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
Estimated asymptotic type I and type II error rates can be found here.
- Empirical Likelihood
- mean
- quantile
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
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)
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)
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)
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)
As implemented, all functions depend on the
X is normally distributed with mu equal to 3 and standard deviation
equal to 2. The two intervals for
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.