Statistical Analysis and Power of Detecting Differences in Failure Rates Between Methods
In the context of quality control, it is often crucial to determine whether two methods yield the same results. Suppose that Method 1 resulted in 20 defects out of 100 produced, whereas Method 2 resulted in 12 defects out of 100. Can we conclude at the 10% significance level that the two methods are equivalent? This article delves into the application of statistical tests to address this question and discusses the importance of power analysis in such scenarios.
Two-Sample Z-Test for Proportions
To answer this question, we can use a two-sample z-test for proportions. This test is particularly useful when comparing two independent proportions to determine if they are significantly different. Let's examine the results of two separate tests: one with continuity correction and another without.
With Continuity Correction
The R prop.test function with continuity correction was used as follows:
X - 20 n - 100 alternative - "" prop.test(X, n, alternativealternative)
The output of this test was:
2-sample test for equality of proportions with continuity correction data: X out of n X-squared 1.8229, df 1, p-value 0.177 alternative hypothesis: 95 percent confidence interval: -0.03100948 0.19100948 sample estimates: prop 1 prop 2 0.20 0.12
This test indicates that the difference between the proportions is not statistically significant at the 10% level, as the p-value is 0.177, which is greater than 0.10. The confidence interval for the difference in proportions also includes 0, suggesting that the rates could be the same.
Without Continuity Correction
Another test without continuity correction was conducted to further illustrate the importance of this correction:
prop.test(X, n, alternativealternative, correctFALSE)
The output was:
2-sample test for equality of proportions without continuity correction data: X out of n X-squared 2.381, df 1, p-value 0.1228 alternative hypothesis: 95 percent confidence interval: -0.02100948 0.18100948 sample estimates: prop 1 prop 2 0.20 0.12
While the p-value in this case is slightly lower at 0.1228, it still does not reach the 10% significance level. The confidence interval is very similar, further supporting the conclusion that the difference is not significant.
Continuity Correction and Its Impact
The textbook used a continuity correction in its analysis, which slightly adjusted the test statistic to improve accuracy. This adjustment is crucial when working with small sample sizes, as it accounts for the discrete nature of the binomial distribution. Without this correction, the p-values and confidence intervals can be misleading.
Power Analysis and Its Relevance
Power analysis is important when deciding whether a difference is likely to be detected by a given sample size and significance level. In this scenario, a power analysis was conducted using the following R code:
n - 100 p1 - 0.2 p2 - c(0.15, 0.10, 0.075) sig.level - 0.05 power - (nn, p1p1, p2p2, sig.levelsig.level)
The results of this power analysis are as follows:
Two-sample comparison of proportions power calculation n 100 p1 0.2 p2 0.150 0.100 0.075 sig.level 0.05 power 0.1511027 0.5081911 0.7313550 alternative
The power of the experiment varies with the degree of difference between the proportions. With a 5% difference, the power is 15.1%, meaning that only 15.1% of the time will you correctly reject the null hypothesis if it is false. With a 10% difference, the power increases to 50.8%, and with a 12.5% difference, the power reaches 73.1%. These low powers indicate that the study is underpowered to detect smaller differences unless the differences are quite large.
The appropriate conclusion, based on these results, is to “Fail to reject the null hypothesis.” This means that the data are not inconsistent with the idea that the failure rates are the same. However, the experiment lacks the power to detect small but potentially meaningful differences.
Conclusion
The confidence intervals and p-values provide a similar cautionary message: the 95% confidence interval for the difference in proportions is -0.031 to 0.191, and includes 0, indicating that it is possible that the difference could be due to random variation.
Understanding the limitations of statistical tests and the importance of power analysis is crucial for making informed decisions in quality control and other fields where such comparisons are necessary.