Empirical likelihood multiple tests

Tests multiple linear hypotheses simultaneously.

# S4 method for EL
elmt(object, rhs = NULL, lhs = NULL, alpha = 0.05, control = NULL)

Arguments

object: An object that inherits from EL.
rhs: A numeric vector (column matrix) or a list of numeric vectors for the right-hand sides of hypotheses. Defaults to NULL. See ‘Details’.
lhs: A list or a numeric matrix for the left-hand sides of hypotheses. For a list lhs, each element must be specified as a single instance of the lhs in elt(). For a matrix lhs, each row gives a linear combination of the parameters in object. The number of columns must be equal to the number of parameters. Defaults to NULL. See ‘Details’.
alpha: A single numeric for the overall significance level. Defaults to 0.05.
control: An object of class ControlEL constructed by el_control(). Defaults to NULL and inherits the control slot in object.

Value

An object of class of ELMT.

Details

elmt() tests multiple hypotheses simultaneously. Each hypothesis corresponds to the constrained empirical likelihood ratio described in CEL. rhs and lhs cannot be both NULL. The right-hand side and left-hand side of each hypothesis must be specified as described in elt().

For specifying linear contrasts more conveniently, rhs and lhs also take a numeric vector and a numeric matrix, respectively. Each element of rhs and each row of lhs correspond to a contrast (hypothesis).

The vector of empirical likelihood ratio statistics asymptotically follows a multivariate chi-square distribution under the complete null hypothesis. The multiple testing procedure asymptotically controls the family-wise error rate at the level alpha. Based on the distribution of the maximum of the test statistics, the adjusted p-values are estimated by Monte Carlo simulation.

References

Kim E, MacEachern SN, Peruggia M (2023). ``Empirical likelihood for the analysis of experimental designs.'' Journal of Nonparametric Statistics, 35(4), 709--732. doi:10.1080/10485252.2023.2206919 .

Kim E, MacEachern SN, Peruggia M (2024). ``melt: Multiple Empirical Likelihood Tests in R.'' Journal of Statistical Software, 108(5), 1--33. doi:10.18637/jss.v108.i05 .

Examples

## Bivariate mean (list `rhs` & no `lhs`)
set.seed(143)
data("women")
fit <- el_mean(women, par = c(65, 135))
rhs <- list(c(64, 133), c(66, 140))
elmt(fit, rhs = rhs)
#> 
#> 	Empirical Likelihood Multiple Tests
#> 
#> Overall significance level: 0.05 
#> 
#> Calibration: Multivariate chi-square 
#> 
#> Hypotheses:
#>   Chisq Df
#> 1 2.069  2
#> 2 1.255  2
#> 

## Pairwise comparison (no `rhs` & list `lhs`)
data("clothianidin")
fit2 <- el_lm(clo ~ -1 + trt, clothianidin)
lhs2 <- list(
  "trtNaked - trtFungicide",
  "trtFungicide - trtLow",
  "trtLow - trtHigh"
)
elmt(fit2, lhs = lhs2)
#> 
#> 	Empirical Likelihood Multiple Tests
#> 
#> Overall significance level: 0.05 
#> 
#> Calibration: Multivariate chi-square 
#> 
#> Hypotheses:
#>                             Estimate Chisq Df
#> trtNaked - trtFungicide = 0  -1.0525 5.510  1
#> trtFungicide - trtLow = 0    -0.6269 1.062  1
#> trtLow - trtHigh = 0         -1.4932 3.774  1
#> 

## Arbitrary hypotheses (list `rhs` & list `lhs`)
data("mtcars")
fit3 <- el_lm(mpg ~ wt + qsec, data = mtcars)
lhs3 <- list(c(1, 4, 0), rbind(c(0, 1, 0), c(0, 0, 1)))
rhs3 <- list(0, c(-6, 1))
elmt(fit3, rhs = rhs3, lhs = lhs3)
#> 
#> 	Empirical Likelihood Multiple Tests
#> 
#> Overall significance level: 0.05 
#> 
#> Calibration: Multivariate chi-square 
#> 
#> Hypotheses:
#>   Chisq Df
#> 1 0.037  1
#> 2 2.790  2
#>