tTOmaxlik {tTOlr}

Maximum Likelihood Under H1, Given T-statistic

Description

Given the t-statistic for a difference in means, or for a mean difference, and degrees of freedom, determine the maximum likelihood under the alternative H1, and the $t$-statistic for the difference in means that makes the likelihood under H1 a maximum. Additionally, return the likelihood under H0.

Usage

tTOmaxlik(t, df)

Arguments

t

t-statistic.

df

Degrees of freedom.

Details

#' @details The function returns the maximum likelihood estimate of the maximum likelihood on the scale of the $t$-statistic, for the likelihood under the alternative, when the when the $t$-statistic is used as non-centrality parameter. This results in a value for the likelihood ratio that differs from (and is smaller than) the standard likelihood ratio statistic. Additionally, return the likelihoods under H0 and H1.

Value

List, with elements

• maxlik - Maximum likelihood under H1

• tmax - t-statistic for difference in means that makes likelihood a maximum under H1

• lik0 - Density (one-sided) under H0

References

van Aubel, A; Gawronski, W (2003). Analytic properties of noncentral distributions. Applied Mathematics and Computation. 141: 3–12. doi:10.1016/S0096-3003(02)00316-8.

Examples

stats <- tTOmaxlik(t=2, df=5)
## Likelihood ratio, 1-sided test and 2-sided test, p=0.05
tvals1 <- qt(0.05, df=c(2,5,20), lower.tail=FALSE)
tvals2 <- qt(0.025, df=c(2,5,20), lower.tail=FALSE)
likrat1 <- likrat2 <- numeric(3)
for(i in 1:3){
stats1 <- tTOmaxlik(t=tvals1[i], df=c(2,5,20)[i])
likrat1[i] <- stats1[['maxlik']]/stats1[['lik0']]
stats2 <- tTOmaxlik(t=tvals2[i], df=c(2,5,20)[i])
likrat2[i] <- stats2[['maxlik']]/(2*stats2[['lik0']])
# NB: 2*stats2[['lik0']] in denominator.
}
likrat <- rbind('One-sided'=likrat1, 'Two-sided'=likrat2)
colnames(likrat) <- paste0('df=',c(2,5,20))
likrat