Design and Analysis of dose-finding studies

Description

The crossdes package provides functions for the construction of of carryover balanced crossover designs. In addition contains functions to check given designs for balance.

Details

The main functions are:
get.plan: Menu-driven selection of crossover designs
allcombs: Construct crossover design with all possible treatment orders
williams: Construct a Williams design
williams.BIB: Construct a crossover design based on a combination of balanced incomplete block designs and Williams designs.
des.MOLS: Construct crossover design based on mutually orthogonal Latin Squares
balmin.RMD: Construct balanced minimal repeated measurements crossover design
isCbalanced: Check whether a crossover design is balanced for first order carryover effects
isGYD: Check whether a crossover design is balanced

Author(s)

Martin Oliver Sailer

References

Earlier version of the package:
Sailer, O. (2005): crossdes: A package for design and randomization in crossover studies. Rnews 5/2, 24-27.

Overview on cross-over designs:
Jones, B. and Kenward, M.G. (1989): Design and Analysis of Cross-Over Trials. Chapman and Hall, London.

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

Construction of Complete Sets of Mutually Orthogonal Latin Squares

Description

The function constructs sets of mutually othogonal latin squares (MOLS) using Galois fields. The construction works for prime powers only.

Usage

MOLS(p, n, primpol = GF(p, n)[[2]][1, ])

Arguments

Details

If trt = p^n is a prime power, then trt-1 latin squares of order trt are constructed. The elements of the squares are numbered 1,...,trt. These squares are mutually orthogonal, i.e. if any two of them are superimposed, the resulting array will contain each ordered pair (i,j), i,j in {1,..., trt} exactly once. The squares are in standard order, i.e. the first row is always equal to (1,...,trt). A primitive polynomial may be constructed automatically using the internal function GF.

Value

For trt = p^n, an array that contains trt-1 latin squares is returned.

Author(s)

Oliver Sailer

References

Cherowitzo, W.: http://www-math.cudenver.edu/~wcherowi/courses/finflds.html

Street, A.P. and Street, D.J. (1987): Combinatorics of experimental design. Oxford University Press, Oxford.

See Also

des.MOLS

Examples

MOLS(7,1) # 6 mutually orthogonal latin squares of order 7
MOLS(2,3) # 7 mutually orthogonal latin squares of order 8

Balanced Row-Column Design with all Possible Treatment Orders

Description

The function constructs a row-column design with subjects as rows and periods as columns. Each subject gets each treatment at most once. All possible treatment orders are assigned to the subjects.

Usage

allcombs(trt, k)

Arguments

Details

The design is a carryover balanced generalized Youden design that is uniform on the columns. The treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

Value

A matrix with \frac{trt!}{(trt-k)!} rows and k columns representing the experimental design.

Note

Requires the package gtools.

Author(s)

Oliver Sailer

References

Patterson, H.D. (1952): The construction of balanced designs for experiments involving sequences of treatments. Biometrika 39, 32-48.

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

See Also

get.plan

Examples

# Design for 4 treatments assigned in 3 periods.
# All possible treatment orders occur.
allcombs(4,3) 

Function to construct the balanced minimal repeated measurements designs of Afsarinejad (1983)

Description

The function constructs a row-column design with subjects as rows and periods as columns. The design is incomplete, i.e. no subject gets all the treatments. The design is balanced for carryover effects but will in general not be a balanced block design.

Usage

balmin.RMD(trt, n, p)

Arguments

Details

A necessary and sufficient condition for the existence of such a design is that \frac{(trt-1)}{(p-1)} be a positive integer. In this case n = \frac{trt(trt-1)}{(p-1)}. In the resulting design the treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

Value

A matrix with n rows and p columns representing the experimental design.

Author(s)

Oliver Sailer

References

Afsarinejad, K. (1983): Balanced repeated measurements designs. Biometrika 70, 199-204.

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

See Also

get.plan

Examples

balmin.RMD(10,30,4) # a balanced minimal RMD
balmin.RMD(11,22,6) # another balanced minimal RMD

Experimental Design for the Beer Testing Data in Kunert (1998)

Description

The design is a balanced incomplete block design with rows as blocks. If one assumes that there is a preperiod, i.e. that each assessor is given the treatment of the fifth period before the experiment starts, then the design is carryover balanced.

Usage

data(biertest.d4)

Format

A matrix with 12 rows corresponding to the assessors, 5 columns corresponding to the five periods.

Details

The five treatments (products) to be tested are numbered 1,...,5.

Source

Kunert, J. (1998): Sensory experiments as crossover studies. Food Quality and Preference 9, 243-253 (design d4).

Examples

data(biertest.d4)

The Beer Testing Data in Kunert (1998)

Description

The data comes from a beer testing experiment where assessors had to rate the bitterness of five beers.

Usage

data(biertest.dat)

Format

A matrix with 12 rows corresponding to the assessors, 5 columns corresponding to the five periods.

Details

The possible values for the ratings lie in the interval [0, 12]. Note that the data were reordered after the experiment.

Source

Kunert, J. (1998): Sensory experiments as crossover studies. Food Quality and Preference 9, 243-253 (table 1).

Examples

data(biertest.dat)

Experimental Design for the Beer Testing Data in Kunert (1998)

Description

The design is a row-column design. It is actually a generalized latin square design that is also carryover balanced.

Usage

data(biertest.d4)

Format

A matrix with 10 rows corresponding to the first 10 assessors in a fictional uniformity trial, 5 columns corresponding to the five periods.

Details

There are five products to be tested, numbered 1,...,5.

Source

Kunert, J. (1998): Sensory experiments as crossover studies. Food Quality and Preference 9, 243-253 (design f2).

Examples

data(biertest.f2)

Internal Functions in crossdes

Description

For internal use only.

Author(s)

Oliver Sailer

Construction of Designs Based on Mutually Orthogonal Latin Squares

Description

The function constructs row-column designs based on complete sets of mutually orthogonal latin squares. Each subject may get each tratment at most once. The design is a generalized Youden design that is also balanced for carryover effects.

Usage

des.MOLS(trt, k = trt)

Arguments

Details

A complete set of mutually orthogonal latin squares is constructed using Galois Fields. The rows of the designs represent the treatment orders for the subjects. If an incomplete design with k columns is needed, only the first k columns of the designs are considered. The treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

Value

A matrix with trt(trt-1) rows and k columns representing the experimental design.

Author(s)

Oliver Sailer

References

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

Williams, E. J. (1949): Experimental designs balanced for the estimation of residual effects of treatments. Australian Journal of Scientific Research, Ser. A 2, 149-168.

See Also

get.plan, MOLS

Examples

des.MOLS(7,7)
des.MOLS(8,5)

Generation of Balanced Incomplete Block Designs Using the Package AlgDesign

Description

The function optBlock of the library AlgDesign is used to search for balanced incomplete block designs (BIBDs). The design is assigned to a matrix where rows represent blocks (subjects) and columns represent periods.

Usage

find.BIB(trt, b, k, iter = 30)

Arguments

Details

The function optBlock tries to find a D-optimal block design for the specified parameters. The resulting design need not be a BIBD. The necessary conditions for the existence are that \frac{bk}{trt} and \frac{bk(k-1)}{trt(trt-1)} positive integers. They are NOT checked automatically. Even if they are fulfilled, there need not be a BIBD. If no BIBD is found, the function is iterated. If no BIBD is found after iter iterations, the search is terminated. The resulting design should be checked by the user applying isGYD.

Value

A matrix that represents the experimental design.

Note

As indicated above, the returned design is not necessarily a BIBD design.

Author(s)

Oliver Sailer

References

Wheeler, R.E. (2004). optBlock. AlgDesign. The R project for statistical computing http://www.r-project.org/

See Also

get.plan, optBlock

Examples

find.BIB(10,30,4)       # BIBD
find.BIB(3,3,3)         # BIBD 
find.BIB(5,5,3, 100)    # There is no BIBD for these parameters   
# isGYD(find.BIB(5,5,3, 100))

Menu-Driven Construction of Carryover Balanced Experimental Designs

Description

This menu based function constructs simple experimental designs for repeated measurements with one or two block variables. It is assumed that each subject is assigned to each treatment at most once. A maximum number of subjects in the study is also requested. Five possible construction methods available. These construction methods and the characteristics of the resulting designs are described in Wakeling and MacFie (1995). See also Jones and Kenward (1989), Ch. 5, for a discussion of these designs. The function is demonstrated in more detail in Sailer (2005).

Usage

get.plan(trt, k = trt, maxsub = 1000)

Arguments

Details

The five types of designs are: designs based on all possible treatment orders ("all.combin"), Williams designs ("williams"), designs based on mutually orthogonal latin squares ("des.MOLS"), a combination of balanced incomplete block designs (BIBDs) and Williams designs ("williams.BIB") by Patterson (1951) and the balanced minimal designs of Afsarinejad ("balmin.RMD"). Some designs are only available for special combinations of treatment number and number of periods. Other designs may require too many subjects. Therefore, the possible choices available for the submitted values of trt, k and maxsub are determined. If there is no design available, the parameters may be changed interactively. If more than one design type is available the user has to choose one. The minimum number of subjects required for the designs is given and maybe a criterion for selecting a design. All types of designs are balanced for first-order carryover effects. All types except the balanced minimal RMDs are also balanced block designs. The user may want to construct a design for a multiple of the minimum number of subjects required to get closer to the preferred number of subjects. Once the design is chosen the design is displayed. In practice the labels for the treatments and subjects should be randomized before the design is used. The treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

Value

A matrix representing the experimental design.

Warning

For the construction of designs that combine BIBDs with Williams designs, the function find.BIB is called to search for a BIBD. If the necessary conditions for the existence of a BIBD are fulfilled, this approach always returns a design. This design will however not always be a BIBD! When using the Patterson approach, please check the resulting design for balance using isGYD and isCbalanced.

It should be noted that this is a computational problem only, not a problem of the theoretical approach of Patterson (1951).

Note

The "All combinations" approach requires the package gtools.

Author(s)

Oliver Sailer

References

Afsarinejad, K. (1983): Balanced repeated measurements designs. Biometrika 70, 199-204.

Jones, B. and Kenward, M.G. (1989): Design and Analysis of Cross-Over Trials. Chapman and Hall, London.

Patterson, H.D. (1951): Change-over trials. Journal of the Royal Statistical Society B 13, 256-271.

Patterson, H.D. (1952): The construction of balanced designs for experiments involving sequences of treatments. Biometrika 39, 32-48.

Sailer, O. (2005): crossdes: A package for design and randomization in crossover studies. Rnews 5, 24-27.

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

Williams, E. J. (1949): Experimental designs balanced for the estimation of residual effects of treatments. Australian Journal of Scientific Research, Ser. A 2, 149-168.

See Also

allcombs, balmin.RMD, des.MOLS, williams, williams.BIB

Examples

## Not run: 
get.plan(10,4,60)  
  # "williams.BIB" or "balmin.RMD"
get.plan(7,7,7000) 
  # "all.combin", "williams" or "des.MOLS", "williams" requires 
  # only 14 subjects, "all.combin" requires 5040.   
get.plan(5,5,5)    
  # Increase maxsub

## End(Not run)

Checking Block Designs for Carryover Balance

Description

The function checks whether a block design is balanced for first order carryover effects (residual effects). The user specifies whether there is a preperiod. The design is checked and the left neighbour incidence matrix is given.

Usage

isCbalanced(d, preperiod = FALSE)

Arguments

Details

The design is said to be carryover balanced (balanced for first order carryover effects), if each treatment is preceeded by all other treatments equally often and if no treatment is preceeded by itself. If the design is balanced, this is stated.

Value

Author(s)

Oliver Sailer

See Also

isGYD

Examples

d1 <- matrix( c(1,2,3,4,1,1,1,1), 4,2)
d2 <- matrix( c(1:4,2:4,1,4,1:3,3,4,1,2),ncol=4)
d3 <- matrix( rep(1:3,each=2), ncol=2)
isCbalanced(d1)
isCbalanced(d1,TRUE)  
isCbalanced(d2)
isCbalanced(d3,TRUE)

Checking Simple Block and Row-Column Designs for Balance

Description

A function to check a simple block or a row-column design for balance. The rows and columns of the design are blocking variables. It is checked which type of balance the design fulfills. Optionally, incidence and concurrence matrices are given.

Usage

isGYD(d, tables=FALSE, type=TRUE)

Arguments

Details

A design is said to be a balanced block design if the following three conditions hold: i) Each treatment appears equally often in the design. ii) The design is binary in the sense that each treatment appears in each block either n or n+1 times where n is an integer. iii) The number of concurrences of treatments i and j is the same for all pairs of distinct treatments (i,j). Here the blocks are either rows or columns.

A design that has less columns (rows) than treatments is said to be incomplete with respect to rows (columns). A design that is balanced with respect to both rows and columns is called a generalized Youden design (GYD). A GYD for which each treatment occurs equally often in each row (column) is called uniform on the rows (columns). If both conditions hold, it is called a generalized latin square. A design where each treatment occurs exactly once in each row and column is called a latin square.

Value

A list containing information about balance in rows and columns as well as incidence and concurrence matrices for the design.

Author(s)

Oliver Sailer

See Also

isCbalanced

Examples

d1 <- matrix( c(1,2,3,4,1,1,1,1), 4,2)          
# d1 is not balanced
d2 <- matrix( c(1:4,2:4,1,4,1:3,3,4,1,2),ncol=4)    
# d2 is a latin square
d3 <- matrix( rep(1:3,each=2), ncol=2)          
# d3 is a balanced incomplete block design.
d1
isGYD(d1,tables=TRUE)  
d2
isGYD(d2,tables=TRUE)
d3
isGYD(d3,tables=TRUE)    

Construction of Williams Designs

Description

The function constructs williams designs. Williams designs are row-column designs. They are used if each of the treatments in the study is given to each of the subjects. If the number of treatments to be tested is even, the design is a latin square, otherwise it consists of two latin squares.

Usage

williams(trt)

Arguments

Details

The resulting design is a (generalized) latin square that is also balanced for first order carryover effects. Carryover balance is achieved with very few subjects. In the experimental design the treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

Value

A matrix representing the experimental design.

Author(s)

Oliver Sailer

References

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

Williams, E. J. (1949): Experimental designs balanced for the estimation of residual effects of treatments. Australian Journal of Scientific Research, Ser. A 2, 149-168.

See Also

get.plan

Examples

williams(3)
williams(10)

Construction of Carryover Balanced Designs Based on Balanced Incomplete Block Designs

Description

Patterson (1951) combined balanced incomplete block designs (BIBDs) with Williams designs to get carryover balanced generalized Youden designs.

Usage

williams.BIB(d)

Arguments

Details

For each row of the design, a Williams design is constructed using the treatments of that row. The rows of the resulting designs are then combined. The treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

Value

A matrix representing the experimental design.

Warning

The resultig design is only balanced properly if the input design actually is a BIBD. This is NOT checked automatically. You have to do this by yourself, e.g. by applying isGYD to your design.

Note

BIBDs may be generated using find.BIB.

Author(s)

Oliver Sailer

References

Patterson, H.D. (1951): Change-over trials. Journal of the Royal Statistical Society B 13, 256-271.

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

See Also

get.plan, isGYD, find.BIB, williams

Examples

d <- matrix( rep(1:3,each=2), ncol=2)
# # check for balance
# isGYD(d)
williams.BIB(d)