Likelihood Ratio Tests and Confidence Intervals.

LRTesteR Overview

LRTesteR provides likelihood ratio tests and associated confidence intervals for many common distributions. All functions match popular tests in R. If you are familiar with t.test and binom.test, you already know how to use these functions. All tests and confidence intervals rely on the $\chi^2$ approximation even when exact sampling distributions are known.

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

Nonparametric Tests and Confidence Intervals

• Empirical Likelihood
  • mean
  • quantile

Parametric Tests and Confidence Intervals

Parametric tests require a sample size of at least 50.

• Beta
  • shape 1
  • shape 2
• Binomial
  • p
• Exponential
  • rate
• Gamma
  • rate
  • scale
  • shape
• Gaussian
  • mu
  • variance
• Negative Binomial
  • p
• Poisson
  • lambda
• Cauchy
  • location
  • scale
• Inverse Gaussian
  • mean
  • shape
  • dispersion

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 $\chi^2$ approximation. To get a sense of accuracy of this approximation, lets compare the likelihood method to the exact method.

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 = 50, 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$conf.int)
#> [1] 2.728337 3.673456
likelihoodTest$conf.int
#> [1] 2.735731 3.666063

The confidence intervals for variance are similar as well.

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$conf.int)
#> [1] 1.929274 4.293414
likelihoodTest$conf.int
#> [1] 1.875392 4.121238

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

exactTest <- stats::binom.test(x = 10, n = 50, p = .50, alternative = "two.sided", conf.level = .95)
likelihoodTest <- binomial_p_one_sample(x = 10, n = 50, p = .50, alternative = "two.sided", conf.level = .95)
as.numeric(exactTest$conf.int)
#> [1] 0.1003022 0.3371831
likelihoodTest$conf.int
#> [1] 0.1056842 0.3242910

When exact methods are known, use them. The utility of the likelihood based approach is its generality. Many tests in this package don’t have other well known options.