1. ASHOK THAKKAR© 2002 Prentice-Hall, Inc.Chap 9-1 Basic Statistics Fundamentals of Hypothesis Testing: One-Sample, Two-Sample Tests PREPARED BY: ASHOK THAKKAR Mail id: ashokmba2000@yahoo.co.in

2. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-2 What is biostatistics Statistics is the science and art of collecting, summarizing, and analyzing data that are subject to random variation. Biostatistics is the application of statistics and mathematical methods to the design and analysis of health, biomedical, and biological studies.

3. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-3 Different Tests of Significance 1. One-Sample z-test or t-test a. Compares one sample mean versus a population mean 2. Two-Sample t-test a. Compares one sample mean versus another sample mean a. Independent t-tests (equal samples) b. Dependent t-tests (dependent/paired samples) 3. One-way analysis of variance (ANOVA) a. Comparing several sample means

4. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-4 How to properly use Biostatistics Develop an underlying question of interest Generate a hypothesis Design a study (Protocol) Collect Data Analyze Data Descriptive statistics Statistical Inference

5. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-5 Relationship between population and sample (Simple random sampling)

6. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-6 Sampling Techniques Population Simple Random Sample Systematic Sampling Stratified Random Sample Convenience Sampling Cluster Sampling Bias free sample Bias free sample Biased sample Bias free sample Biased sample

7. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-7 Example How are my 10 patients doing after I put them on an anti-hypertensive medications? Describe the results of your 10 patients

8. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-8 Example What is the in hospital mortality rate after open heart surgery at SAL hospital so far this year Describe the mortality What is the in hospital mortality after open heart surgery likely to be this year, given results from last year Estimate probability of death for patients like those seen in the previous year.

9. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-9 Misuse of statistics About 25% of biological research is flawed because of incorrect conclusions drawn from confounded experimental designs and misuse of statistical methods

10. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-10 What is a Hypothesis? A hypothesis is a claim (assumption) about the population parameter Difference between the value of sample statistic and the corresponding hypothesized parameter value is called hypothesis testing. I claim that mean CVD in the INDIA is atleast 3! © 1984-1994 T/Maker Co.

11. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-11 Hypothesis Testing Process Identify the Population Assume the population mean age is 50. ( ) REJECT Take a Sample Null Hypothesis No, not likely!

12. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-12 Sampling Distribution of = 50 It is unlikely that we would get a sample mean of this value...... Therefore, we reject the null hypothesis that m = 50. Reason for Rejecting H 0  20 If H 0 is true... if in fact this were the population mean.

13. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-13 Biostatistics Descriptive Statistical Inference Estimation Confidence Intervals Hypothesis Testing P-values Components of Biostatistics

14. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-14 Normal Distribution

15. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-15 Normal distribution bell-shaped symmetrical about the mean (No skewness) total area under curve = 1 approximately 68% of distribution is within one standard deviation of the mean approximately 95% of distribution is within two standard deviations of the mean approximately 99.7% of distribution is within 3 standard deviations of the mean Mean = Median = Mode

16. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-16 Empirical Rule About 95% of the area lies within 2 standard deviations About 99.7% of the area lies within 3 standard deviations of the mean 68% About 68% of the area lies within 1 standard deviation of the mean

17. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-17

18. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-18 Level of Significance, Is designated by, (level of significance) Typical values are.01,.05,.10 Is selected by the researcher at the beginning Provides the critical value(s) of the test

19. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-19 The z-Test for Comparing Population Means Critical values for standard normal distribution

20. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-20 Level of Significance and the Rejection Region H 0 :   3 H 1 :  < 3 0 0 0 H 0 :   3 H 1 :  > 3 H 0 :   3 H 1 :   3    /2 Critical Value(s) Rejection Regions I claim that mean CVD in the INDIA is atleast 3!

21. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-21 Hypothesis Testing 1. State the research question. 2. State the statistical hypothesis. 3. Set decision rule. 4. Calculate the test statistic. 5. Decide if result is significant. 6. Interpret result as it relates to your research question.

22. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-22 Rejection & Nonrejection Regions = I claim that mean CVD in the INDIA is atleast 3!

23. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-23 The Null Hypothesis, H 0 States the assumption (numerical) to be tested e.g.: The average number of CVD in INDIA is at least three ( ) Is always about a population parameter ( ), not about a sample statistic ( )

24. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-24 The Null Hypothesis, H 0 Begins with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty (continued)

25. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-25 The Alternative Hypothesis, H 1 Is the opposite of the null hypothesis e.g.: The average number of CVD in INDIA is less than 3 ( ) Never contains the “=” sign May or may not be accepted

26. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-26 General Steps in Hypothesis Testing e.g.: Test the assumption that the true mean number of of CVD in INDIA is at least three ( Known) 1.State the H 0 2.State the H 1 3.Choose 4.Choose n 5.Choose Test

27. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-27 100 persons surveyed Computed test stat =-2, p-value =.0228 Reject null hypothesis The true mean number of CVD is less than 3 in human population. (continued) Reject H 0  -1.645 Z 6.Set up critical value(s) 7. Collect data 8. Compute test statistic and p-value 9. Make statistical decision 10. Express conclusion General Steps in Hypothesis Testing

28. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-28 The z-Test for Comparing Population Means Critical values for standard normal distribution

29. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-29 p-Value Approach to Testing Convert Sample Statistic (e.g. ) to Test Statistic (e.g. Z, t or F –statistic) Obtain the p-value from a table or computer Compare the p-value with If p-value, do not reject H 0 If p-value, reject H 0

30. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-30 Comparison of Critical-Value & P-Value Approaches

31. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-31 Result Probabilities H 0 : Innocent The Truth Verdict InnocentGuilty Decision H 0 TrueH 0 False Innocent CorrectError Do Not Reject H 0 1 -  Type II Error (  ) Guilty Error Correct Reject H 0 Type I Error (  ) Power (1 -  ) Jury Trial Hypothesis Test

32. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-32 Type I & II Errors Have an Inverse Relationship   If you reduce the probability of one error, the other one increases so that everything else is unchanged.

33. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-33 Critical Values Approach to Testing Convert sample statistic (e.g.: ) to test statistic (e.g.: Z, t or F –statistic) Obtain critical value(s) for a specified from a table or computer If the test statistic falls in the critical region, reject H 0 Otherwise do not reject H 0

34. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-34 One-tail Z Test for Mean ( Known) Assumptions Population is normally distributed If not normal, requires large samples Null hypothesis has or sign only Z test statistic

35. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-35 Rejection Region Z 0 Reject H 0 Z 0 H 0 :  0 H 1 :  <  0 H 0 :  0 H 1 :  >  0 Z Must Be Significantly Below 0 to reject H 0 Small values of Z don’t contradict H 0 Don’t Reject H 0 !

36. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-36 Example: One Tail Test Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified  to be 15 grams. Test at the  0.05 level. 368 gm. H 0 :  368 H 1 :  > 368

37. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-37 Finding Critical Value: One Tail Z.04.06 1.6.9495.9505.9515 1.7.9591.9599.9608 1.8.9671.9678.9686.9738.9750 Z 0 1.645. 05 1.9.9744 Standardized Cumulative Normal Distribution Table (Portion) What is Z given  = 0.05?  =.05 Critical Value = 1.645.95

38. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-38 Example Solution: One Tail Test  = 0.5 n = 25 Critical Value: 1.645 Test Statistic: Decision: Conclusion: Do Not Reject at  =.05 No evidence that true mean is more than 368 Z 0 1.645.05 Reject H 0 :  368 H 1 :  > 368 1.50

39. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-39 p -Value Solution Z 0 1.50 P-Value =.0668 Z Value of Sample Statistic From Z Table: Lookup 1.50 to Obtain.9332 Use the alternative hypothesis to find the direction of the rejection region. 1.0000 -.9332.0668 p-Value is P(Z  1.50) = 0.0668

40. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-40 p -Value Solution (continued) 0 1.50 Z Reject (p-Value = 0.0668)  (  = 0.05) Do Not Reject. p Value = 0.0668  = 0.05 Test Statistic 1.50 is in the Do Not Reject Region 1.645

41. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-41 Example: Two-Tail Test Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified  to be 15 grams. Test at the  0.05 level. 368 gm. H 0 :  368 H 1 :  368

42. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-42  = 0.05 n = 25 Critical Value: ±1.96 Example Solution: Two-Tail Test Test Statistic: Decision: Conclusion: Do Not Reject at  =.05 No Evidence that True Mean is Not 368 Z 0 1.96.025 Reject -1.96.025 H 0 :  368 H 1 :  368 1.50

43. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-43 p-Value Solution (p Value = 0.1336)  (  = 0.05) Do Not Reject. 0 1.50 Z Reject  = 0.05 1.96 p Value = 2 x 0.0668 Test Statistic 1.50 is in the Do Not Reject Region Reject

44. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-44 Connection to Confidence Intervals

45. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-45 What is a t Test? Commonly Used Definition: Comparing two means to see if they are significantly different from each other Technical Definition: Any statistical test that uses the t family of distributions

46. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-46 Independent Samples t Test Use this test when you want to compare the means of two independent samples on a given variable “Independent” means that the members of one sample do not include, and are not matched with, members of the other sample Example: Compare the average height of 50 randomly selected men to that of 50 randomly selected women Compare using t test Independent Mean #1 #2

47. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-47 Dependent Samples t Test Used to compare the means of a single sample or of two matched or paired samples Example: If a group of students took a math test in March and that same group of students took the same math test two months later in May, we could compare their average scores on the two test dates using a dependent samples t test

48. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-48 Comparing the Two t Tests Independent Samples Tests the equality of the means from two independent groups (diagram below) Relies on the t distribution to produce the probabilities used to test statistical significance Dependent Samples Tests the equality of the means between related groups or of two variables within the same group (diagram below) Relies on the t distribution to produce the probabilities used to test statistical significance Before treatment Person #1 After treatment Person #1 Treatment group Person #1 Control group Person #2

49. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-49 Types One sample compare with population Unpaired compare with control Paired same subjects: pre-post Z-test large samples >30

50. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-50 Compare Means (or medians)Example: Compare blood presures of two or more groups, or compare BP of one group with a theoretical value. 1 Group: 1. One Sample t test 2. Wilcoxon rank sum test 2 Groups: 1. Unpaired t test 2. Paired t test 3. Mann-Whitney t test 4. Welch’s corrected t test 5. Wilcoxon matched pairs test

51. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-51 3-26 Groups: 1. One-way ANOVA 2. Repeated measures ANOVA 3. Kruskal-Wallis test 4. Friedman test (All with post tests) Raw data Average data Mean, SD, & NAverage data Mean, SEM, & N

52. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-52 Is there a difference? between you…means, who is meaner?

53. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-53 Statistical Analysis control group mean treatment group mean Is there a difference? Slide downloaded from the Internet

54. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-54 What does difference mean? medium variability high variability low variability The mean difference is the same for all three cases Slide downloaded from the Internet

55. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-55 What does difference mean? medium variability high variability low variability Which one shows the greatest difference? Slide downloaded from the Internet

56. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-56 t Test: Unknown Assumption Population is normally distributed If not normal, requires a large sample T test statistic with n-1 degrees of freedom

57. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-57 Example: One-Tail t Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and  s  15. Test at the  0.01 level. 368 gm. H 0 :  368 H 1 :  368  is not given

58. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-58 Example Solution: One-Tail  = 0.01 n = 36, df = 35 Critical Value: 2.4377 Test Statistic: Decision: Conclusion: Do Not Reject at  =.01 No evidence that true mean is more than 368 t 35 0 2.437 7.01 Reject H 0 :  368 H 1 :  368 1.80

59. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-59 The t Table Since it takes into account the changing shape of the distribution as n increases, there is a separate curve for each sample size (or degrees of freedom). However, there is not enough space in the table to put all of the different probabilities corresponding to each possible t score. The t table lists commonly used critical regions (at popular alpha levels).

60. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-60 Z-distribution versus t-distribution

61. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-61 The z-Test for Comparing Population Means Critical values for standard normal distribution

62. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-62 Summary We can use the z distribution for testing hypotheses involving one or two independent samples To use z, the samples are independent and normally distributed The sample size must be greater than 30 Population parameters must be known

63. ASHOK THAKKAR© 2002 Prentice-Hall, Inc. Chap 9-63