# Model fitting

## Fitting an EL object

The melt package provides several functions to construct an EL object or an object that inherits from EL:

• el_mean() for the mean.
• el_sd() for the standard deviation.
• el_lm() for linear models.
• el_glm() for generalized linear models.

We illustrate the usage of el_mean() with the faithful data set.

data("faithful")
str(faithful)
#> 'data.frame':    272 obs. of  2 variables:
#>  $eruptions: num 3.6 1.8 3.33 2.28 4.53 ... #>$ waiting  : num  79 54 74 62 85 55 88 85 51 85 ...
summary(faithful)
#>    eruptions        waiting
#>  Min.   :1.600   Min.   :43.0
#>  1st Qu.:2.163   1st Qu.:58.0
#>  Median :4.000   Median :76.0
#>  Mean   :3.488   Mean   :70.9
#>  3rd Qu.:4.454   3rd Qu.:82.0
#>  Max.   :5.100   Max.   :96.0

Suppose we are interested in evaluating empirical likelihood at (3.5, 70).

fit <- el_mean(faithful, par = c(3.5, 70))
class(fit)
#> [1] "EL"
#> attr(,"package")
#> [1] "melt"
showClass("EL")
#> Class "EL" [package "melt"]
#>
#> Slots:
#>
#> Name:         optim         logp         logl        loglr    statistic
#> Class:         list      numeric      numeric      numeric      numeric
#>
#> Name:            df         pval         nobs         npar      weights
#> Class:      integer      numeric      integer      integer      numeric
#>
#> Name:  coefficients       method         data      control
#> Class:      numeric    character          ANY    ControlEL
#>
#> Known Subclasses:
#> Class "CEL", directly
#> Class "SD", directly
#> Class "LM", by class "CEL", distance 2
#> Class "GLM", by class "CEL", distance 3
#> Class "QGLM", by class "CEL", distance 4

The faithful data frame is coerced to a numeric matrix. Simple print method shows essential information on fit.

fit
#>
#>  Empirical Likelihood
#>
#> Model: mean
#>
#> Maximum EL estimates:
#> eruptions   waiting
#>     3.488    70.897
#>
#> Chisq: 8.483, df: 2, Pr(>Chisq): 0.01439
#> EL evaluation: converged

Note that the maximum empirical likelihood estimates are the same as the sample average. The chi-square value shown corresponds to the minus twice the empirical log-likelihood ratio. It has an asymptotic chi-square distribution of 2 degrees of freedom under the null hypothesis. Hence the $$p$$-value here is not exact. The convergence status at the bottom can be used to check the convex hull constraint.

Weighted data can be handled by supplying the weights argument. For non-NULL weights, weighted empirical likelihood is computed. Any valid weights is re-scaled for internal computation to add up to the total number of observations. For simplicity, we use faithful$waiting as our weight vector. w <- faithful$waiting
(wfit <- el_mean(faithful, par = c(3.5, 70), weights = w))
#>
#>  Weighted Empirical Likelihood
#>
#> Model: mean
#>
#> Maximum EL estimates:
#> eruptions   waiting
#>     3.684    73.494
#>
#> Chisq: 25.41, df: 2, Pr(>Chisq): 3.039e-06
#> EL evaluation: converged

We get different results, where the estimates are now the weighted sample average. The chi-square value and the associated $$p$$-value are based on the same limit theorem, but care must be taken when interpreting the results since they are largely affected by the limiting behavior of the weights.