Adjust a predictor matrix for the presence of another matrix

Description

adjust adjusts a predictor matrix X for the presence of another predictor matrix Y, by orthogonalizing X against Y.

Usage

adjust(X, Y)

Arguments

Details

The function can handle rank deficient matrices.

Value

A matrix with an orthogonal basis for the adjusted predictor matrix.

Author(s)

Øyvind Langsrud

Conversion between matrices and partitioned matrices

Description

Functions to convert a matrix to a list of partitioned matrices, and back again.

Usage

c2df(CC)

c2m(CC)

m2c(M, df = rep(1, dim(M)[2]))

Arguments

Details

m2c partitions a matrix into a list of matrices, by putting the first df[1] coloumns into the first matrix, the next df[2] coloumns into the second, etc.

c2m joins a partitioned matrix back into a single matrix. c2m(m2c(X, df)) equals X.

c2df takes a list of matrices and returns a vector with the number of coloumns of the matrices.

Value

m2c returns a list of matrices.

c2m returns a matrix.

c2df returns a numeric vector.

Note

sum(df) must equal ncol(X).

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

ffmanova

Dressing data

Description

A dataset from an experiment studying structural and rheological properties of a full fat dressing.

Usage

data(dressing)

Format

A data frame with 29 observations on the following 7 variables.

press

a numeric vector with values 75, 125 and 225. The homogenisation pressure.

stab

a numeric vector with values 0.3, 0.4 and 0.5. Amount of stabiliser.

emul

a numeric vector with values 0.1, 0.2 and 0.35. Amount of emulsifier.

day

a factor with levels 1, ..., 5. The day the experimental run was performed on.

visc

a numeric vector. The measured viscosity of the dressing.

rheo

a matrix with 9 columns. Nine different response-parameters derived from rheological measuring. These parameters contain information about the physics of the dression (more general that viscosity).

pvol

a matrix with 241 columns. Particle-volume curves. Using a coulter-counter instrument fat particles were counted and their volumes were registered. These data are presented as smoothed histograms (equally spaced bins on log-scale). The total area under the curve represents the total volume of the counted fat particles. The shape of the curve reflects how the total fat volume is distributed among the different particle sizes.

Details

The data comes from an experiment in which full fat dressings were produced with different amount of stabiliser and emulsifier, and with different homogenisation pressure (se above).

A full factorial 3^3 design with two additional center points was used. The experiment was run over five days. It was unknown up front how many experimental runs could be performed each day, so the order of the runs was randomised.

For each dressing, viscosity, rheology and particle volume measurements were taken (se above).

The day is stored as a factor. The other design variables are stored as numerical variables. If one wants to use them as factors, one can use e.g. factor(press) in the model formula, or dressing$press <- factor(dressing$press) prior to calling the modelling function.

Source

The data is taken from a research project financed by a grant (131472/112) from the Norwegian Research Council. The project was managed by Stabburet, which is a major manufacturer of dressing in Norway.

Type II* Anova

Description

Analysis of variance table for a linear model using type II* sums of squares, which are described in Langsrud et al. (2007). Type II* extends the type II philosophy to continuous variables. The results are invariant to scale changes and pitfalls are avoided.

Usage

ffAnova(formula, data = NULL)

Arguments

Details

This function is a variant of ffmanova for the univariate special case. The two input parameters will be interpreted by model.frame.

Value

An object of class "anova" (see anova).

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

References

Langsrud, Ø., Jørgensen, K., Ofstad, R. and Næs, T. (2007): “Analyzing Designed Experiments with Multiple Responses”, Journal of Applied Statistics, 34, 1275-1296.

Examples

# Generate example data
set.seed(123)
a <- c(0, 0, 0, 10, 10, 10, 1, 1, 1)
A <- as.character(a)  # A is categorical
b <- 1:9
y <- rnorm(9)/10 + a  # y depends strongly on a (and A)
a100 <- a + 100  # change of scale (origin)
b100 <- b + 100  # change of scale (origin)

# Four ways of obtaining the same results
ffAnova(y ~ A * b)
ffAnova(y ~ A * b100)
ffAnova(lm(y ~ A * b))
ffAnova(y ~ A * b, data.frame(A = A, y = y, b = 1:9))

# Second order continuous variable
ffAnova(y ~ a + I(a^2))

# Model equivalent to 'y ~ A * b'
ffAnova(y ~ (a + I(a^2)) * b)

# Demonstrating similarities and differences using package car
if (!require(car))        # Package car is loaded if available 
  Anova <- function(...) {  # Replacement function if car not available
    warning("No results since package car is not available")}

lm_Ab <- lm(y ~ A * b)
lm_Ab100 <- lm(y ~ A * b100)

# Type II same as type II* in this case
Anova(lm_Ab)      # Type II
Anova(lm_Ab100)   # Type II
ffAnova(lm_Ab)    # Type II*
ffAnova(lm_Ab100) # Type II*

# Type III depends on scale
Anova(lm_Ab, type = 3)
Anova(lm_Ab100, type = 3)

lm_a <- lm(y ~ a + I(a^2))
lm_a100 <- lm(y ~ a100 + I(a100^2))

# Now Type II depends on scale
Anova(lm_a)      # Type II
Anova(lm_a100)   # Type II
ffAnova(lm_a)    # Type II*
ffAnova(lm_a100) # Type II*

Fifty-fifty MANOVA

Description

General linear modeling of fixed-effects models with multiple responses is performed. The function calculates 50-50 MANOVA p-values, ordinary univariate p-values and adjusted p-values using rotation testing.

Usage

ffmanova(
  formula,
  data = NULL,
  stand = TRUE,
  nSim = 0,
  verbose = TRUE,
  returnModel = TRUE,
  returnY = FALSE,
  returnYhat = FALSE,
  returnYhatStd = FALSE,
  newdata = NULL,
  linComb = NULL,
  nonEstimableAsNA = TRUE,
  outputClass = "ffmanova"
)

Arguments

Details

An overall p-value for all responses is calculated for each model term. This is done using the 50-50 MANOVA method, which is a modified variant of classical MANOVA made to handle several highly correlated responses.

Ordinary single response p-values are produced. By using rotation testing these can be adjusted for multiplicity according to familywise error rates or false discovery rates. Rotation testing is a Monte Carlo simulation framework for doing exact significance testing under multivariate normality. The number of simulation repetitions (nSim) must be chosen.

Unbalance is handled by a variant of Type II sums of squares, which has several nice properties:

1. Invariant to ordering of the model terms.

2. Invariant to scale changes.

3. Invariant to how the overparameterization problem of categorical variable models is solved (how constraints are defined).

4. Whether two-level factors are defined to be continuos or categorical does not influence the results.

5. Analysis of a polynomial model with a single experimental variable produce results equivalent to the results using an orthogonal polynomial.

In addition to significance testing an explained variance measure, which is based on sums of sums of squares, is computed for each model term.

Value

An object of class "ffmanova", which consists of the concatenated results from the underlying functions manova5050, rotationtests and unitests:

The matrices stat, pRaw, pAdjusted and pAdjFDR have one row for each model term and one column for each response.

According to the input parameters, additional elements can be included in output.

Note

The model is specified with formula, in the same way as in lm (except that offsets are not supported). See lm for details. Input parameters formula and data will be interpreted by model.frame.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

References

Langsrud, Ø. (2002) 50-50 Multivariate Analysis of Variance for Collinear Responses. The Statistician, 51, 305–317.

Langsrud, Ø. (2003) ANOVA for Unbalanced Data: Use Type II Instead of Type III Sums of Squares. Statistics and Computing, 13, 163–167.

Langsrud, Ø. (2005) Rotation Tests. Statistics and Computing, 15, 53–60.

Moen, B., Oust, A., Langsrud, Ø., Dorrell, N., Gemma, L., Marsden, G.L., Hinds, J., Kohler, A., Wren, B.W. and Rudi, K. (2005) An explorative multifactor approach for investigating global survival mechanisms of Campylobacter jejuni under environmental conditions. Applied and Environmental Microbiology, 71, 2086-2094.

See also https://www.langsrud.com/stat/program.htm.

See Also

ffAnova and predict.ffmanova.

Examples

data(dressing)

# An ANOVA model with all design variables as factors 
# and with visc as the only response variable.
# Classical univariate Type II test results are produced.
ffmanova(visc ~ (factor(press) + factor(stab) + factor(emul))^2 + day,
         data = dressing) 

# A second order response surface model with day as a block factor. 
# The properties of the extended Type II approach is utilized. 
ffmanova(visc ~ (press + stab + emul)^2 + I(press^2)+ I(stab^2)+ I(emul^2)+ day,
         data = dressing)

# 50-50 MANOVA results with the particle-volume curves as 
# multivariate responses. The responses are not standardized.
ffmanova(pvol ~ (press + stab + emul)^2 + I(press^2)+ I(stab^2)+ I(emul^2)+ day,
         stand = FALSE, data = dressing)

# 50-50 MANOVA results with 9 rheological responses (standardized).
# 99 rotation simulation repetitions are performed. 
res <- ffmanova(rheo ~ (press + stab + emul)^2 + I(press^2)+ I(stab^2)+ I(emul^2)+ day,
                nSim = 99, data = dressing)
res$pRaw      #  Unadjusted single responses p-values 
res$pAdjusted #  Familywise error rate adjusted p-values 
res$pAdjFDR   #  False discovery rate adjusted p-values

# As above, but this time 9999 rotation simulation repetitions 
# are performed, but only for the model term stab^2. 
res <- ffmanova(rheo ~ (press + stab + emul)^2 + I(press^2)+ I(stab^2)+ I(emul^2)+ day,
                nSim = c(0,0,0,0,0,9999,0,0,0,0,0), data = dressing)
res$pAdjusted[6,] # Familywise error rate adjusted p-values for stab^2
res$pAdjFDR[6,]   # False discovery rate adjusted p-values for stab^2

# Note that the results of the first example above can also be 
# obtained by using the car package.
## Not run: 
   require(car)
   Anova(lm(visc ~ (factor(press) + factor(stab) + factor(emul))^2 + day,
         data = dressing), type = "II")
## End(Not run)

# The results of the second example differ because Anova does not recognise 
# linear terms (emul) as being contained in quadratic terms (I(emul^2)).
# A consequence here is that the clear significance of emul disappears.
## Not run: 
   require(car)
   Anova(lm(visc ~ (press + stab + emul)^2 + I(press^2)+ I(stab^2)+ I(emul^2)+ day,
         data = dressing), type="II")
## End(Not run)

50-50 MANOVA testing

Description

The function performs 50-50 MANOVA testing based on a matrix of hypothesis observations and a matrix of error observations.

Usage

ffmanovatest(
  modelData,
  errorData,
  stand = 0,
  part = c(0.9, 0.5),
  partBufDim = 0.5,
  minBufDim = 0,
  maxBufDim = 1e+08,
  minErrDf = 3,
  cp = -1
)

Arguments

Details

modelData and errorObs correspond to hypObs and errorObs calculated by xy_Obj.

Value

A list with components

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

ffmanova

Fix the "factor" matrix of a terms object.

Description

The function takes the factor matrix of the terms object corresponding to a model formula and changes it so that model hierarchy is preserved also for powers of terms (e.g., I(a^2)).

Usage

fixModelMatrix(mOld)

Arguments

Details

The ordinary model handling functions in do not treat powers of terms (a^n) as being higher order terms (like interaction terms). fixModelMatrix takes the "factor" attribute of a terms object (usually created from a model formula) and changes it such that power terms can be treated hierarchically just like interaction terms.

The factor matrix has one row for each variable and one coloumn for each term. Originally, an entry is 0 if the term does not contain the variable. If it contains the variable, the entry is 1 if the variable should be coded with contrasts, and 2 if it should be coded with dummy variables. See terms.object for details.

The changes performed by fixModelMatrix are:

• Any 2's are changed to 1.

• In any coloumn corresponding to a term that contains I(a^n), where a is the name of a variable and n is a positive integer, the element in the row corresponding to a is set to n. For instance, the entry of row D and coloumn C:I(D^2) is set to 2.

• Rows corresponding to I(a^n) are deleted.

Note that this changes the semantics of the factor matrix: 2 no longer means ‘code via dummy variables’.

Value

A factor matrix.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

terms, terms.object

Examples

mt <- terms(y ~ a + b + a:b + a:c + I(a^2) + I(a^3) + I(a^2):b)
print(mOld <- attr(mt, "factor"))
fixModelMatrix(mOld)

Linear regression estimation

Description

Function that performs multivariate multiple linear regression modelling (Y = XB + E) according to a principal component regression (PCR) approach where the number of components equals the number of nonzero eigenvalues (generalised inverse).

Usage

linregEnd(Umodel, Y)

linregEst(X, Y)

linregStart(X, rank_lim = 1e-09)

Arguments

Details

The function linregEst performs the calculations in two steps by calling linregStart and linregEnd. The former functions function makes all calculations that can be done without knowing Y. The singular value decomposition (SVD) is an essential part of the calculations and some of the output variables are named according to SVD (‘⁠U⁠’, ‘⁠S⁠’ and ‘⁠V⁠’).

Value

linregEst returns a list with seven components. The first three components is returned by linregStart - the rest by linregEnd.

Note

When the number of error degrees of freedom exceeds the number of linearly independent responses, then the matrix of error observations is made so that several rows are zero. In this case the zero rows are omitted and a list with components errorObs and df_error is returned.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

ffmanova

Computation of 50-50 MANOVA results

Description

The function takes a design-with-responses object created by xy_Obj and produces 50-50 MANOVA output. Results are produced for each term in the model.

Usage

manova5050(xyObj, stand)

Arguments

Details

Classical multivariate ANOVA (MANOVA) are useless in many practical cases. The tests perform poorly in cases with several highly correlated responses and the method collapses when the number of responses exceeds the number of observations. 50-50 MANOVA is made to handle this problem. Principal component analysis (PCA) is an important part of this methodology. Each test is based on a separate PCA.

Value

A list with components

Note

The 50-50 MANOVA p-values are based on the Hotelling-Lawley Trace Statistic. The number of components for testing and the number of buffer components are chosen according to default rules.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

References

Langsrud, Ø. (2002) Rotation Tests. The Statistician, 51, 305–317.

See Also

ffmanova

Simulate Matlab's ‘:’

Description

A function to simulate Matlab's ‘:’ operator.

Usage

matlabColon(from, to)

Arguments

Details

matlabCode(a,b) returns a:b ('s version) unless a > b, in which case it returns integer(0).

Value

A numeric vector, possibly empty.

Author(s)

Bjørn-Helge Mevik

See Also

seq

Examples

identical(3:5, matlabColon(3, 5)) ## => TRUE
3:1 ## => 3 2 1
matlabColon(3, 1) ## => integer(0)

p-values from MANOVA test statistics

Description

p-values from the four MANOVA test statistics are calculated according to the traditional F-distribution approximations (exact in some cases).

Usage

multiPvalues(D, E, A, M, dim, dimX, dimY)

Arguments

Details

The parameters dim, dimX and dimY corresponds to a situation where the test statistics are calculated from two data matrices with zero mean (test of independence).

Value

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

ffmanova

MANOVA test statistics

Description

The four classical MANOVA test statistics are calculated from a set of eigenvalues.

Usage

multiStatistics(ss)

Arguments

Details

These eigenvalues are also known as the squared canonical correlation coefficients.

Value

A list with elements

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

F-test p-values

Description

This is simply a wrapper around pf: my_pValueF(f, ny1, ny2) is equivalent to pf(f, ny1, ny2, lower.tail = FALSE).

Usage

my_pValueF(f, ny1, ny2)

Arguments

Value

A p-value.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

pf

Rank and orthonormal basis

Description

myorth(X) makes an orthonormal basis for the space spanned by the columns of X. The number of columns returned equals myrank(X), which is the rank of X.

Usage

myorth(X, tol_ = 1e-09)

Arguments

Details

The calculations are based on the singular value decomposition (svd). And myrank(X) is the number of singular values of X that are larger than max(dim(X))*svd(x)$d[1]*tol_.

Value

myorth returns a matrix, whose columns form an orthonormal basis.

myrank returns a single number, which is the rank of X.

Note

In the special case where X has a single column, myorth(X) returns c*X where c is a positive constant.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

svd

Matrix norm.

Description

norm(X) returns the largest singular value of X; it is equivalent to svd(X, nu = 0, nv = 0)$d[1].

Usage

norm(X)

Arguments

Value

The largest singular value of X.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

svd

Making adjusted design matrix data

Description

The function takes the output from modelData as input and calculates adjusted data

Usage

orth_D(D, model, method)

Arguments

Details

The "test" method adjusts data according to Type II* sums of squares. This is an extension of the traditional Type II method. The "om" method orthogonalises terms according to the model hierarchy. The result is a non-overparameterised representation of the model.

Value

An adjusted version of D is returned.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

Predictions, mean predictions, adjusted means and linear combinations

Description

The same predictions as lm can be obtained. With some variables missing in input, adjusted means or mean predictions are computed (Langsrud et al., 2007). Linear combinations of such predictions, with standard errors, can also be obtained.

Usage

## S3 method for class 'ffmanova'
predict(object, newdata = NULL, linComb = NULL, nonEstimableAsNA = TRUE, ...)

Arguments

Value

A list of two matrices:

References

Langsrud, Ø., Jørgensen, K., Ofstad, R. and Næs, T. (2007): “Analyzing Designed Experiments with Multiple Responses”, Journal of Applied Statistics, 34, 1275-1296.

Examples

# Generate data
x1 <- 1:6
x2 <- rep(c(100, 200), each = 3)
y1 <- x1 + rnorm(6)/10
y2 <- y1 + x2 + rnorm(6)/10

# Create ffmanova object
ff <- ffmanova(cbind(y1, y2) ~ x1 + x2)

# Predictions from the input data
predict(ff)

# Rows 1 and 5 from above predictions
predict(ff, data.frame(x1 = c(1, 5), x2 = c(100, 200)))

# Rows 1 as above and row 2 different
predict(ff, data.frame(x1 = c(1, 5), x2 = 100))

# Three ways of making the same mean predictions
predict(ff, data.frame(x1 = c(1, 5), x2 = 150))
predict(ff, data.frame(x1 = c(1, 5), x2 = NA))
predict(ff, data.frame(x1 = c(1, 5)))

# Using linComb input specified to produce regression coefficients 
# with std. As produced by summary(lm(cbind(y1, y2) ~ x1 + x2))
predict(ff, data.frame(x1 = c(1, 2)), matrix(c(-1, 1), 1, 2))
predict(ff, data.frame(x2 = c(101, 102)), matrix(c(-1, 1), 1, 2))

# Above results by a 2*4 linComb matrix and with rownames
lC <- t(matrix(c(-1, 1, 0, 0, 0, 0, -1, 1), 4, 2))
rownames(lC) <- c("x1", "x2")
predict(ff, data.frame(x1 = c(1, 2, 1, 1), x2 = c(100, 100, 101, 102)), lC)

Print method for ffmanova

Description

Print method for objects of class "ffmanova". It prints an ANOVA table.

Usage

## S3 method for class 'ffmanova'
print(x, digits = max(getOption("digits") - 3, 3), ...)

Arguments

Details

The function constructs an anova table, and prints it using printCoefmat with tailored arguments.

Value

Invisibly returns the original object.

Author(s)

Bjørn-Helge Mevik

See Also

ffmanova, printCoefmat

Rotation testing

Description

The functions perform rotation testing based on a matrix of hypothesis observations and a matrix of error observations. Adjusted p-values according to familywise error rates and false discovery rates are calculated.

Usage

rotationtests(xyObj, nSim, verbose = TRUE)

rotationtest(modelData, errorData, simN = 999, dfE = -1, dispsim = TRUE)

Arguments

Details

modelData and errorObs correspond to hypObs and errorObs calculated by xy_Obj. These matrices are efficient representations of sums of squares and cross-products (see xy_Obj for details). This means that rotationtest can be viewed as a generalised F-test function.

rotationtests is a wrapper function that calls rotationtest for each term in the xyObj and collects the results.

Value

Both functions return a list with components

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

References

Langsrud, Ø. (2005) Rotation Tests. Statistics and Computing, 15, 53–60.

Moen, B., Oust, A., Langsrud, Ø., Dorrell, N., Gemma, L., Marsden, G.L., Hinds, J., Kohler, A., Wren, B.W. and Rudi, K. (2005) An explorative multifactor approach for investigating global survival mechanisms of Campylobacter jejuni under environmental conditions. Applied and Environmental Microbiology, 71, 2086-2094.

See Also

unitest, unitests

Centering and scaling of matrices

Description

Function to center and/or scale the coloumns of a matrix in various ways. The coloumns can be centered with their means or with supplied values, and they can be scaled with their standard deviations or with supplied values.

Usage

stdize(x, center = TRUE, scale = TRUE, avoid.zero.divisor = FALSE)

stdize3(x, center = TRUE, scale = TRUE, avoid.zero.divisor = FALSE)

Arguments

Details

stdize standardizes the coloumns of a matrix by subtracting their means (or the supplied values) and dividing by their standard deviations (or the supplied values).

If avoid.zero.divisor is TRUE, division-by-zero is guarded against by substituting any 0 in center (either calculated or supplied) with 1 prior to division.

The main difference between stdize and scale is that stdize divides by the standard deviations even when center is not TRUE.

Value

A matrix.

Note

stdize3 is a variant with a three-element list as output (x, center, scale) and where avoid.zero.divisor is also used to avoid centring (constant term in model matrix is unchanged).

Author(s)

Bjørn-Helge Mevik and Øyvind Langsrud

See Also

scale

Examples

A <- matrix(rnorm(15, mean = 1), ncol = 3)
stopifnot(all.equal(stdize(A), scale(A), check.attributes = FALSE))

## These are different:
stdize(A, center = FALSE)
scale(A, center = FALSE)

Univariate F or t testing

Description

The functions perform F or t testing for several responses based on a matrix of hypothesis observations and a matrix of error observations.

Usage

unitests(xyObj)

unitest(modelData, errorData, dfError = dim(errorData)[1])

Arguments

Details

modelData and errorObs correspond to hypObs and errorObs calculated by xy_Obj. These matrices are efficient representations of sums of squares and cross-products (see xy_Obj for details). This means the univariate F-statistics can be calculated straightforwardly from these input matrices. Furthermore, in the single-degree-of-freedom case, t-statistics with correct sign can be obtained.

unitests is a wrapper function that calls unitest for each term in the xyObj (see xy_Obj for details) and collects the results.

Value

unitest returns a list with components

unitests returns a list with components

Note

The function calculates the p-values by making a call to pf.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

rotationtest, rotationtests

Creation of a design matrix object

Description

The function takes design/model information as input and performs initial computations for prediction and testing.

Usage

x_Obj(D, model)

Arguments

Details

See the source code of ffmanova to see how D and model are created.

Value

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

See Also

linregEst, xy_Obj.

Creation of a design-with-responses object

Description

The function takes an object created by x_Obj as input and add response values. Further initial computations for prediction and testing is made.

Usage

xy_Obj(xObj, Y)

ffModelObj(
  xObj,
  Y,
  modelMatrix,
  modelTerms,
  model,
  xlev,
  scaleY,
  scaleX,
  centerX,
  isIntercept,
  returnY = FALSE,
  returnYhat = FALSE,
  returnYhatStd = FALSE
)

Arguments

Details

Traditionally, sums of squares and cross-products (SSC) is the multivariate generalisation of sums of squares. When there is a large number of responses this representation is inefficient and therefore linear combinations of observations (Langsrud, 2002) is stored instead, such as errorObs. The corresponding SSC matrix can be obtained by t(errorObs)%*%errorObs. When there is a large number of observations the errorObs representation is also inefficient, but it these cases it is possible to chose a representation with several zero rows. Then, errorObs is stored as a two-component list: A matrix containing the nonzero rows of errorObs and an integer representing the degrees of freedom for error (number of rows in the full errorObs matrix).

Value

A list with components

Note

ffModelObj is a rewrite of xy_Obj with additional elements in output corresponding to the additional parameters in input. Furthermore, Y and YhatStd is by default not included in output.

Author(s)

Øyvind Langsrud and Bjørn-Helge Mevik

References

Langsrud, Ø. (2002) 50-50 Multivariate Analysis of Variance for Collinear Responses. The Statistician, 51, 305–317.