Profile Likelihoods

For this example, we’ll explore the curve_lik() function, which can help generate profile likelihood functions, and deviance functions with the help of the ProfileLikelihood package.¹. For an introduction to what profile likelihoods are, see the following paper.²

library(ProfileLikelihood)
#> Loading required package: nlme
#> Loading required package: MASS

We’ll use a simple example taken directly from the ProfileLikelihood documentation where we’ll calculate the likelihoods from a glm model

data(dataglm)
xx <- profilelike.glm(y ~ x1 + x2,
  data = dataglm, profile.theta = "group",
  family = binomial(link = "logit"), length = 500, round = 2
)
#> Warning message: provide lo.theta and hi.theta

Then, we’ll use curve_lik() on the object that the ProfileLikelihood package produced.

lik <- curve_lik(xx, data = dataglm)

Next, we’ll plot four functions, the relative likelihood, the log-likelihood, the likelihood, and the deviance function.

ggcurve(lik[[1]], type = "l1", nullvalue = TRUE)

ggcurve(lik[[1]], type = "l2")

ggcurve(lik[[1]], type = "l3")

ggcurve(lik[[1]], type = "d")

The obvious advantage of using reduced likelihoods is that they are free of nuisance parameters

\[L_{t_{n}}(\theta)=f_{n}\left(F_{n}^{-1}\left(H_{p i v}(\theta)\right)\right)\left|\frac{\partial}{\partial t} \psi\left(t_{n}, \theta\right)\right|=h_{p i v}(\theta)\left|\frac{\partial}{\partial t} \psi(t, \theta)\right| /\left.\left|\frac{\partial}{\partial \theta} \psi(t, \theta)\right|\right|_{t=t_{n}}\] thus, giving summaries of the data that can be incorporated into combined analyses.

Cite R Packages

Please remember to cite the packages that you use.

citation("concurve")
#> 
#> Rafi Z, Vigotsky A (2020). _concurve: Computes and Plots Compatibility
#> (Confidence) Intervals, P-Values, S-Values, & Likelihood Intervals to
#> Form Consonance, Surprisal, & Likelihood Functions_. R package version
#> 2.7.7, <URL: https://CRAN.R-project.org/package=concurve>.
#> 
#> Rafi Z, Greenland S (2020). "Semantic and Cognitive Tools to Aid
#> Statistical Science: Replace Confidence and Significance by
#> Compatibility and Surprise." _BMC Medical Research Methodology_, *20*,
#> 244. ISSN 1471-2288, doi: 10.1186/s12874-020-01105-9 (URL:
#> https://doi.org/10.1186/s12874-020-01105-9), <URL:
#> https://doi.org/10.1186/s12874-020-01105-9>.
#> 
#> To see these entries in BibTeX format, use 'print(<citation>,
#> bibtex=TRUE)', 'toBibtex(.)', or set
#> 'options(citation.bibtex.max=999)'.
citation("ProfileLikelihood")
#> 
#> To cite package 'ProfileLikelihood' in publications use:
#> 
#>   Leena Choi (2011). ProfileLikelihood: Profile Likelihood for a
#>   Parameter in Commonly Used Statistical Models. R package version 1.1.
#>   https://CRAN.R-project.org/package=ProfileLikelihood
#> 
#> A BibTeX entry for LaTeX users is
#> 
#>   @Manual{,
#>     title = {ProfileLikelihood: Profile Likelihood for a Parameter in Commonly Used Statistical
#> Models},
#>     author = {Leena Choi},
#>     year = {2011},
#>     note = {R package version 1.1},
#>     url = {https://CRAN.R-project.org/package=ProfileLikelihood},
#>   }
#> 
#> ATTENTION: This citation information has been auto-generated from the
#> package DESCRIPTION file and may need manual editing, see
#> 'help("citation")'.

References

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1. Choi L. ProfileLikelihood: Profile likelihood for a parameter in commonly used statistical models. 2011. https://CRAN.R-project.org/package=ProfileLikelihood.

2. Cole SR, Chu H, Greenland S. Maximum Likelihood, Profile Likelihood, and Penalized Likelihood: A Primer. American Journal of Epidemiology. 2013;179(2):252-260. doi:10/f5mx4q