1. Statistics: Unlocking the Power of Data Lock 5 Inference for Proportions STAT 250 Dr. Kari Lock Morgan Chapter 6.1, 6.2, 6.3, 6.7, 6.8, 6.9 Formulas for standard errors Normal based inference

2. Statistics: Unlocking the Power of Data Lock 5 Confidence Interval Formula From original data From bootstrap distribution From N(0,1) IF SAMPLE SIZES ARE LARGE…

3. Statistics: Unlocking the Power of Data Lock 5 Formula for p-values From randomization distribution From H 0 From original data Compare z to N(0,1) for p-value IF SAMPLE SIZES ARE LARGE…

4. Statistics: Unlocking the Power of Data Lock 5 Standard Error Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations? We can!!!

5. Statistics: Unlocking the Power of Data Lock 5 ParameterDistributionStandard Error Proportion Normal Difference in Proportions Normal Meant, df = n – 1 Difference in Meanst, df = min(n 1, n 2 ) – 1 Correlationt, df = n – 2 Standard Error Formulas

6. Statistics: Unlocking the Power of Data Lock 5 SE Formula Observations n is always in the denominator (larger sample size gives smaller standard error) Standard error related to square root of 1/n Standard error formulas use population parameters… (uh oh!) For intervals, plug in the sample statistic(s) as your best guess at the parameter(s) For testing, plug in the null value for the parameter(s), because you want the distribution assuming H 0 true

7. Statistics: Unlocking the Power of Data Lock 5 Hormone Replacement Therapy Until 2002, hormone replacement therapy (HRT), estrogen and/or progesterone, was commonly prescribed to post-menopausal women. This changed in 2002, when the results of a large clinical trial were published 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer Does hormone replacement therapy cause increased risk of breast cancer?

8. Statistics: Unlocking the Power of Data Lock 5 Interval for Proportion First: What are the chances a woman not taking HRT develops invasive breast cancer? Give a 90% confidence interval. What is the sample statistic? What is z* for a 90% interval? What is the standard error?

9. Statistics: Unlocking the Power of Data Lock 5 Sample Statistic 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer

10. Statistics: Unlocking the Power of Data Lock 5 z*

11. Statistics: Unlocking the Power of Data Lock 5 Standard Error

12. Statistics: Unlocking the Power of Data Lock 5 Standard Error

13. Statistics: Unlocking the Power of Data Lock 5 Interval for Proportion

14. Statistics: Unlocking the Power of Data Lock 5 Bootstrap Interval

15. Statistics: Unlocking the Power of Data Lock 5 Interpretation “We are 90% confident that the true proportion of post-menopausal women not taking HRT who develop invasive breast cancer is between 0.013 and 0.018” Do you believe this sentence is accurate? a) Yes b) No

16. Statistics: Unlocking the Power of Data Lock 5 Your Turn! 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer Give a 95% confidence interval for the proportion of women taking HRT who develop invasive breast cancer: a) (0.013, 0.027) b) (0.015, 0.025) c) (0.017, 0.023) d) (0.019, 0.021)

17. Statistics: Unlocking the Power of Data Lock 5 Testing When testing, use the null value for p, rather than the sample statistic So, if testing whether the proportion of people not taking HRT who develop invasive breast cancer is less than 0.1, the standard error would be

18. Statistics: Unlocking the Power of Data Lock 5 Hormone Replacement Therapy 8506 women were randomized to take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo women developed invasive breast cancer Does hormone replacement therapy cause increased risk of breast cancer?

19. Statistics: Unlocking the Power of Data Lock 5 Hypothesis Test State hypotheses. Calculate z-statistic. Compare to standard normal distribution to find p-value. Make a conclusion.

20. Statistics: Unlocking the Power of Data Lock 5 Hypotheses Does hormone replacement therapy cause increased risk of invasive breast cancer? p 1 = proportion of women taking HRT who get invasive breast cancer p 2 = proportion of women not taking HRT who get invasive breast cancer a) H 0 : p 1 = p 2, H a : p 1 ≠ p 2 b) H 0 : p 1 = p 2, H a : p 1 > p 2 c) H 0 : p 1 = p 2, H a : p 1 < p 2

21. Statistics: Unlocking the Power of Data Lock 5 Calculate z-statistic null value = 0 What to use for p 1 and p 2 ??? Have to assume null is true…

22. Statistics: Unlocking the Power of Data Lock 5 Null Values Testing a difference in proportions: H 0 : p 1 = p 2 Use the overall sample proportion from both groups (called the pooled proportion) as an estimate for both p 1 and p 2 Note that this is in between the sample proportions for each group.

23. Statistics: Unlocking the Power of Data Lock 5 Standard Error

24. Statistics: Unlocking the Power of Data Lock 5 Standard Error

25. Statistics: Unlocking the Power of Data Lock 5 z-statistic

26. Statistics: Unlocking the Power of Data Lock 5 p-value

27. Statistics: Unlocking the Power of Data Lock 5 p-value

28. Statistics: Unlocking the Power of Data Lock 5 Conclusion Does this provide evidence that hormone replacement therapy increases risk of invasive breast cancer? a) Yes b) No

29. Statistics: Unlocking the Power of Data Lock 5 Your Turn! Same trial, different variable of interest. 8506 women were randomized to take HRT, 8102 were randomized to placebo. 502 HRT and 458 placebo women developed any kind of cancer. Does hormone replacement therapy cause increased risk of cancer in general? a) Yes b) No

30. Statistics: Unlocking the Power of Data Lock 5 Margin of Error For a single proportion, what is the margin of error? a) b) c)

31. Statistics: Unlocking the Power of Data Lock 5 Margin of Error You can choose your sample size in advance, depending on your desired margin of error! Given this formula for margin of error, solve for n.

32. Statistics: Unlocking the Power of Data Lock 5 Margin of Error

33. Statistics: Unlocking the Power of Data Lock 5 Margin of Error Suppose we want to estimate a proportion with a margin of error of 0.03 with 95% confidence. How large a sample size do we need? (a)About 100 (b)About 500 (c)About 1000 (d) About 5000

34. Statistics: Unlocking the Power of Data Lock 5 To Do Read Sections 6.1, 6.2, 6.3, 6.7, 6.8, 6.9 Do HW 6a (due Friday, 4/10)