Statistical model for count data

1. Statistical model for count data Speaker : Tzu-Chun Lo Advisor : Yao-Ting Huang

2. Outline • Why use statistical model • Target • Gene expression • Binomial distribution • Poisson distribution • Over dispersion • Negative binomial • Chi-square approximation • Conclusion

3. Statistics model • A statistical model is a probability distribution constructed to enable inferences to be drawn or decisions made from data. Information : sample Height, weight, etc. Population Inference We We have to choose a statistics model for sample (mean, variance) Make a decision : Hypothesis testing (mean, variance) size designer consumer

4. Target • Gene expression • We like to use statistical model to test an observed difference in read counts is significant. Look like a significant region How about this Can we sure ? Noise or not

5. Count data • A type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking. • An individual piece of count data is often termed a count variable. Poisson All of them are this type Binomial Negative binomial

6. Binomial distribution • The number of successes in a sequence ofnindependent yes/no experiments, each of which yields success with probability p. • Notation :

7. Binomial distribution Ex : p=0.8 , (1-p)=0.2 , times : 3 , success : 2 (1 1 0) (1 0 1) (0 1 1) f(2)=0.384 33 goals 110 shots in this season Success : 0.3 Fail : 0.7 What is the probability if he scored 6 goals in 10 shots

8. Binomial distribution • Exactly six goals • Most three goals 0 1 2 3 4 5 6 7 8 9 10 6

9. Poisson distribution • Expresses the probability of a given number of events occurring in a fixed interval. • Notation :

11. e = 2.718281828… Poisson distribution • Suppose interval : goals per game

12. Poisson Games • Total : 11 games • Score : 33 goals • (33/11) = 3 goals per game • Poisson : • Raw data : • We could test inaccurately in this case by poisson goals

13. Overdispersion • The presence of greater variability (statistical dispersion) in a data set than would be expected based on a given simple statistical model.

14. Negative binomial • Gamma-poisson (mixture) distribution

15. Negative binomial

16. Parameter estimation

17. Approximate control limits • Chi-square approximation

18. Example = 67.0

21. Conclusion • Conclusion • Thanks for attention

22. Statistics model • Suitable type • Which distribution should we use • Parameters • Get some information from data • Inference • What do we want to know • How could we make a decision • Hypothesis testing

23. Statistics model • Suitable type • Binomial distribution • Parameters • n = 10, p = 0.7 • Inference • 2 successes

24. Multinomial distribution • The analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number k of possible outcomes. • http://en.wikipedia.org/wiki/Multinomial_distribution

25. Trinomial distribution

26. Count data • A type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking. • We tend to use fixed fractions of genes. The probability that reads appeared in this region The number of read counts in this interval (Binomial distribution) (Poisson distribution)

28. Poisson example

29. Negative binomial