Tests a linear hypothesis.

## Usage

# S4 method for EL
elt(
object,
rhs = NULL,
lhs = NULL,
alpha = 0.05,
calibrate = "chisq",
control = NULL
)

## Arguments

object

An object that inherits from EL.

rhs

A numeric vector or a column matrix for the right-hand side of hypothesis, with as many entries as the rows in lhs. Defaults to NULL. See ‘Details’.

lhs

A numeric matrix or a vector (treated as a row matrix) for the left-hand side of a hypothesis. Each row gives a linear combination of the parameters in object. The number of columns must be equal to the number of parameters. Or a character vector with a symbolic description of the hypothesis is allowed. Defaults to NULL. See ‘Details’.

alpha

A single numeric for the significance level. Defaults to 0.05.

calibrate

A single character for the calibration method. It is case-insensitive and must be one of "chisq", "boot", or "f". Defaults to "chisq". See ‘Details’.

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 ELT. If lhs is non-NULL, the optim slot corresponds to that of CEL. Otherwise, it corresponds to that of EL.

## Details

elt() performs the constrained minimization of $$l(\theta)$$ described in CEL. rhs and lhs cannot be both NULL. For non-NULL lhs, it is required that lhs have full row rank $$q \leq p$$ and $$p$$ be equal to the number of parameters in the object.

Depending on the specification of rhs and lhs, we have the following three cases:

1. If both rhs and lhs are non-NULL, the constrained minimization is performed with the right-hand side $$r$$ and the left-hand side $$L$$ as $$\inf_{\theta: L\theta = r} l(\theta).$$

2. If rhs is NULL, $$r$$ is set to the zero vector as $$\inf_{\theta: L\theta = 0} l(\theta)$$.

3. If lhs is NULL, $$L$$ is set to the identity matrix and the problem reduces to evaluating at $$r$$ as $$l(r)$$.

calibrate specifies the calibration method used. Three methods are available: "chisq" (chi-square calibration), "boot" (bootstrap calibration), and "f" ($$F$$ calibration). "boot" is applicable only when lhs is NULL. The nthreads, seed, and B slots in control apply to the bootstrap procedure. "f" is applicable only to the mean parameter when lhs is NULL.

## References

Adimari G, Guolo A (2010). “A Note on the Asymptotic Behaviour of Empirical Likelihood Statistics.” Statistical Methods & Applications, 19(4), 463--476. doi:10.1007/s10260-010-0137-9 .

Qin J, Lawless J (1995). “Estimating Equations, Empirical Likelihood and Constraints on Parameters.” Canadian Journal of Statistics, 23(2), 145--159. doi:10.2307/3315441 .

EL, ELT, elmt(), el_control()

## Examples

## F calibration for the mean
data("precip")
fit <- el_mean(precip, 32)
elt(fit, rhs = 32, calibrate = "f")
#>
#> 	Empirical Likelihood Test
#>
#> Hypothesis:
#> par = 32
#>
#> Significance level: 0.05, Calibration: F
#>
#> Statistic: 2.997, Critical value: 3.98
#> p-value: 0.0879
#> EL evaluation: converged
#>

## Test of no treatment effect
data("clothianidin")
contrast <- matrix(c(
1, -1, 0, 0,
0, 1, -1, 0,
0, 0, 1, -1
), byrow = TRUE, nrow = 3)
fit2 <- el_lm(clo ~ -1 + trt, clothianidin)
elt(fit2, lhs = contrast)
#>
#> 	Empirical Likelihood Test
#>
#> Hypothesis:
#> trtNaked - trtFungicide = 0
#> trtFungicide - trtLow = 0
#> trtLow - trtHigh = 0
#>
#> Significance level: 0.05, Calibration: Chi-square
#>
#> Statistic: 26.6, Critical value: 7.815
#> p-value: 7.148e-06
#> Constrained EL: converged
#>

## A symbolic description of the same hypothesis
elt(fit2, lhs = c(
"trtNaked - trtFungicide",
"trtFungicide - trtLow",
"trtLow - trtHigh"
))
#>
#> 	Empirical Likelihood Test
#>
#> Hypothesis:
#> trtNaked - trtFungicide = 0
#> trtFungicide - trtLow = 0
#> trtLow - trtHigh = 0
#>
#> Significance level: 0.05, Calibration: Chi-square
#>
#> Statistic: 26.6, Critical value: 7.815
#> p-value: 7.148e-06
#> Constrained EL: converged
#>