1. Canadian Bioinformatics Workshops www.bioinformatics.ca

2. 2Module #: Title of Module

3. Module 2 Exploratory Data Analysis (EDA) D EPARTMENT OF B IOCHEMISTRY D EPARTMENT OF M OLECULAR G ENETICS † Odysseus listening to the song of the sirens. Late Archaic (500–480 BCE) Exploratory Data Analysis of Biological Data using R Boris Steipe Toronto, May 23. and 24. 2013 This workshop includes material originally developed by Raphael Gottardo, FHCRC and by Sohrab Shah, UBC †

4. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Exploratory Data Analysis (EDA) EDA is an approach to data analysis without a particular statistical model or hypothesis. It is therefore often the first step of data analysis.

5. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Exploratory Data Analysis (EDA) The objectives of EDA include: uncovering underlying structure and identifying trends and patterns; extracting important variables; detecting outliers and anomalies; testing underlying assumptions; developing statistical models.

6. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Exploratory Data Analysis (EDA) The practice of EDA emphasizes looking at data in different ways through: computing and tabulating basic descriptors of data properties such as ranges, means, and variances; generating graphics, such as boxplots, histograms, scatter plots; applying transformations, such as log or rank; comparing observations to statistical models, such as the QQ-plot, or regression; identifying underlying structure through clustering; simplifying data through dimension reduction...... all with the final goal of defining a statistical model and using the model for hypothesis testing and prediction.

7. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Graphics Good graphics are immensely valuable. Poor graphics are worse than none. If you want to learn more about good graphics and information design, find a copy of Edward Tufte's The Visual Display of Quantitative Information. You can also visit his Web site to get a sense of the field (www.edwardtufte.com). Fundamentally, there is one simple rule. The rule has many corollaries. Use less ink.

8. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Use less ink Make sure that all elements on your graphics are necessary. Make sure that all elements on your graphics are informative. Make sure that all information in your data is displayed. Not all of R's defaults use as little ink as possible...

9. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca refined graphics... for a popular alternative to R's defaults, check out the ggplot2 package (http://www.ggplot2.org). > install.packages("ggplot2")... > library(ggplot2)... Example: Bubble chart. http://ygc.name/2010/12/01/bubble-chart-by-using-ggplot2/ crime <- read.csv("crimeRatesByState2008.csv", header=TRUE, sep="\t") p <- ggplot(crime, aes(murder,burglary,size=population, label=state)) p <- p+geom_point(colour="red") +scale_area(to=c(1,20))+geom_text(size=3) p + xlab("Murders per 100,000 population") + ylab("Burglaries per 100,000")

10. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Graphics R Graph Gallery: http://gallery.r-enthusiasts.com/

11. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca exploring data When teaching (or learning new procedures) I prefer to work with synthetic data. Synthetic data has the advantage that I know what the outcome of the analysis should be. Typically one would create values acording to a function and then add noise. R has several functions to create sequences of values – or you can write your own... 0:10 seq(0, pi, 5*pi/180) rep(1:3, each=3, times=2) for (i in 1:10) { print(i*i) }

12. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca synthetic data Function... Explore functions and noise. Noise... Noisy Function...

13. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca discrete data TGGGACGATCAACTCGAAGCGGATGCCCCTATTGATCCTCTCCGAACACAATGAGAATGCTCTCAAAG GCTGTAAGCTCAGTGAACGGGACTAAATCAGACGTTTGTCGTCGATACGGGTGCCCTACCTCTACCTA ATCTGATCGAGGCCAAGTATCTGGAAGGACTGCTAATTTTGCCGTAAACCCCGTTCTTGTTCAGTATAA TCTAAGTAGATGTGTTAGCAGTCATATGACGATAGTTTGGGGGTCCTCCTATAGGAAAAAGAAAGAGTC GCCACACCAGAGCCTGTAGCGCTTTCTAGATAGTGCTGCATATTTATATGTCGGCCCCGAACCAGAGC GCTCCAATGGTAGCCCCTTTAATCTTCGTATCTTACCTTTTATGAGTGCACAGGTTTCCATCGAGGGAG AAAAACCTCAGCAACGTGGGTGGGTAGAAGAGCCCTAGTTTGAAAGCCGCACCATAACCCGCACATC GTCAGATCAGTAACCCAAGATCGGTGGGCTGTAACTAGGCTCCGTGACACAGCGTGGTATTCCGAGT TCCCGAAATCGTTTCACCTATAGAACGCCACCCCGGACGGGGTTGTTAGTTTTTCTACCTTTTAAGAAG AAAAGCAAAGTGTGTGGACACGAGAACTAGTGTGAGTACGGTTTTGTATGTGGCCCTACTGTGGAAAC TCAGTAGTACGAAGGGGATAGCGAGACTTAGCTTTGCCCCAACTGCCGTCACGCACCCGCTTGTGCC GGTACGCAGAGCTCCGCCGGGTGTCCAAGTGCCGTTCTACGATAAGAACCTGTGTATCTAGCGCGCC CGATATGAATAAAGCCTACTCTTATCCAGATTTTGCGGACTGGTAAGCGTGACAATTATTGCGCAGCTT CGACTTAGTTCTCCTTGCCTTGCTTTAGGGGAGTTCTCCACTCAAAAGTCGTTGACGTACAATCGCAG ATTTTGTAATCCCTTAAACCTCTGATTAGTCTCAGCCGTATTCACTA Exercise: Write a function to output random nucleotides.

14. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca random nucleotides: solution 1 Generating uniformly distributed random nucleotides using sample(). rNucUnif <- function(n) { nuc <- c("A", "C", "G", "T") rSeq <- sample(nuc, n, replace=TRUE) return(rSeq) } Task: Explore a function that samples according to arbitrary probabilities.

15. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca sample() Random samples from a vector with/without replacement. Sampling a vector over its length without replacement is a permutation or shuffle of the vector. sums <- rep(0, 10) digits <- 0:9 rsd <- vector() for (i in 1:10000) { perm <- sample(digits, 10) sums <- sums + perm if (i % 1000 == 0) { rsd <- c(rsd, sd(sums)/mean(sums)) } Task: Test whether the permutation is biased.

16. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca probability distributions Can be either discrete or continuous (uniform, Bernoulli, normal, etc) Defined by a density function, p(x) or f(x) Bernoulli distribution Be(p): flip a coin (T=0, H=1). Assume probability of H is 0.1... x <- 0:1 f <- dbinom(x, size=1, prob=0.1) plot(x, f, xlab="x", ylab="density", type="h", lwd=5)

17. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca probability distributions Random sampling: Generate 100 observations from a Be(0.1) set.seed(100) x <- rbinom(100, size=1, prob=0.1) hist(x)

18. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca probability distributions Normal distribution N(μ,σ 2 ) μ is the mean and σ 2 is the variance. Extremely important because of the Central Limit Theorem: if a random variable is the sum of a large number of small random variables, it will be normally distributed. x <- seq(-4, 4, 0.1) f <- dnorm(x, mean=0, sd=1) plot(x, f, xlab="x", ylab="density", lwd=5, type="l") The area under the curve is the probability of observing a value between 0 and 2.

19. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca probability distributions Normal distribution N(μ,σ 2 ) μ is the mean and σ 2 is the variance. Extremely important because of the Central Limit Theorem: if a random variable is the sum of a large number of small random variables, it will be normally distributed. x <- seq(-4, 4, 0.1) f <- dnorm(x, mean=0, sd=1) plot(x, f, xlab="x", ylab="density", lwd=5, type="l") The area under the curve is the probability of observing a value between 0 and 2. Task: Explore line parameters

20. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca probability distributions Random sampling: Generate 100 observations from a N(0,1) set.seed(100) x <- rnorm(100, mean=0, sd=1) hist(x) lines(seq(-3,3,0.1),50*dnorm(seq(-3,3,0.1)), col="red") Histograms can be used to estimate densities!

21. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca quantiles (Theoretical) Quantiles: The p-quantile has the property that there is a probability p of getting a value less than or equal to it. The 50% quantile is called the median. 90% of the probability (area under the curve) is to the left of the red vertical line. q90 <- qnorm(0.90, mean = 0, sd = 1) x <- seq(-4, 4, 0.1) f <- dnorm(x, mean=0, sd=1) plot(x, f, xlab="x", ylab="density", type="l", lwd=5) abline(v=q90, col=2, lwd=5)

22. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca descriptive statistics Empirical Quantiles: The p-quantile has the property that p% of the observations are less than or equal to it. Empirical quantiles can be easily obtained in R. > set.seed(100) > x <- rnorm(100, mean=0, sd=1) > quantile(x) 0% 25% 50% 75% 100% -2.2719255 -0.6088466 -0.0594199 0.6558911 2.5819589 > quantile(x, probs=c(0.1, 0.2, 0.9)) 10% 20% 90% -1.1744996 -0.8267067 1.3834892

23. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca descriptive statistics We often need to quickly 'quantify' a data set, and this can be done using a set of summary statistics (mean, median, variance, standard deviation). > mean(x) [1] 0.002912563 > median(x) [1] -0.0594199 > IQR(x) [1] 1.264738 > var(x) [1] 1.04185 > summary(x) Min. 1st Qu. Median Mean 3rd Qu. Max. -2.272000 -0.608800 -0.059420 0.002913 0.655900 2.582000 Exercise: what are the units of variance and standard deviation?

24. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Box-plot Descriptive statistics can be intuitively summarized in a Box-plot. > boxplot(x) IQR 1.5 x IQR Everything above and below 1.5 x IQR is considered an "outlier". 75% quantile Median 25% quantile IQR = Inter Quantile Range = 75% quantile – 25% quantile

25. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Violinplot Internal structure of a data-vector can be made visible in a violin plot. The principle is the same as for a boxplot, but a width is calculated from a smoothed histogram. p <- ggplot(X, aes(1,x)) p + geom_violin()

26. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca plotting data in R Task: Explore types of plots.

27. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca QQ–plot One of the first things we may ask about data is whether it deviates from an expectation e.g. to be normally distributed. The quantile-quantile plot provides a way to visually verify this. The QQ-plot shows the theoretical quantiles versus the empirical quantiles. If the distribution assumed (theoretical one) is indeed the correct one, we should observe a straight line. R provides qqnorm() and qqplot().

28. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca QQ–plot: sample vs. Normal Only valid for the normal distribution! qqnorm(x) qqline(x, col=2)

29. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Plotting: lines and legends Example: compare the t-distribution with the Normal distribution. x <- seq(-4, 4, 0.1) f1 <- dnorm(x, mean=0, sd=1) f2 <- dt(x, df=2) plot(x, f1, xlab="x", ylab="density", lwd=5, type="l") lines(x, f2, lwd=5, col=2) legend(-4,.4, c("Normal", "t2"), col=1:2, lwd=5)

30. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca QQ–plot: sample vs. Normal Clearly the t distribution with two degrees of freedom is not Normal. set.seed(100) t <- rt(100, df=2) qqnorm(t) qqline(t, col=2)

31. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca QQ–plot set.seed(101) generateVariates <- function(n) { Nvar <- 10000 Vout <- c() for (i in 1:n) { x <- runif(Nvar, -0.01, 0.01) Vout <- c(Vout, sum(x) ) } return(Vout) } x <- generateVariates(1000) y <- rnorm(1000, mean=0, sd=1) qqnorm(x) qqline(x, y, col=2) Verify the CLT.

32. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca QQ–plot: sample vs. sample Comparing two samples: are their distributions the same?... or... compare a sample vs. a synthetic dataset. set.seed(100) x <- rt(100, df=2) y <- rnorm(100, mean=0, sd=1) qqplot(x, y) Exercise: try different values of df for rt() and compare the vectors.

33. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca scatter plots Biological data sets often contain several variables, they are multivariate. Scatter plots allow us to look at two variables at a time. This can be used to assess independence and identify subgroups. # GvHD flow cytometry data gvhd <- read.table("GvHD.txt", header=TRUE) # Only extract the CD3 positive cells gvhdCD3p 280, 3:6]) plot(gvhdCD3p[, 1:2])

34. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca scatter plots Task: Explore scatter plots.

35. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca trellis graphics Trellis Graphics is a family of techniques for viewing complex, multi-variable data sets. plot(gvhdCD3p, pch=".") Tip: Many more possibilities are available in the "lattice" package. See ?Lattice

36. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Boxplots The boxplot function can be used to display several variables at a time. boxplot(gvhdCD3p) Exercise: Interpret this plot.

37. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Histograms We discover a mix of distinct cell subpopulations! par(mfrow=c(2, 2)) hist(gvhdCD3p[, 1], 50, main=names(gvhdCD3p)[1], xlab="fluorescent intensity", ylim=c(0, 120)) hist(gvhdCD3p[, 2], 50, main=names(gvhdCD3p)[2], xlab="fluorescent intensity", ylim=c(0, 120)) hist(gvhdCD3p[, 3], 50, main=names(gvhdCD3p)[3], xlab="fluorescent intensity", ylim=c(0, 120)) hist(gvhdCD3p[, 4], 50, main=names(gvhdCD3p)[4], xlab="fluorescent intensity", ylim=c(0, 120))

38. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: HIV microarray data HIV Data: The expression levels of 7680 genes were measured in CD4-T-cell lines at time t = 24 hours after infection with HIV type 1 virus. 12 positive controls (HIV genes). 4 replicates (2 with a dye swap)

39. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: HIV microarray data We get the data after the image analysis is done. For each spot we have an estimate of the intensity in both channels. The data matrix therefore is of size 7680 x 8. One of the array chips.

40. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: HIV microarray data data <- read.table(file="hiv.raw.data.24h.txt", sep="\t", header=TRUE) summary(data) boxplot(data)

41. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: HIV microarray data opar <- par(mfrow=c(2, 2)) hist(data[, 1], 50, main=names(data)[1], xlab="fluorescent intensity") hist(data[, 2], 50, main=names(data)[2], xlab="fluorescent intensity") hist(data[, 5], 50, main=names(data)[5], xlab="fluorescent intensity") hist(data[, 6], 50, main=names(data)[6], xlab="fluorescent intensity") par(oPar)

42. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: HIV microarray data # 'apply' will apply the function to all rows of the data matrix mean <- apply(data[, 1:4], 1, "mean") sd <- apply(data[, 1:4], 1, "sd") plot(mean, sd) trend <- lowess(mean, sd) lines(trend, col=2, lwd=5) lowess fit LOcally WEighted Scatter plot Smoother; used to estimate the trend in a scatter plot. Non parametric!

43. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: Transformations Observations: The data are highly skewed. The standard deviation is not constant, as it increases with the mean. Solution: Look for a transformation that will make the data more symmetric and the variance more constant. With positive data the log transformation is often appropriate.

44. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: Transformations data <- log(read.table(file="hiv.raw.data.24h.txt", sep="\t", header=TRUE)) summary(data) boxplot(data)

45. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: HIV microarray data opar <- par(mfrow=c(2, 2)) hist(data[, 1], 50, main=names(data)[1], xlab="fluorescent intensity") hist(data[, 2], 50, main=names(data)[2], xlab="fluorescent intensity") hist(data[, 5], 50, main=names(data)[5], xlab="fluorescent intensity") hist(data[, 6], 50, main=names(data)[6], xlab="fluorescent intensity") par(oPar)

46. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA: Transformations The standard deviation has become almost independent of the mean. # 'apply' will apply the function to all rows of the data matrix mean <- apply(data[, 1:4], 1, "mean") sd <- apply(data[, 1:4], 1, "sd") plot(mean, sd) trend <- lowess(mean, sd) lines(trend, col=2, lwd=5)

47. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA for microarray data: always log Makes the data more symmetric, large observations are not as influential The variance is (more) constant Turns multiplication into addition (log(ab)=log(a)+log(b)) In practice use log base 2, log2(x)=log(x)/log(2)

48. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA for gene expression Is this a good way to look at the data? # scatter plot plot(data[, 1], data[, 5], xlab=names(data)[1], ylab=names(data)[5])

49. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA for gene expression: MA plots M (minus) is the log ratio. A (average) is overall intensity. # MA plot A <- (data[, 1]+data[, 5])/2 M <- (data[, 1]-data[, 5]) plot(A, M, xlab="A", ylab="M")

50. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca EDA for gene expression: MA plots The data suggests that a small group of genes are interesting. How do we define/find such differentially expressed genes? More on this in Module 6: Hypothesis Testing. # MA plots per replicate par(mfrow=c(2, 2)) A1 <- (data[, 1]+data[, 5])/2 M1 <- (data[, 1]-data[, 5]) plot(A1, M1, xlab="A", ylab="M", main="rep 1") trend <- lowess(A1, M1) lines(trend, col=2, lwd=5) A2 <- (data[, 2]+data[, 6])/2 M2 <- (data[, 2]-data[, 6]) plot(A2, M2, xlab="A", ylab="M", main="rep 2") trend <- lowess(A2, M2) lines(trend, col=2, lwd=5) A3 <- (data[, 3]+data[, 7])/2 M3 <- (data[, 3]-data[, 7]) plot(A3, M3, xlab="A", ylab="M", main="rep 3") trend <- lowess(A3, M3) lines(trend, col=2, lwd=5) A4 <- (data[, 4]+data[, 8])/2 M4 <- (data[, 4]-data[, 8]) plot(A4, M4, xlab="A", ylab="M", main="rep 4") trend <- lowess(A4, M4) lines(trend, col=2, lwd=5) par(OldPar)

51. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca More plots Task: Explore 3D plots.

52. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca Summary EDA should be the first step in any statistical analysis! Good modeling starts and ends with EDA. R provides a great framework for EDA.

53. Module 2: Exploratory Data Analysis (EDA) bioinformatics.ca boris.steipe@utoronto.ca