1. Hypothesis Testing Hypothesis Testing Topic 11

2. Hypothesis Testing Another way of looking at statistical inference in which we want to ask a question like: “Is the mean glucose level of infants small-for-gestational age (SGA) the same as those for normal infants?” i.e.  SGA =  normal ?

3. Hypothesis Testing “Is the mean systolic blood pressure of people taking calcium supplement the same as those in the general population?” i.e.  calcium =  gen ? “Is the mean systolic pulse rate of people taking a stress test the same as those in the control group?” i.e.  stress =  control ?

4. Null and alternative hypothesis Null Hypothesis H 0 is usually a statement of no effect or no difference. The aim of testing is to assess the strength of evidence presented in the data against the null hypothesis. Alternative Hypothesis H 1 is the hypothesis that we hope or suspect to be true instead of H 0 in order to demonstrate the presence of some physical phenomenon or the usefulness of a treatment (e.g., new drug better than old)

5. Asymmetric treatment of H 0 and H 1 H 0 is usually the more conservative hypothesis (no change, no effect, etc) which is assumed to be true until sufficient evidence is obtained to warrant its rejection H 1 is usually the scientifically more interesting hypothesis that we hope or suspect is true (e.g., new drug better than existing drug, smokers more at risk) The burden of proof is placed on H 1. Substantial supporting evidence is required before it will be accepted The consequence of wrongly rejecting H 0 (Type I error) is considered more severe than that of wrongly accepting H 0 (Type II error)

6. Type I and Type II errors Type I error Correct action Type II error No difference (H 0 is correct) Difference exists (H 0 is incorrect) True situation Conclusion from Hypothesis test Do not reject H 0 Reject H 0

7. An analogy

8. Another analogy

9. The 4 steps to hypothesis testing 1. Formulate H o and H 1. 2. Select an appropriate test statistic which is usually some kind of standardized difference between the estimated and hypothesized value or between the expected and observed. 3. Use the null distribution (the sampling distribution of the test statistic under H o ) to calculate the probability due to chance alone of getting a difference larger than or equal to that actually observed in the data. This probability is called the p-value. 4. Draw the appropriate conclusion in the context of the medical problem Note: A small p-value means it is difficult to attribute the observed difference to chance alone and this can be taken as evidence against the null hypothesis of no difference.

10. How small is small?

11. Some people prefer to report the p-value rather than using terms like “accept/reject” the hypothesis at level 0.05. The acceptance/rejection rule is too rigid and exaggerate the difference between, say, p-value = 0.049 and 0.051. More informative to report the p-value. Suppose we are told that H o is rejected at level 0.05, we wouldn’t know whether H o will still be rejected at level 0.01. The p-value gives us the whole story, e.g., if p-value = 0.012, then we know the hypothesis will be rejected at level 0.05, 0.02 but not 0.01.

12. Birthweight example revisited Suppose a random sample of 50 Malay male livebirths gave a sample mean birthweight of 3.55kg and a sample standard deviation of 0.92 kg. Question of interest: What is the likelihood that the mean birth-weight from the sampled population (ie all Malay male livebirths) is the same as the mean birth-weight of all male livebirths in the general population which is known to be 3.27kg, after taking into consideration sampling error?

13. 2. Define and compute test statistic The 4 steps to hypothesis testing 1. Formulate hypotheses Null Hypothesis H o :  =  0 = 3.27 kg Alternative Hypothesis H 1 :  3.27 kg Difference between observed sample mean and hypothesized value = 3.55 - 3.27 = 0.28 kg Test statistic = T = standardized difference =.28/.13 = 2.15 ^

14. Distribution of the standardized difference under the null hypothesis ^

15. Two-sided t test 3. Calculate p-value T = standardized difference Observed T = 2.15 More extreme difference means |T|>2.15. Now H 1 is 2-sided, so p-value = Pr( |T| >= 2.15 | H 0 ) = Pr( | t 49 | >= 2.15) = 0.036 ^

16. 4. Interpretation of results There is only a 3.6% chance of observing a standardized difference of magnitude 2.15 or more if the random sample of 50 Malay males had come from a population with the same mean birthweight as that of the general population. This difference is statistically significant as it cannot be reasonably attributed to chance. Thus there is strong evidence that the mean birthweight for male Malay livebirth is different from the general population.

17. An equivalent way to test H 0 :  =  0 at level 0.05 is to see if  0 is contained in a 95% C.I. for . A 95% C.I. for the mean Malay male birthweight was calculated earlier as (3.29kg, 3.81kg). Since the general population mean 3.27kg is not in this C.I., we reach the same conclusion of rejecting H 0 :  =3.27 at level 0.05. The 95% C.I. (3.29, 3.81) tells us more. It tells us that we can reject  =3.27 but not  =3.3 at level 0.05. In fact, the 95% C.I. consists of all those  0 such that H 0 :  =  0 cannot be rejected at level 0.05 by the 2-sided t-test. Relationship between C.I. and hypothesis testing Recommendation: Report both, p-value and C. I.

18. One-sided vs Two-sided Alternatives One-sided alternative is appropriate when we can anticipate a priori the direction of the difference or when we are primarily interested in detecting difference in one direction only, e.g., mean glucose level lower for SGA infants, new drug better than old. If H 1 is 1-sided, the test should also be 1-sided, i.e., reject H o only for large difference in the same direction as indicated by the alternative. For example, if H 1 :  >  0, then p-value=Pr(T>observed t | H o )=(1/2) p-value for 2-sided test Two-sided alternatives are conventionally used because, most of the time, we are not sure of the direction of the difference.

19. Example of a one-sided test

21. What to do with non-normal data? If sample size is large, the t-test/C.I. is still approximately valid by virtue of the Central Limit Theorem Transform the data so that the transformed data is closer to normally distributed Use nonparametric or distribution- free procedures

22. How large is large? n >= 15 should suffice if the underlying distribution is not too skewed and there are no outliers A smaller sample size may suffice if the data are closed to normally distributed n >= 40 should suffice even for fairly skewed populations