Introduction to Biostatistics (BIO/EPI 540) Lecture 11: Hypothesis Testing

1. Introduction to Biostatistics(BIO/EPI 540) Lecture 11: Hypothesis Testing Acknowledgement: Thanks to Professor Pagano (Harvard School of Public Health) for lecture material

2. Testing No human investigation can be called true science without passing through mathematical tests. Leonardo da Vinci (1452-1519) (in Treatise on Painting)

3. Sampling Paradigm Inference μ, σ Population ,S Sample

4. Inference • Sample mean is an estimate of • Sample variance (S) is an estimate • of • Confidence intervals and • hypothesis tests are equivalent • techniques to quantify uncertainty • in sample derived inferences • regarding population parameters μ σ2

5. Confidence Interval - Illustration We know that cholesterol levels in US men 20-24 yrs are normally distributed with σX 46 mg/100ml. We obtain a sample of n=25 and want to infer μ.

6. Use of C.I. to infer value value of μ

7. Population mean = 211?

8. Population mean = 211?

9. Population mean = 211?

10. If true • Alternatively IF • = 211 and  = 46 and we take a sample of size n=25 from this pop., then the Central Limit Theorem says that the sample mean is approx. normal with mean  = 211 and std. dev. 46/5; i.e.

11. Hypothesis Testing Hypothesis TestingTrial by jury

12. Hypothesis Testing & Trial by jury Individual on trial. Is he/she innocent? Evidence Trial

13. Hypothesis Testing & Trial by jury Individual on trial. Is he/she innocent? Evidence Trial

14. Hypothesis Testing & Trial by jury Individual on trial. Is he/she innocent? Evidence Trial

15. Hypothesis Testing & Trial by jury Individual on trial. Is he/she innocent? Evidence Trial

16. Hypothesis Testing & Trial by jury Individual on trial. Is he/she innocent? Evidence Trial

17. Hypothesis Testing Test of Hypothesis that  = 0? Evidence Trial Evidence Trial

18. Hypothesis Testing Test of Hypothesis that  = 0? Sample Trial Trial

19. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

20. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

21. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

22. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

23. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

24. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

25. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

26. Hypothesis Testing Test of Hypothesis that  = 0? Analysis Sample

27. Possible errors in analysis results Probability of Type I error is  i.e. the probability of rejecting the null hypothesis when it is true. Probability of Type II error is  i.e the probability of not rejecting the null hypothesis when it is false. 1- is the power of the test.

28. Hypothesis testing about  :

29. 2 sided hypothesis test -Illustration We know that cholesterol levels in US men 20-74 yrs are normally distributed with σX 46 mg/100ml and μ = 211. We obtain a random sample of 12 hypertensive smokers and obtain a sample mean of 217 mg/100ml. We want to test whether their population mean is the same as that of the general population?

30. 2 sided hypothesis test -Illustration  = 46 mg/100ml 12 hypertensive smokers have:

31. P-value Some prefer to quote the p-value. The p-value answers the question, “What is the probability of get- ting as large, or larger, a Discrepancy given the null hypothesis is true?” Question: Do hypertensive smokers have the same mean as the general population?

32. Rejecting the null hypothesis • Assume a specific threshold of Type I error, α • Typically α = 0.05 • If p value < α Reject null

33. P-value Some prefer to quote the p-value. The p-value answers the question, “What is the probability of get- ting as large, or larger, a Discrepancy given the null hypothesis is true?” Answer: Do not reject the null hypothesis. No evidence that hypertensive smokers have a different mean than general population

34. Summary Decide on statistic: Determine which values of are consonant with the hypothesis that  = 0 and which ones are not. Look at and decide.

35. Alternative hypothesis Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities:

36. Example - One-sided alternative Blood glucose level of healthy persons has  = 9.7 mmol/L and  = 2.0 mmol/L Sample of 64 diabetics yields Do diabetics have blood glucose levels that are higher on average when compared to the general population?

37. Example - One-sided alternative Blood glucose level of healthy persons has  = 9.7 mmol/L and  = 2.0 mmol/L n = 64 p-value << 0.001 Answer: Reject the null hypothesis. Significant evidence that diabetics have a higher mean level of glucose when compared to the general population

38. Alternative hypothesis Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities:

39. Summary • Hypothesis testing: • Type I and II errors • Power • Two sided hypothesis test • One sided hypothesis test