X Treatment population Control population 0 Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx  Let X = decrease (–) in cholesterol level (mg/dL); Patients satisfying inclusion criteria RANDOMIZERANDOMIZE Treatment Arm Control Arm RANDOM SAMPLES End of Study T-test F-test (ANOVA ) Experiment significant? possible expected distributions:

S(t) = P(T > t) 0 1 T Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx  Let T = Survival time (months); End of Study Log-Rank Test, Cox Proportional Hazards Model Kaplan-Meier estimates population survival curves: significant? S 2 (t) Control S 1 (t) Treatment AUC difference survival probability

Case-Control studies Cohort studies

E+ vs. E– statistically significant Observational study designs that test for a statistically significant association between a disease D and exposure E to a potential risk (or protective) factor, measured via “odds ratio,” “relative risk,” etc. Lung cancer / Smoking PRESENT E+ vs. E– ? D+ vs. D– ? Case-Control studies Cohort studies Both types of study yield a 2  2 “contingency table” of data: D+D+D–D– E+E+ aba + b E–E– cdc + d a + cb + dn relatively easy and inexpensive relatively easy and inexpensive subject to faulty records, “recall bias” subject to faulty records, “recall bias” D+ vs. D– FUTUREPAST measures direct effect of E on D expensive, extremely lengthy expensive, extremely lengthy… Example: Framingham, MA study where a, b, c, d are the numbers of individuals in each cell. cases controls reference group End of Study Chi-squared Test McNemar Test H 0 : No association between D and E.

Surveys, prevalence studies, etc.

The analytical techniques that apply to longitudinal studies (i.e., observations over time) are also appropriate for these cross-sectional studies (i.e., observations at a fixed time). PRESENTPASTFUTURE

But what if the two variables – say, X and Y – are numerical measurements? As seen, testing for association between categorical variables – such as disease D and exposure E – can generally be done via a Chi-squared Test. Furthermore, if sample data does suggest that one exists, what is the nature of that association, and how can it be quantified, or modeled via Y = f (X)? JAMA. 2003;290: correlation coefficient regression methods Other statistical issues along the way… BIASBIAS - Sources? What can we do about it? How do we check if standard assumptions are valid, and what do we do if they are violated, or we can’t tell? Do these techniques generalize, and if so, how? What are some other applications?