1. 1 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing General Idea How unusual is the result? Test statistics Type I error (alpha level) p-value Type II error (beta level) –Power=1-beta

2. 2 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea Total cholesterol (mg/dl) is measured on a simple random sample of 32 women over the age of 60. Is there evidence that mean cholesterol is different in women of this age group as compared with women under age 50? Estimate of TC: (see ejs09b540p36.sas) Plot histogram of SRS of n=32 from women <50. (see ejs09b540p37.sas) 244 Result is very unusual relative to what we’d expect from sampling. Conclude the mean is differnet.

3. 3 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 244 Histogram of distribution of sample means/standardized value- –need to know mean and variance of TC for women < 50. –use Z if variance is known, t if variance is estimated for women < 50:

4. 4 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 244 Plot Histogram of distribution of sample means under Null H or … histogram of standardized values of the difference of the sample mean from the mean TC for women < 50 –need to know mean and variance of TC for women < 50. for women < 50: –use Z if variance is known, t if variance is estimated

5. 5 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 244

6. 6 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea z=8.08

7. 7 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea z=8.3z=8.32

8. 8 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea Is the result ‘unusual’? –Decide a level of ‘unusualness’ – usually set at values so that 5% of time, sample mean would be further away (also called TYPE 1 Error) –If in either direction, then 2.5% on either side, and test is called ‘2-sided’ Called 2-sided test –If unusual is important only in one direction (drug lowers cholesterol), then put all 5% on one side Called 1-sided test –Null hypothesis is ‘usual’ or commonly accepted position. –Alternative hypothesis is what you want to ‘prove’

9. 9 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea Null Hypothesis:

10. 10 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea Alternative Hypothesis

11. 11 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 2-sided test Unusual Critical region

12. 12 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 1-sided test Unusual Critical region

13. 13 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 2-sided test Unusual

14. 14 Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea 1-sided test Unusual

15. 15 Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea 1-sided test Unusual

16. 16 Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea 1-sided test Unusual

17. 17 Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea 1-sided test Unusual 1-sided test

18. 18 Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea 1-sided test Unusual 1-sided test

19. 19 Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea 1-sided test Unusual 1-sided test

20. 20 Introduction to Biostatistics (PUBHLTH 540) Power- example Assume in one population, we know TC for males is normally distributed with mean 220, and variance 1524. Our interest is in mean TC for men in a different population. We would like to know whether TC is less in the other population (vs a null hypothesis that it is equal to or greater than 220). Consider a one sided test of the null hypothesis. Suppose we select a sample of n=25 subjects from the new population. Let us test the null hypothesis that the mean is 220, versus an alternative hypothesis that the mean is 205 based on a one sided test with n=25. What is the power of the test? Figure out the rejection region under the null hypothesis in terms of the distribution of sample means. Make a sketch indicating the critical region (on the scale of TC). Use the z-applet with an assumption that the alternative hypothesis is true to figure the power.

21. 21 Introduction to Biostatistics (PUBHLTH 540) Power- example Figure out the rejection region under the null hypothesis in terms of the distribution of sample means.

22. 22 Introduction to Biostatistics (PUBHLTH 540) Power- example Figure out the rejection region under the null hypothesis in terms of the distribution of sample means.

23. 23 Introduction to Biostatistics (PUBHLTH 540) Power- example Make a sketch.