Statistics in Biology and Medicine

1. Pharmamatrix Workshop 2010 Statistics in Biology and Medicine Richard Tseng July 14, 2010

2. The goal of statistics is to analyze, interpret and present data collected to study systems of interest!!

3. Outline • Descriptive statistics • Inferential statistics • Probability theory • Hypothesis test • Regression • Some other tools • Tools for component analysis • Bayesian statistics • Summary

4. Descriptive statistics • Definitions • Set: A well-defined collection of objects and each object is called an element • Operation of sets: union and intersection For example, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}

5. Data type • Interval scale • For example, body weight (g): 1, 1.5, 2, 3 … • Ordinal scale • For example, scores for patient responses to treatment • Nominal scale • Categorical data. For example, factors to influence treatments

6. How large are the numbers? • Mean • Median [1]

7. How variable are the numbers? • Standard deviation (SD) • Coefficient of variance (CV = SD/mean) [1]

8. Inferential Statistics: Probability theory • Law of large numbers • The mean of elements in a set converges to the expected value when the number of elements close to infinite • Law of small numbers •  There are not enough small numbers to satisfy all the demands placed on them

9. Central limit theorem • states conditions under which the mean of a sufficiently large number of independent random variable, each with finite mean and variance, will be approximately normally distributed http://www.stat.sc.edu/~west/javahtml/CLT.html

10. Probability • Meaning • Frequency interpretation: A number are associated with the rate of occurrence of an event in a well defined random physical systems • Bayesian interpretation: A number assigned to any statement whatsoever, even when no random process is involved, as a way to represent the degree to which the statement is supported by the available evidence

11. Probability • Basic rules • Subtraction • Addition • Multiplication

12. Probability • Bayesian rule Posterior Prior Likelihood

13. Probability • Maximum entropy principle: The most honest probability distribution assignment to a system is the one that maximizes the entropy of the system subject to any information available in hand.

14. Inferential Statistics: Regression • Goal: To correlate the study outcomes of systems of interest and possible factors. • Model: • Linear model • Logistic model

15. Optimization Suppose there are n outcomes di of a study • Least-square method • Maximum Likelihood estimate: Supoose a likelihood function is given by L(a,b|d)

16. Regression tests • Residual analysis • Standard errors of regression coefficients • Coefficient of determination

17. Example 1: Linear regression

18. Example 2: MLE solution of Emax and EC50 in Michales-Menten equation Likelihood function MLE solution

19. Inferential Statistics: Hypothesis test

20. Goal: Test of significance • Rationale • Null hypothesis: H0, outcomes of a study purely result from chance • Alternative hypothesis: H1, outcomes of a study are influenced from non-random sources • Appropriate model: Normal distribution, t-distribution…

22. Rationale • Appropriate analysis method • P-value: The probability of observing a sample statistic as extreme as the test statistic, assuming the null hypothesis is true. • Parametric method: t-test, F-test, Chi-square test • Non-parametric method: Kolmogorov-Smirnov test, Mann-Whitney test

23. P-value for significant test: • What is the probability of a test value from a random population? One or two tailed? • If p-value is less than the confidence level a, the null hypothesis is rejected t-distribution http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html

24. Parametric test

25. Two sample t-test: (Online calculator http://www.usablestats.com/calcs/2samplet) Observed difference (Sample 1 - Sample 2): -0.298Standard Deviation of Difference : 0.0764Unequal VariancesDF : 2795% Confidence Interval for the Difference ( -0.4548 , -0.1412 )T-Value -3.9005 Population 1 ≠ Population 2: P-Value = 0.0006 Population 1 > Population 2: P-Value = 0.9997 Population 1 < Population 2: P-Value = 0.0003 Equal VariancesPooled Standard Deviation: 0.2093 Pooled DF: 2895% Confidence Interval for the Difference ( -0.4545 , -0.1415 ) T-Value -3.8992 Population 1 ≠ Population 2: P-Value = 0.0006 Population 1 > Population 2: P-Value = 0.9997 Population 1 < Population 2: P-Value = 0.0003

26. Some Statistics Worth to Know • Tool for component analysis: • Principle Component Analysis (PCA): A way to identify patterns in data, and express in a way to highlight their similarities and differences • Independent Component Analysis (ICA): A way to separate independent components in data • Variable and model selection: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) • Bayesian statistics

27. Summary • What is “right” null hypothesis? • What is the appropriate distribution function? • What is the appropriate test statistics? “Know” your data before analyze that!!

28. Information theory based statistics: Bayesian statstics • Goal: Using Bayesian method to design and analyze data • Bayesian inference • Appropriate distribution functions • Appropriate sampling techniques • Maximum entropy method based inference • Appropriate form of entropy • Appropriate constriants

29. Information theory based statistics: Method of maximum entropy

30. Reference [1] P. Rowe, Essential Statistics for Pharmaceutical Sciences, Wiley 2007.