Introduction to statistics ii
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This document discusses approaches for analyzing microarray gene expression data with few observations. It suggests assuming a canonical distribution based on prior knowledge, like the log-normal distribution often seen in this type of data. Parameters of this distribution can then be estimated from the available observations using techniques like estimating the mean and variance. A t-statistic can be used to determine if two gene conditions have the same distribution parameters and thus identify differentially expressed genes. Quality control plots of the data are recommended to validate distribution assumptions.
![Benford’s Law
• Frequency distribution of first significant digit
Pr(d<=x<d+1 )= log10(1+d)-log10(d), log10(x) is uniformly distributed in [0,1]](https://image.slidesharecdn.com/introductiontostatistics-ii-120725071908-phpapp01/85/Introduction-to-statistics-ii-14-320.jpg)
Introduction to statistics ii
- 1. Statistics for Microarray Data
- 2. Background μ, σ2 • Few observations made by a black box • What is the distribution behind the black box? • E.g., with what probability will it output a number bigger than 5?
- 3. Approach • Easy to determine with many observations • With few observations.. • Assume a canonical distribution based on prior knowledge • Determine parameters of this distribution using the observations, e.g., mean, variance
- 5. Estimating the variance σ2 Chi-Square if the original distribution was Normal
- 6. Microarray Data • Many genes, 25000 • 2 conditions (or more), many replicates within each condition • Which genes are differentially expressed between the two conditions?
- 7. More Specifically • For a particular gene – Each condition is a black box – Say 3 observations from each black box • Do both black boxes have the same distribution? – Assume same canonical distribution – Do both have the same parameters?
- 8. Which Canonical Distribution • Use data with many replicates • 418.0294, 295.8019, 272.1220, 315.2978, 294.2242, 379.8320, 392.1817, 450.4758, 335.8242, 265.2478, 196.6982, 289.6532, 274.4035, 246.6807, 254.8710, 165.9416, 281.9463, 246.6434, 259.0019, 242.1968 • Distribution??
- 9. What is a QQ Plot
- 10. Distribution of log raw intensities across genes on a single array
- 11. The QQ plot of log scale intensities (i.e., actual vs simulated from normal)
- 12. QQ Plot against a Normal Distribution • 10 + 10 replicates in two groups • Single group QQ plot • Combined 2 groups QQ plot • Combined log-scale QQ plot Shapiro- Wilk Test
- 13. Which Canonical Distribution • Assume log normal distribution
- 14. Benford’s Law • Frequency distribution of first significant digit Pr(d<=x<d+1 )= log10(1+d)-log10(d), log10(x) is uniformly distributed in [0,1]
- 15. Differential Expression μ1,σ12 μ2,σ22 Group 1 Group 2 Is μ1= μ2? σ1 = σ2 ? Is variance a function of mean?
- 16. SD increases linearly with Mean SD vs Mean across 3 replicates plotted for all genes
- 17. SD is flat now, except for very low values Another reason to work on the log scale SD vs Mean across 3 replicates computed for all genes after log-transformation
- 18. Differential Expression μ1,σ12 μ2,σ22 Group 1 Group 2 Is μ1= μ2? σ1 = σ2 ? Sort-of YES
- 19. The T-Statistic
- 20. The T-Statistic
- 21. The T-Statistic
- 22. The T-Statistic Flattened Normal or T- Distribution
- 23. A Problem
- 24. The curve fit here may be a better estimate Lots of false positives can Not much be avoided difference here here SD vs Mean across 3 replicates computed for all genes after log-transformattion
- 25. Thank You