constrained through the origins, for either the drop in dispersion permutation test or the rank.score test option. The save.test = TRUE option allows the Monte Carlo resampled test statistics to be saved into a single column variable as part of the LAD object, where the first value is always the observed test statistic value. These can be retrieved using the command ResampVals.

lad returns an object of class LADObj.

Marian Talbert

Cade, B.S., and J.D. Richards. 1996. Permutation tests for least absolute deviation regression. Biometrics 52, 886–902.

Cade, B.S. 2005. Linear models: Permutation methods. Pages 1049–1054 in B. Everitt and D. Howell, eds. Encyclopedia of Statistics in the Behavioral Science. Vol. 2. John Wiley and Sons.

Cade, B.S., J.D. Richards, and P.W. Mielke, Jr. 2006. Rank score and permutation testing alternatives for regression quantile estimates. Journal of Statistical Computation and Simulation 76, 331–355.

Cade, B.S., and J.D. Richards. 2006. A permutation test for quantile regression. Journal of Agricultural, Biological, and Environmental Statistics 11, 106–126.

hypothesis.test, LADObj and summary.

Out <- lad(lcc~-1+apico+picopha,data = fraserf,number.perms = 10000,
 test = TRUE)
 
summary(Out)
residuals(Out)
predict(Out)
coefficients(Out)

Out<-lad(lnlctm~widrat,data = lahontan,all.quants = TRUE)
summary(Out)
AllQuants<-QuantValues(Out)