LRT {LBI}

Likelihood Ratio Test

Description

Likelihood ratio test with given fitting results, sample size, number of parameters, log-likelihoods, and alpha

Usage

  LRT(n, pFull, pReduced, logLikFull, logLikReduced, alpha=0.05, Wilks=FALSE)

Arguments

n

number of observations

pFull

number of parameters of full model

pReduced

number of parameters of reduced model

logLikFull

log likelihood of full model

logLikReduced

log likelihood of reduced model

alpha

alpha value for type I error, significance level

Wilks

if TRUE, Wilks theorem (chi-square distribution) will be used, otherwise F distribution will be used.

Details

It performs likelihood ratio test with given fitting results. The default test is using F distribution. For small n (i.e. less than 100), you need to use F distribution. If the residuals are normally distributed, the delta -2 log likelihood (the difference between -2LL, the objective function value of each model) follows exactly an F-distribution, independent of sample size. When the distribution of the residuals is not normal (no matter what the distribution of the residuals is), it approaches a chi-square distribution as sample size increases (Wilks' theorem). The extreme distribution of the F-distribution (when the degrees of freedom in the denominator go to infinity) is chi-square distribution. The p-value from the F-distribution is slightly larger than the p-value from the chi-square distribution, meaning the F-distribution is more conservative. The difference decreases as sample size increases.

Value

n

number of observations

paraFull

number of parameters of full model

paraReduced

number of parameters of reduced model

deltaPara

difference of parameter counts

cutoff

cutoff, threshold, critical value of log-likelihood for the test

deltaLogLik

difference of log likelihood, if negative 0 is used.

Chisq or Fval

statistics according to the used distribution Chi-square of F

pval

p-value of null hypothesis. i.e. the reduced model is better.

Verdict

the model preferred.

Author(s)

Kyun-Seop Bae [e-mail]

References

1. Ruppert D, Cressie N, Carroll RJ. A Transformation/Weighting Model For Estimating Michaelis-Menten Parameters. School of Operations Research and Industrial Engineering, College of Engineering, Cornell University. Technical Report No. 796. May 1988.

2. Scheffé H. The Analysis of Variance. Wiley. 1959.

3. Wilks SS. The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. Annals Math. Statist. 1938;9:60-62

Examples

  LRT(20, 4, 2, -58.085, -60.087)
  LRT(20, 4, 2, -58.085, -60.087, Wilks=TRUE)
  LRT(20, 4, 2, -57.315, -66.159)
  LRT(20, 4, 2, -57.315, -66.159, Wilks=TRUE)

  r1 = lm(mpg ~ disp + drat + wt, mtcars)
  r2 = lm(mpg ~ disp + drat, mtcars)
  anova(r2, r1)
  LRT(nrow(mtcars), r1$rank, r2$rank, logLik(r1), logLik(r2))