1. Introduction Introduction Basics Simple Statistics More on S - - PowerPoint PPT Presentation

Introduction Basics Simple Statistics More on S

Using R for Data Analysis and Graphics

1. Introduction

Introduction Basics Simple Statistics More on S What is R?

1.1 What is R?

R is a software environment for statistical computing. R is based on commands. Implements the S language. There is an inofficial menu based interface called R-Commander. Drawbacks of menus: difficult to store what you do. A script of commands

documents the analysis and allows for easy repetition with changed data, options, ...

R is free software. http://www.r-project.org Supported operating systems: Linux, Mac OS X, Windows Language for exchanging statistical methods among researchers

Introduction Basics Simple Statistics More on S Other Statistical Software

1.2 Other Statistical Software

S-Plus: same programming language, commercial. Features a GUI. SPSS: good for standard procedures. SAS: all-rounder, good for large data sets, complicated analyses. Systat: Analysis of Variance, easy-to-use graphics system. Excel: Very limited collection of statistical methods. Good for getting the dataset ready. Matlab: Mathematical methods. Statistical methods limited. Similar “paradigm”, less flexible structure.

Introduction Basics Simple Statistics More on S Introductory examples

1.3 Introductory examples

A dataset that we have stored before in the system is called

d.sport

weit kugel hoch disc stab speer punkte OBRIEN 7.57 15.66 207 48.78 500 66.90 8824 BUSEMANN 8.07 13.60 204 45.04 480 66.86 8706 DVORAK 7.60 15.82 198 46.28 470 70.16 8664 : : : : : : : : : : : : : : : : : : : : : : : : CHMARA 7.75 14.51 210 42.60 490 54.84 8249

Draw a histogram of the results of variable kugel ! We type

hist(d.sport[,"kugel"])

The graphics window is opened automatically. We have called the S-function hist with argument

d.sport[,"kugel"] . [,] is used to select the column.

Introduction Basics Simple Statistics More on S Introductory examples

1.3 Introductory examples

Scatter plot: type

plot(d.sport[,"kugel"], d.sport[,"speer"])

First argument: x coordinates; second: y coordinates Many optional arguments!

plot(d.sport[,"kugel"], d.sport[,"speer"], xlab="ball push", ylab="javelin", pch=7)

Scatter plot matrix

pairs(d.sport)

Every column of d.sport is plotted against all other columns.

Introduction Basics Simple Statistics More on S Introductory examples

1.3 Introductory examples

Get a dataset from a text file and assign it to a name:

d.sport <- read.table(...) "http://stat.ethz.ch/Teaching/Datasets /WBL/sport.dat", header=TRUE)

Start browser of operating system to get a file:

d.sport <- read.table(file....())

Introduction Basics Simple Statistics More on S Using R

1.4 Using R

Within a window running R, you will see the prompt >. You type a command and get a result and a new prompt.

> hist(d.sport[,"kugel"]) >

An incomplete statement can be continued on the next line

> plot(d.sport[,"kugel"], + d.sport[,"speer"])

R stores “objects” in your workspace

> d.sport <- read.table(...)

Objects have names like

a, fun, d.sport

R provides a huge number of functions and other objects

Introduction Basics Simple Statistics More on S Using R

1.4 Using R

An R statement consists of a name of an object − → object is displayed

> d.sport

a call to a function − → graphical or numerical result

> hist(d.sport[,"kugel"])

an assignment

> a <- 2*pi/360 > mn <- mean(d.sport[,"kugel"])

stores the mean of d.sport[,"kugel"] under the name mn

Introduction Basics Simple Statistics More on S Using R

1.4 Using R

Some special and useful functions (more details later): documentation on the arguments etc. of a function (or dataset provided by the system):

> help(hist)

• r

?hist

list all “objects” (names) in the workspace:

> objects()

leave the R session:

> q()

You get the question:

Save workspace image? [y/n/c]:

If you answer ”y”, your objects will be available for your next session.

Introduction Basics Simple Statistics More on S Scripts and Editors

1.5 Scripts and Editors

Instead of typing commands into the R window, you can generate commands by an editor and then “send” them to the R window. ... and later modify (correct) them and send again. Text Editors supporting R WinEdt: http://www.winedt.com/ Emacs: http://www.gnu.org/software/emacs/ ESS: http://stat.ethz.ch/ESS/ Tinn-R: http://www.sciviews.org/Tinn-R/

Introduction Basics Simple Statistics More on S Scripts and Editors

1.5 Scripts and Editors

The Tinn-R Window

Introduction Basics Simple Statistics More on S Scripts and Editors

1.5 Scripts and Editors

Define Tinn-R Keyboard Shortcuts: Use dialog R / Hotkeys of R

Introduction Basics Simple Statistics More on S

Using R for Data Analysis and Graphics

2. Basics

Introduction Basics Simple Statistics More on S Vectors

2.1 Vectors

Functions and operations are usually applied to whole “collections” instead of single numbers, including “vectors”, “matrices”, “data.frames” ( d.sport ) Numbers can be combined into “vectors” using the function c() (“combine”)

> t.v <- c(4,2,7,8,2) > t.a <- c(3.1, 5, -0.7, 0.9, 1.7) > t.u <- c(t.v,t.a) > t.u

Introduction Basics Simple Statistics More on S Vectors

2.1 Vectors

Generate a sequence of consecutive integers:

> seq(1, 9) [1] 1 2 3 4 5 6 7 8 9

Since sequences of integers are needed very often, this can be abbreviated to 1:9 . Equally spaced numbers: Use argument by (default: 1)

> seq(0, 3, by=0.5) [1] 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Repetition:

> rep(0.7, 5) [1] 0.7 0.7 0.7 0.7 0.7 > rep(c(1, 3, 5), length=8) [1] 1 3 5 1 3 5 1 3

Introduction Basics Simple Statistics More on S Vectors

2.1 Vectors

Basic functions for vectors: Call, Example Description

length(t.v)

Length of a vector, number of elements

sum(t.v)

Sum of all elements

mean(t.v)

arithmetic mean

var(t.v)

empirical variance

range(t.v)

range

Introduction Basics Simple Statistics More on S Arithmetic

2.2 Arithmetic

Simple arithmetic is as expected:

> 2+5 [1] 7

Operations:

+

• *

/ ˆ (Exponentiation)

These operations are applied to vectors elementwise.

> (2:5) ˆ c(2,3,1,0) [1] 4 27 4 1

Priorities as usual. Use parentheses!

> (2:5) ˆ 2 [1] 4 9 16 25

Introduction Basics Simple Statistics More on S Arithmetic

2.2 Arithmetic

Elements are recycled:

> (1:6)*(1:2) [1] 1 4 3 8 5 12 > (1:5)-(0:1) [1] 1 1 3 3 5 Warning message: longer object length is not a multiple

• f

shorter object length in: (1:5) - (0:1) > (1:6)-(0:1) [1] 1 1 3 3 5 5

Be careful, there is no warning in this case!

Introduction Basics Simple Statistics More on S Character Vectors

2.3 Character Vectors

Character strings:

"abc" , ’nut 999’

Combine strings into vector of “mode” character:

> t.names <- c("Urs", "Anna", "Max", "Pia")

Length of strings:

> nchar(t.names) [1] 3 4 3 5

String manipulations:

> substring(t.names,3,4)

[1] "s" "na" "x" "ud"

> paste(t.names,"Z.")

[1] "Urs Z." "Anna Z." "Max Z." "Pia Z."

> paste("X",1:3, sep="")

[1] "X1" "X2" "X3"

Introduction Basics Simple Statistics More on S Logical Vectors

2.4 Logical Vectors

Logical vectors contain elements TRUE

• r FALSE

> rep(c(TRUE, FALSE), length=6) [1] TRUE FALSE TRUE FALSE TRUE FALSE

• ften result from comparisons:

< <= > >= == != > (1:5)>=3 [1] FALSE FALSE TRUE TRUE TRUE

Logical operations: & (and), | (or), ! (not).

> t.i <- (t.a>2)&(t.a<5) > t.i [1] TRUE FALSE FALSE FALSE FALSE

Introduction Basics Simple Statistics More on S Selecting elements

2.5 Selecting elements

Select elements from vectors or data.frames: [ ] , [,]

> t.v[c(1,3,5)]

[1] 15.66 15.82 16.32

> d.sport[c(1,3,5),1:3]

weit kugel hoch OBRIEN 7.57 15.66 207 DVORAK 7.60 15.82 198 HAMALAINEN 7.48 16.32 198

For data.frames, use names of columns or rows:

> d.sport[c("OBRIEN","DVORAK"), c("kugel","speer","punkte")]

kugel speer punkte OBRIEN 15.66 66.90 8824 DVORAK 15.82 70.16 8664

Introduction Basics Simple Statistics More on S Selecting elements

2.5 Selecting elements

Using logical vectors:

> t.a[c(TRUE,FALSE,TRUE,TRUE,FALSE,FALSE)] [1] 3.1 -0.7 0.9 > d.sport[d.sport[,"kugel"] > 16, c(2,7)]

kugel punkte HAMALAINEN 16.32 8613 PENALVER 16.91 8307 SMITH 16.97 8271

Introduction Basics Simple Statistics More on S Matrices

2.6 Matrices

Matrices are “data tables” like data.frames, but they can

• nly contain data of a single type (numeric or character)

Generate a matrix:

> t.m1 <- matrix(1:10, nrow=2, ncol=5) > t.m1

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

> t.m2 <- matrix(1:10, ncol=2, + byrow=TRUE)

Transpose: t(t.m1) equals t.m2 .

Introduction Basics Simple Statistics More on S Matrices

2.6 Matrices

Selection of elements as with data.frames:

> t.m1[2,1:3] [1] 2 4 6

Matrix multiplication:

> t.m1 %*% t.m2

[,1] [,2] [1,] 95 220 [2,] 110 260 Vectors are treated as 1-row or 1-column matrices (mostly) Functions for linear algebra are available.

Introduction Basics Simple Statistics More on S

Using R for Data Analysis and Graphics

3. Simple Statistics

Introduction Basics Simple Statistics More on S Simple Statistical Functions

3.1 Simple Statistical Functions

Count number of cases with same value:

> table(d.blast[,"loc"])

L1 L2 L3 L4 L5 L6 14 10 14 10 24 24 Cross-table

> table(d.blast[,"loc"], + d.blast[,"loading"])

2.08 2.18 2.5 2.6 3.12 3.33 3.64 L1 2 2 1 5 1 2 1 L2 2 4 3 1 ...

Introduction Basics Simple Statistics More on S Simple Statistical Functions

3.1 Simple Statistical Functions

Estimation of a “location parameter”:

mean(x) median(x)

Variance:

var(x) ;

correlation:

> cor(d.sport[,"kugel"], d.sport[,"speer"])

Correlation matrix:

> t.cor <- cor(d.sport[,1:3]) > round(100*t.cor)

weit kugel hoch weit 100

• 63

34 kugel

• 63

100

• 9

hoch 34

• 9

100

Introduction Basics Simple Statistics More on S Hypothesis Tests

3.2 Hypothesis Tests

Do two groups differ in their “location”? − → Wilcoxon’s Rank Sum Test

> t.y1 <- sleep[sleep[,’group’]==1,’extra’] > t.y2 <- sleep[sleep[,’group’]==2,’extra’] > wilcox.test(t.y1, t.y2, paired=FALSE)

Wilcoxon rank sum test with continuity correction data: t.y1 and t.y2 W = 25.5, p-value = 0.06933 alternative hyp.: true location shift not equal to 0

Introduction Basics Simple Statistics More on S Hypothesis Tests

3.2 Hypothesis Tests

More well-known: t-test. Assumes normal distributions.

> t.test(t.y2,t.y1,alternative="two.sided", + paired=F)

Welch Two Sample t-test data: t.y1 and t.y2 t = -1.8608, df = 17.776, p-value = 0.0794 alternative hyp.: true diff. in means not equal to 0 95 percent confidence interval:

• 3.365

0.205 sample estimates: mean of x mean of y 0.75 2.33

− → Confidence interval!

Introduction Basics Simple Statistics More on S Two Groups

3.3 Two Groups

Plots for two samples of data.

> boxplot(t.y1,t.y2,ylab="extra") > plot(sleep[,"group"],sleep[,"extra"], + xlab="group", ylab="extra")

Introduction Basics Simple Statistics More on S Statistical Models, Formula Objects

3.4 Statistical Models, Formula Objects

Statistics is concerned with relations between “variables”. Prototype: Relationship between target variable Y and explanatory variables X1, X2, ... − → Regression. Symbolic notation of such a relation:

Y ˜ X1 + X2

This symbolic notation is an S object (of class formula ) (The notation is also used in other statistical packages.) Use of formula :

> plot(punkte ˜ kugel + speer, + data = d.sport)

gives 2 scatterplots, punkte (vertical) against

kugel and speer , respectively (horizontal axis).

Introduction Basics Simple Statistics More on S Statistical Models, Formula Objects

3.4 Statistical Models, Formula Objects

Grouping or nominal or categorical variables, e.g., location, type, group, species, plot, ... Role in models different from continuous variables − → S must know! − → stores them as factor s – Character variables enter data.frame as factor s – Grouping var. with numerical “labels” can be declared as factor

> sleep[,’group’] <- + factor(sleep[,’group’]) > plot(extra ˜ group, data = sleep)

produces two box plots.