# STAT 250 Dr Kari Lock Morgan Inference for

• Slides: 34

STAT 250 Dr. Kari Lock Morgan Inference for Proportions Chapter 6. 1, 6. 2, 6. 3, 6. 7, 6. 8, 6. 9 • Formulas for standard errors • Normal based inference Statistics: Unlocking the Power of Data Lock 5

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

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

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

Standard Error Formulas Parameter Distribution Proportion Standard Error Normal Difference in Proportions Normal Mean t, df = n – 1 Difference in Means t, df = min(n 1, n 2) – 1 Correlation t, df = n – 2 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 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? 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? 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 Statistics: Unlocking the Power of Data Lock 5

z* Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

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

Bootstrap Interval 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 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) b) c) d) (0. 013, 0. 027) (0. 015, 0. 025) (0. 017, 0. 023) (0. 019, 0. 021) 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 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? 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. 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, Ha: p 1≠ p 2 H 0: p 1= p 2, Ha: p 1> p 2 c) H 0: p 1= p 2, Ha: p 1< p 2 b) 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… 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. Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

Standard Error Statistics: Unlocking the Power of Data Lock 5

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

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

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

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

Margin of Error 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 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 6 a (due Friday, 4/10) Statistics: Unlocking the Power of Data Lock 5