Clinical Statistics for Non-Statisticians – Part II

1. Clinical Statistics for Non-Statisticians – Part II Kay M. Larholt, Sc.D. Vice President, Biometrics & Clinical Operations Abt Bio-Pharma Solutions

2. Topics • Review of Statistical Concepts • Hypothesis Testing • Power and Sample Size • Interim Analysis

3. Basic Statistical Concepts

4. Statistics Per the American Heritage dictionary - “The mathematics of the collection, organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling.” • Two broad areas • Descriptive – Science of summarizing data • Inferential – Science of interpreting data in order to make estimates, hypothesis testing, predictions, or decisions from the sample to target population.

5. Introduction to Clinical Statistics • Statistics - The science of making decisions in the face of uncertainty • Probability - The mathematics of uncertainty • The probability of an event is a measure of how likely the event is to happen

6. Sample versus Population

7. Measures of central tendency Mean, Median, Mode Measures of dispersion Range, Variance, Standard deviation Measures of relative standing Lower quartile (Q1) Upper quartile (Q3) Interquartile range (IQR) range (IQR) Descriptive Statistics for Continuous Variables

8. Basic Probability Concepts Sample spaces and events Simple probability Joint probability

9. Probability 1 Certain • Probability is the numerical measure of the likelihood that an event will occur • Value is between 0 and 1 .5 0 Impossible

10. Number of event outcomes P( E ) = Total number of possible outcomes in the sample space Computing Probabilities The probability of an event E: Assumes each of the outcomes in the sample space is equally likely to occur

11. Gaussian or Normal Distribution aka “Bell Curve” • Most important probability distribution in the statistical analysis of experimental data. • Data from many different types of processes follow a “normal” distribution: • Heights of American women • Returns from a diversified asset portfolio • Even when the data do not follow a normal distribution, the normal distribution provides a good approximation

12. Gaussian or Normal Distribution aka “Bell Curve” The Normal Distribution is specified by two parameters • The mean,  • The standard deviation, 

13. m=0 Standard Normal Distribution =1

14. Characteristics of the Standard Normal Distribution • Mean µ of 0 and standard deviation σof 1. • It is symmetric about 0(the mean, median and the mode are the same). • The total area under the curve is equal to one. One half of the total area under the curve is on either side of zero.

15. Area in the Tails of Distribution • The total area under the curve that is more than 1.96 units away from zero is equal to 5%. Because the curve is symmetrical, there is 2.5% in each tail.

16. Normal Distribution • 68% of observations lie within ± 1 std dev of mean • 95% of observations lie within ± 2 std dev of mean • 99% of observations lie within ± 3 std dev of mean

17. Study Design

18. Sample versus Population • A population is a whole, and a sample is a fraction of the whole. • A population is a collection of all the elements we are studying and about which we are trying to draw conclusions. • A sample is a collection of some, but not all, of the elements of the population

19. Sample versus Population

20. Sample versus Population • To make generalizations from a sample, it needs to be representative of the larger population from which it is taken. • In the ideal scientific world, the individuals for the sample would be randomly selected. This requires that each member of the population has an equal chance of being selected each time a selection is made.

21. Randomisation • To guard against any use of judgement or systematic arrangements i.e to avoid bias • To provide a basis for the standard methods of statistical analysis such as significance tests • Assures that treatment groups are balanced (on average) in all regards. • i.e. balance occurs for known prognostic variables and for unknown or unrecorded variables

22. Inferential statistics calculated from a clinical trial make an allowance for differences between patients and that this allowance will be correct on average if randomisation has been employed.

23. Hypothesis Testing

24. Hypothesis Testing • Steps in hypothesis testing: state problem, define endpoint, formulating hypothesis, - choice of statistical test, decision rule, calculation, decision, and interpretation • Statistical significance: types of errors, p-value, one-tail vs. two-tail tests, confidence intervals

25. Descriptive and inferential statistics • Descriptive statistics is devoted to the summarization and description of data (population or sample) . • Inferential statistics uses sample data to make an inference about a population .

26. Objectives and Hypotheses • Objectives are questions that the trial was designed to answer • Hypotheses are more specific than objectives and are amenable to explicit statistical evaluation

27. Examples of Objectives • To determine the efficacy and safety of Product ABC in diabetic patients • To evaluate the efficacy of Product DEF in the prevention of disease XYZ • To demonstrate that images acquired with product GHI are comparable to images acquired with product JKL for the diagnosis of cancer

28. How do you measure the objectives? • Endpoints need to be defined in order to measure the objectives of a study.

29. Endpoints: Examples: • Primary Effectiveness Endpoint – • Percentage of patients requiring intervention due to pain, where an intervention is defined as : • Change in pain medication • Early device removal

30. Endpoints: Examples: • Primary Endpoint: Percentage of patients with a reduction in pain: • Reduction in the Brief Pain Inventory (BPI) worst pain scores of ≥ 2 points at 4 weeks over baseline.

31. Endpoints: Examples • Patient Survival • Proportion of patients surviving two years post-treatment • Average length of survival of patients post-treatment

32. Objectives and Hypotheses • Primary outcome measure • greatest importance in the study • used for sample size • More than one primary outcome measure - multiplicity issues

33. Hypothesis Testing • Null Hypothesis (H0) • Status Quo • Usually Hypothesis of no difference • Hypothesis to be questioned/disproved • Alternate Hypothesis (HA) • Ultimate goal • Usually Hypothesis of difference • Hypothesis of interest

34. Decision Making “Truth” Decision Type I Error

35. Decision Making “Truth” Decision Type I Error

36. Decision Making “Truth” Decision Type I Error

37. Decision Making “Truth” Test Type I Error

38. Decision Making “Truth” Decision Type I Error

39. Hypothesis Testing Type I Error – Society’s Risk Type II Error – Sponsor’s Risk

40. The Type I Erroroccurs when we conclude from an experiment that a difference between groups exists when in truth it does not rejecting H0 when H0 is in Fact True Investigators reject H0 and declare that a real effect exists when the chance of this decision being wrong is less than 5%. Two Possible Errors of Hypothesis Testing

41. The Type II Error occurs when we conclude that there is no difference between treatments when in truth there is a difference fail to reject H0 when H0 is in fact False Two Possible Errors of Hypothesis Testing

42. In many circumstances a type I error is often regarded as more serious than a type II error. Example: H0: innocent vs. H1: guilty Type I error = declaring an innocent man guilty Type II error = declaring a guilty man innocent Presumption of innocence Negative test result means "There is not enough evidence to convict“ rather than "innocence" Two Possible Errors of Hypothesis Testing

43. One will never know whether one has committed either error unless data are available for the entire population. The only thing we are able to do is to assign α and β as the probabilities of making either type of error. It is important to keep in mind the difference between the truth and the decision that is being made as a result of the experiment. Review of errors in hypothesis testing

44. Hypothesis testing • Null Hypothesis • No difference between Treatment and Control • Type I error, alpha, , p-value • The probability of declaring a difference between treatment and control groups even though one does not exist (ie treatment is not statistically different from control in this experiment) • As this is “society’s risk” it is conventionally set at 0.05 (5%)

45. Hypothesis testing • Type II error, beta,  • The probability of not declaring a difference between treatment and control groups even though one does exist (ie treatment is statistically different from control in this experiment) • 1 - is the power of the study • Often set at 0.8 (80% power) however many companies use 0.9 (90% power) • Underpowered studies have less probability of showing a difference if one exists

46. Steps in Hypothesis Testing • Choose the null hypothesis (H0) that is to be tested • Choose an alternative hypothesis (HA) that is of interest • Select a test statistic, define the rejection region for decision making about when to reject H0 • Draw a random sample by conducting a clinical trial

47. Steps in Hypothesis Testing • Calculate the test statistic and its corresponding p-value • Make conclusion according to the pre-determined rule specified in step 3

48. Hypothesis Testing - How to test a hypothesis • Assume that we believe that we have a fair coin – equal chance of getting H or T when we flip the coin • Test the hypothesis by carrying out an experiment.

49. Hypothesis Testing - How to test a hypothesis • Flip the coin 4 times, each time is H. What is the likelihood of getting 4 H if this is a fair coin?

50. Remember the Binomial Probability Function Let X be the event of getting a H X ~ Binomial (n = 4, p=0.5) In this case, we want x=4 = 0.0625 = 6.25%