1. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.1 Counts and Proportions

2. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.2 Binary Data  Often in medical and public health studies, our endpoint of interest is binary or dichotomous  Examples  disease vs. no disease  response vs. no response  death vs. no death  Success vs. failure

3. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.3 Binary Data  Only 2 possible responses  Often, continuous endpoints are dichotomized into a binary endpoint  For example, in a study of the effect of a drug on LDL levels, for each subject, the LDL measurement at the end of the study (a continuous measure) may be dichotomized into “response” vs. “no response” based on a cutpoint defining whether the LDL level has been reduced to acceptable, normal, or safe levels.

4. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.4 Binary Data  Similar to hypothesis testing with continuous data, one may perform hypothesis tests on binary data:  1-sample test of a proportion  H 0 : p=p 0  H A : p  p 0  2-sample test comparing proportions  H 0 : p 1 =p 2  H A : p 1  p 2

5. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.5 Binary Data  Also, similar to continuous data, we may derive confidence intervals for  A single proportion  The difference between two proportions

6. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.6 Note  We may use the CLT for binary data also (as the CLT applies to all distributions)  But note that the CLT is an asymptotic result (as n   )  Thus, we must be careful when n is small

7. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.7 Example  Etoposide for the treatment of relapsed or progressed Kaposi’s Sarcoma  Insert Etoposide.pdf

8. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.8 Binary Data  Insert Binary1.pdf

9. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.9 Exact Confidence Intervals  “Exact” confidence intervals for a binomial parameter are possible  These do not rely on the normal approximation to the binomial (i.e., use of the CLT)  Computationally very intensive (particularly for large N)  May require special programming/software

10. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.10 Exact Confidence Intervals  General Rule:  Use exact confidence intervals whenever software is available and is feasible given the computing resources  If N is large then it is OK to use normal approximation (as CLT kicks in)  If N is small:  the normal approximation may not be appropriate  Use exact CIs if possible

11. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.11 Example  The proportion of students against the Iraq war

12. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.12 Example  Special Case: Observe 0 events or responses. How do we get a CI for the response rate when the variability is 0?  Insert A5129_example.pdf

13. Introduction to Biostatistics, Harvard Extension School, Fall, 2005 © Scott Evans, Ph.D.13 Example  CI for the difference between two proportions  Insert HIV_HCV.pdf  Comparing HIV+/HCV+ with HIV+/HCV- individuals with respect to depressive symptomatology