Likelihood Ratio Tests and Confidence Intervals.

LRTesteR Overview

LRTesteR provides likelihood ratio tests and confidence intervals for many common distributions.

Example 1: Test lambda of a poisson distribution

To test lambda, simply call poisson_lambda_one_sample.

library(LRTesteR)

set.seed(1)
x <- rpois(n = 100, lambda = 1)
poisson_lambda_one_sample(x = x, lambda = 1, alternative = "two.sided")
#> Log Likelihood Statistic: 0.01
#> p value: 0.92
#> Confidence Level: 95%
#> Confidence Interval: (0.826, 1.22)

Example 2: Confidence Interval

To get a confidence interval, set the conf.level to the desired confidence. Below gets a two sided 90% confidence interval for scale from a Cauchy random variable.

set.seed(1)
x <- rcauchy(n = 100, location = 3, scale = 5)
cauchy_scale_one_sample(x = x, scale = 5, alternative = "two.sided", conf.level = .90)
#> Log Likelihood Statistic: 1.21
#> p value: 0.271
#> Confidence Level: 90%
#> Confidence Interval: (4.64, 7.284)

Setting alternative to “less” gets a lower one sided interval.

cauchy_scale_one_sample(x = x, scale = 5, alternative = "less", conf.level = .90)
#> Log Likelihood Statistic: 1.1
#> p value: 0.865
#> Confidence Level: 90%
#> Confidence Interval: (0, 6.93)

Setting it to “greater” gets an upper one sided interval.

cauchy_scale_one_sample(x = x, scale = 5, alternative = "greater", conf.level = .90)
#> Log Likelihood Statistic: 1.1
#> p value: 0.135
#> Confidence Level: 90%
#> Confidence Interval: (4.878, Inf)

Example 3: One-way Analysis

One-way ANOVA is generalized to all distributions. Here gamma random variables are created with different shapes. The one way test has a small p value and provides confidence intervals with 95% confidence for the whole set.

set.seed(1)
x <- c(rgamma(n = 50, shape = 1, rate = 2), rgamma(n = 50, shape = 2, rate = 2), rgamma(n = 50, shape = 3, rate = 2))
fctr <- c(rep(1, 50), rep(2, 50), rep(3, 50))
fctr <- factor(fctr, levels = c("1", "2", "3"))
gamma_shape_one_way(x = x, fctr = fctr, conf.level = .95)
#> Log Likelihood Statistic: 68.59
#> p value: 0
#> Confidence Level Of Set: 95%
#> Individual Confidence Level: 98.3%
#> Confidence Interval For Group 1: (0.65, 1.515)
#> Confidence Interval For Group 2: (1.376, 3.376)
#> Confidence Interval For Group 3: (1.691, 4.192)

Example 4: Empirical Likelihood

The empirical likelihood tests do not require any distributional assumptions and work with less data.

set.seed(1)
x <- rnorm(n = 25, mean = 1, sd = 1)
empirical_mu_one_sample(x = x, mu = 1, alternative = "two.sided")
#> Log Likelihood Statistic: 0.73
#> p value: 0.392
#> Confidence Level: 95%
#> Confidence Interval: (0.752, 1.501)

The $\chi^2$ approximation

As implemented, all functions depend on the asymptotic $\chi^2$ approximation. To get a sense of accuracy of this approximation for large samples, the likelihood tests are compared to the exact tests.

X is normally distributed with mu equal to 3 and standard deviation equal to 2. The two intervals for $\mu$ are similar.

set.seed(1)
x <- rnorm(n = 500, mean = 3, sd = 2)
exactTest <- t.test(x = x, mu = 2.5, alternative = "two.sided", conf.level = .95)
likelihoodTest <- gaussian_mu_one_sample(x = x, mu = 2.5, alternative = "two.sided", conf.level = .95)
as.numeric(exactTest$conf.int)
#> [1] 2.867461 3.223115
likelihoodTest$conf.int
#> [1] 2.867729 3.222847

The confidence intervals for variance are similar as well.

set.seed(1)
x <- rnorm(n = 500, mean = 3, sd = 2)
sigma2 <- 1.5^2 # Variance, not standard deviation.
exactTest <- EnvStats::varTest(x = x, sigma.squared = sigma2, alternative = "two.sided", conf.level = .95)
likelihoodTest <- gaussian_variance_one_sample(x = x, sigma.squared = sigma2, alternative = "two.sided", conf.level = .95)
as.numeric(exactTest$conf.int)
#> [1] 3.631734 4.655834
likelihoodTest$conf.int
#> [1] 3.620303 4.639384

Changing to p for a binomial random variable, the confidence intervals are similar yet again.

exactTest <- stats::binom.test(x = 250, n = 500, p = .50, alternative = "two.sided", conf.level = .95)
likelihoodTest <- binomial_p_one_sample(x = 250, n = 500, p = .50, alternative = "two.sided", conf.level = .95)
as.numeric(exactTest$conf.int)
#> [1] 0.4552856 0.5447144
likelihoodTest$conf.int
#> [1] 0.4562579 0.5437421

When sample size is small, similarity will decrease. When exact methods are available, they are the better option. The utility of the likelihood based approach is its generality. Many tests in this package don’t have other well known options.

Estimated asymptotic type I and type II error rates can be found here.