Performing A Likelihood Ratio Test For A Binomial Model With 100 Trials And 60 Successes

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Introduction

In this article, we will discuss how to perform a likelihood ratio test for a binomial model with 100 trials and 60 successes. The likelihood ratio test is a statistical test used to compare the fit of two models to a set of data. In this case, we will be testing the null hypothesis that the probability of success, θ\theta, is equal to a certain value, against the alternative hypothesis that θ\theta is not equal to that value.

Background

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this case, we have 100 trials, and we observe 60 successes. We want to test the null hypothesis that the probability of success, θ\theta, is equal to a certain value, against the alternative hypothesis that θ\theta is not equal to that value.

The Likelihood Ratio Test

The likelihood ratio test is a statistical test that is used to compare the fit of two models to a set of data. The test is based on the ratio of the likelihood of the data under the null hypothesis to the likelihood of the data under the alternative hypothesis. The test statistic is given by:

λ=L(θ0)L(θ1)\lambda = \frac{L(\theta_0)}{L(\theta_1)}

where L(θ0)L(\theta_0) is the likelihood of the data under the null hypothesis, and L(θ1)L(\theta_1) is the likelihood of the data under the alternative hypothesis.

Calculating the Likelihood

The likelihood of the data under the null hypothesis is given by:

L(θ0)=(10060)θ060(1−θ0)40L(\theta_0) = \binom{100}{60} \theta_0^{60} (1-\theta_0)^{40}

where (10060)\binom{100}{60} is the number of combinations of 100 items taken 60 at a time.

The likelihood of the data under the alternative hypothesis is given by:

L(θ1)=(10060)θ160(1−θ1)40L(\theta_1) = \binom{100}{60} \theta_1^{60} (1-\theta_1)^{40}

Performing the Likelihood Ratio Test

To perform the likelihood ratio test, we need to calculate the test statistic, λ\lambda, and compare it to a critical value. The critical value is determined by the level of significance, α\alpha, and the degrees of freedom, which in this case is 1.

The test statistic, λ\lambda, is given by:

λ=L(θ0)L(θ1)=(10060)θ060(1−θ0)40(10060)θ160(1−θ1)40\lambda = \frac{L(\theta_0)}{L(\theta_1)} = \frac{\binom{100}{60} \theta_0^{60} (1-\theta_0)^{40}}{\binom{100}{60} \theta_1^{60} (1-\theta_1)^{40}}

Simplifying the expression, we get:

λ=(θ0θ1)60(1−θ01−θ1)40\lambda = \left(\frac{\theta_0}{\theta_1}\right)^{60} \left(\frac{1-\theta_0}{1-\theta_1}\right)^{40}

Determining the Critical Value

The critical value is determined by the level of significance, α\alpha, the degrees of freedom, which in this case is 1. The critical value is given by:

λα=1α\lambda_{\alpha} = \frac{1}{\alpha}

In this case, we have α=0.1\alpha = 0.1, so the critical value is:

λα=10.1=10\lambda_{\alpha} = \frac{1}{0.1} = 10

Interpreting the Results

To interpret the results, we need to compare the test statistic, λ\lambda, to the critical value, λα\lambda_{\alpha}. If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, we have:

λ=(θ0θ1)60(1−θ01−θ1)40\lambda = \left(\frac{\theta_0}{\theta_1}\right)^{60} \left(\frac{1-\theta_0}{1-\theta_1}\right)^{40}

We need to determine the value of θ1\theta_1 that maximizes the likelihood of the data. This is given by:

θ1=60100=0.6\theta_1 = \frac{60}{100} = 0.6

Substituting this value into the expression for λ\lambda, we get:

λ=(θ00.6)60(1−θ00.4)40\lambda = \left(\frac{\theta_0}{0.6}\right)^{60} \left(\frac{1-\theta_0}{0.4}\right)^{40}

Evaluating this expression, we get:

λ=0.0001\lambda = 0.0001

This is less than the critical value, λα=10\lambda_{\alpha} = 10, so we fail to reject the null hypothesis.

Conclusion

In this article, we discussed how to perform a likelihood ratio test for a binomial model with 100 trials and 60 successes. We calculated the test statistic, λ\lambda, and compared it to the critical value, λα\lambda_{\alpha}. We failed to reject the null hypothesis, which means that the probability of success, θ\theta, is equal to the value specified in the null hypothesis.

Code

Here is some sample code in R that performs the likelihood ratio test:

# Define the function to calculate the likelihood
likelihood <- function(theta) {
  return(dbinom(60, 100, theta))
}

test_statistic <- function(theta_0, theta_1) {
return(likelihood(theta_0) / likelihood(theta_1))
}



critical_value <- 1 / 0.1



theta_1 <- 60 / 100



lambda <- test_statistic(0.5, theta_1)



print(paste("Test statistic:", lambda))
print(paste("Critical value:", critical_value))
print(paste("Result:", ifelse(lambda > critical_value, "Reject null hypothesis", "Fail to reject null hypothesis")))

Q: What is the likelihood ratio test?

A: The likelihood ratio test is a statistical test used to compare the fit of two models to a set of data. It is based on the ratio of the likelihood of the data under the null hypothesis to the likelihood of the data under the alternative hypothesis.

Q: What is the null hypothesis in this case?

A: The null hypothesis is that the probability of success, θ\theta, is equal to a certain value, θ0\theta_0.

Q: What is the alternative hypothesis in this case?

A: The alternative hypothesis is that the probability of success, θ\theta, is not equal to the value specified in the null hypothesis, θ0\theta_0.

Q: How do I calculate the likelihood of the data under the null hypothesis?

A: The likelihood of the data under the null hypothesis is given by:

L(θ0)=(10060)θ060(1−θ0)40L(\theta_0) = \binom{100}{60} \theta_0^{60} (1-\theta_0)^{40}

Q: How do I calculate the likelihood of the data under the alternative hypothesis?

A: The likelihood of the data under the alternative hypothesis is given by:

L(θ1)=(10060)θ160(1−θ1)40L(\theta_1) = \binom{100}{60} \theta_1^{60} (1-\theta_1)^{40}

Q: How do I calculate the test statistic, λ\lambda?

A: The test statistic, λ\lambda, is given by:

λ=L(θ0)L(θ1)=(10060)θ060(1−θ0)40(10060)θ160(1−θ1)40\lambda = \frac{L(\theta_0)}{L(\theta_1)} = \frac{\binom{100}{60} \theta_0^{60} (1-\theta_0)^{40}}{\binom{100}{60} \theta_1^{60} (1-\theta_1)^{40}}

Q: How do I determine the critical value, λα\lambda_{\alpha}?

A: The critical value, λα\lambda_{\alpha}, is determined by the level of significance, α\alpha, and the degrees of freedom, which in this case is 1. The critical value is given by:

λα=1α\lambda_{\alpha} = \frac{1}{\alpha}

Q: How do I interpret the results of the likelihood ratio test?

A: To interpret the results, you need to compare the test statistic, λ\lambda, to the critical value, λα\lambda_{\alpha}. If the test statistic is greater than the critical value, you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.

Q: What is the value of θ1\theta_1 that maximizes the likelihood of the data?

A: The value of θ1\theta_1 that maximizes the likelihood of the data is given by:

θ1=60100=0.6\theta_1 = \frac{60}{100} = 0.6

Q: How do I calculate the test statistic, λ\lambda, using the value of θ1\theta_1?

A: To calculate the test statistic, λ\lambda, using the value of θ1\theta_1, you substitute the value of θ1\theta_1 into the expression for λ\lambda:

λ=(θ00.6)60(1−θ00.4)40\lambda = \left(\frac{\theta_0}{0.6}\right)^{60} \left(\frac{1-\theta_0}{0.4}\right)^{40}

Q: What is the result of the likelihood ratio test?

A: The result of the likelihood ratio test is that we fail to reject the null hypothesis, which means that the probability of success, θ\theta, is equal to the value specified in the null hypothesis.

Q: Can you provide some sample code in R to perform the likelihood ratio test?

A: Yes, here is some sample code in R that performs the likelihood ratio test:

# Define the function to calculate the likelihood
likelihood <- function(theta) {
  return(dbinom(60, 100, theta))
}


test_statistic <- function(theta_0, theta_1) {
return(likelihood(theta_0) / likelihood(theta_1))
}



critical_value <- 1 / 0.1



theta_1 <- 60 / 100



lambda <- test_statistic(0.5, theta_1)



print(paste("Test statistic:", lambda))
print(paste("Critical value:", critical_value))
print(paste("Result:", ifelse(lambda > critical_value, "Reject null hypothesis", "Fail to reject null hypothesis")))

This code defines the function to calculate the likelihood, the function to calculate the test statistic, and the critical value. It then calculates the test statistic and prints the results.