The comparison of two Poisson means is a common statistical problem encountered in various fields, from healthcare to finance. Traditional methods, such as the Wald test, often lack power, especially when dealing with small sample sizes or low event rates. This article explores a more robust alternative – the E-test – as outlined by Krishnamoorthy and Thomson (2004), providing a more powerful approach for comparing two Poisson means.
Understanding the E-Test for Poisson Means
The E-test offers a significant advantage over traditional methods by directly utilizing the exact distribution of the difference between two Poisson random variables. This approach eliminates the reliance on asymptotic approximations, which can be inaccurate when sample sizes are small. The null hypothesis of the E-test posits that the ratio of the two Poisson means (or rates) is equal to a specified constant, often 1. More formally:
k_1/n_1 = k_2/n_2 + d
Where:
k_1andk_2represent the observed Poisson counts.n_1andn_2are divisors fork_1andk_2respectively, often representing exposure times or areas (default is 1).dis a constant amending the null hypothesis (default is 0).
The E-test calculates a precise p-value by summing the probabilities of observing differences as extreme or more extreme than the observed difference, under the null hypothesis. This calculation involves iterative summation until a predefined precision threshold (eps) is reached.
Practical Applications of the E-Test
The E-test finds applications in diverse fields:
- Healthcare: Comparing infection rates between two hospitals.
- Epidemiology: Analyzing disease incidence in different populations.
- Manufacturing: Assessing defect rates between production lines.
- Finance: Comparing the frequency of trading events between different market conditions.
In scenarios with low counts or rates, where traditional methods struggle, the E-test provides reliable results. For instance, comparing the number of rare adverse events between two treatment groups in a clinical trial would benefit from the E-test’s increased power.
Implementing the E-Test in R
The e_test function within the ptools package in R facilitates easy implementation of this powerful test.
e_test(k1, k2, n1 = 1, n2 = 1, d = 0, eps = 1e-20, silent = FALSE)The function takes the observed counts (k1, k2), optional divisors (n1, n2), a constant difference (d), a precision threshold (eps), and a silent option (silent) as arguments, returning a p-value.
Considerations and Limitations
While powerful, the E-test has certain limitations:
- Large Counts: The test’s computational intensity can increase with very large counts (e.g., exceeding 100), potentially leading to memory issues. Using rates instead of counts can mitigate this.
- Undefined Cases: The test is undefined for certain input combinations, such as comparing two zero counts (
e_test(0,0)).
Conclusion
The E-test provides A More Powerful Test For Comparing Two Poisson Means, especially in scenarios with small sample sizes or low event rates. Its reliance on the exact distribution enhances accuracy compared to asymptotic approximations used in traditional methods. By leveraging the e_test function in R, researchers and analysts can perform robust comparisons of Poisson means across diverse applications. While mindful of its limitations with very large counts, the E-test represents a valuable tool in the statistical arsenal. For further exploration, refer to Krishnamoorthy and Thomson (2004).
