1. Introduction to Biostatistics II Jane L. Meza, Ph.D. October 24, 2005

2. Outline Hypothesis testing  Comparing 2 groups Paired t-test 2 Independent Samples t-test Wilcoxon Signed Ranks test Wilcoxon Rank Sum test  Comparing 3 or more groups ANOVA  One-Way  Bonferroni Comparisons  Repeated Measures  Kruskal-Wallis Chi-square Regression  Linear Correlation  Linear Regression

3. Deck of Cards If you randomly select a card, what is the probability the card is red? If we draw 10 cards, how many of the 10 cards do we expect to be red? Are we guaranteed that 5 of the cards will be red?

4. Deck of Cards Experiment Suppose we draw 10 cards from a deck of 52 cards, and all 10 cards are red. Is it possible that we could draw 10 red cards from a standard deck of cards? Is it very likely that we could draw 10 red cards from a standard deck of cards? We have conflicting information – we assumed that 50% of the cards were red, but in our sample 100% of the cards were red. What should we conclude?

5. Experiment Why did you make that conclusion? What assumptions are you making? Is there a possibility that your conclusion is incorrect?

6. Hypothesis Testing Start with an assumption (Null Hypothesis)  50% of the cards are red Gather data  Draw 10 cards

7. Hypothesis Testing Find the probability of the results under your assumptions  Find the probability of drawing 10 red cards, assuming that 50% of the 52 cards are red.  Probability of drawing 10 cards in a row is highly unlikely if 50% of the 52 cards are red (<0.001).

8. Hypothesis Testing State your conclusion.  Either we experienced a rare event, or one of our assumptions is incorrect.  Since the probability of drawing 10 red cards is small, we conclude that our assumptions are probably incorrect.  We conclude that more than 50% of the cards are red.

9. Hypothesis Testing Example: Is There a Difference? Compare treatments or groups Psoriasis Example:  Some studies have suggested that psoriasis is more common among heavy alcohol drinkers.  Case-control study of men age 19-50. Cases were men who had psoriasis. Controls were men who did not have psoriasis.  All subjects completed questionnaires regarding life style and alcohol consumption.  Is the mean alcohol intake for men with psoriasis (cases) greater than men without psoriasis (controls)?  Cases: mean=43, SD=85.8, n=142  Controls: mean=21, SD=34.2, n=265 Poikolainen et al Br Med J 1990; 300:780-783

10. Hypothesis Testing: Is There a Difference? Null Hypothesis: H O  Often a statement of no treatment effect  Example 1: The proportion of red cards is the same as the proportion of black cards (50%).  Example 2: There is no association between alcohol intake and psoriasis. In other words, the mean alcohol intake for men with psoriasis is the same as the mean alcohol intake for men without psoriasis.

11. Hypothesis Testing: Is There a Difference? Alternative Hypothesis: H A  May be one-sided or two-sided  Example 1: One-sided: The proportion of red cards is larger than the proportion of black cards. Two-sided: The proportion of red cards is different than the proportion of black cards.  Example 2: One-sided: Mean alcohol intake for cases (with psoriasis) is larger than mean alcohol intake for controls (without psoriasis) Two-sided: Mean alcohol intake for cases is different than the mean alcohol intake for controls

12. Hypothesis Testing: Conclusions The null hypothesis is assumed true until evidence suggests otherwise. 2 possible conclusions:  Reject the null hypothesis in favor of the alternative.  Do not reject the null hypothesis.

13. Hypothesis Testing: Errors Significance level:   Probability of rejecting a true null hypothesis   Probability of not rejecting a false null hypothesis Power: 1-   Probability of detecting a true difference Type I Error (  ) Type II Error (  ) Correct Decision DECISION Reject H O Do not Reject H O TRUTH H O is False H O is True

14. Hypothesis Testing: Steps Assume the null hypothesis is true. Determine a test statistic based on the observed data. Using the test statistic, how likely is it that we observe the outcome or something more extreme if the null hypothesis is true? If the test statistic is unlikely under the null hypothesis, we reject the null hypothesis in favor of the alternative hypothesis.

15. Hypothesis Testing: P-value Measures how likely is it that we observe the outcome or something more extreme, assuming the null hypothesis is true. Small p-value is evidence against the null hypothesis and we reject the null hypothesis. Large p-value suggests the data are likely if the null hypothesis is true and we do not reject the null hypothesis.

16. Hypothesis Testing: P-value Method If p < , Reject the null in favor of the alternative hypothesis. If p >= , Do Not Reject the null hypothesis. p <.05 is generally considered statistically significant. Determining the p-value requires making assumptions about the data.

17. Hypothesis Testing: Psoriasis Example Ho: There is no association between alcohol intake and psoriasis. Ha: The mean alcohol intake is different for cases and controls. Using the test statistic, the p-value was 0.004. Conclusion: Since the p-value is less than 0.05, Reject Ho. There is evidence that the mean alcohol intake is higher for cases (mean=43) than controls (mean=21).

18. Hypothesis Testing: Antihypertensive Example Aim: Compare two antihypertensive strategies for lowering blood pressure  Double-blind, randomized study  Enalapril + Felodipine vs. Enalapril  6-week treatment period  217 patients Outcome of interest: diastolic blood pressure Based on AJH, 1999;12:691-696.

19. Hypothesis Testing: Antihypertensive Example After 6 weeks of therapy, the average change in DBP was: 10.6 mm Hg in the Enalapril + Felodipine group (n=109, SD=8.1) compared to 7.4 mm Hg in the Enalapril group (n=108, SD=6.9) The authors used a hypothesis test to help determine which therapy was more effective.

20. Hypothesis Testing: Antihypertensive Example Statement from AJH  “The group randomized to 5 mg enalapril + 5 mg felodipine had a significantly greater reduction in trough DBP after 6 weeks of blinded therapy (10.6 mm Gh) than the group randomized to 10 mg enalapril (7.4 mm Hg, P<0.01).”  What does P<0.01 mean? Assuming that the 2 therapies are equally effective, there is less than a 1% chance that we would have observed treatment differences as large or larger than what was observed.

21. Hypothesis Testing Parametric methods make assumptions about the distribution of the observations. Non-parametric methods do not make assumptions about the distribution of the observations. The distribution of the data and the design of the study should be carefully considered when choosing the statistical test to be used.

22. Comparing 2 Groups - Continuous Data Paired Data For each observation in the first group, there is a corresponding observation in the second group. Example: “Before and After” Pairing eliminates some of the variability among individuals, since measurements are made on the same (or similar) subjects. Paired groups are called dependent.

23. Comparing 2 Groups - Continuous Data Paired t-test Two paired groups Sample size is large (30 or more pairs)

24. Normal Distribution Data follows a normal distribution if the histogram is approximately symmetric and bell shaped. Described by two parameters  mean (  )  SD (  )

25. Normal Distribution Z-score measures how many SDs an observation is away from the mean Z=(x-  )/  About 95% of the values fall within 2 SDs of the mean

26. Comparing 2 Groups - Continuous Data Paired t-test Example In 40 subjects, blood pressure was measured before and after taking Captopril. Outcome of interest: change in blood pressure after taking the drug H O : No association between Captopril and blood pressure. H A : Mean blood pressure is lower after patients take Captopril. P-value < 0.001. Reject H O in favor of H A. There is evidence that mean blood pressure is lower after taking Captopril.  Based on MacGregor et al., British Medical Journal, Vol. 2

27. Comparing 2 Groups - Continuous Data Wilcoxon Signed Ranks Test Two paired groups Sample size is small (less than 30 pairs). Wilcoxon Signed Ranks Test compares medians rather than means. Non-parametric test.

28. Comparing 2 Groups - Continuous Data Wilcoxon Signed Ranks Test Example In 10 postcoronary patients, maximum oxygen uptake was measured before and after a 6 month exercise program. Outcome of interest: change in oxygen uptake after a 6 month exercise program

29. Comparing 2 Groups - Continuous Data Wilcoxon Signed Ranks Test Example H O : There is no association between exercise and oxygen uptake. H A : Median oxygen uptake is higher after exercise program. p-value =.09. Do not reject H O. There is not enough evidence to conclude that oxygen uptake is higher after the exercise program.

30. Comparing 2 Groups - Continuous Data Independent Samples t-test Two independent groups Sample size is large (30 or more in each group).

31. Comparing 2 Groups - Continuous Data Independent Samples t-test Example 30 women with pregnancy-induced hypertension are given low-dose aspirin 42 women with pregnancy-induced hypertension given a placebo Outcome of interest: blood pressure  Based on Schiff, E et al., Obstetrics and Gynecology, Vol 76, Nov 1990, 742-744.

32. Comparing 2 Groups - Continuous Data Independent Samples t-test Example H O : No association between low- dose aspirin and blood pressure. H A : Mean blood pressure is lower for the aspirin group P-value = 0.15. Do not reject H O. There is not enough evidence to conclude that the mean blood pressure is lower for the aspirin group.

33. Comparing 2 Groups - Continuous Data Wilcoxon Rank Sum Test Two independent groups Sample size is small (less than 30). Wilcoxon Rank Sum Test compares medians rather than means Nonparametric test

34. Comparing 2 Groups - Continuous Data Wilcoxon Rank Sum Test Example 13 patients randomized to placebo 15 randomized to receive calcium supplements Outcome of interest: blood pressure H O : No association between calcium supplements and blood pressure. H A : Median blood pressure in calcium supplement group is different than placebo group. P-value =.79. Do not reject H O. There is not enough evidence to conclude that median blood pressure for the calcium group is different than the placebo group.  Based on Lyle et al., JAMA, Vol 257, No 13.

35. Comparing 3 or more groups Chi-square Test for categorical data Analysis of Variance (ANOVA) for continuous data Common uses:  Compare an outcome for 3 or more treatments  Compare a characteristic in 3 or more populations

36. Chi-Square Test Compare 2 or more groups Categorical data Example: To study effectiveness of bicycle helmets, individuals who were in an accident were studied. Outcome of interest: Compare proportion of persons suffering a head injury while wearing a helmet to proportion of persons suffering a head injury while not wearing helmet

37. Chi-Square Test 2x2 Table 12% (17/147) of those wearing a helmet had a head injury 34% (218/646) of those not wearing a helmet had a head injury Wearing Helmet InjuryYesNo Yes No 17 (12%) 130 (88%) 218 (34%) 428 (66%) Total147646

38. Chi-Square Test Ho: The proportion suffering a head injury is the same for accident victims who wore helmets vs. accident victims who did not wear helmets. Ha: The proportion suffering a head injury is different for accident victims who wore helmets vs. accident victims who did not wear helmets. p-value < 0.001 Conclusion: Reject Ho. The proportion of individuals suffering head injuries was higher for accident victims who did not wear helmets (34%) compared to those who did wear helmets (12%). Among persons in an accident, wearing a helmet appears to lower incidence of head injury.

39. ANOVA (Analysis of variance) Used to compare a continuous variable among three or more groups H O : The group (or treatment) means are the same. H A : At least one mean is different from the others.

40. One-Way ANOVA One factor (characteristic) is being studied  Example: treatment group Placebo experimental treatment 1 experimental treatment 2 3 or more independent groups The distribution for each group is not heavily skewed. Group variances or sample sizes are approximately equal.

41. One-Way ANOVA Example Aim: Compare microbiological growth under 3 different CO 2 pressure levels. Factor of interest: 3 different CO 2 pressure levels Outcome of interest: average microbiological growth in each treatment group H O : The mean microbiological growth for the 3 treatments (CO 2 level) is the same H A : At least one of the means is different. p-value =.001 Reject H O in favor of H A. There is evidence that mean growth is different for the three treatment groups.

42. One-Way ANOVA Example Mean microbiological growth under 3 different CO 2 pressure levels.  Group 1 mean: 56.2  Group 2 mean : 22.5  Group 3 mean: 26.1

43. Bonferroni Comparisons Use when ANOVA yields a significant p-value. If we perform several t-tests to compare each pair of means, the probability of a Type I error is > 0.05. The Bonferroni method modifies the p-value to account for multiple comparisons so that, overall, the probability of making a Type I error is 0.05.

44. Bonferroni Comparisons Example Is the mean for group 1 different from the mean for group 2?  P=.001  Conclusion: The mean for group 1 is different from the mean for group 2. Is the mean for group 1 different from the mean for group 3?  P=.02  Conclusion: The mean for group 1 is different from the mean for group 3. Is the mean for group 2 different from the mean for group 3?  P=.34  Conclusion: The mean for group 2 is different from the mean for group 3. Therefore, the difference in the 3 group means can primarily be explained by the higher mean for group 1 compared to groups 2 and 3.

45. Repeated Measures ANOVA Subjects are measured at more than one time point Since multiple measurements are taken for the same subject over time, the observations are not independent

46. Repeated Measures ANOVA Example 12 rabbits receive in random order 3 different dose levels of a drug to increase blood pressure, with a washout period between treatments. Outcome of interest: average blood pressure for the three dose levels H O : Average blood pressure is the same for the 3 dose levels H A : At least one of the means is different. P=0.01 Reject H O. There is evidence of a difference in mean blood pressure for the 3 dose levels.

47. Kruskal-Wallis ANOVA Nonparametric ANOVA Use when the distribution for one or more groups is heavily skewed.

48. Linear Regression Is there a linear relation between 2 continuous variables? If so, what line best fits the data? Use the line to predict a value for a new observation  Example: Can we predict muscle based on a woman’s age? Explore relationship between 2 numerical variables  Example: What is the relation between muscle mass and age?

49. Linear Correlation (r) Is There an Association? Measures linear relationship between 2 continuous variables. Interpreting r : Absolute Value Linear of r Relationship 0 -.25poor.25 -.50fair.50 -.75good.75 – 1.0very good

50. Linear Correlation (r) Examples r =.55r = 0 r =.85 r = -.85

51. Linear Correlation (r) Examples r = 1 r = -1

52. Linear Regression Least Squares Regression Line Estimate the best line to fit the data Y = b 0 + b 1 X  Y is the dependent variable Example: Muscle mass  X is the independent variable Example: Age of woman  b 0 is the intercept  b 1 is the slope

53. Linear Regression Example Predict the muscle mass of a 60 year old woman  148 - 60 = 80

54. Linear Regression Example On average, what is the difference in muscle mass for women who differ in age by 1 year?  b 1 = -1  For women whose age differs by one year, we expect the average muscle mass will be one unit lower for the older women

55. Linear Regression Notes Significant correlation does not necessarily imply causation. Do not use a line to predict new observations if there is not significant linear correlation. When predicting new observations, stay within the domain of the sample data.

56. References Dawson-Saunders, B and Trapp RG (1994). Basic and Clinical Biostatistics. Appleton and Lange. Norwalk, CT. Lane, DM. (2000). Hyperstat Online. On-line text, www.statistics.com.www.statistics.com MacGregor GA, Markandu ND, Roulston JE and Jones JC (1979). “ Essential Hypertension: Effect of an Oral Inhibitor of Angiotensin-Converting Enzyme”. British Medical Journal, Nov 3; Vol 2, 1106-9. Neter, J., Wasserman W. and Kutner, MH. (1990). Applied Linear Statistical Models. Irwin. Burr Ridge, IL. Pagano M and Gauvreau, K. (1993). Principles of Biostatistics. Duxbury Press. Belmont, CA. Schiff E, Barkai G, Ben-Baruch G and Mashiach S. (1990). “Low-Dose Aspirin Does Not Influence the Clinical Course of Women with Mild Pregnancy-Induced Hypertension”. Obstetrics and Gynecology, Vol 76, November, 742-744. Swinscow, TDV. (1997). Statistics at Square One. BMJ Publishing Group. On-line text, www.statistics.com. Triola MF (1998), Elementary Statistics. Addison-Wesley. Reading, MS.