1. XIAO WU XIAO.WU@YALE.EDU DATA ANALYSIS & BASIC STATISTICS

2. PURPOSE OF THIS WORKSHOP Statistics as a useful tool to analyze results Basic terminology and most commonly used tests Exposure to more advanced statistical tools

3. WHY DO WE NEED STATISTICS?

4. Summary Classification Interpretation Pattern searching Abnormality identification Prediction Intrapolation Extrapolation

5. SUMMARY http://www.mymarketresearchmethods.com/descriptive-inferential-statistics-difference/

6. SUMMARY Mean, median, mode Variance, standard deviation Max, min values and range Quartiles http://www.mymarketresearchmethods.com/descriptive-inferential-statistics-difference/

7. EXAMPLE Firm A Mean: $5,800 Firm B Mean: $5,000

8. EXAMPLE Firm A Mean: $5,800 Median: $4,000 SD: $7,270 3 rd Quartile: $4,000 1 st Quartile: $500 Firm B Mean: $5,000 Median: $5,000 SD: $203 3 rd Quartile: $5,175 1 st Quartile: $4,825

9. EXAMPLE #Salary ($) 1 4650 2 4700 3 4750 4 4800 5 4850 6 4900 7 4950 8 5000 9 5050 10 5100 11 5150 12 5200 13 5250 14 5300 15 5350 #Salary ($) 120000 24000 3 4500 5

10. CLASSIFICATION Identification of variable Independent vs. dependent Numeric vs. categorical Variable Categorical Nominal Ordinal Numeric Continuous Discrete

11. PATTERN SEARCHING Distribution of data Some commonly used distributions Uniform Binomial Poisson … Central limit theorem http://www.mathwave.com/img/art/graphs_pdf2.gif

12. UNIFORM Every outcome has equal chance Example: Flipping a coin Rolling a dice What if you need to flip multiple times?

13. BINOMIAL Two outcomes, probability p and 1- p Multiple trials: n Example: Flipping a coin 100 times Germination of multiple seeds https://onlinecourses.science.psu.edu/stat414/sites/onlinecourses.science.p su.edu.stat414/files/lesson09/graph_n15_p02.gif

14. POISSON Counts of rare, independent events Each with probability, or average rate p Example: radioactive decay http://kaffee.50webs.com/Science/images/alpha_decay.gif

15. THE MOST IMPORTANT DISTRIBUTION

16. NORMAL DISTRIBUTION Central limit theorem Every distribution converges to a normal distribution Large sample size  normal distribution Parameters: mean standard deviation https://www.mathsisfun.com/data/images/normal-distrubution-large.gif

17. PATTERN SEARCHING Hypothesis testing Difference between two populations Z-test or t-test? What does p-value mean? Family-wise error – Bonferroni correction More than two possibilities Chi square test Fisher’s exact test More than two variables ANOVA

18. EXAMPLE 1 SAT score is related to gender Null hypothesis Alternative hypothesis (3 possibilities) One or two tail? Z or T test? p=0.07, conclusion?

19. EXAMPLE 2 Predictors of stroke Age Hypertension Gender …

20. EXAMPLE 3 Genome-wide association studies Scanning markers across the DNA of many people to find genetic variations associated with certain diseases

21. PATTERN SEARCHING Hypothesis testing One variable Z-test or t-test? What does p-value mean? Family-wise error – Bonferroni correction Compare two categorical variables Chi square test Fisher’s exact test More than two variables ANOVA

22. CHI SQUARE Punnett Square A cross between two pea plants yields 880 plants, 639 green, 241 yellow Hypothesis: The green allele is dominant and both parents are heterozygous. http://www2.lv.psu.edu/jxm57/irp/chisquar.html

23. CHI SQUARE Gg G GG (green) Gg(green) g gg (yellow) 75% green 25% yellow

24. CHI SQUARE GreenYellow Observed (o)639241 Expected (e)660220 Deviation (d=o – e)-2121 Deviation squared (d^2) 441 d^2/e0.6682 Sum2.669 Degree of freedom: number of categories – 1 = 1

25. CHI SQUARE

26. PREDICTION Regression Linear regression Multiple linear regression Accuracy vs. simplicity Validation leave-k-out http://2.bp.blogspot.com/-W7Ptp8uB02U/T8UAGm4Uw5I/AAAAAAAAC08/DcHCtLWXv- U/s1600/actnactn+1.png

27. EXAMPLE Use brain structural measurements to predict a subject’s performance on picture vocabulary test 144 total structural measurements 521 subjects First step: eliminate unnecessary variables All zeros? Highly correlated pairs Variables that do not correlate well with performance score

28. EXAMPLE Run regression Validation: leave 1 out and leave 10 out Principle component analysis …

29. PREDICTION More complicated models: Baysian approach Use prior knowledge to update prediction Diffusion weights Use local structure to predict neighboring values

30. STATISTICAL TOOLS EXCEL MatLab R MiniTab …

31. QUESTIONS?

32. MY OWN RESEARCH Cost-effectiveness analysis Mathematical modeling in medicine Simulate iterations rather than actual patients

33. RECENT RESULTS

34. RESULTS

35. GROUP EXERCISE