An R Companion Package for the Book "A Course in Statistics with R"

Description

"A Course in Statistics with R" has been designed to meet the requirements of masters students.

Details

Author(s)

Prabhanjan Tattar

Maintainer: Prabhanjan Tattar <prabhanjannt@gmail.com>

References

Tattar, P. N., Suresh, R., and Manjunath, B. G. (2016). A Course in Statistics with R. J. Wiley.

Examples

hist(rnorm(100))

Simulation for Binomial Distribution

Description

A simple function to understand the algorithm to simulate psuedo-observations from binomial distribution. It is an implementation of the algorithm given in Section 11.3.1. This function is not an alternative to the rbinom function.

Usage

Binom_Sim(size, p, N)

Arguments

Note

This function is to simply explain the algorithm described in the text. For efficient results, the user should use the rbinom function.

Author(s)

Prabhanjan N. Tattar

See Also

rbinom

Examples

Binom_Sim(10,0.5,100)

Disease Outbreak Study

Description

The purpose of this health study is investigation of an epidemic outbreak due to mosquitoes. A random sample from two sectors of the city among the individuals has been tested to determine if the individual had contracted the disease forming the binary outcome.

Usage

data(Disease)

Format

A data frame with 98 observations on the following 5 variables.

x1

age

x2

socioeconomic status of three categories between x2 and x3

x3

socioeconomic status of three categories between x2 and x3

x4

sector of the city

y

if the individual had contracted the disease forming the binary outcome

References

Kutner, M. H., Nachtsheim, C. J., Neter, J., and Li, W. (1974-2005). Applied Linear Statistical Models, 5e. McGraw-Hill.

Examples

data(Disease)
DO_LR <- glm(y~.,data=Disease,family='binomial')
LR_Residuals <- data.frame(Y = Disease$y,Fitted = fitted(DO_LR),
Hatvalues = hatvalues(DO_LR),Response = residuals(DO_LR,"response"), Deviance = 
residuals(DO_LR,"deviance"), Pearson = residuals(DO_LR,"pearson"), 
Pearson_Standardized = residuals(DO_LR,"pearson")/sqrt(1-hatvalues(DO_LR)))
LR_Residuals

Generate transition probability matrix of Ehrenfest model

Description

The Ehrenfest model is an interesting example of a Markov chain. Though the probabilities in decimals are not as interesting as expressed in fractions, the function will help the reader generate the transition probability matrices of 2n balls among two urns.

Usage

Ehrenfest(n)

Arguments

Details

In this experiment there are i balls in Urn I, and remaining 2n-i balls in Urn II. Then at any instance, the probability of selecting a ball from Urn I and placing it in Urn II is i/2n, and the other way of placing a ball from Urn II to Urn I is (2n-i)/2n. At each instant we let the number i of balls in the Urn I to be the state of the system. Thus, the state space is S = 0, 1, 2, ..., 2n . Then we can pass from state i only to either of the states i-1 or i+1. Here, S = 0, 1, ..., 2n.

Author(s)

Prabhanjan N. Tattar

Examples

Ehrenfest(2)
Ehrenfest(3)

Simulation for Geometric Distribution

Description

A simple function to understand the algorithm to simulate (psuedo-)observations from binomial distribution. It is an implementation of the algorithm given in Section 11.3.1 "Simulation from Discrete Distributions", and as such the function is not an alternative to the "rgeom" function.

Usage

Geom_Sim(p, n)

Arguments

Details

To simulate a random number from geometric RV, we make use of the algorithm described in the book.

Author(s)

Prabhanjan N. Tattar

See Also

rgeom

Examples

mean(Geom_Sim(0.01,10))

Likelihood Ratio Test for Equality of Means when Variance Unknown

Description

This function sets up the likelihood ratio test for equality of means when the variance term is unknown. Refer Chapter 7 for more details.

Usage

LRNormal2Mean(x, y, alpha)

Arguments

Details

Likelihood ratio test is setup through this function. For more details, refer Chapter 7 of the book.

Author(s)

Prabhanjan N. Tattar

See Also

t.test

Examples

lisa <- c(234.26, 237.18, 238.16, 259.53, 242.76, 237.81, 250.95, 277.83)
mike <- c(187.73, 206.08, 176.71, 213.69, 224.34, 235.24)
LRNormal2Mean(mike,lisa,0.05)

Likelihood ratio test for equality of mean when the variance is known

Description

Likelihood ratio test for equality of mean when the variance is known for a sample from normal distribution is setup here. For details, refer Chapter 7 of the book.

Usage

LRNormalMean_KV(x, mu0, alpha, sigma)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

t.test

Examples

data(hw)
LRNormalMean_KV(hw$Height,mu0=70, alpha=0.05, sigma=sqrt(20))

Likelihood ratio test for mean when variance is unknown

Description

Likelihood ratio test for mean when variance is unknown for a sample from normal distribution is setup here.

Usage

LRNormalMean_UV(x, mu0, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

LRNormalMean_KV

Likelihood ratio test for the variance of normal distribution with mean is unknown

Description

This function returns the LR test for the variance of normal distribution with the mean being unknown. Refer Chapter 7 for more details.

Usage

LRNormalVariance_UM(x, sigma0, alpha)

Arguments

Author(s)

Prabhanjan Tattar

Examples

LRNormalVariance_UM(rnorm(20),1,0.05)

Most Powerful Test for Normal Distribution

Description

The most powerful test for a sample from normal distribution is given here. The test is obtained by an application of the Neyman-Pearson lemma.

Usage

MPNormal(mu0, mu1, sigma, n, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

t.test

Most Powerful Test for Poisson Distribution

Description

The most powerful test for a sample from Poisson distribution is given here. The test is obtained by an application of the Neyman-Pearson lemma.

Usage

MPPoisson(Hlambda, Klambda, alpha, n)

Arguments

Author(s)

Prabhanjan N. Tattar

Most Powerful Binomial Test

Description

The function returns the level alpha MP test for the testing the hypothesis H:p=p0 against K:p=p_1 as ensured by the application of Neyman-Pearson lemma.

Usage

MPbinomial(Hp, Kp, alpha, n)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

binom.test

Mucociliary Clearance

Description

Table 6.1 of Hollander and Wolfe (1999) lists the data for Half-Time of Mucociliary Clearance. We need to test if the time across various treatments is equal or not.

Usage

data(Mucociliary)

Format

A data frame with 14 observations on the following 2 variables.

Treatment

treatment levels Asbestosis Normal Subjects Obstructive Airways Disease

Time

half-time of mucociliary clearance

References

Hollander, M., and Wolfe, D. A. (1973-99). Nonparametric Statistical Methods, 2e. J. Wiley.

Examples

data(Mucociliary)
Mucociliary$Rank <- rank(Mucociliary$Time)
aggregate(Mucociliary$Rank,by=list(Mucociliary$Treatment),sum)
kruskal.test(Time~Treatment,data=Mucociliary)

Simulation for Poisson Distribution

Description

A simple function to understand the algorithm to simulate (psuedo-)observations from binomial distribution. It is an implementation of the algorithm given in Section 11.3.1 "Simulation from Discrete Distributions". This function is not an alternative to the "rpois" function.

Usage

Poisson_Sim(lambda, n)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

rpois

Examples

set.seed(123)
mean(Poisson_Sim(4,1000))

Quesenberry-Hurst Simultaneous Confidence Interval

Description

Quesenberry and Hurst (1964) have obtained the "simultaneous confidence intervals" for the vector of success in a multinomial distribution.

Usage

QH_CI(x, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

prop.test

Understanding Strength of Paper with a Three Factorial Experiment

Description

The strength of a paper depends on three variables: (i) the percentage of hardwood concentration in the raw pulp, (ii) the vat pressure, and (iii) the cooking time of the pulp. The hardwood concentration is tested at three levels of 2, 4, and 8 percentage, the vat pressure at 400, 500, and 650, while the cooking time is at 3 and 4 hours. For each combination of the these three factor variables, 2 observations are available, and thus a total of 3.3.2.2 = 36 observations. The goal of the study is investigation of the impact of the three factor variables on the strength of the paper, and the presence of interaction effect, if any.

Usage

data(SP)

Format

A data frame with 36 observations on the following 4 variables.

Hardwood

a factor with levels 2 4 8

Pressure

a factor with levels 400 500 650

Cooking_Time

a factor with levels 3 4

Strength

a numeric vector

References

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(SP)
summary(SP.aov <- aov(Strength~.^3,SP))

Simulating Random Observations from an Arbitrary Distribution

Description

An implementation of the algorithm for simulation of observations from an arbitrary discrete distribution is provided here.

Usage

ST_Ordered(N, x, p_x)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

sample

Examples

N <- 1e4
x <- 1:10
p_x <- c(0.05,0.17,0.02,0.14,0.11,0.06,0.05,0.04,0.17,0.19)
table(ST_Ordered(N, x, p_x))

Simulating Random Observations from an Arbitrary Distribution (ordered probabilities)

Description

Simulation observations from an arbitrary discrete distribution with probabilities arranged in desending/ascending order.

Usage

ST_Unordered(N, x, p_x)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

sample

Examples

N <- 1e2
x <- 1:10
p_x <- c(0.05,0.17,0.02,0.14,0.11,0.06,0.05,0.04,0.17,0.19)
ST_Unordered(N,x,p_x)

Trimmed Mean

Description

The trimean can be viewed as the average of median and average of the lower and upper quartiles. A fairly simply function is defined here.

Usage

TM(x)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

TMH, mean, median

Trimean based on hinges instead of quartiles

Description

The trimean is modified and defined based on hinges instead of the quartiles.

Usage

TMH(x)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

TM

Uniformly Most Powerful Test for Exponential Distribution

Description

A function is defined here which will return the uniformly most powerful test for exponential distribution. The function needs a simple use of the "qgamma" function.

Usage

UMPExponential(theta0, n, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

Uniformly Most Powerful Test for Normal Distribution

Description

The "UMPNormal" function returns the critical points required for the UMP test for a sample from normal distribution. The standard deviation is assumed to be known.

Usage

UMPNormal(mu0, sigma, n, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

Uniformly Most Powerful Test for Uniform Sample

Description

A simple and straightforward function for obtaining the UMP test for a random sample from uniform distribution.

Usage

UMPUniform(theta0, n, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

Examples

UMPUniform(0.6,10,0.05)

Wilson Confidence Interval

Description

The Wilson confidence interval for a sample from binomial distribution is a complex formula. This function helps the reader in easily obtaining the required confidence interval as discussed and detailed in Section 16.5.

Usage

WilsonCI(x, n, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

Examples

WilsonCI(x=10658,n=15000,alpha=0.05)
prop.test(x=10658,n=15000)$conf.int

Abrasion Index for the Tire Tread

Description

To understand the relationship between the abrasion index for the tire tread, the output y, as a linear function of the hydrated silica level x1, silane coupling agent level x2 and the sulfur level x3, Derringer and Suich (1980) collected data on 14 observation points.

Usage

data("abrasion_index")

Format

A data frame with 14 observations on the following 4 variables.

x1

hydrated silica level

x2

silane coupling agent level

x3

sulfur level

y

abrasion index for the tire tread

References

Derringer, G., and Suich, R. (1980). Simultaneous Optimization of Several Response Variables. Journal of Quality Technology, 12, 214-219.

Examples

data(abrasion_index)
ailm <- lm(y~x1+x2+x3,data=abrasion_index)
pairs(abrasion_index)

A Dataset for Factor Analysis

Description

The data set is obtained from Rencher (2002). Here, a 12-year old girl rates 7 of her acquaintances on a differential grade of 1-9 for five adjectives kind, intelligent, happy, likable, and just.

Usage

data(adjectives)

Format

A data frame with 7 observations on the following 6 variables.

People

a factor with levels FATHER FSM1a FSM2 FSM3 MSMb SISTER TEACHER

Kind

a numeric vector

Intelligent

a numeric vector

Happy

a numeric vector

Likeable

a numeric vector

Just

a numeric vector

References

Rencher, A.C. (2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(adjectives)
adjectivescor <- cor(adjectives[,-1])
round(adjectivescor,3)
adj_eig <- eigen(adjectivescor)
cumsum(adj_eig$values)/sum(adj_eig$values)
adj_eig$vectors[,1:2]
loadings1 <- adj_eig$vectors[,1]*sqrt(adj_eig$values[1])
loadings2 <- adj_eig$vectors[,2]*sqrt(adj_eig$values[2])
cbind(loadings1,loadings2)
communalities <- (adj_eig$vectors[,1]*sqrt(adj_eig$values[1]))^2+
(adj_eig$vectors[,2]*sqrt(adj_eig$values[2]))^2
round(communalities,3)
specific_variances <- 1-communalities
round(specific_variances,3)
var_acc_factors <- adj_eig$values
round(var_acc_factors,3)
prop_var <- adj_eig$values/sum(adj_eig$values)
round(prop_var,3)
cum_prop <- cumsum(adj_eig$values)/sum(adj_eig$values)
round(cum_prop,3)

Japanese atomic bomb survivors

Description

Gore, et al. (2006) consider the frequencies of cancer deaths of Japanese atomic bomb survivors by extent of exposure, years after exposure, etc. This data set has appeared in the journal "Statistical Sleuth".

Usage

data("atombomb")

Format

A data frame with 84 observations on the following 4 variables.

Radians

Extent of exposure to the radian levels

Count.Type

the type of count At Risk Death Count

Count.Age.Group

age group with levels '0-7' '12-15' '16-19' '20-23' '24-27' '28-41' '8-11'

Frequency

the count of deaths

References

Gore, A.P., Paranjape, S. A., and Kulkarni, M.B. (2006). 100 Data Sets for Statistics Education. Department of Statistics, University of Pune.

Examples

data(atombomb)
atombombxtabs <- xtabs(Frequency~Radians+Count.Type+Count.Age.Group,data=atombomb)
atombombxtabs

Two Factorial Experiment for Battery Data

Description

An experiment where the life of a battery is modeled as a function of the extreme variations in temperature of three levels 15, 70, and 1250 Fahrenheit and three type of plate material. Here, the engineer has no control on the temperature variations once the device leaves the factory. Thus, the task of the engineer is to investigate two major problems: (i) The effect of material type and temperature on the life of the device, and (ii) Finding the type of material which has least variation among the varying temperature levels. For each combination of the temperature and material, 4 replications of the life of battery are tested.

Usage

data(battery)

Format

A data frame with 36 observations on the following 3 variables.

Life

battery life

Material

the type of plate material

Temperature

three extreme variations of temperature

Source

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(battery)
names(battery) <- c("L","M","T")
battery$M <- as.factor(battery$M)
battery$T <- as.factor(battery$T)
battery.aov <- aov(L~M*T,data=battery)
model.matrix(battery.aov)
summary(battery.aov)

A Three Factorial Experiment for Bottling Data

Description

The height of the fills in the soft drink bottle is required to be as consistent as possible and it is controlled through three factors: (i) the percent carbonation of the drink, (ii) the operating pressure in the filler, and (iii) the line speed which is the number of bottles filled per minute. The first factor variable of the percent of carbonation is available at three levels of 10, 12, and 14, the operating pressure is at 25 and 30 psi units, while the line speed are at 200 and 250 bottles per minute. Two complete replicates are available for each combination of the three factor levels, that is, 24 total number of observations. In this experiment, the deviation from the required height level is measured.

Usage

data(bottling)

Format

A data frame with 24 observations on the following 4 variables.

Deviation

deviation from required height level

Carbonation

the percent carbonation of the drink

Pressure

the operating pressure in the filler

Speed

the number of bottles filled per minute

Source

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(bottling)
summary(bottling.aov <- aov(Deviation~.^3,bottling))
# Equivalent way
summary(aov(Deviation~ Carbonation + Pressure + Speed+ (Carbonation*Pressure)+
(Carbonation*Speed)+(Pressure*Speed)+(Carbonation*Speed*Pressure),data=bottling)) 

Simulated Sample from Binomial Distribution

Description

The data set is used to understand the sampling variation of the score function. The simulated data is available in Pawitan (2001).

Usage

data(bs)

Format

A data frame with 10 observations on the following 20 variables.

Sample.1

a numeric vector

Sample.2

a numeric vector

Sample.3

a numeric vector

Sample.4

a numeric vector

Sample.5

a numeric vector

Sample.6

a numeric vector

Sample.7

a numeric vector

Sample.8

a numeric vector

Sample.9

a numeric vector

Sample.10

a numeric vector

Sample.11

a numeric vector

Sample.12

a numeric vector

Sample.13

a numeric vector

Sample.14

a numeric vector

Sample.15

a numeric vector

Sample.16

a numeric vector

Sample.17

a numeric vector

Sample.18

a numeric vector

Sample.19

a numeric vector

Sample.20

a numeric vector

Source

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

References

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

Examples

data(bs)
n <- 10
sample_means <- colMeans(bs)
binomial_score_fn <- function(p,xbar)
      n*(xbar-10*p)/(p*(1-p))
p <- seq(from=0,to=1,by=0.02)
plot(p,sapply(p,binomial_score_fn,xbar=sample_means[1]),"l",xlab=expression(p),
ylab=expression(S(p)))
title(main="C: Score Function Plot of Binomial Model")
for(i in 2:20) lines(p,sapply(p,
binomial_score_fn,xbar=sample_means[i]),"l")
abline(v=4)
abline(h=0)

British Doctors Smoking and Coronary Heart Disease

Description

The problem is to investigate the impact of smoking tobacco among British doctors, refer Example 9.2.1 of Dobson. In the year 1951, a survey was sent across among all the British doctors asking them whether they smoked tobacco and their age group Age_Group. The data also collects the person-years Person_Years of the doctors in the respective age group. A follow-up after ten years reveals the number of deaths Deaths, the smoking group indicator Smoker_Cat.

Usage

data(bs1)

Format

A data frame with 10 observations on the following 9 variables.

Age_Group

a factor variable of age group with levels 35-44 45-54 55-64 65-74 75-84

Age_Cat

slightly re-coded to extract variables with Age_Cat taking values 1-5 respectively for the age groups 35-44, 45-54, 55-64, 65-74, and 75-84

Age_Square

square of the variable Age_Cat

Smoker_Cat

the smoking group indicator NO YES

Smoke_Ind

a numeric vector

Smoke_Age

takes the Age_Cat values for the smokers group and 0 for the non-smokers

Deaths

a follow-up after ten years revealing the number of deaths

Person_Years

the number of deaths standardized to 100000

Deaths_Per_Lakh_Years

a numeric vector

Source

Dobson (2002)

References

Dobson, A. J. (1990-2002). An Introduction to Generalized Linear Models, 2e. Chapman & Hall/CRC.

Examples

library(MASS)
data(bs1)
BS_Pois <-  glm(Deaths~Age_Cat+Age_Square+Smoke_Ind+Smoke_Age,offset=
log(Person_Years),data=bs1,family='poisson')
logLik(BS_Pois)
summary(BS_Pois)
with(BS_Pois, pchisq(null.deviance - deviance,df.null - 
df.residual,lower.tail = FALSE)) 
confint(BS_Pois)

The Cesarean Cases

Description

An increasing concern has been the number of cesarean deliveries, especially in the private hospitals. Here, we know the number of births, the type of hospital (private or Government hospital), and the number of cesareans. We would like to model the number of cesareans as a function of the number of births and the type of hospital. A Poisson regression model is fitted for this data set.

Usage

data(caesareans)

Format

A data frame with 20 observations on the following 3 variables.

Births

total number of births

Hospital_Type

type of hospital, private or government

Caesareans

number of caesareans

Source

http://www.oxfordjournals.org/our_journals/tropej/online/ma_chap13.pdf

Examples

data(caesareans)
names(caesareans)
cae_pois <- glm(Caesareans~Hospital_Type+Births,data=caesareans,family='poisson')
summary(cae_pois)

The Calcium in Soil

Description

Kramer and Jensen (1969) collected data on three variables at 10 different locations. The variables of interest are available calcium in the soil, y1, exchangeable soil calcium, y2, and turnip green calcium, y3. The hypothesis of interest is whether the mean vector is [15.0 6.0 2.85].

Usage

data(calcium)

Format

A data frame with 10 observations on the following 4 variables.

Location.Number

a numeric vector

y1

a numeric vector

y2

a numeric vector

y3

a numeric vector

Source

Kramer, C. Y., and Jensen, D. R. (1969). Fundamentals of Multivariate Analysis, Part I. Inference about Means. Journal of Quality Technology, 1 (2), 120-133.

References

Rencher, A.C. (1990-2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(calcium)
n <- nrow(calcium)
meanx <- colMeans(calcium[,-1])
varx <- var(calcium[,-1])
mu0 <- c(15,6,2.85)
t2 <- n*t(meanx-mu0)
t2

Car Data

Description

The data is used to show the effectiveness of Chernoff faces.

Usage

data(cardata)

Format

A data frame with 74 observations on the following 14 variables.

Model

various car models

P

Price

M

Mileage (in miles per gallon)

R78

Repair record 1978

R77

Repair record 1977

H

Headroom (in inches)

R

Rear seat clearance

Tr

Trunk space

W

Weight (in pound)

L

Length (in inches)

T

Turning diameter

D

Displacement (in cubic inches)

G

Gear ratio for high gear

C

Company headquarter

Examples

data(cardata)
pairs(cardata)

Coronary Heart Disease

Description

A well known explanation of the heart disease is that as the age increases, the risk of coronary heart disease also increase. The current data set and the example may be found in Chapter 1 of Hosmer and Lemeshow (1990-2013).

Usage

data(chdage)

Format

A data frame with 100 observations on the following 3 variables.

ID

patient ID

AGE

age of the patient

CHD

Coronary Heart Disease indicator

Source

Hosmer and Lemeshow (1990-2013).

References

Hosmer, D.W., and Lemeshow, S. (1990-20013). Applied Logistic Regression, 3e. Wiley.

Examples

data(chdage)
plot(chdage$AGE,chdage$CHD,xlab="AGE",ylab="CHD Indicator", 
main="Scatter plot for CHD Data")
agegrp <- cut(chdage$AGE,c(19,29,34,39,44,49,54,59,69),include.lowest=TRUE,
labels=c(25,seq(31.5,56.5,5),64.5))
mp <- c(25,seq(31.5,56.5,5),64.5) # mid-points
chd_percent <- prop.table(table(agegrp,chdage$CHD),1)[,2]
points(mp,chd_percent,"l",col="red")

Chemical Reaction Experiment

Description

This data set is used to illustrate the concept of canonical correlations. Here, temperature, concentration, and time have influence on three yield variables, namely outputs, while the percentage of unchanged starting material, the percentage converted to the desired product, and the percentage of unwanted by-product form another set of related variables.

Usage

data(chemicaldata)

Format

A data frame with 19 observations on the following 6 variables.

y1

the percentage of unchanged starting material

y2

the percentage converted to the desired product

y3

the percentage of unwanted by-product

x1

temperature

x2

concentration

x3

time

Source

Box, G. E. P., and Youle, P. V. (1955). The Exploration of Response Surfaces: An Example of the Link between the Fitted Surface and the Basic Mechanism of the System. Biometrics, 11, 287-323.

References

Rencher, A.C. (2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(chemicaldata)
names(chemicaldata)
chemicaldata$x12 <- chemicaldata$x1*chemicaldata$x2; 
chemicaldata$x13 <- chemicaldata$x1*chemicaldata$x3; 
chemicaldata$x23 <- chemicaldata$x2*chemicaldata$x3
chemicaldata$x1sq <- chemicaldata$x1^{2}
chemicaldata$x2sq <- chemicaldata$x2^{2}
chemicaldata$x3sq <- chemicaldata$x3^{2}
S_Total <- cov(chemicaldata)
cancor_xy <- sqrt(eigen(solve(S_Total[1:3,1:3])%*%S_Total[1:3,
4:12]%*%solve(S_Total[4:12,4:12])%*%S_Total[4:12,1:3])$values)
cancor_xy
cancor(chemicaldata[,1:3],chemicaldata[,4:12])

The Militiamen's Chest Dataset

Description

Militia means an army composed of ordinary citizens and not of professional soldiers. This data set is available in an 1846 book published by the Belgian statistician Adolphe Quetelet, and the data is believed to have been collected some thirty years before that.

Usage

data(chest)

Format

A data frame with 16 observations on the following 2 variables.

Chest

chest width measured in inches

Count

the corresponding frequencies

References

Velleman, P.F., and Hoaglin, D.C. (2004). ABC of Exploratory Data Analysis. Duxbury Press, Boston.

Examples

data(chest)
attach(chest)
names(chest)
militiamen <- rep(Chest,Count)
length(militiamen)
bins <- seq(33,48)
bins
bin.mids <- (bins[-1]+bins[-length(bins)])/2
par(mfrow=c(1,2))
h <- hist(militiamen, breaks = bins, xlab= "Chest Measurements (Inches)",
main= "A: Histogram for the Militiamen")
h$counts <- sqrt(h$counts)
plot(h,xlab= "Chest Measurements (Inches)",ylab= "ROOT FREQUENCY",
main= "B: Rootogram for the Militiamen")

The Cloud Seeding Data

Description

Chambers, et al. (1983), page 381, contains the cloud seeding data set. Rainfall in acre-feet for 52 clouds are measured, 50% of which have natural rain (control group) whereas the others are seeded. We need to visually compare whether seeding the clouds lead to increase in rainfall in acre-feet.

Usage

data(cloud)

Format

A data frame with 26 observations on the following 2 variables.

Control

Rainfall in acre-feet for 26 clouds are measured which had natural rain, that is, control group

Seeded

Rainfall in acre-feet for 26 clouds are measured which had seeded rain

References

Chambers, J.M., Cleveland, W.S., Kleiner, B., and Tukey, P.A. (1983). Graphical Methods for Data Analysis. Wadsworth and Brooks/Cole.

Examples

data(cloud)
stem(log(cloud$Seeded),scale=1)
stem(log(cloud$Control),scale=1)

The Cork Dataset

Description

Thickness of cork borings in four directions of North, South, East, and West are measured for 28 trees. The problem here is to examine if the bark deposit is same in all the directions.

Usage

data(cork)

Format

A data frame with 28 observations on the following 4 variables.

North

thickness of cork boring in the North direction

East

thickness of cork boring in the East direction

South

thickness of cork boring in the South direction

West

thickness of cork boring in the West direction

References

Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2e. J. Wiley.

Examples

data(cork)
corkcent <- cork*0
corkcent[,1] <- cork[,1]-mean(cork[,1])
corkcent[,2] <- cork[,2]-mean(cork[,2])
corkcent[,3] <- cork[,3]-mean(cork[,3])
corkcent[,4] <- cork[,4]-mean(cork[,4])
corkcentsvd <- svd(corkcent)
t(corkcentsvd$u)%*%corkcentsvd$u
t(corkcentsvd$v)%*%corkcentsvd$v
round(corkcentsvd$u %*% diag(corkcentsvd$d) %*% t(corkcentsvd$v),2)
round(corkcent,2)
corkcentsvd$d

Random Samples from Cauchy Distribution

Description

The data set is used to understand the sampling variation of the score function. The simulated data is available in Pawitan (2001).

Usage

data(cs)

Format

A data frame with 10 observations on the following 20 variables.

Sample.1

a numeric vector

Sample.2

a numeric vector

Sample.3

a numeric vector

Sample.4

a numeric vector

Sample.5

a numeric vector

Sample.6

a numeric vector

Sample.7

a numeric vector

Sample.8

a numeric vector

Sample.9

a numeric vector

Sample.10

a numeric vector

Sample.11

a numeric vector

Sample.12

a numeric vector

Sample.13

a numeric vector

Sample.14

a numeric vector

Sample.15

a numeric vector

Sample.16

a numeric vector

Sample.17

a numeric vector

Sample.18

a numeric vector

Sample.19

a numeric vector

Sample.20

a numeric vector

References

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

Examples

data(cs)
n <- 10
cauchy_score_fn  <-  function(mu,x)
      sum(2*(x-mu)/(1+(x-mu)^{2}))
mu <- seq(from=-15,to=20,by=0.5)
plot(mu,sapply(mu,cauchy_score_fn,x=cs[,1]),"l",xlab=expression(mu),
ylab=expression(S(mu)),ylim=c(-10,10))
title(main="D: Score Function Plot of Cauchy Model")
for(i in 2:20) lines(mu,sapply(mu,
cauchy_score_fn,x=cs[,i]),"l")
abline(v=4)
abline(h=0)

The Hamilton Depression Scale Factor

Description

Hamilton depression scale factor IV is a measurement of mixed anxiety and depression and it is named after its inventor. In a double-blind experiment, this scale factor is obtained for 9 patients on their entry in a study, denoted by X. Post a tranquilizer T, the scale factor IV is again obtained for the same set of patients, which is denoted by Y. Here, an improvement due to tranquilizer T corresponds to a reduction in factor IV values.

Usage

data(depression)

Format

A data frame with 9 observations on the following 3 variables.

Patient_No

Patient ID

X

measurement of depression at entry in a study

Y

measurement of depression post a tranquilizer

References

Sheshkin, D. J. (1997-2011). Handbook of Parametric and Nonparametric Statistical Procedures, 5e. Chapman and Hall/CRC.

Examples

data(depression)
attach(depression)
names(depression)
wilcox.test(Y-X, alternative = "less")
wilcox.test(Y-X, alternative = "less",exact=FALSE,correct=FALSE)

Injuries in Airflights

Description

Injuries in airflights, road accidents, etc, are instances of rare occurrences which are appropriately modeled by a Poisson distribution. Two models, before and after transformation, are fit and it is checked if the transformation led to a reduction to the variance.

Usage

data(flight)

Format

A data frame with 9 observations on the following 2 variables.

Injury_Incidents

Count of injury incidents

Total_Flights

Total number of flights

References

Chatterjee, S., and Hadi, A. S. (1977-2006). Regression Analysis by Examples, 4e. J. Wiley.

Examples

data(flight)
names(flight)
injurylm <- lm(Injury_Incidents~Total_Flights,data=flight)
injurysqrtlm <- lm(sqrt(Injury_Incidents)~Total_Flights,data=flight)
summary(injurylm)
summary(injurysqrtlm)

Strength Data Set of a Girder Experiment

Description

The shear strength of steel plate girders need to be modeled as a function of the four methods and nine girders.

Usage

data(girder)

Format

A data frame with 9 observations on the following 5 variables.

Girder

The row names, varying from S1.1 to S4.2, represent the nine type of girders, S1.1 S1.2 S2.1 S2.2 S3.1 S3.2 S4.1 S4.2 S5.1

Aarau

one of the four methods of preparation of the steel plates

Karisruhe

one of the four methods of preparation of the steel plates

Lehigh

one of the four methods of preparation of the steel plates

Cardiff

one of the four methods of preparation of the steel plates

References

Wu, C.F.J. and M. Hamada (2000-9). Experiments: Planning, Analysis, and Parameter Design Optimization, 2e. J. Wiley.

Examples

data(girder)
girder
boxplot(girder[,2:5])

Hardness and a Block Experiment

Description

Four types of tip are used which form the blocks in this experiment. The variable of interest is the hardness which further depends on the type of metal coupon. For each type of the tip, the hardness is observed for 4 different types the metal coupon.

Usage

data(hardness)

Format

A data frame with 16 observations on the following 3 variables.

Tip_Type

Four types of tip which form the blocks

Test_Coupon

Four different types of metal coupons

Hardness

Hardness of the coupon

References

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(hardness)
hardness$Tip_Type <- as.factor(hardness$Tip_Type)
hardness$Test_Coupon <- as.factor(hardness$Test_Coupon)
hardness_aov <- aov(Hardness~Tip_Type+Test_Coupon,data=hardness)
summary(hardness_aov)

Hearing Loss Data

Description

A study was carried in the Eastman Kodak Company which involved the measurement of hearing loss. Such studies are called as audiometric study. This data set contains 100 males, each aged 39, who had no indication of noise exposure or hearing disorders. Here, the individual is exposed to a signal of a given frequency with an increasing intensity till the signal is perceived.

Usage

data(hearing)

Format

A data frame with 100 observations on the following 9 variables.

Sl_No

Serial Number

L500

Observation for 500Hz in the left ear

L1000

Observation for 1000Hz in the left ear

L2000

Observation for 2000Hz in the left ear

L4000

Observation for 4000Hz in the left ear

R500

Observation for 500Hz in the right ear

R1000

Observation for 1000Hz in the right ear

R2000

Observation for 2000Hz in the right ear

R4000

Observation for 4000Hz in the right ear

References

Jackson, J.E. (1991). A User's Guide to Principal Components. New York: Wiley.

Examples

data(hearing)
round(cor(hearing[,-1]),2)
round(cov(hearing[,-1]),2)
hearing.pc <- princomp(hearing[,-1])
screeplot(hearing.pc,main="B: Scree Plot for Hearing Loss Data")

Height-Weight Covariance Study

Description

The data set highlights the importance of handling covariance when such information is available. If the covariance is not incorporated, hypothesis testing may lead to entirely difference conclusion.

Usage

data(hw)

Format

A data frame with 20 observations on the following 2 variables.

Height

the height of an individual

Weight

the weight of an individual

References

Rencher, A.C. (2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(hw)
sigma0 <- matrix(c(20, 100, 100, 1000),nrow=2)
sigma <- var(hw)
v <- nrow(hw)-1
p <- ncol(hw)
u <- v*(log(det(sigma0))-log(det(sigma)) + sum(diag(sigma%*%solve(sigma0)))-p)
u1 <- (1- (1/(6*v-1))*(2*p+1 - 2/(p+1)))*u
u;u1;qchisq(1-0.05,p*(p+1)/2)

Insurance Claims Data

Description

Montgomery (2005), page 42, describes this data set in which the number of days taken by the company to process and settle the claims of employee health insurance customers. The data is recorded for the number of days for settlement from the first to fortieth claim.

Usage

data(insurance)

Format

A data frame with 40 observations on the following 2 variables.

Claim

Claim number

Days

Days to settle the claim amount

References

Montgomery, D.C. (1985-2012). Introduction to Statistical Quality Control, 7e. J. Wiley.

Examples

data(insurance)
plot(insurance$Claim,insurance$Days,"l",xlab="Claim Sequence",
   ylab="Time to Settle the Claim")
title("B: Run Chart for Insurance Claim Settlement")

Blocking for Intensity Data Set

Description

The intent of this experiment is to help the engineer in improving the ability of detecting targets on a radar system. The two variables chosen which are believed to have the most impact on the detecting abilities of the radar system are marked as the amount of the background noise and the type of filter on the screen.

Usage

data(intensity)

Format

A data frame with 24 observations on the following 4 variables.

Intensity

intensity of targets

Operator

different operators who form the blocks 1 2 3 4

Filter

two types of filter 1 2

Ground

the type of background noise high low medium

References

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(intensity)
intensity.aov <- aov(Intensity~Ground*Filter+Error(Operator),intensity)
summary(intensity.aov)
intensity.aov

Coefficient of Kurtosis

Description

A simple function to obtain the coefficient of kurtosis on numeric variables.

Usage

kurtcoeff(x)

Arguments

Details

A straight-forward implementation of the formula is give here. A complete function "kurtosis" is available in the "e1071" package.

Author(s)

Prabhanjan N. Tattar

See Also

e1071::kurtosis

Life Expectancies

Description

This data set consists of life expectancy in years by country, age, and sex.

Usage

data(life)

Format

A data frame with 31 observations on the following 8 variables.

m0

life expectancy for males at age 0

m25

life expectancy for males at age 25

m50

life expectancy for males at age 50

m75

life expectancy for males at age 75

w0

life expectancy for females at age 0

w25

life expectancy for females at age 25

w50

life expectancy for females at age 50

w75

life expectancy for females at age 75

References

Everitt, B. S., and Hothorn, T. (2011). An Introduction to Applied Multivariate Analysis with R. Springer.

Examples

data(life)
factanal(life,factors=1)$PVAL
factanal(life,factors=2)$PVAL
factanal(life,factors=3)

The Low-Birth Weight Problem

Description

Low birth weight of new-born infants is a serious concern. If the weight of the new-born is less than 2500 grams, we consider that instance as a low-birth weight case. A study was carried out at Baystate Medical Center in Springfield, Massachusetts.

Usage

data(lowbwt)

Format

A data frame with 189 observations on the following 10 variables.

LOW

Low Birth Weight

AGE

Age of Mother

LWT

Weight of Mother at Last Menstrual Period

RACE

Race 1 2 3

SMOKE

Smoking Status During Pregnancy

PTL

History of Premature Labor

HT

History of Hypertension

UI

Presence of Uterine Irritability

FTV

Number of Physician Visits During the First Trimester

BWT

Birth Weight

References

Hosmer, D.W., and Lemeshow, S. (1989-2000). Applied Logistic Regression, 2e. J. Wiley.

Examples

data(lowbwt)
lowglm <- glm(LOW~AGE+LWT+RACE+FTV,data=lowbwt,family='binomial') 
lowglm$coefficients

Letter Values

Description

This function is adapted from Prof. Jim Albert's "LearnEDA" package. It returns the letter values as discussed in Chapter 4.

Usage

lval(x, na.rm = TRUE)

Arguments

Author(s)

Prabhanjan Tattar

See Also

LearnEDA

Memory Recall Times

Description

A test had been conducted with the purpose of investigating if people recollect pleasant memories associated with a word earlier than some unpleasant memory related with the same word. The word is flashed on the screen and the time an individual takes to respond via keyboard is recorded for both type of the memories.

Usage

data(memory)

Format

A data frame with 20 observations on the following 2 variables.

Pleasant.memory

time to recollect pleasant memory

Unpleasant.memory

time to recollect unpleasant memory

References

Dunn, and Master. (1982). Obtained from

http://openlearn.open.ac.uk/mod/resource/view.php?id=165509

Examples

data(memory)
lapply(memory,fivenum)
lapply(memory,mad)
lapply(memory,IQR)

Psychological Tests for Males and Females

Description

A psychological study consisting of four tests was conducted on males and females group and the results were noted. Since the four tests are correlated and each one is noted for all the individuals, one is interested to know if the mean vector of the test scores is same across the gender group.

Usage

data(mfp)

Format

A data frame with 32 observations on the following 8 variables.

M_y1

pictorial inconsistencies for males

M_y2

paper form board test for males

M_y3

tool recognition test for males

M_y4

vocabulary test for males

F_y1

pictorial inconsistencies for females

F_y2

paper form board test for females

F_y3

tool recognition test for females

F_y4

vocabulary test for females

Examples

data(mfp)
males <- mfp[,1:4]; females <- mfp[,5:8]
nm <- nrow(males);nf <- nrow(females)
p <- 4; k <- 2
vm <- nm-1; vf <- nf-1
meanm <- colMeans(males); meanf <- colMeans(females)
sigmam <- var(males); sigmaf <- var(females)
sigmapl <- (1/(nm+nf-2))*((nm-1)*sigmam+(nf-1)*sigmaf)
ln_M <- .5*(vm*log(det(sigmam))+vf*log(det(sigmaf))) -.5*(vm+vf)*log(det(sigmapl))
exact_test <- -2*ln_M # the Exact Test
exact_test

m-step Transition Probability Matrix Computation

Description

The m-step transition probability matrix computation is provided in this function. The equation is based on the well-known "Chapman-Kolmogorov equation".

Usage

msteptpm(TPM, m)

Arguments

Author(s)

Prabhanjan N. Tattar

Examples

EF2 <- Ehrenfest(2)
msteptpm(as.matrix(EF2),4)

The Nerve Data

Description

The Nerve data set has been popularized by Cox and Lewis (1966). In this experiment 799 waiting times are recorded for successive pulses along a nerve fiber.

Usage

data(nerve)

Format

The format is: num [1:799] 0.21 0.03 0.05 0.11 0.59 0.06 0.18 0.55 0.37 0.09 ...

Source

Cox, D. and Lewis, P. (1966). The Statistical Analysis of Series of Events. Chapman & Hall.

Examples

data(nerve)
nerve_ecdf <- ecdf(nerve)
knots(nerve_ecdf) # Returns the jump points of the edf
summary(nerve_ecdf) # the usual R summaries
nerve_ecdf(nerve) # returns the percentiles at the data points

Simulated Sample from Normal Distribution

Description

The data set is used to understand the sampling variation of the score function. The simulated data is available in Pawitan (2001).

Usage

data(ns)

Format

A data frame with 10 observations on the following 20 variables.

Sample.1

a numeric vector

Sample.2

a numeric vector

Sample.3

a numeric vector

Sample.4

a numeric vector

Sample.5

a numeric vector

Sample.6

a numeric vector

Sample.7

a numeric vector

Sample.8

a numeric vector

Sample.9

a numeric vector

Sample.10

a numeric vector

Sample.11

a numeric vector

Sample.12

a numeric vector

Sample.13

a numeric vector

Sample.14

a numeric vector

Sample.15

a numeric vector

Sample.16

a numeric vector

Sample.17

a numeric vector

Sample.18

a numeric vector

Sample.19

a numeric vector

Sample.20

a numeric vector

Source

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

References

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

Examples

library(stats4)
data(ns)
x <- ns[,1]
nlogl <- function(mean,sd) { -sum(dnorm(x,mean=mean,sd=sd,log=TRUE)) }
norm_mle <- mle(nlogl,start=list(mean=median(x),sd=IQR(x)),nobs=length(x))
summary(norm_mle)

The Olson Heart Lung Dataset

Description

We need to determine the effect of the number of revolutions per minute (rpm) of the rotary pump head of an Olson heart-lung pump on the fluid flow rate Liters_minute. The rpm's are replicated at 50, 75, 100, 125, and 150 levels with respective frequencies 5, 3, 5, 2, and 5. The fluid flow rate is measured in litters per minute.

Usage

data(olson)

Format

A data frame with 20 observations on the following 4 variables.

Observation

observation number

rpm

rmp levels at 50, 75, 100, 125, and 150

Level

the rpm levels

Liters_minute

litters per minute

References

Dean, A., and Voss, D. (1999). Design and Analysis of Experiments. Springer.

Examples

data(olson)
par(mfrow=c(2,2))
plot(olson$rpm,olson$Liters_minute,xlim=c(25,175),xlab="RPM",
  ylab="Flow Rate",main="Scatter Plot")
boxplot(Liters_minute~rpm,data=olson,main="Box Plots")
aggregate(olson$Liters_minute,by=list(olson$rpm),mean)
olson_crd <- aov(Liters_minute ~ as.factor(rpm), data=olson)

Pareto density

Description

A simple function is given here which returns the density function values for a Pareto RV. A more efficient implementation is obtainable in the function "dpareto" from the "VGAM" package.

Usage

pareto_density(x, scale, shape)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

VGAM::dpareto

Examples

m <- 9184
n <- 103
b <- 10000
K <- 10
theta <- seq(1000,20000,500)
plot(theta,as.numeric(sapply(theta,pareto_density,scale=b,shape=K)),"l",
     xlab=expression(theta),ylab="The Posterior Density")
(n+1)*m/n

Quantile of Pareto RV

Description

A simple function is given here which returns the quantiles for a Pareto RV. A more efficient implementation is obtainable in the function "qpareto" from the "VGAM" package.

Usage

pareto_quantile(p, scale, shape)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

VGAM::qpareto

Examples

pareto_quantile(c(0.05,0.95),scale=10000,shape=10)

A Function to Plot the Power of a UMP Test for Normal Distribution

Description

A simple function for obtaining the plot of power of UMP test.

Usage

powertestplot(mu0, sigma, n, alpha)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

t.test

Examples

UMPNormal <- function(mu0, sigma, n,alpha)  {
  qnorm(alpha)*sigma/sqrt(n)+mu0
}
UMPNormal(mu0=0, sigma=1,n=1,alpha=0.5)
powertestplot <- function(mu0,sigma,n,alpha)	{
  mu0seq <- seq(mu0-3*sigma, mu0+3*sigma,(6*sigma/100))
  betamu <- pnorm(sqrt(n)*(mu0seq-mu0)/sigma-qnorm(1-alpha))
  plot(mu0seq,betamu,"l",xlab=expression(mu),ylab="Power of UMP Test",
    main = expression(paste("H:",mu <= mu[0]," vs K:",mu>mu[0])))
  abline(h=alpha)
  abline(v=mu0)
}
powertestplot(mu0=0,sigma=1,n=10,alpha=0.05)
# H:mu > mu_0 vs K: mu <= mu_0
UMPNormal <- function(mu0, sigma, n,alpha)	{
  mu0-qnorm(alpha)*sigma/sqrt(n)
}
UMPNormal(mu0=0, sigma=1,n=1,alpha=0.5)
powertestplot <- function(mu0,sigma,n,alpha)	{
  mu0seq <- seq(mu0-3*sigma, mu0+3*sigma,(6*sigma/100))
  betamu <- pnorm(sqrt(n)*(mu0-mu0seq)/sigma-qnorm(1-alpha))
  plot(mu0seq,betamu,"l",xlab=expression(mu),ylab="Power of UMP Test",
    main=expression(paste("H:",mu >= mu[0]," vs K:",mu<mu[0])))
  abline(h=alpha)
  abline(v=mu0)
}
powertestplot(mu0=0,sigma=1,n=10,alpha=0.05)

Simulated Sample from Poisson Distribution

Description

The data set is used to understand the sampling variation of the score function. The simulated data is available in Pawitan (2001).

Usage

data(ps)

Format

A data frame with 10 observations on the following 20 variables.

Sample.1

a numeric vector

Sample.2

a numeric vector

Sample.3

a numeric vector

Sample.4

a numeric vector

Sample.5

a numeric vector

Sample.6

a numeric vector

Sample.7

a numeric vector

Sample.8

a numeric vector

Sample.9

a numeric vector

Sample.10

a numeric vector

Sample.11

a numeric vector

Sample.12

a numeric vector

Sample.13

a numeric vector

Sample.14

a numeric vector

Sample.15

a numeric vector

Sample.16

a numeric vector

Sample.17

a numeric vector

Sample.18

a numeric vector

Sample.19

a numeric vector

Sample.20

a numeric vector

Source

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

References

Pawitan, Y. (2001). In All Likelihood. Oxford Science Publications.

Examples

data(ps)
n <- 10
sample_means <- colMeans(ps)
poisson_score_fn <- function(theta,xbar) n*(xbar-theta)/theta
theta <- seq(from=2,to=8,by=0.2)
plot(theta,sapply(theta,poisson_score_fn,xbar=sample_means[1]),"l",xlab=
  expression(lambda),ylab=expression(S(lambda)),ylim=c(-5,15))
title(main="B: Score Function Plot of the Poisson Model")
for(i in 2:20) 
lines(theta,sapply(theta,poisson_score_fn,xbar=sample_means[i]),"l")
abline(v=4)
abline(h=0)

The Linguistic Probe Word Analysis

Description

Probe words are used to test the recall ability of words in various linguistic contexts. In this experiment the response time to five different probe words are recorded for 11 individuals. The interest in the experiment is to examine if the response times to the different words are independent or not.

Usage

data(pw)

Format

A data frame with 11 observations on the following 6 variables.

Subject.Number

a numeric vector

y1

a numeric vector

y2

a numeric vector

y3

a numeric vector

y4

a numeric vector

y5

a numeric vector

References

Rencher, A.C. (2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(pw)
sigma <- var(pw[2:6])
p <- ncol(pw)-1; v <- nrow(pw)-1
u <- p^p*(det(sigma))/(sum(diag(sigma))^p)
u1 <- -(v-(2*p^2+p+2)/(6*p))*log(u)
u;u1

Chemical Reaction Experiment

Description

For a chemical reaction experiment, the blocks arise due to the Batch number, Catalyst of different types form the treatments, and the reaction time is the output. Due to a restriction, all the catalysts cannot be analysed within each batch and hence we need to look at the BIBD model.

Usage

data("reaction")

Format

A data frame with 16 observations on the following 3 variables.

Catalyst

different types forming the treatments

Batch

batch number

Reaction

reaction time

Examples

data(reaction)

Resistant Line EDA Regression Technique

Description

"Resistant Line" is an important EDA way of fitting a regression model. The function here develops the discussion in Section 4.5.1 Resistant Line. An alternative for this function is available in "rline" function of the "LearnEDA" package.

Usage

resistant_line(x, y, iterations)

Arguments

Author(s)

Prabhanjan N. Tattar

References

Velleman, P.F., and Hoaglin, D.C. (2004). ABC of Exploratory Data Analysis. Duxbury Press, Boston. Republished in 2004 by The Internet-First University Press.

See Also

LearnEDA::rline

Rocket Propellant

Description

Five different formulations of a rocket propellant x1 may be used in an aircrew escape systems on the observed burning rate Y. Here, each of the formulation is prepared by mixing from a batch of raw materials x2 which can support only five formulations required for the purpose of testing.

Usage

data(rocket)

Format

A data frame with 25 observations on the following 4 variables.

y

burning rate

batch

raw materials batch

op

experience of the operator

treat

formulation type of the propellant A B C D E

References

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(rocket)
matrix(rocket$treat,nrow=5)
par(mfrow=c(1,3))
plot(y~factor(op)+factor(batch)+treat,rocket)
rocket_aov <- aov(y~factor(op)+factor(batch)+treat,rocket)

Rocket Propellant Example Extended

Description

In continuation of Example 13.4.7 of the Rocket Propellant data, we now have the added blocking factor in test assemblies.

Usage

data(rocket_Graeco)

Format

A data frame with 25 observations on the following 5 variables.

y

burning rate

batch

raw materials batch

op

experience of the operator

treat

formulation type of the propellant A B C D E

assembly

test assemblies a b c d e

References

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(rocket_Graeco)
plot(y~op+batch+treat+assembly,rocket_Graeco)
rocket.glsd.aov <- aov(y~factor(op)+factor(batch)+treat+assembly,rocket_Graeco)
summary(rocket.glsd.aov)

Apple of Different Rootstock

Description

The goal is to test if the mean vector of the four variables is same across 6 stratas of the experiment.

Usage

data(rootstock)

Format

A data frame with 48 observations on the following 5 variables.

rootstock

Six different rootstocks

y1

trunk girth at 4 years

y2

extension growth at 4 years

y3

trunk girth at 15 years

y4

weight of tree above ground at 15 years

References

Rencher, A.C. (2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(rootstock)
attach(rootstock)
rs <- rootstock[,1]
rs <- factor(rs,ordered=is.ordered(rs)) # Too important a step
root.manova <- manova(cbind(y1,y2,y3,y4)~rs)
summary(root.manova, test = "Wilks")

Simulated Dataset

Description

In the data set sample, we have data from five different probability distributions. Histograms are used to intuitively understand the underlying probability model.

Usage

data(sample)

Format

A data frame with 100 observations on the following 5 variables.

Sample_1

A sample 1

Sample_2

A sample 2

Sample_3

A sample 3

Sample_4

A sample 4

Sample_5

A sample 5

Examples

data(sample)
layout(matrix(c(1,1,2,2,3,3,0,4,4,5,5,0), 2, 6, byrow=TRUE),respect=FALSE) 
matrix(c(1,1,2,2,3,3,0,4,4,5,5,0), 2, 6, byrow=TRUE)
hist(sample[,1],main="Histogram of Sample 1",xlab="sample1", ylab="frequency")
hist(sample[,2],main="Histogram of Sample 2",xlab="sample2", ylab="frequency")
hist(sample[,3],main="Histogram of Sample 3",xlab="sample3", ylab="frequency")
hist(sample[,4],main="Histogram of Sample 4",xlab="sample4", ylab="frequency")
hist(sample[,5],main="Histogram of Sample 5",xlab="sample5", ylab="frequency")

The Seishu Wine Study

Description

The odor and taste of wines are recorded in a study. It is believed that the variables such as the pH concentration, alcohol content, total sugar, etc, explain the odor and taste of the wine.

Usage

data(sheishu)

Format

A data frame with 30 observations on the following 10 variables.

Taste

taste

Odor

odor

pH

pH concentration

Acidity_1

Acidity 1

Acidity_2

Acidity 2

Sake_meter

Sake meter

Direct_reducing_sugar

Direct reducing sugar

Total_sugar

Total sugar

Alcohol

type of alcohol

Formyl_nitrogen

Formyl nitrogen

References

Rencher, A.C. (2002). Methods of Multivariate Analysis, 2e. J. Wiley.

Examples

data(sheishu)
noc <- c(2,3,3,2)
nov <- 10
v <- nrow(sheishu)-1
varsheishu <- var(sheishu)
s11 <- varsheishu[1:2,1:2]
s22 <- varsheishu[3:5,3:5]
s33 <- varsheishu[6:8,6:8]
s44 <- varsheishu[9:10,9:10]
u <- det(varsheishu)/(det(s11)*det(s22)*det(s33)*det(s44))
a2 <- nov^2 - sum(noc^2)
a3 <- nov^3 - sum(noc^3)
f <- a2/2
cc <- 1 - (2*a3 + 3*a2)/(12*f*v)
u1 <- -v*cc*log(u)
u; a2; a3; f; cc; u1
qchisq(1-0.001,37)

The Shelf-Stocking Data

Description

A merchandiser stocks soft-drink on a shelf as a multiple number of the number of cases. The time required to put the cases in the shelves is recorded as a response. Clearly, if there are no cases to be stocked, it is natural that the time to put them on the shelf will be 0.

Usage

data("shelf_stock")

Format

A data frame with 15 observations on the following 2 variables.

Time

time required to put the cases in the shelves

Cases_Stocked

number of cases

Examples

data(shelf_stock)
sslm <- lm(Time ~ Cases_Stocked -1, data=shelf_stock)

Siegel-Tukey Nonparametric Test

Description

This function provided an implementation of the nonparametric discussed in "Section 8.5.3 The Siegel-Tukey Test".

Usage

siegel.tukey(x, y)

Arguments

Details

For more details, refer Section 8.5.3 The Siegel-Tukey Test.

Author(s)

Prabhanjan N. Tattar

Examples

x <- c(0.028, 0.029, 0.011, -0.030, 0.017, -0.012, -0.027,-0.018, 0.022, -0.023)
y <- c(-0.002, 0.016, 0.005, -0.001, 0.000, 0.008, -0.005,-0.009, 0.001, -0.019)
siegel.tukey(x,y)

A simple and straightforward function to compute the coefficient of skewness

Description

The function is fairly easy to follow.

Usage

skewcoeff(x)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

e1071::skewness

Scatter Plots for Understanding Correlations

Description

A cooked data tailor made for the use of scatter plots towards understanding correlations.

Usage

data(somesamples)

Format

A data frame with 200 observations on the following 12 variables.

x1

x of Sample 1

y1

y of Sample 1

x2

x of Sample 2

y2

y of Sample 2

x3

x of Sample 3

y3

y of Sample 3

x4

x of Sample 4

y4

y of Sample 4

x5

x of Sample 5

y5

y of Sample 5

x6

x of Sample 6

y6

y of Sample 6

Examples

data(somesamples)
attach(somesamples)
par(mfrow=c(2,3))
plot(x1,y1,main="Sample I",xlim=c(-4,4),ylim=c(-4,4))
plot(x2,y2,main="Sample II",xlim=c(-4,4),ylim=c(-4,4))
plot(x3,y3,main="Sample III",xlim=c(-4,4),ylim=c(-4,4))
plot(x4,y4,main="Sample IV",xlim=c(-4,4),ylim=c(-4,4))
plot(x5,y5,main="Sample V",xlim=c(-4,4),ylim=c(-4,4))
plot(x6,y6,main="Sample VI",xlim=c(-4,4),ylim=c(-4,4))

A function which will return the stationary distribution of an ergodic Markov chain

Description

This function returns the stationary distribution of an ergodic Markov chain. For details, refer Chapter 11 of the book.

Usage

stationdistTPM(M)

Arguments

Author(s)

Prabhanjan N. Tattar

See Also

eigen

Examples

P <- matrix(nrow=3,ncol=3) # An example
P[1,] <- c(1/3,1/3,1/3)
P[2,] <- c(1/4,1/2,1/4)
P[3,] <- c(1/6,1/3,1/2)
stationdistTPM(P)

The Board Stiffness Dataset

Description

Four measures of stiffness of 30 boards are available. The first measure of stiffness is obtained by sending a shock wave down the board, the second measure is obtained by vibrating the board, and remaining are obtained from static tests.

Usage

data(stiff)

Format

A data frame with 30 observations on the following 4 variables.

x1

first measure of stiffness is obtained by sending a shock wave down the board

x2

second measure is obtained by vibrating the board

x3

third measure is obtained by a static test

x4

fourth measure is obtained by a static test

References

Johnson, R.A., and Wichern, D.W. (1982-2007). Applied Multivariate Statistical Analysis, 6e. Pearson Education.

Examples

data(stiff)
colMeans(stiff)
var(stiff)
pairs(stiff)

Forged Swiss Bank Notes

Description

The swiss data set consists of measurements on the width of bottom margin and image diagonal length for forged and real notes. Histogram smoothing method is applied to understand the width of bottom margins for the forged notes.

Usage

data(swiss)

Format

A data frame with 100 observations on the following 4 variables.

Bottforg

bottom margin of forged notes

Diagforg

diagonal margin of forged notes

Bottreal

bottom margin of real notes

Diagreal

diagonal margin of real notes

References

Simonoff, J.S. (1996). Smoothing Methods in Statistics. Springer.

Examples

data(swiss)
par(mfrow=c(1,3))
hist(swiss$Bottforg,breaks=28,probability=TRUE,col=0,ylim=c(0,.5),
  xlab="Margin width (mm)",ylab="Density")
hist(swiss$Bottforg,breaks=12,probability=TRUE,col=0,ylim=c(0,.5),
  xlab="Margin width (mm)",ylab="Density")
hist(swiss$Bottforg,breaks=6,probability=TRUE,col=0,ylim=c(0,.5),
  xlab="Margin width (mm)",ylab="Density")

The Toluca Company Labour Hours against Lot Size

Description

The Toluca Company manufactures equipment related to refrigerator. The company, in respect of a particular component of a refrigerator, has data on the labor hours required for the component in various lot sizes. Using this data, the officials wanted to find the optimum lot size for producing this part.

Usage

data("tc")

Format

A data frame with 25 observations on the following 2 variables.

Lot_Size

size of the lot

Labour_Hours

the labor hours required

References

Kutner, M. H., Nachtsheim, C. J., Neter, J., and Li, W. (1974-2005). Applied Linear Statistical Models, 5e. McGraw-Hill.

Examples

data(tc)
tclm <- lm(Labour_Hours~Lot_Size,data=tc)
tclm$coefficients

The Tensile Strength Experiment

Description

An engineer wants to find out if the cotton weight percentage in a synthetic fiber effects the tensile strength. Towards this, the cotton weight percentage is fixed at 5 different levels of 15, 20, 25, 30, and 35. Each level of the percentage is assigned 5 experimental units and the tensile strength is measured on each of them. The randomization is specified in the Run_Number column. The goal of the engineer is to investigate if the tensile strength is same across the cotton weight percentage.

Usage

data(tensile)

Format

A data frame with 25 observations on the following 4 variables.

Test_Sequence

the test sequence

Run_Number

the run number

CWP

cotton weight percentage

Tensile_Strength

the tensile strength

References

Montgomery, D. C. (1976-2012). Design and Analysis of Experiments, 8e. J.Wiley.

Examples

data(tensile)
tensile$CWP <- as.factor(tensile$CWP)
tensile_aov <- aov(Tensile_Strength~CWP, data=tensile)
summary(tensile_aov)
model.matrix(tensile_aov)

A transition probability matrix

Description

A transition probaility matrix for understanding Markov chains.

Usage

data(testtpm)

Format

A matrix of transition probability matrix

A

transitions probabilities from State A

B

transitions probabilities from State B

C

transitions probabilities from State C

D

transitions probabilities from State D

E

transitions probabilities from State E

F

transitions probabilities from State F

Examples

data(testtpm)

A matrix of transition probability matrix, second example

Description

A transition probaility matrix for understanding Markov chains.

Usage

data(testtpm2)

Format

A matrix of transition probability matrix.

A

transitions probabilities from State A

B

transitions probabilities from State B

C

transitions probabilities from State C

D

transitions probabilities from State D

E

transitions probabilities from State E

F

transitions probabilities from State F

Examples

data(testtpm2)

A matrix of transition probability matrix, third example

Description

A transition probaility matrix for understanding Markov chains

Usage

data(testtpm3)

Format

A data frame with 7 observations on the following 7 variables.

A

transitions probabilities from State A

B

transitions probabilities from State B

C

transitions probabilities from State C

D

transitions probabilities from State D

E

transitions probabilities from State E

F

transitions probabilities from State F

G

transitions probabilities from State G

Examples

data(testtpm3)

US Crime Data

Description

Data is available on the crime rates across 47 states in USA, and we have additional information on 13 more explanatory variables.

Usage

data(usc)

Format

A data frame with 47 observations on the following 14 variables.

R

Crime rate - the number of offenses known to the police per 1,000,000 population

Age

Age distribution - the number of males aged 14 to 24 years per 1000 of total state population

S

Binary variable distinguishing southern states (S = 1) from the rest

Ed

Educational level - mean number of years of schooling times 10 of the population 25 years old and over

Ex0

Police expenditure - per capita expenditure on police protection by state and local governments in 1960

Ex1

Police expenditure - as Ex0, but for 1959

LF

Labour force participation rate per 1000 civilian urban males in the age group 14 to 24 years

M

Number of males per 1000 females

N

State population size in hundred thousands

NW

Number of non-whites per 1000

U1

Unemployment rate of urban males per 1000 in the age group 14 to 24 years

U2

Unemployment rate of urban males per 1000 in the age group 35 to 39 years

W

Wealth, as measured by the median value of transferable goods and assets. or family income (unit 10 dollars)

X

Income inequality: the number of families per 1000 earning below one half of the median income

References

Der, G., and Everitt, B.S. (2002). A Handbook of Statistical Analysis using SAS, 2e. CRC.

Examples

data(usc)
pairs(usc)
round(cor(usc),2)

The Box-Cox Transformation for Viscosity Dataset

Description

The goal of this study is to find the impact of temperature on the viscosity of toluence-tetralin blends.

Usage

data(viscos)

Format

A data frame with 8 observations on the following 2 variables.

Temperature

temperature

Viscosity

viscosity of toluence-tetralin blends

References

Montgomery, D.C., Peck, E.A., and Vining, G.G. (1983-2012). Introduction to Linear Regression Analysis, 5e. J. Wiley.

Examples

data(viscos)
names(viscos)
viscoslm <- lm(Viscosity~Temperature,data=viscos)

von Neumann Random Number Generator

Description

The "vonNeumann" function implements the von Neumann random generator as detailed in Section 11.2.

Usage

vonNeumann(x, n)

Arguments

Author(s)

Prabhanjan N. Tattar

Examples

vonNeumann(x=11,n=10)
vonNeumann(x=675248,n=10)
vonNeumann(x=8653,n=100)

Testing for Physico-chemical Properties of Water in 4 Cities

Description

Water samples from four cities are collected and their physico-chemical properties for ten variables, such as pH, Conductivity, Dissolution, etc., are measured. We would then like to test if the properties are same across the four cities and in which case a same water treatment approach can be adopted across the cities.

Usage

data(waterquality)

Format

A data frame with 63 observations on the following 10 variables.

City

four cities City1 City2 City3 City4

pH

the pH concentration

Conductivity

water conductivity

Dissolution

water dissolution

Alkalinity

alkalinity of the water sample

Hardness

water hardness

Calcium.Hardness

calcium hardness of the water

Magnesium.Hardness

magnesium hardness of the water

Chlorides

chloride content

Sulphates

sulphate content

References

Gore, A.P., Paranjape, S. A., and Kulkarni, M.B. (2006). 100 Data Sets for Statistics Education. Department of Statistics, University of Pune.

Examples

data(waterquality)

Wald-Wolfowitz Nonparametric Test

Description

The "ww.test" function is an implementation of the famous Wald-Wolfowitz nonparametric test as discussed in Section 8.5.

Usage

ww.test(x, y)

Arguments

Author(s)

Prabhanjan N. Tattar

Understanding kernel smoothing through a simulated dataset

Description

This is a simulated dataset with two modes at -2 and 2 and we have 400 observations.

Usage

data(x_bimodal)

Format

The format is: num [1:400] -4.68 -4.19 -4.05 -4.04 -4.02 ...

Examples

data(x_bimodal)
h <- 0.5; n <- length(x_bimodal)
dens_unif <- NULL; dens_triangle <- NULL; dens_epanechnikov <- NULL
dens_biweight <- NULL; dens_triweight <- NULL; dens_gaussian <- NULL
for(i in 1:n)  {
  u <- (x_bimodal[i]-x_bimodal)/h
  xlogical <- (u>-1 & u <= 1)
  dens_unif[i] <- (1/(n*h))*(sum(xlogical)/2)
  dens_triangle[i] <- (1/(n*h))*(sum(xlogical*(1-abs(u))))
  dens_epanechnikov[i] <- (1/(n*h))*(sum(3*xlogical*(1-u^2)/4))
  dens_biweight[i] <- (1/(n*h))*(15*sum(xlogical*(1-u^2)^2/16))
  dens_triweight[i] <- (1/(n*h))*(35*sum(xlogical*(1-u^2)^3/32))
  dens_gaussian[i] <- (1/(n*h))*(sum(exp(-u^2/2)/sqrt(2*pi)))
}
plot(x_bimodal,dens_unif,"l",ylim=c(0,.25),xlim=c(-5,7),xlab="x",
     ylab="Density",main="B: Applying Kernel Smoothing")
points(x_bimodal,dens_triangle,"l",col="red")
points(x_bimodal,dens_epanechnikov,"l",col="green")
points(x_bimodal,dens_biweight,"l",col="blue")
points(x_bimodal,dens_triweight,"l",col="yellow")
points(x_bimodal,dens_gaussian,"l",col="orange")
legend(4,.23,legend=c("rectangular","triangular","epanechnikov","biweight",
                      "gaussian"),col=c("black","red","green","blue","orange"),lty=1)

Youden and Beale's Data on Lesions of Half-Leaves of Tobacco Plant

Description

A simple and innovative design is often priceless. Youden and Beale (1934) sought to find the effect of two preparations of virus on tobacco plants. One half of a tobacco leaf was rubbed with cheesecloth soaked in one preparation of the virus extract and the second half was rubbed with the other virus extract. This experiment was replicated on just eight leaves, and the number of lesions on each half leaf was recorded.

Usage

data(yb)

Format

A data frame with 8 observations on the following 2 variables.

Preparation_1

a numeric vector

Preparation_2

a numeric vector

References

Youden, W. J., and Beale, H. P. (1934). A Statistical Study of the Local Lesion Method for Estimating Tobacco Mosaic Virus. Contrib. Boyce Thompson Inst, 6, 437-454.

Examples

data(yb)
summary(yb)
quantile(yb$Preparation_1,seq(0,1,.1)) # here seq gives 0, .1, .2, ...,1
quantile(yb$Preparation_2,seq(0,1,.1))
fivenum(yb$Preparation_1)
fivenum(yb$Preparation_2)