1. 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. 2 No human investigation can be called true science without passing through mathematical tests. Leonardo da Vinci (1452-1519) (in Treatise on Painting) Testing

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

4. 4 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 σ2σ2

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

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

7. 7 IF95% C.I.  200 mg/100ml (182,218) Population mean = 211?

8. 8 IF95% C.I.  200 mg/100ml (182,218)  190 mg/100ml (172,208) Population mean = 211?

9. 9 IF95% C.I.  200 mg/100ml (182,218)  190 mg/100ml (172,208)  175 mg/100ml (157,193) Population mean = 211?

10. 10 Alternatively IF m= 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. If true

11. 11 Hypothesis Testing Hypothesis Testing Trial by jury

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

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

14. 14 Individual on trial. Is he/she innocent? Evidence Trial Jury Person InnocentGuilty Not Guilty Guilty Hypothesis Testing & Trial by jury

15. 15 Individual on trial. Is he/she innocent? Evidence Trial Jury Person InnocentGuilty Not Guilty  Guilty  Hypothesis Testing & Trial by jury

16. 16 Individual on trial. Is he/she innocent? Evidence Trial Jury Person InnocentGuilty Not Guilty  x Guiltyx  Hypothesis Testing & Trial by jury

17. 17 Evidence Trial Jury Person InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? EvidenceTrial Hypothesis Testing

18. 18 Trial Jury Person InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? SampleTrial Hypothesis Testing

19. 19 Jury Person InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

20. 20 Jury Population InnocentGuilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

21. 21 Jury Population  =  0 Guilty Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

22. 22 Jury Population  =  0   0  0 Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

23. 23 Us Population  =  0   0  0 Not Guilty  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

24. 24 Us Population  =  0   0  0 Not reject  x Guiltyx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

25. 25 Us Population  =  0   0  0 Not reject  x Rejectx  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

26. 26 Us Population  =  0   0  0 Not reject  Type II RejectType I  Test of Hypothesis that  =  0 ? Sample Analysis Hypothesis Testing

27. 27 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. Possible errors in analysis results

28. 28 1o1o Hypothesize a value (  0 ) 2o2o Take a random sample (n). 3o3o Is it likely that the sample came from a population with mean  0 (  = 0.05) ? Hypothesis testing about  :

29. 29 We know that cholesterol levels in US men 20-74 yrs are normally distributed with σ X  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? 2 sided hypothesis test - Illustration H 0 :  = 211 H A :   211

30. 30 H 0 :  = 211 H A :   211  = 46 mg/100ml 12 hypertensive smokers have: 2 sided hypothesis test - Illustration

31. 31 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?” P-value Question: Do hypertensive smokers have the same mean as the general population?

32. 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. 33 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?” P-value Answer: Do not reject the null hypothesis. No evidence that hypertensive smokers have a different mean than general population

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

35. 35 Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities: 1. Two-sided H 0 :  =  0 H A :    0 Alternative hypothesis

36. 36 Blood glucose level of healthy persons has  = 9.7 mmol/L and  = 2.0 mmol/L H 0 :   9.7 H A :  > 9.7 Sample of 64 diabetics yields Example - One-sided alternative Do diabetics have blood glucose levels that are higher on average when compared to the general population?

37. 37 Blood glucose level of healthy persons has  = 9.7 mmol/L and  = 2.0 mmol/L H 0 :   9.7 H A :  > 9.7 n = 64 p-value << 0.001 Example - One-sided alternative Answer: Reject the null hypothesis. Significant evidence that diabetics have a higher mean level of glucose when compared to the general population

38. 38 Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities: Two-sided H 0 :  =  0 H A :    0 One-sided H 0 :    0 H A :  <  0 One-sided H 0 :    0 H A :  >  0 Alternative hypothesis

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