Empirical likelihood for linear models

Fits a linear model with empirical likelihood.

Usage

el_lm(
  formula,
  data,
  weights = NULL,
  na.action,
  offset,
  control = el_control(),
  ...
)

Arguments

formula

An object of class formula (or one that can be coerced to that class) for a symbolic description of the model to be fitted.

data

An optional data frame, list or environment (or object coercible by as.data.frame() to a data frame) containing the variables in formula. If not found in data, the variables are taken from environment(formula).

weights

An optional numeric vector of weights to be used in the fitting process. Defaults to NULL, corresponding to identical weights. If non-NULL, weighted empirical likelihood is computed.

na.action

A function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset.

offset

An optional expression for specifying an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector or matrix of extents matching those of the response. One or more offset terms can be included in the formula instead or as well, and if more than one are specified their sum is used.

control

An object of class ControlEL constructed by el_control().

...

Additional arguments to be passed to the low level regression fitting functions. See ‘Details’.

Value

An object of class of LM.

Details

Suppose that we observe \(n\) independent random variables \({Z_i} \equiv {(X_i, Y_i)}\) from a common distribution, where \(X_i\) is the \(p\)-dimensional covariate (including the intercept if any) and \(Y_i\) is the response. We consider the following linear model: $$Y_i = X_i^\top \theta + \epsilon_i,$$ where \(\theta = (\theta_0, \dots, \theta_{p-1})\) is an unknown \(p\)-dimensional parameter and the errors \(\epsilon_i\) are independent random variables that satisfy \(\textrm{E}(\epsilon_i | X_i)\) = 0. We assume that the errors have finite conditional variances. Then the least square estimator of \(\theta\) solves the following estimating equations: $$\sum_{i = 1}^n(Y_i - X_i^\top \theta)X_i = 0.$$ Given a value of \(\theta\), let \({g(Z_i, \theta)} = {(Y_i - X_i^\top \theta)X_i}\) and the (profile) empirical likelihood ratio is defined by $$R(\theta) = \max_{p_i}\left\{\prod_{i = 1}^n np_i : \sum_{i = 1}^n p_i g(Z_i, \theta) = \theta,\ p_i \geq 0,\ \sum_{i = 1}^n p_i = 1 \right\}.$$ el_lm() first computes the parameter estimates by calling lm.fit() (with ... if any) with the model.frame and model.matrix obtained from the formula. Note that the maximum empirical likelihood estimator is the same as the the quasi-maximum likelihood estimator in our model. Next, it tests hypotheses based on asymptotic chi-square distributions of the empirical likelihood ratio statistics. Included in the tests are overall test with $$H_0: \theta_1 = \theta_2 = \cdots = \theta_{p-1} = 0,$$ and significance tests for each parameter with $$H_{0j}: \theta_j = 0,\ j = 0, \dots, p-1.$$

References

Owen A (1991). “Empirical Likelihood for Linear Models.” The Annals of Statistics, 19(4), 1725–1747. doi:10.1214/aos/1176348368 .

See also

EL, LM, el_glm(), elt(), el_control()

Examples

## Linear model
data("thiamethoxam")
fit <- el_lm(fruit ~ trt, data = thiamethoxam)
summary(fit)
#> 
#> 	Empirical Likelihood
#> 
#> Model: lm 
#> 
#> Call:
#> el_lm(formula = fruit ~ trt, data = thiamethoxam)
#> 
#> Number of observations: 165 
#> Number of parameters: 4 
#> 
#> Parameter values under the null hypothesis:
#> (Intercept)    trtSpray   trtFurrow     trtSeed 
#>       6.016       0.000       0.000       0.000 
#> 
#> Lagrange multipliers:
#> [1] -0.03994 -0.29622  0.59777 -0.15872
#> 
#> Maximum EL estimates:
#> (Intercept)    trtSpray   trtFurrow     trtSeed 
#>      5.6667     -0.7167      2.7798     -0.3452 
#> 
#> logL: -875.9 , logLR: -33.38 
#> Chisq: 66.76, df: 3, Pr(>Chisq): 2.113e-14
#> Constrained EL: converged 
#> 
#> Coefficients:
#>             Estimate   Chisq Pr(>Chisq)    
#> (Intercept)   5.6667 413.766    < 2e-16 ***
#> trtSpray     -0.7167   1.978      0.160    
#> trtFurrow     2.7798  19.259   1.14e-05 ***
#> trtSeed      -0.3452   0.416      0.519    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

## Weighted data
wfit <- el_lm(fruit ~ trt, data = thiamethoxam, weights = visit)
summary(wfit)
#> 
#> 	Weighted Empirical Likelihood
#> 
#> Model: lm 
#> 
#> Call:
#> el_lm(formula = fruit ~ trt, data = thiamethoxam, weights = visit)
#> 
#> Number of observations: 165 
#> Number of parameters: 4 
#> 
#> Parameter values under the null hypothesis:
#> (Intercept)    trtSpray   trtFurrow     trtSeed 
#>       6.358       0.000       0.000       0.000 
#> 
#> Lagrange multipliers:
#> [1] -0.06482 -0.21994  0.51894 -0.19381
#> 
#> Maximum EL estimates:
#> (Intercept)    trtSpray   trtFurrow     trtSeed 
#>     5.71940    -0.10153     2.94806    -0.07815 
#> 
#> logL: -847.5 , logLR: -31.59 
#> Chisq: 63.18, df: 3, Pr(>Chisq): 1.229e-13
#> Constrained EL: converged 
#> 
#> Coefficients:
#>             Estimate   Chisq Pr(>Chisq)    
#> (Intercept)  5.71940 415.486    < 2e-16 ***
#> trtSpray    -0.10153   0.028      0.867    
#> trtFurrow    2.94806  19.830   8.46e-06 ***
#> trtSeed     -0.07815   0.020      0.887    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

## Missing data
fit2 <- el_lm(fruit ~ trt + scb, data = thiamethoxam,
  na.action = na.omit, offset = NULL
)
summary(fit2)
#> 
#> 	Empirical Likelihood
#> 
#> Model: lm 
#> 
#> Call:
#> el_lm(formula = fruit ~ trt + scb, data = thiamethoxam, na.action = na.omit, 
#>     offset = NULL)
#> 
#> Number of observations: 162 
#> Number of parameters: 5 
#> 
#> Parameter values under the null hypothesis:
#> (Intercept)    trtSpray   trtFurrow     trtSeed         scb 
#>       6.043       0.000       0.000       0.000       0.000 
#> 
#> Lagrange multipliers:
#> [1] -0.017410 -0.301618  0.595467 -0.153307 -0.008558
#> 
#> Maximum EL estimates:
#> (Intercept)    trtSpray   trtFurrow     trtSeed         scb 
#>     5.62981    -0.74024     2.82886    -0.24460     0.01551 
#> 
#> logL: -857 , logLR: -32.79 
#> Chisq: 65.58, df: 4, Pr(>Chisq): 1.939e-13
#> Constrained EL: converged 
#> 
#> Coefficients:
#>             Estimate   Chisq Pr(>Chisq)    
#> (Intercept)  5.62981 405.317    < 2e-16 ***
#> trtSpray    -0.74024   2.160      0.142    
#> trtFurrow    2.82886  23.410   1.31e-06 ***
#> trtSeed     -0.24460   0.235      0.628    
#> scb          0.01551   0.141      0.707    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>   (3 observations deleted due to missingness)
#>