• Review of syllabus
• Random variables and distributions
• Introduction to the R language

• Introduction to dplyr

• Book sections:
  • Chapter 0
  • Chapter 1 - section Random Variables

A built html version of this lecture is available.

The source R Markdown is also available from Github.

• High-dimensional statistics (more variables than observations)
• Predictive modeling and methodology for validation
• Metagenomic profiling of the human microbiome
• Cancer genomics
• HIV treatment effectiveness
• http://www.waldronlab.org

• A random variable: any characteristic that can be measured or categorized, and where any particular outcome is determined at least partially by chance.

• Examples:
  • number of new diabetes cases in NYC in a given year
  • The weight of a randomly selected individual in NYC

• Types:
  • Categorical random variable (e.g. disease / healthy)
  • Discrete random variable (e.g. sequence read counts)
  • Continuous random variable (e.g. normalized qPCR intensity)

• We use probability distributions to describe the probability of all possible realizations of a random variable
• In public health we use probability distributions to describe hypotheses or inference about the population
  • because we normally cannot observe the entire population
• In practice we study a sample selected from the population

Normally distributed random variable with mean \(\mu = 0\) / standard deviation \(\sigma = 1\), and a sample of \(n=100\)

Poisson distributed random variable (\(\lambda = 2\)), and a sample of \(n=100\).

Negative Binomially distributed random variable (\(size=30, \mu=2\)), and a sample of \(n=100\).

• Binomial Distribution random variable (\(size=20, prob=0.25\)), and a sample of \(n=100\).
  • We will only use for binary outcomes

Slide credit: Marcel Ramos

• See the Bioconductor site for more info

Pseudo code:

source("https://bioconductor.org/biocLite.R")
packages <- c("packagename", "githubuser/repository", "biopackage")
BiocInstaller::biocLite(packages)

• Works for CRAN, GitHub, and Bioconductor packages!

• Useful for building packages
• Download and install from GitHub, directly or via BiocInstaller::biocLite()
• Installation dependent on OS (Rtools for Windows)

• 5 + 2 #addition
• 5 - 2 #subtraction
• 5 * 2 #multiplication
• 5 / 2 #division
• 5 ^ 2 #exponentiation
• 5 ** 2 #exponentiation
• 5 %% 2 #modulus (a.k.a. remainder)

• 5 < x #less than
• 5 <= x #less than or equal to
• 5 > x #greater than
• 5 >= x #greater than or equal to
• 5 == x #equal to
• 5 != x #not equal to
• !x #logical NOT
• True || False #stepwise logical OR
• True && False #stepwise logical AND

• Letters, numbers, dots, underscores
• Must start with a letter or a dot not followed by a number
• No reserve words, No spaces

x <- 5
x * 2

## [1] 10

x <- x + 1
y <- 4
x * y

## [1] 24

• numeric (set seed to sync random number generator):

set.seed(1)
rnorm(5)

## [1] -0.6264538  0.1836433 -0.8356286  1.5952808  0.3295078

• integer:

1:5

## [1] 1 2 3 4 5

sample( 1:5 )

## [1] 2 1 3 4 5

• character:

c("yes", "no")

## [1] "yes" "no"

• factor (play with this to show character/integer properties):

factor(c("yes", "no"))

## [1] yes no 
## Levels: no yes

• ordered factor:

factor(c("good", "very good", "poor"), 
       levels=c("poor", "good", "very good"), 
       ordered=TRUE)

## [1] good      very good poor     
## Levels: poor < good < very good

• logical:

1:5 %in% 4:5

## [1] FALSE FALSE FALSE  TRUE  TRUE

• Missing Values and others - IMPORTANT

c(NA, NaN, -Inf, Inf)

## [1]   NA  NaN -Inf  Inf

class() to find the class of a variable.

c( 1, "2", FALSE)

## [1] "1"     "2"     "FALSE"

c( 1, FALSE )

## [1] 1 0

• One element

x <- 1:4
x[ 2 ]

## [1] 2

• A slice of a vector

x <- 1:10
x[ 4:7 ]

## [1] 4 5 6 7

• Multiple elements ( not contiguous )

x <- c( "a", "b", "c", "d", "e", "f" )
x[ c(5,3,1) ]

## [1] "e" "c" "a"

• Removing elements

x[ -1 ]

## [1] "b" "c" "d" "e" "f"

x[-1:-2]

## [1] "c" "d" "e" "f"

• Using logical vector

x <- 1:10
y <- x%%2 == 0
x[y]

## [1]  2  4  6  8 10

matrix( 1:20, nrow = 5, ncol = 4 )

##      [,1] [,2] [,3] [,4]
## [1,]    1    6   11   16
## [2,]    2    7   12   17
## [3,]    3    8   13   18
## [4,]    4    9   14   19
## [5,]    5   10   15   20

• matrix[ r, c ]

boring.matrix <- matrix( 1:20, nrow = 5, ncol = 4 )
dim( boring.matrix )

## [1] 5 4

boring.matrix[ ,1 ]

## [1] 1 2 3 4 5

boring.matrix[ 2, 1 ]

## [1] 2

boring.matrix[ 2, ]

## [1]  2  7 12 17

boring.matrix

##      [,1] [,2] [,3] [,4]
## [1,]    1    6   11   16
## [2,]    2    7   12   17
## [3,]    3    8   13   18
## [4,]    4    9   14   19
## [5,]    5   10   15   20

boring.matrix[ boring.matrix[ ,1 ] ==3,]

## [1]  3  8 13 18

• Transpose

boring.matrix <- matrix(1:9, nrow = 3)
boring.matrix

##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9

t(boring.matrix)

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6
## [3,]    7    8    9

• Adding (note vectorization)

boring.matrix + 1

##      [,1] [,2] [,3]
## [1,]    2    5    8
## [2,]    3    6    9
## [3,]    4    7   10

boring.matrix + 1:3

##      [,1] [,2] [,3]
## [1,]    2    5    8
## [2,]    4    7   10
## [3,]    6    9   12

• Adding

boring.matrix

##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9

boring.matrix + boring.matrix

##      [,1] [,2] [,3]
## [1,]    2    8   14
## [2,]    4   10   16
## [3,]    6   12   18

• Multiplying

boring.matrix * boring.matrix

##      [,1] [,2] [,3]
## [1,]    1   16   49
## [2,]    4   25   64
## [3,]    9   36   81

boring.matrix %*% boring.matrix

##      [,1] [,2] [,3]
## [1,]   30   66  102
## [2,]   36   81  126
## [3,]   42   96  150

colnames(boring.matrix) <- c("col.1", "col.2", "col.3")
rownames(boring.matrix) <- c("row.1", "row.2", "row.3")
boring.matrix

##       col.1 col.2 col.3
## row.1     1     4     7
## row.2     2     5     8
## row.3     3     6     9

boring.matrix["row.1", ]

## col.1 col.2 col.3 
##     1     4     7

• e.g. if we have 5 medical measurements, 10 self-reported measurements, a sex, two parent names:

measurements <- c( 1.3, 1.6, 3.2, 9.8, 10.2 )
self.reporting <- c( 13, 6, 4, 7, 6, 5, 8, 9, 7, 4 )
sex <- FALSE
parents <- c( "Parent1.name", "Parent2.name" )

my.person <- list( measurements, self.reporting, 
                   sex, parents)
my.person

## [[1]]
## [1]  1.3  1.6  3.2  9.8 10.2
## 
## [[2]]
##  [1] 13  6  4  7  6  5  8  9  7  4
## 
## [[3]]
## [1] FALSE
## 
## [[4]]
## [1] "Parent1.name" "Parent2.name"

• Single bracket accessing

my.person[1:2]

## [[1]]
## [1]  1.3  1.6  3.2  9.8 10.2
## 
## [[2]]
##  [1] 13  6  4  7  6  5  8  9  7  4

• Double bracket accessing

my.person[[1]]

## [1]  1.3  1.6  3.2  9.8 10.2

my.person <- list( measure = measurements, 
                   parents = parents )
my.person

## $measure
## [1]  1.3  1.6  3.2  9.8 10.2
## 
## $parents
## [1] "Parent1.name" "Parent2.name"

my.person$parents

## [1] "Parent1.name" "Parent2.name"

• a data.frame is:
• a matrix with columns of potentially different data types, and
• a list with vector elements of equal length

x <- 11:16
y <- seq(0,1,.2)
z <- c( "one", "two", "three", "four", "five", "six" )
a <- factor( z )

test.dataframe <- data.frame(x,y,z,a)

test.dataframe[[4]]

## [1] one   two   three four  five  six  
## Levels: five four one six three two

test.dataframe$parents

## NULL

class( test.dataframe[[1]] )

## [1] "integer"

class( test.dataframe[[2]] )

## [1] "numeric"

class( test.dataframe[[3]] )

## [1] "factor"

• binding columns with common row names

mini.frame.one <- data.frame( "one" = 1:5 )
mini.frame.two <- data.frame( "two" = 6:10 )

cbind( mini.frame.one, mini.frame.two )

##   one two
## 1   1   6
## 2   2   7
## 3   3   8
## 4   4   9
## 5   5  10

Alternatively: c( mini.frame.one, mini.frame.two )

• rbind for binding rows ( with common column names )

test.dataframe[[1]]

## [1] 11 12 13 14 15 16

test.dataframe[[1]] = 21:26
test.dataframe

##    x   y     z     a
## 1 21 0.0   one   one
## 2 22 0.2   two   two
## 3 23 0.4 three three
## 4 24 0.6  four  four
## 5 25 0.8  five  five
## 6 26 1.0   six   six

• Bioconductor (www.bioconductor.org) defines the DataFrame
• like a data.frame but more flexible: columns can be any atomic vector type such as:
  • GenomicRanges objects
  • Rle (run-length encoding)

• Paths start at where you opened R in the terminal
• Home directory for RStudio
• "/" inside a folder
• "parent_folder/inside_folder"
• ".." move up one
• "../.." move up two
• getwd()
• setwd()

• Traditional:
  • read.table
  • read.csv
  • read.delim
• New:
  • biocLite(data.table) # fread() function for very large tables
  • biocLite(hadley/readr) # fast / convenient / advanced
  • biocLite(hadley/haven) # data from other statistical packages

• dplyr convention aims to ease cognitive burden
• Function names are easy to remember:

1. select (Y)
2. mutate/transmute (add Ys / new Y)
3. filter (get Xs based on condition)
4. slice (get Xs specified)
5. summarise (reduce to single observation)
6. arrange (re-order observations)

library(nycflights13)
library(dplyr)
delays <- flights %>% 
  filter(!is.na(dep_delay)) %>%
  group_by(year, month, day, hour) %>%
  summarise(delay = mean(dep_delay), n = n()) %>%
  filter(n > 10)

hist(delays$delay, main="Mean hourly delay", xlab="Delay (hours)")

1. Getting Started
2. dplyr exercises
3. random variables exercises