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Stat 528 (Autumn 2008) Comparing two proportions
Reading: Section 8.2.
- Comparing two proportions
- Mean and variance of the difference in sample proportions
- Sampling distribution for the difference in sample proportions
- The significance test for a difference in proportions
- The variance of the difference under H 0
- Approximations for the standard error of the difference
- A confidence interval for the difference in proportions.
Comparing two proportions – a motivating example
A study was designed to find reason why patients leave a health maintenance organization (HMO). Patients were classified as to whether or not they had filed a complaint with the HMO. We want to compare the proportion of complainers who leave the HMO with the proportion of those who do not file complaints. In the year of the study, 639 patients filed complaints, and 54 of these patients left the HMO voluntarily. For comparison, the HMO chose an SRS of 743 patients who did not file complaints. Twenty-two of these patients left voluntarily.
- Is there a difference in the two proportions?
- Provide a 95% confidence interval for the difference in the two proportions.
Mean and variance for the difference in sample pro- portions
- By the rules for means μD = μ (^) p̂ 1 −̂p 2 = μ (^) p̂ 1 − μ (^) p̂ 2 = p 1 − p 2.
- Thus D is an unbiased estimator for the difference in the population proportions.
- By independence of the samples and the rules for variances σ^2 D = σ p^2 ̂ 1 −̂p 2 = σ p^2 ̂ 1 + (−1)^2 σ p^2 ̂ 2 = σ p^2 ̂ 1 + σ^2 ̂ p 2 = p^1 (1 n^ − 1 p^1 ) + p^2 (1 n^ − 2 p^2 ).
Sampling distribution for the difference in sample proportions
- For large n 1 and n 2 , D has an approximate N
p 1 − p 2 , p^1 (1 n^ − 1 p^1 ) + p^2 (1 n^ − 2 p^2 )
distribution.
- If SE(D) did not depend on p 1 and p 2 , a test would be based on Z = p̂^1 − SE(^ p̂^2 D^ −)^ δ^0
- A 100(1 − α)% CI for the difference in the population pro- portions, p 1 − p 2 , would be p̂ 1 − p̂ 2 ± zα/ 2 SE(D).
- Since p 1 and p 2 are unknown in practice we need to approxi- mate SE(D).
The variance under H 0
- Under H 0 : p 1 = p 2 ,
- let p = p 1 = p 2 be the common population parameter.
- Then X 1 is a B(n 1 , p) RV and X 2 is a B(n 2 , p) RV.
- These two RVs are independent, and so X 1 + X 2 is a B(n 1 + n 2 , p) RV.
- An estimate of p is p̂ = X n^11 ++^ Xn 22 , and so SÊ (Dp) =
p̂ (1 − p̂ )
n 1 +
n 2
The test for a difference in proportions (cont.)
- Check:
- n 1 p̂ 1 ≥ 10 and n 1 (1 − p̂ 1 ) ≥ 10.
- n 2 p̂ 2 ≥ 10, and n 2 (1 − p̂ 2 ) ≥ 10.
- Then under H 0 , the test statistic approximately follows a N(0, 1) distribution.
- P-value: Using Table A, calculate the area under the Z distribution curve. - For Ha : p 1 − p 2 < 0, the P-value is P (Z ≤ z). - For Ha : p 1 − p 2 > 0, the P-value is P (Z ≥ z). - For Ha : p 1 − p 2 6 = 0, the P-value is 2P (Z ≥ |z|).
Testing in the HMO example
- We expect a higher proportion of complainers to leave. Do the data support this belief?
From tests to intervals
- Hypothesis tests.
- The hypothesis test for “no difference” was simplified by the presumption of a common proportion under the null.
- Other hypothesized differences are more difficult.
- How would you estimate p 1 and p 2 under the restriction that p 2 − p 1 = δ for some specified δ?
- Confidence intervals.
- Without the base of a solid family of hypothesis tests, the construction of a confidence interval becomes fuzzy.
- Idea: Since the plug-in method worked for a single pro- portion, we could try the same for two proportions.
- This idea motivates the most commonly used confidence interval for the difference between two proportions.
- Alternatively, we could patch up the interval a little with a parallel to the “plus four” method.
The confidence interval
- An approximate 100(1 − α)% confidence interval for p 1 − p 2 is given by: p̂ 1 − p̂ 2 ± zα/ 2 SÊ (D), where SÊ (D) is given by the plug-in method.
- Alternatively (rarely used), an approximate 100(1−α)% con- fidence interval for p 1 − p 2 is given by: p ˜ 1 − p˜ 2 ± zα/ 2 SÊ (D), where SÊ (D) is given by the plus four method.
A confidence interval in the HMO example
- Form a 95% confidence interval for the difference in the two proportions.
