Help for package msos

Type:

Package

Title:

Data Sets and Functions Used in Multivariate Statistics: Old School by John Marden

Version:

1.2.0

Description:

Multivariate Analysis methods and data sets used in John Marden's book Multivariate Statistics: Old School (2015) <ISBN:978-1456538835>. This also serves as a companion package for the STAT 571: Multivariate Analysis course offered by the Department of Statistics at the University of Illinois at Urbana-Champaign ('UIUC').

URL:

https://github.com/coatless/msos, https://coatless.github.io/msos/

BugReports:

https://github.com/coatless/msos/issues

Depends:

R (≥ 3.0.0), mclust, tree

License:

MIT + file LICENSE

LazyData:

true

Encoding:

UTF-8

RoxygenNote:

7.1.1

NeedsCompilation:

Packaged:

2020-10-30 01:10:55 UTC; ronin

Author:

John Marden [aut, cph], James Balamuta

[cre, ctb, com]

Maintainer:

James Balamuta <james.balamuta@gmail.com>

Repository:

CRAN

Date/Publication:

2020-10-31 06:10:07 UTC

msos: Data Sets and Functions Used in Multivariate Statistics: Old School by John Marden

Description

Author(s)

Maintainer: James Balamuta james.balamuta@gmail.com (ORCID) [contributor, compiler]

Authors:

John Marden jimarden@illinois.edu [copyright holder]

South African Hearth Disease Data

Description

A retrospective sample of males in a heart-disease high-risk region of the Western Cape, South Africa.

Usage

SAheart

Format

A data frame with 462 observations on the following 10 variables.

sbp: systolic blood pressure
tobacco: cumulative tobacco (kg)
ldl: low density lipoprotein cholesterol
adiposity: a numeric vector
famhist: family history of heart disease, a factor with levels "Absent" and "Present"
typea: type-A behavior
obesity: a numeric vector
alcohol: current alcohol consumption
age: age at onset
chd: response, coronary heart disease

Details

A retrospective sample of males in a heart-disease high-risk region of the Western Cape, South Africa. There are roughly two controls per case of CHD. Many of the CHD positive men have undergone blood pressure reduction treatment and other programs to reduce their risk factors after their CHD event. In some cases the measurements were made after these treatments. These data are taken from a larger dataset, described in Rousseauw et al, 1983, South African Medical Journal.

Source

Rousseauw, J., du Plessis, J., Benade, A., Jordaan, P., Kotze, J. and Ferreira, J. (1983). Coronary risk factor screening in three rural communities, South African Medical Journal 64: 430–436.

ElemStatLearn, R-Package

Spam

Description

In the Hewlett-Packard spam data, a set of n = 4601 emails were classified according to whether they were spam, where "0" means not spam, "1" means spam. Fifty-seven explanatory variables based on the content of the emails were recorded, including various word and symbol frequencies. The emails were sent to George Forman (not the boxer) at Hewlett-Packard labs, hence emails with the words "George" or "hp" would likely indicate non-spam, while "credit" or "!" would suggest spam. The data were collected by Hopkins et al. [1999], and are in the data matrix Spam. ( They are also in the R data frame spam from the ElemStatLearn package [Halvorsen, 2009], as well as at the UCI Machine Learning Repository [Frank and Asuncion, 2010].)

Usage

Spam

Format

A double matrix with 4601 observations on the following 58 variables.

WFmake: Percentage of words in the e-mail that match make.
WFaddress: Percentage of words in the e-mail that match address.
WFall: Percentage of words in the e-mail that match all.
WF3d: Percentage of words in the e-mail that match 3d.
WFour: Percentage of words in the e-mail that match our.
WFover: Percentage of words in the e-mail that match over.
WFremove: Percentage of words in the e-mail that match remove.
WFinternet: Percentage of words in the e-mail that match internet.
WForder: Percentage of words in the e-mail that match order.
WFmail: Percentage of words in the e-mail that match mail.
WFreceive: Percentage of words in the e-mail that match receive.
WFwill: Percentage of words in the e-mail that match will.
WFpeople: Percentage of words in the e-mail that match people.
WFreport: Percentage of words in the e-mail that match report.
WFaddresses: Percentage of words in the e-mail that match addresses.
WFfree: Percentage of words in the e-mail that match free.
WFbusiness: Percentage of words in the e-mail that match business.
WFemail: Percentage of words in the e-mail that match email.
WFyou: Percentage of words in the e-mail that match you.
WFcredit: Percentage of words in the e-mail that match credit.
WFyour: Percentage of words in the e-mail that match your.
WFfont: Percentage of words in the e-mail that match font.
WF000: Percentage of words in the e-mail that match 000.
WFmoney: Percentage of words in the e-mail that match money.
WFhp: Percentage of words in the e-mail that match hp.
WFgeorge: Percentage of words in the e-mail that match george.
WF650: Percentage of words in the e-mail that match 650.
WFlab: Percentage of words in the e-mail that match lab.
WFlabs: Percentage of words in the e-mail that match labs.
WFtelnet: Percentage of words in the e-mail that match telnet.
WF857: Percentage of words in the e-mail that match 857.
WFdata: Percentage of words in the e-mail that match data.
WF415: Percentage of words in the e-mail that match 415.
WF85: Percentage of words in the e-mail that match 85.
WFtechnology: Percentage of words in the e-mail that match technology.
WF1999: Percentage of words in the e-mail that match 1999.
WFparts: Percentage of words in the e-mail that match parts.
WFpm: Percentage of words in the e-mail that match pm.
WFdirect: Percentage of words in the e-mail that match direct.
WFcs: Percentage of words in the e-mail that match cs.
WFmeeting: Percentage of words in the e-mail that match meeting.
WForiginal: Percentage of words in the e-mail that match original.
WFproject: Percentage of words in the e-mail that match project.
WFre: Percentage of words in the e-mail that match re.
WFedu: Percentage of words in the e-mail that match edu.
WFtable: Percentage of words in the e-mail that match table.
WFconference: Percentage of words in the e-mail that match conference.
CFsemicolon: Percentage of characters in the e-mail that match SEMICOLON.
CFparen: Percentage of characters in the e-mail that match PARENTHESES.
CFbracket: Percentage of characters in the e-mail that match BRACKET.
CFexclam: Percentage of characters in the e-mail that match EXCLAMATION.
CFdollar: Percentage of characters in the e-mail that match DOLLAR.
CFpound: Percentage of characters in the e-mail that match POUND.
CRLaverage: Average length of uninterrupted sequences of capital letters.
CRLlongest: Length of longest uninterrupted sequence of capital letters.
CRLtotal: Total number of capital letters in the e-mail
spam: Denotes whether the e-mail was considered spam (1) or not (0), i.e. unsolicited commercial e-mail.

Source

Mark Hopkins, Erik Reeber, George Forman, and Jaap Suermondt. Spam data. Hewlett-Packard Labs, 1501 Page Mill Rd., Palo Alto, CA 94304, 1999.

Birthrates throughout the day in four Hospitals

Description

The data on average number of births for each hour of the day for four hospitals.

Usage

births

Format

A double matrix with 24 observations on the following 4 variables.

Hospital1: Average number of births for each hour of the day within Hospital 1
Hospital2: Average number of births for each hour of the day within Hospital 2
Hospital3: Average number of births for each hour of the day within Hospital 3
Hospital4: Average number of births for each hour of the day within Hospital 4

Source

To be determined

Calculate the least squares estimates

Description

This function fits the model using least squares. It takes an optional pattern matrix P as in (6.51), which specifies which \beta _{ij}'s are zero.

Usage

bothsidesmodel(x, y, z = diag(qq), pattern = matrix(1, nrow = p, ncol = l))

Arguments

x

An N \times P design matrix.

y

The N \times Q matrix of observations.

z

A Q \times L design matrix

pattern

An optional N \times P matrix of 0's and 1's indicating which elements of \beta are allowed to be nonzero.

Value

A list with the following components:

Beta: The least-squares estimate of \beta.
SE: The P \times L matrix with the ijth element being the standard error of \hat{\beta}_ij.
T: The P \times L matrix with the ijth element being the t-statistic based on \hat{\beta}_{ij}.
Covbeta: The estimated covariance matrix of the \hat{\beta}_{ij}'s.
df: A p-dimensional vector of the degrees of freedom for the t-statistics, where the jth component contains the degrees of freedom for the jth column of \hat{\beta}.
Sigmaz: The Q \times Q matrix \hat{\Sigma}_z.
Cx: The Q \times Q residual sum of squares and crossproducts matrix.

Examples

# Mouth Size Example from 6.4.1
data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(c(1, 1, 1, 1), c(-3, -1, 1, 3), c(1, -1, -1, 1), c(-1, 3, -3, 1))
bothsidesmodel(x, y, z)

Test subsets of `\beta` are zero

Description

Tests the null hypothesis that an arbitrary subset of the \beta _{ij}'s is zero, based on the least squares estimates, using the \chi^2 test as in Section 7.1. The null and alternative are specified by pattern matrices P_0 and P_A, respectively. If the P_A is omitted, then the alternative will be taken to be the unrestricted model.

Usage

bothsidesmodel.chisquare(
  x,
  y,
  z,
  pattern0,
  patternA = matrix(1, nrow = ncol(x), ncol = ncol(z))
)

Arguments

x

An N \times P design matrix.

y

The N \times Q matrix of observations.

z

A Q \times L design matrix.

pattern0

An N \times P matrix of 0's and 1's specifying the null hypothesis.

patternA

An optional N \times P matrix of 0's and 1's specifying the alternative hypothesis.

Value

A 'list' with the following components:

Theta: The vector of estimated parameters of interest.
Covtheta: The estimated covariance matrix of the estimated parameter vector.
df: The degrees of freedom in the test.
chisq: T^2 statistic in (7.4).
pvalue: The p-value for the test.

Examples

# TBA - Submit a PR!

Obtain the degrees of freedom for a model.

Description

Determines the denominators needed to calculate an unbiased estimator of \Sigma_R.

Usage

bothsidesmodel.df(xx, n, pattern)

Arguments

xx

Result of (X^T * X), where T denotes tranpose.

n

Number of rows in observation matrix given

pattern

An N \times P matrix of 0's and 1's indicating which elements of \beta are allowed to be nonzero.

Value

A numeric matrix of size N \times N containing the degrees of freedom for the test.

Examples

# Find the DF for a likelihood ratio test statistic.
x <- cbind(
  1, c(-2, -1, 0, 1, 2), c(2, -1, -2, -1, 2),
  c(-1, 2, 0, -2, 1), c(1, -4, 6, -4, 1)
)
# or x <- cbind(1, poly(1:5, 4))
data(skulls)
x <- kronecker(x, rep(1, 30))
y <- skulls[, 1:4]
z <- diag(4)
pattern <- rbind(c(1, 1, 1, 1), 1, 0, 0, 0)
xx <- t(x) %*% x
bothsidesmodel.df(xx, nrow(y), pattern)

Test blocks of `\beta` are zero.

Description

Performs tests of the null hypothesis H0 : \beta^* = 0, where \beta^* is a block submatrix of \beta as in Section 7.2.

Usage

bothsidesmodel.hotelling(x, y, z, rows, cols)

Arguments

x

An N \times P design matrix.

y

The N \times Q matrix of observations.

z

A Q \times L design matrix

rows

The vector of rows to be tested.

cols

The vector of columns to be tested.

Value

A list with the following components:

Hotelling: A list with the components of the Lawley-Hotelling T^2 test (7.22)

T2: The T^2 statistic (7.19).
F: The F version (7.22) of the T^2 statistic.
df: The degrees of freedom for the F.
pvalue: The p-value of the F.

Wilks: A list with the components of the Wilks \Lambda test (7.37)

lambda: The \Lambda statistic (7.35).
Chisq: The \chi ^2 version (7.37) of the \Lambda statistic, using Bartlett's correction.
df: The degrees of freedom for the \chi ^2

pvalue: The p-value of the \chi ^2

Examples

# Finds the Hotelling values for example 7.3.1
data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(c(1, 1, 1, 1), c(-3, -1, 1, 3), c(1, -1, -1, 1), c(-1, 3, -3, 1))
bothsidesmodel.hotelling(x, y, z, 1:2, 3:4)

Test subsets of `\beta` are zero.

Description

Tests the null hypothesis that an arbitrary subset of the \beta _{ij}'s is zero, using the likelihood ratio test as in Section 9.4. The null and alternative are specified by pattern matrices P_0 and P_A, respectively. If the P_A is omitted, then the alternative will be taken to be the unrestricted model.

Usage

bothsidesmodel.lrt(
  x,
  y,
  z,
  pattern0,
  patternA = matrix(1, nrow = ncol(x), ncol = ncol(z))
)

Arguments

x

An N \times P design matrix.

y

The N \times Q matrix of observations.

z

A Q \times L design matrix.

pattern0

An N \times P matrix of 0's and 1's specifying

patternA

An optional N \times P matrix of 0's and 1's specifying the alternative hypothesis.

Value

A list with the following components:

chisq: The likelihood ratio statistic in (9.44).
df: The degrees of freedom in the test.
pvalue: The p-value for the test.

Examples


# Load data
data(caffeine)

# Matrices
x <- cbind(
  rep(1, 28),
  c(rep(-1, 9), rep(0, 10), rep(1, 9)),
  c(rep(1, 9), rep(-1.8, 10), rep(1, 9))
)
y <- caffeine[, -1]
z <- cbind(c(1, 1), c(1, -1))
pattern <- cbind(c(rep(1, 3)), 1)

# Fit model
bsm <- bothsidesmodel.lrt(x, y, z, pattern)

Calculate the maximum likelihood estimates

Description

This function fits the model using maximum likelihood. It takes an optional pattern matrix P as in (6.51), which specifies which \beta _{ij}'s are zero.

Usage

bothsidesmodel.mle(x, y, z = diag(qq), pattern = matrix(1, nrow = p, ncol = l))

Arguments

x

An N \times P design matrix.

y

The N \times Q matrix of observations.

z

A Q \times L design matrix

pattern

An optional N \times P matrix of 0's and 1's indicating which elements of \beta are allowed to be nonzero.

Value

A list with the following components:

Beta: The least-squares estimate of \beta.
SE: The P \times L matrix with the ijth element being the standard error of \hat{\beta}_{ij}.
T: The P \times L matrix with the ijth element being the t-statistic based on \hat{\beta}_{ij}.
Covbeta: The estimated covariance matrix of the \hat{\beta}_{ij}'s.
df: A p-dimensional vector of the degrees of freedom for the t-statistics, where the jth component contains the degrees of freedom for the jth column of \hat{\beta}.
Sigmaz: The Q \times Q matrix \hat{\Sigma}_z.
Cx: The Q \times Q residual sum of squares and crossproducts matrix.
ResidSS: The dimension of the model, counting the nonzero \beta _{ij}'s and components of \Sigma _z.
Deviance: Mallow's C_p Statistic.
Dim: The dimension of the model, counting the nonzero \beta _{ij}'s and components of \Sigma_z
AICc: The corrected AIC criterion from (9.87) and (aic19)
BIC: The BIC criterion from (9.56).

Examples

data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(1, c(-3, -1, 1, 3), c(-1, 1, 1, -1), c(-1, 3, -3, 1))
bothsidesmodel.mle(x, y, z, cbind(c(1, 1), 1, 0, 0))

Helper function to determine `\beta` estimates for MLE regression with patterning.

Description

Generates \beta estimates for MLE using a conditioning approach with patterning support.

Usage

bsm.fit(x, y, z, pattern)

Arguments

x

An N \times (P + F) design matrix, where F is the number of columns conditioned on. This is equivalent to the multiplication of xyzb.

y

The N \times (Q - F) matrix of observations, where F is the number of columns conditioned on. This is equivalent to the multiplication of Yz_a.

z

A (Q - F) \times L design matrix, where F is the number of columns conditioned on.

pattern

An optional N-F x F matrix of 0's and 1's indicating which elements of \beta are allowed to be nonzero.

Value

A list with the following components:

Beta: The least-squares estimate of \beta.
SE: The (P+F)\times L matrix with the ijth element being the standard error of \hat{\beta}_ij.
T: The (P+F)\times L matrix with the ijth element being the t-statistic based on \hat{\beta}_ij.
Covbeta: The estimated covariance matrix of the \hat{\beta}_ij's.
df: A p-dimensional vector of the degrees of freedom for the t-statistics, where the jth component contains the degrees of freedom for the jth column of \hat{\beta}.
Sigmaz: The (Q - F) \times (Q - F) matrix \hat{\Sigma}_z.
Cx: The Q \times Q residual sum of squares and crossproducts matrix.

Examples

# NA

Helper function to determine `\beta` estimates for MLE regression.

Description

Generates \beta estimates for MLE using a conditioning approach.

Usage

bsm.simple(x, y, z)

Arguments

x

An N \times (P + F) design matrix, where F is the number of columns conditioned on. This is equivalent to the multiplication of xyzb.

y

The N \times (Q - F) matrix of observations, where F is the number of columns conditioned on. This is equivalent to the multiplication of Yz_a.

z

A (Q - F) \times L design matrix, where F is the number of columns conditioned on.

Details

The technique used to calculate the estimates is described in section 9.3.3.

Value

A list with the following components:

Beta: The least-squares estimate of \beta.
SE: The (P + F) \times L matrix with the ijth element being the standard error of \hat{\beta}_ij.
T: The (P + F) \times L matrix with the ijth element being the t-statistic based on \hat{\beta}_ij.
Covbeta: The estimated covariance matrix of the \hat{\beta}_ij's.
df: A p-dimensional vector of the degrees of freedom for the t-statistics, where the jth component contains the degrees of freedom for the jth column of \hat{\beta}.
Sigmaz: The (Q - F) \times (Q - F) matrix \hat{\Sigma}_z.
Cx: The Q \times Q residual sum of squares and crossproducts matrix.

Examples

# Taken from section 9.3.3 to show equivalence to methods.
data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(1, c(-3, -1, 1, 3), c(-1, 1, 1, -1), c(-1, 3, -3, 1))
yz <- y %*% solve(t(z))
yza <- yz[, 1:2]
xyzb <- cbind(x, yz[, 3:4])
lm(yza ~ xyzb - 1)
bsm.simple(xyzb, yza, diag(2))

The Effects of Caffeine

Description

Henson et al. [1996] conducted an experiment to see whether caffeine has a negative effect on short-term visual memory. High school students were randomly chosen: 9 from eighth grade, 10 from tenth grade, and 9 from twelfth grade. Each person was tested once after having caffeinated Coke, and once after having decaffeinated Coke. After each drink, the person was given ten seconds to try to memorize twenty small, common objects, then allowed a minute to write down as many as could be remembered. The main question of interest is whether people remembered more objects after the Coke without caffeine than after the Coke with caffeine.

Usage

caffeine

Format

A double matrix with 28 observations on the following 3 variables.

Grade: Grade of the Student, which is either 8th, 10th, or 12th
Without: Number of items remembered after drinking Coke without Caffine
With: Number of items remembered after drinking Coke with Caffine

Source

Claire Henson, Claire Rogers, and Nadia Reynolds. Always Coca-Cola. Technical report, University Laboratory High School, Urbana, IL, 1996.

Automobile Data from Consumer Reports

Description

The data set cars [Consumers' Union, 1990] contains 111 models of automobile. The original data can be found in the S-Plus? [TIBCO Software Inc., 2009] data frame cu.dimensions. In cars, the variables have been normalized to have medians of 0 and median absolute deviations (MAD) of 1.4826 (the MAD for a N(0, 1)).

Usage

cars

Format

A double matrix with 111 observations on the following 11 variables.

Length: Overall length, in inches, as supplied by manufacturer.
Wheelbase: Length of wheelbase, in inches, as supplied by manufacturer.
Width: Width of car, in inches, as supplied by manufacturer.
Height: Height of car, in inches, as supplied by manufacturer
FrontHd: Distance between the car's head-liner and the head of a 5 ft. 9 in. front seat passenger, in inches, as measured by CU.
RearHd: Distance between the car's head-liner and the head of a 5 ft 9 in. rear seat passenger, in inches, as measured by CU
FrtLegRoom: Maximum front leg room, in inches, as measured by CU.
RearSeating: Rear fore-and-aft seating room, in inches, as measured by CU.
FrtShld: Front shoulder room, in inches, as measured by CU.
RearShld: Rear shoulder room, in inches, as measured by CU.
Luggage: Luggage Area in Car.

Source

Consumers' Union. Body dimensions. Consumer Reports, April 286 - 288, 1990.

Cereal

Description

Chakrapani and Ehrenberg [1981] analyzed people's attitudes towards a variety of breakfast cereals. The data matrix cereal is 8 ? 11, with rows corresponding to eight cereals, and columns corresponding to potential attributes about cereals. The original data consisted of the percentage of subjects who thought the given cereal possessed the given attribute. The present matrix has been doubly centered, so that the row means and columns means are all zero. (The original data can be found in the S-Plus [TIBCO Software Inc., 2009] data set cereal.attitude.)

Usage

cereal

Format

A double matrix with 8 observations on the following 11 variables.

Return: A cereal one would come back to
Tasty: Tastes good
Popular: Popular with the entire family
Nourishing: Cereal is fulfilling
NaturalFlavor: Cereal lacks flavor additives
Affordable: Cereal is priced well for the content
GoodValue: Quantity for Price
Crispy: Stays crispy in milk
Fit: Keeps one fit
Fun: Fun for children

Source

T. K. Chakrapani and A. S. C. Ehrenberg. An alternative to factor analysis in marketing research part 2: Between group analysis. Professional Marketing Research Society Journal, 1:32-38, 1981.

Morphological Measurements on Leptograpsus Crabs

Description

The crabs data frame has 200 rows and 8 columns, describing 5 morphological measurements on 50 crabs each of two colour forms and both sexes, of the species Leptograpsus variegatus collected at Fremantle, W. Australia.

Usage

crabs

Format

This data frame contains the following columns:

sp: species - "B" or "O" for blue or orange.
sex: "M" (Male) or "F" (Female).
index: index 1:50 within each of the four groups.
FL: frontal lobe size (mm).
RW: rear width (mm).
CL: carapace length (mm).
CW: carapace width (mm).
BD: body depth (mm).

Source

Campbell, N.A. and Mahon, R.J. (1974) A multivariate study of variation in two species of rock crab of genus Leptograpsus. Australian Journal of Zoology 22, 417–425.

MASS, R-Package

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

Decathlon Event Data from 2008 Olympics.

Description

The decathlon data set has scores on the top 24 men in the decathlon (a set of ten events) at the 2008 Olympics. The scores are the numbers of points each participant received in each event, plus each person's total points.

Usage

decathlon08

Format

A double matrix with 24 observations on the following 11 variables.

X_100meter: Individual point score for 100 Meter event.
LongJump: Individual point score for Long Jump event.
ShotPut: Individual point score for Shot Put event.
HighJump: Individual point score for High Jump event.
X_400meter: Individual point score for 400 Meter event.
Hurdles: Individual point score for 110 Hurdles event.
Discus: Individual point score for Discus event.
PoleVault: Individual point score for Pole Vault event.
Javelin: Individual point score for Javelin event.
X_1500meter: Individual point score for 1500 Meter event.
Total: Individual total point score for events participated in.

Source

NBC's Olympic site

Decathlon Event Data from 2012 Olympics.

Description

The decathlon data set has scores on the top 26 men in the decathlon (a set of ten events) at the 2012 Olympics. The scores are the numbers of points each participant received in each event, plus each person's total points.

Usage

decathlon12

Format

A double matrix with 26 observations on the following 11 variables.

Meter100: Individual point score for 100 Meter event.
LongJump: Individual point score for Long Jump event.
ShotPut: Individual point score for Shot Put event.
HighJump: Individual point score for High Jump event.
Meter400: Individual point score for 400 Meter event.
Hurdles110: Individual point score for 110 Hurdles event.
Discus: Individual point score for Discus event.
PoleVault: Individual point score for Pole Vault event.
Javelin: Individual point score for Javelin event.
Meter1500: Individual point score for 1500 Meter event.
Total: Individual total point score for events participated in.

Source

NBC's Olympic site

Presidential Election Data

Description

The data set election has the results of the first three US presidential races of the 2000's (2000, 2004, 2008). The observations are the 50 states plus the District of Columbia, and the values are the (D - R)/(D + R) for each state and each year, where D is the number of votes the Democrat received, and R is the number the Republican received.

Usage

election

Format

A double matrix with 51 observations on the following 3 variables.

2000: Results for 51 States in Year 2000
2004: Results for 51 States in Year 2004
2008: Results for 51 States in Year 2008

Source

Calculated by Prof. John Marden, data source to be announced.

Statistics Students' Scores on Exams

Description

The exams matrix has data on 191 statistics students, giving their scores (out of 100) on the three midterm exams, and the final exam.

Usage

exams

Format

A double matrix with 191 observations on the following 4 variables.

Midterm1: Student score on the first midterm out of 100.
Midterm2: Student score on the second midterm out of 100.
Midterm3: Student score on the third midterm out of 100.
FinalExam: Student score on the Final Exam out of 100.

Source

Data from one of Prof. John Marden's earlier classes

Make a square matrix

Description

The function fillout takes a Q \times (Q - L) matrix Z and fills it out so that it is a square matrix Q \times Q.

Usage

fillout(z)

Arguments

z

A Q \times (Q-L) matrix

Value

A square matrix Q \times Q

Examples

# Create a 3 x 2 matrix
a <- cbind(c(1, 2, 3), c(4, 5, 6))

# Creates a 3 x 3 Matrix from 3 x 2 Data
fillout(a)

Grades

Description

The data set contains grades of 107 students.

Usage

grades

Format

A double matrix with 107 observations on the following 7 variables.

Gender: Sex (0=Male, 1=Female)
HW: Student Score on all Homework.
Labs: Student Score on all Labs.
InClass: Student Score on all In Class work.
Midterms: Student Score on all Midterms.
Final: Student Score on the Final.
Total: Student's Total Score

Source

Data from one of Prof. John Marden's earlier classes

Histamine in Dogs

Description

Sixteen dogs were treated with drugs to see the effects on their blood histamine levels. The dogs were split into four groups: Two groups received the drugmorphine, and two received the drug trimethaphan, both given intravenously. For one group within each pair of drug groups, the dogs had their supply of histamine depleted before treatment, while the other group had histamine intact. (Measurements with the value "0.10" marked data that was missing and, were filled with that value arbitrarily.)

Usage

histamine

Format

A double matrix with 16 observations on the following 4 variables.

Before: Histamine levels (in micrograms per milliliter of blood) before the inoculation.
After1: Histamine levels (in micrograms per milliliter of blood) one minute after inoculation.
After3: Histamine levels (in micrograms per milliliter of blood) three minute after inoculation.
After5: Histamine levels (in micrograms per milliliter of blood) five minutes after inoculation.

Source

Kenny J.Morris and Robert Zeppa. Histamine-induced hypotension due to morphine and arfonad in the dog. Journal of Surgical Research, 3(6):313-317, 1963.

Obtain largest value index

Description

Obtains the index of a vector that contains the largest value in the vector.

Usage

imax(z)

Arguments

z

A vector of any length

Value

The index of the largest value in a vector.

Examples

# Iris example
x.iris <- as.matrix(iris[, 1:4])
# Gets group vector (1, ... , 1, 2, ... , 2, 3, ... , 3)
y.iris <- rep(1:3, c(50, 50, 50))
ld.iris <- lda(x.iris, y.iris)
disc <- x.iris %*% ld.iris$a
disc <- sweep(disc, 2, ld.iris$c, "+")
yhat <- apply(disc, 1, imax)

Linear Discrimination

Description

Finds the coefficients a_k and constants c_k for Fisher's linear discrimination function d_k in (11.31) and (11.32).

Usage

lda(x, y)

Arguments

x

The N \times P data matrix.

y

The N-vector of group identities, assumed to be given by the numbers 1,...,K for K groups.

Value

A list with the following components:

a: A P x K matrix, where column K contains the coefficents a_k for (11.31). The final column is all zero.
c: The K-vector of constants c_k for (11.31). The final value is zero.

Examples

# Iris example
x.iris <- as.matrix(iris[, 1:4])
# Gets group vector (1, ... , 1, 2, ... , 2, 3, ..., 3)
y.iris <- rep(1:3, c(50, 50, 50))
ld.iris <- lda(x.iris, y.iris)

Leprosy Patients

Description

Dataset with leprosy patients found in Snedecor and Cochran [1989]. There were 30 patients, randomly allocated to three groups of 10. The first group received drug A, the second drug D, and the third group received a placebo. Each person had their bacterial count taken before and after receiving the treatment.

Usage

leprosy

Format

A double matrix with 30 observations on the following 3 variables.

Before: Bacterial count taken before receiving the treatment.
After: Bacterial count taken after receiving the treatment.
Group: Group Coding: 0 = Drug A, 1 = Drug B, 2 = Placebo

Source

George W. Snedecor and William G. Cochran. Statistical Methods. Iowa State University Press, Ames, Iowa, eighth edition, 1989.

Log Determinant

Description

Takes the log determinant of a square matrix. Log is that of base e sometimes referred to as ln().

Usage

logdet(a)

Arguments

a

Square matrix (Q \times Q)

Value

A single-value double.

Examples

# Identity Matrix of size 2
logdet(diag(c(2, 2)))

Mouth Sizes

Description

Measurements were made on the size of mouths of 27 children at four ages: 8, 10, 12, and 14. The measurement is the distance from the "center of the pituitary to the pteryomaxillary fissure" in millimeters. These data can be found in Potthoff and Roy [1964]. There are 11 girls (Sex=1) and 16 boys (Sex=0).

Usage

mouths

Format

A data frame with 27 observations on the following 5 variables.

Age8: Measurement on child's month at age eight.
Age10: Measurement on child's month at age ten.
Age12: Measurement on child's month at age twelve.
Age14: Measurement on child's month at age fourteen.
Sex: Sex Coding: Girl=1 and Boys=0

Source

Richard F. Potthoff and S. N. Roy. A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51:313-326, 1964.

Estimating negative entropy

Description

Calculates the histogram-based estimate (A.2) of the negentropy,

Negent(g) = (1/2)*(1 + log(2\pi \sigma ^2)) - Entropy(g),

for a vector of observations.

Usage

negent(x, K = ceiling(log2(length(x)) + 1))

Arguments

x

The n-vector of observations.

K

The number of bins to use in the histogram.

Value

The value of the estimated negentropy.

Examples

# TBA - Submit a PR!

Maximizing negentropy for `q = 2` dimensions

Description

Searches for the rotation that maximizes the estimated negentropy of the first column of the rotated data, for q = 2 dimensional data.

Usage

negent2D(y, m = 100)

Arguments

y

The n \times 2 data matrix.

m

The number of angles (between 0 and \pi) over which to search.

Value

A list with the following components:

vectors: The 2 ? 2 orthogonal matrix G that optimizes the negentropy.
values: Estimated negentropies for the two rotated variables. The largest is first.

Examples

# Load iris data
data(iris)

# Centers and scales the variables.
y <- scale(as.matrix(iris[, 1:2]))

# Obtains Negent Vectors for 2 x 2 matrix
gstar <- negent2D(y, m = 10)$vectors

Maximizing negentropy for `Q = 3` dimensions

Description

Searches for the rotation that maximizes the estimated negentropy of the first column of the rotated data, and of the second variable fixing the first, for q = 3 dimensional data. The routine uses a random start for the function optim using the simulated annealing option SANN, hence one may wish to increase the number of attempts by setting nstart to a integer larger than 1.

Usage

negent3D(y, nstart = 1, m = 100, ...)

Arguments

y

The N \times 3 data matrix.

nstart

The number of times to randomly start the search routine.

m

The number of angles (between 0 and \pi) over which to search to find the second variables.

...

Further optional arguments to pass to the optim function to control the simulated annealing algorithm.

Value

A 'list' with the following components:

vectors: The 3 x 3 orthogonal matrix G that optimizes the negentropy.
values: Estimated negentropies for the three rotated variables, from largest to smallest.

Examples

## Not run: 
# Running this example will take approximately 30s.
# Centers and scales the variables.
y <- scale(as.matrix(iris[, 1:3]))

# Obtains Negent Vectors for 3x3 matrix
gstar <- negent3D(y, nstart = 100)$vectors

## End(Not run)

The Painter's Data of de Piles

Description

The subjective assessment, on a 0 to 20 integer scale, of 54 classical painters. The painters were assessed on four characteristics: composition, drawing, colour and expression. The data is due to the Eighteenth century art critic, de Piles.

Usage

painters

Format

The row names of the data frame are the painters. The components are:

Composition: Composition score.
Drawing: Drawing score.
Colour: Colour score.
Expression: Expression score.
School: The school to which a painter belongs, as indicated by a factor level code as follows: "A": Renaissance; "B": Mannerist; "C": Seicento; "D": Venetian; "E": Lombard; "F": Sixteenth Century; "G": Seventeenth Century; "H": French.

Source

A. J. Weekes (1986) A Genstat Primer. Edward Arnold.

M. Davenport and G. Studdert-Kennedy (1972) The statistical analysis of aesthetic judgement: an exploration. Applied Statistics 21, 324–333.

I. T. Jolliffe (1986) Principal Component Analysis. Springer.

MASS, R-Package

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

BIC for a particular pattern

Description

Find the BIC and MLE from a set of observed eigenvalues for a specific pattern.

Usage

pcbic(eigenvals, n, pattern)

Arguments

eigenvals

The Q-vector of eigenvalues of the covariance matrix, in order from largest to smallest.

n

The degrees of freedom in the covariance matrix.

pattern

The pattern of equalities of the eigenvalues, given by the K-vector (Q_1, ... , Q_K) as in (13.8).

Value

A 'list' with the following components:

lambdaHat: A Q-vector containing the MLE's for the eigenvalues.
Deviance: The deviance of the model, as in (13.13).
Dimension: The dimension of the model, as in (13.12).
BIC: The value of the BIC for the model, as in (13.14).

Examples

# Build cars1
require("mclust")
mcars <- Mclust(cars)
cars1 <- cars[mcars$classification == 1, ]
xcars <- scale(cars1)
eg <- eigen(var(xcars))
pcbic(eg$values, 95, c(1, 1, 3, 3, 2, 1))

Choosing a good pattern

Description

Uses the stepwise procedure described in Section 13.1.4 to find a pattern for a set of observed eigenvalues with good BIC value.

Usage

pcbic.stepwise(eigenvals, n)

Arguments

eigenvals

The Q-vector of eigenvalues of the covariance matrix, in order from largest to smallest.

n

The degrees of freedom in the covariance matrix.

Value

A list with the following components:

Patterns: A list of patterns, one for each value of length K.
BICs: A vector of the BIC's for the above patterns.
BestBIC: The best (smallest) value among the BIC's in BICs.
BestPattern: The pattern with the best BIC.
lambdaHat: A Q-vector containing the MLE's for the eigenvalues for the pattern with the best BIC.

Examples

# Build cars1
require("mclust")
mcars <- Mclust(cars)
cars1 <- cars[mcars$classification == 1, ]
xcars <- scale(cars1)
eg <- eigen(var(xcars))
pcbic.stepwise(eg$values, 95)

Obtain the best subpattern among the patterns.

Description

Obtains the best pattern and its BIC among the patterns obtainable by summing two consecutive terms in pattern0.

Usage

pcbic.subpatterns(eigenvals, n, pattern0)

Arguments

eigenvals

The Q-vector of eigenvalues of the covariance matrix, in order from largest to smallest.

n

The degrees of freedom in the covariance matrix.

pattern0

The pattern of equalities of the eigenvalues, given by the K-vector (Q_1, ... , Q_K) as in (13.8).

Value

A 'list' containing:

pattern: A double matrix containing the pattern evaluated.
bic: A vector containing the BIC for the above pattern matrix.

Examples

# NA

Obtain pattern

Description

Returns the pattern obtained by summing q_i and q_{i+1}.

Usage

pcbic.unite(pattern, index1)

Arguments

pattern

The pattern of equalities of the eigenvalues, given by the K-vector (Q_1, ... , Q_K) as in (13.8).

index1

Index i where 1 \le i < K

Value

A 'vector' containing a pattern.

Examples

# NA

Planets

Description

Six astronomical variables are given on each of the historical nine planets (or eight planets, plus Pluto).

Usage

planets

Format

A double matrix with 9 observations on the following 6 variables.

Distance: Average distance in millions of miles the planet is from the sun.
Day: The length of the planet's day in Earth days
Year: The length of year in Earth days
Diameter: The planet's diameter in miles
Temperature: The planet's temperature in degrees Fahrenheit
Moons: Number of moons

Source

John W. Wright, editor. The Universal Almanac. Andrews McMeel Publishing, Kansas City, MO, 1997.

Quadratic discrimination prediction

Description

The function uses the output from the function qda (Section A.3.2) and a P-vector X, and calculates the predicted group for this X.

Usage

predict_qda(qd, newx)

Arguments

qd

The output from qda.

newx

A P-vector X whose components match the variables used in the qda function.

Value

A K-vector of the discriminant values d_k^Q(X) in (11.48) for the given X.

Examples


# Load Iris Data
data(iris)

# Build data
x.iris <- as.matrix(iris[, 1:4])
n <- nrow(x.iris)
# Gets group vector (1, ... , 1, 2, ..., 2, 3, ... , 3)
y.iris <- rep(1:3, c(50, 50, 50))

# Perform QDA
qd.iris <- qda(x.iris, y.iris)
yhat.qd <- NULL
for (i in seq_len(n)) {
  yhat.qd <- c(yhat.qd, imax(predict_qda(qd.iris, x.iris[i, ])))
}
table(yhat.qd, y.iris)

Prostaglandin

Description

Data from Ware and Bowden [1977] taken at six four-hour intervals (labelled T1 to T6) over the course of a day for 10 individuals. The measurements are prostaglandin contents in their urine.

Usage

prostaglandin

Format

A double matrix with 10 observations on the following 6 variables.

T1: First four-hour interval measurement of prostaglandin.
T2: Second four-hour interval measurement of prostaglandin.
T3: Third four-hour interval measurement of prostaglandin.
T4: Fourth four-hour interval measurement of prostaglandin.
T5: Fifth four-hour interval measurement of prostaglandin.
T6: Sixth four-hour interval measurement of prostaglandin.

Source

J H Ware and R E Bowden. Circadian rhythm analysis when output is collected at intervals. Biometrics, 33(3):566-571, 1977.

Quadratic discrimination

Description

The function returns the elements needed to calculate the quadratic discrimination in (11.48). Use the output from this function in predict_qda (Section A.3.2) to find the predicted groups.

Usage

qda(x, y)

Arguments

x

The N \times P data matrix.

y

The N-vector of group identities, assumed to be given by the numbers 1,...,K for K groups.

Value

A 'list' with the following components:

Mean: A P \times K matrix, where column K contains the coefficents a_k for (11.31). The final column is all zero.
Sigma: A K \times P \times P array, where the Sigma[k,,] contains the sample covariance matrix for group k, \hat{\Sigma_k}.
c: The K-vector of constants c_k for (11.48).

Examples


# Load Iris Data
data(iris)

# Iris example
x.iris <- as.matrix(iris[, 1:4])

# Gets group vector (1, ... , 1, 2, ... , 2, 3, ... , 3)
y.iris <- rep(1:3, c(50, 50, 50))

# Perform QDA
qd.iris <- qda(x.iris, y.iris)

Reverses the matrices in a Kronecker product

Description

This function takes a matrix that is Kronecker product A \otimes B (Definition 3.5), where A is P x Q and B is N x M, and outputs the matrix B \otimes A.

Usage

reverse.kronecker(ab, p, qq)

Arguments

ab

The (NP) \times (QM) matrix A \otimes B.

p

The number of rows of A.

qq

The number of columns of A.

Value

The (NP) \times (QM) matrix B \otimes A.

Examples


# Create matrices
(A <- diag(1, 3))
(B <- matrix(1:6, ncol = 2))

# Perform kronecker
(kron <- kronecker(A, B))

# Perform reverse kronecker product
(reverse.kronecker(kron, 3, 3))

# Perform kronecker again
(kron2 <- kronecker(B, A))

Silhouettes for `K`-Means Clustering

Description

Find the silhouettes (12.9) for K-means clustering from the data and and the groups' centers.

Usage

silhouette.km(x, centers)

Arguments

x

The N \times P data matrix.

centers

The K \times P matrix of centers (means) for the K Clusters, row k being the center for cluster K.

Details

This function is a bit different from the silhouette function in the cluster package, Maechler et al., 2005.

Value

The n-vector of silhouettes, indexed by the observations' indices.

Examples


# Uses sports data.
data(sportsranks)

# Obtain the K-means clustering for sports ranks.
kms <- kmeans(sportsranks, centers = 5, nstart = 10)

# Silhouettes
sil <- silhouette.km(sportsranks, kms$centers)

Egyptian Skulls

Description

The data concern the sizes of Egyptian skulls over time, from Thomson and Randall-MacIver [1905]. There are 30 skulls from each of five time periods, so that n = 150 all together.

Usage

skulls

Format

A double matrix with 150 observations on the following 5 variables.

MaximalBreadth: Maximum length in millimeters
BasibregmaticHeight: Basibregmatic Height in millimeters
BasialveolarLength: Basialveolar Length in millimeters
NasalHeight: Nasal Height in millimeters
TimePeriod: Time groupings

Source

A. Thomson and R. Randall-MacIver. Ancient Races of the Thebaid. Oxford University Press, 1905.

Soft Drinks

Description

A data set that contains 23 peoples' ranking of 8 soft drinks: Coke, Pepsi, Sprite, 7-up, and their diet equivalents

Usage

softdrinks

Format

A double matrix with 23 observations on the following 8 variables.

Coke: Ranking given to Coke
Pepsi: Ranking given to Pepsi
7up: Ranking given to 7-up
Sprite: Ranking given to Sprite
DietCoke: Ranking given to Diet Coke
DietPepsi: Ranking given to Diet Pepsi
Diet7up: Ranking given to Diet 7-up
DietSprite: Ranking given to Diet Sprite

Source

Data from one of Prof. John Marden's earlier classes

Sort the silhouettes by group

Description

Sorts the silhouettes, first by group, then by value, preparatory to plotting.

Usage

sort_silhouette(sil, cluster)

Arguments

sil

The n-vector of silhouette values.

cluster

The n-vector of cluster indices.

Value

The n-vector of sorted silhouettes.

Examples


# Uses sports data.
data(sportsranks)

# Obtain the K-means clustering for sports ranks.
kms <- kmeans(sportsranks, centers = 5, nstart = 10)

# Silhouettes
sil <- silhouette.km(sportsranks, kms$centers)
ssil <- sort_silhouette(sil, kms$cluster)

Sports ranking

Description

Louis Roussos asked n = 130 people to rank seven sports, assigning #1 to the sport they most wish to participate in, and #7 to the one they least wish to participate in. The sports are baseball, football, basketball, tennis, cycling, swimming and jogging.

Usage

sportsranks

Format

A double matrix with 130 observations on the following 7 variables.

Baseball: Baseball's ranking out of seven sports.
Football: Football's ranking out of seven sports.
Basketball: Basketball's ranking out of seven sports.
Tennis: Tennis' ranking out of seven sports.
Cycling: Cycling's ranking out of seven sports.
Swimming: Swimming's ranking out of seven sports.
Jogging: Jogging's ranking out of seven sports.

Source

Data from one of Prof. John Marden's earlier classes

States

Description

A data set containing several demographic variables on the 50 United States, plus D.C.

Usage

states

Format

A double matrix with 51 observations on the following 11 variables.

Population: In thousands
PctCities: The percentage of the population that lives in metripolitan areas.
Doctors: Number per 100,000 people.
SchoolEnroll: The percentage enrollment in primary and secondary schools.
TeacherSalary: The average salary of primary and secondary school teachers.
CollegeEnroll: The percentage full-time enrollment at college
Crime: Violent crimes per 100,000 people
Prisoners: Number of people in prison per 10,000 people.
Poverty: Percentage of people below the poverty line.
Employment: Percentage of people employed
Income: Median household income

Source

United States (1996) Statistical Abstract of the United States. Bureau of the Census.

References

http://www.census.gov/statab/www/ranks.html

Trace of a Matrix

Description

Takes the traces of a matrix by extracting the diagonal entries and then summing over.

Usage

tr(x)

Arguments

x

Square matrix (Q \times Q)

Value

Returns a single-value double.

Examples


# Identity Matrix of size 4, gives trace of 4.
tr(diag(4))

msos: Data Sets and Functions Used in Multivariate Statistics: Old School by John Marden

Description

Author(s)

See Also

South African Hearth Disease Data

Description

Usage

Format

Details

Source

Spam

Description

Usage

Format

Source

Birthrates throughout the day in four Hospitals

Description

Usage

Format

Source

Calculate the least squares estimates

Description

Usage

Arguments

Value

See Also

Examples

Test subsets of \beta are zero

Description

Usage

Arguments

Value

See Also

Examples

Obtain the degrees of freedom for a model.

Description

Usage

Arguments

Value

See Also

Examples

Test blocks of \beta are zero.

Description

Usage

Arguments

Value

See Also

Examples

Test subsets of \beta are zero.

Description

Usage

Arguments

Value

See Also

Examples

Calculate the maximum likelihood estimates

Description

Usage

Arguments

Value

See Also

Examples

Helper function to determine \beta estimates for MLE regression with patterning.

Description

Usage

Arguments

Value

See Also

Examples

Helper function to determine \beta estimates for MLE regression.

Description

Usage

Arguments

Details

Value

See Also

Examples

The Effects of Caffeine

Description

Usage

Test subsets of `\beta` are zero

Test blocks of `\beta` are zero.

Test subsets of `\beta` are zero.

Helper function to determine `\beta` estimates for MLE regression with patterning.

Helper function to determine `\beta` estimates for MLE regression.