Tests a linear hypothesis with various calibration options.
Usage
# S4 method for class '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 toNULL
. 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 toNULL
. See ‘Details’.- alpha
A single numeric for the significance level. Defaults to
0.05
.- calibrate
A single character representing the calibration method. It is case-insensitive and must be one of
"ael"
,"boot"
,"chisq"
, or"f"
. Defaults to"chisq"
. See ‘Details’.- control
An object of class ControlEL constructed by
el_control()
. Defaults toNULL
and inherits thecontrol
slot inobject
.
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:
If both
rhs
andlhs
are non-NULL
, the constrained minimization is performed with the right-hand side \(r\) and the left-hand side \(L\) as $$\inf_{\theta: L\theta = r} l(\theta).$$If
rhs
isNULL
, \(r\) is set to the zero vector as \(\inf_{\theta: L\theta = 0} l(\theta)\).If
lhs
isNULL
, \(L\) is set to the identity matrix and the problem reduces to evaluating at \(r\) as \(l(r)\).
calibrate
specifies the calibration method used. Four methods are
available: "ael"
(adjusted empirical likelihood calibration), "boot"
(bootstrap calibration), "chisq"
(chi-square calibration), and "f"
(\(F\) calibration). When lhs
is not NULL
, only "chisq"
is
available. "f"
is applicable only to the mean parameter. The an
slot in
control
applies specifically to "ael"
, while the nthreads
, seed
,
and B
slots apply to "boot"
.
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 .
Chen J, Variyath AM, Abraham B (2008). “Adjusted Empirical Likelihood and Its Properties.” Journal of Computational and Graphical Statistics, 17(2), 426–443.
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 .
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 .
See also
EL, ELT, elmt()
, el_control()
Examples
## Adjusted empirical likelihood calibration
data("precip")
fit <- el_mean(precip, 32)
elt(fit, rhs = 100, calibrate = "ael")
#>
#> Empirical Likelihood Test
#>
#> Hypothesis:
#> par = 100
#>
#> Significance level: 0.05, Calibration: Adjusted EL
#>
#> Statistic: 45.48, Critical value: 3.841
#> p-value: 1.539e-11
#> EL evaluation: converged
#>
## Bootstrap calibration
elt(fit, rhs = 32, calibrate = "boot")
#>
#> Empirical Likelihood Test
#>
#> Hypothesis:
#> par = 32
#>
#> Significance level: 0.05, Calibration: Bootstrap
#>
#> Statistic: 2.997, Critical value: 3.928
#> p-value: 0.084
#> EL evaluation: converged
#>
## F calibration
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
#>