Information Criteria for averaging models

Description

Functions to extract or recalculate the Akaike Information Criterion and the Bayesian Information Criterion of an averaging model fitted by the rav function.

Usage

AIC(object, ..., k = 2)
BIC(object, ...)

Arguments

Details

The functions AIC and BIC are used, respectively, to extract the Akaike Information Criterion and the Bayesian Information Criterion of a model fitted by the function rav.

AIC is calculated as:

AIC = n \ln \left( \frac{RSS}{n} \right) + k p

where n is the number of data available, k is the penalty per parameter ()usually equal to 2), p is the number of parameters and RSS is the residual sum of squares.

BIC is calculated as:

BIC = n \ln \left( \frac{RSS}{n} \right) + \ln(n) p

As default, when n / p < 40, AIC and BIC are corrected in AICc and BICc:

AICc = AIC + \frac{2 (p+1) p}{n-p-1}

BICc = BIC + \frac{\ln(n) (p+1) p}{n-p-1}

to avoid the correction, set correct = FALSE. On the contrary, if you want the correction, set correct = TRUE. When the argument correct is not specified, the rule n / p < 40 is applied.

As default, the functions extract the indices of the (first) best model. The optional argument whichModel can be specified to extract the indices of another model. Options are:

1. "null": null model

2. "ESM": equal scale values model

3. "SAM": simple averaging model

4. "EAM": equal-weights averaging model

5. "DAM": differential-weight averaging model

6. "IC": information criteria model

Value

A numeric value representing the information criterion of the selected model.

See Also

rav, rAverage-package AIC, BIC

Examples

## Not run: 
data(fmdata1)
fm1 <- rav(fmdata1, lev=c(3,3))
AIC(fm1)
BIC(fm1)

## End(Not run)

Extract coefficients from an averaging model

Description

Function to extract fit model coefficients from an object returned by rav.

Usage

coef(object, ...)

Arguments

Details

Returns the parameters of an averaging model fitted by the rav function, in the order: s0, w0, s(k,j), and w(k,j).

As default, the function extract the cofficients of the (first) best model. The optional argument whichModel can be specified to extract the values of another model. Options are:

1. "null": null model

2. "ESM": equal scale values model

3. "SAM": simple averaging model

4. "EAM": equal-weights averaging model

5. "DAM": differential-weight averaging model

6. "IC": information criteria model

Value

A numeric vector.

See Also

rav, rAverage-package

Examples

## Not run: 
data(fmdata1)
fm1 <- rav(fmdata1, lev=c(3,3))
coef(fm1)
coef(fm1, whichModel="EAM")

## End(Not run)

Generating Noisy Responses for an Averaging Model

Description

This function generates noisy synthetic responses R for an averaging model given the true parameters s0, w0, s(k,j), and w(k,j).

Usage

datgen(param, lev, t.par = FALSE, trials = 1, sd = 0, range = NULL)

Arguments

Value

A matrix object containing the generated responses of the averaging model, in the order: one-way design, two-way design, three way design, etc. See rav function.

See Also

rav, pargen, rav.indices, rAverage-package

Examples

## Not run: 
# Generating random parameters for a 3x4 design:
par <- pargen(lev = c(3,4), s.range = c(0,20))
# Computing the responses:
R <- datgen(param=par, lev=c(3,4), sd=0) ; R
R <- datgen(param=par, lev=c(3,4), sd=1, trials=10, range=c(0,20)) ; R

## End(Not run)

Extract fitted values from an averaging model

Description

Function to extract fitted values from an object returned by rav.

Usage

fitted(object, ...)

Arguments

Details

Returns the expected responses given an averaging model fitted by the rav function.

As default, the function extract the fitted values of the (first) best model. The optional argument whichModel can be specified to extract the values of another model. Options are:

1. "null": null model

2. "ESM": equal scale values model

3. "SAM": simple averaging model

4. "EAM": equal-weights averaging model

5. "DAM": differential-weight averaging model

6. "IC": information criteria model

Value

A matrix of numeric values.

See Also

rav, rAverage-package

Examples

## Not run: 
library(rAverage)
data(fmdata1)
fm1 <- rav(fmdata1, lev=c(3,3))
fitted(fm1)
fitted(fm1, whichModel="EAM")

## End(Not run)

rAverage dataset examples

Description

Examples of dataset for R-Average analysis.
fmdata1: example of a 3x3 design. Original parameters:

fmdata2: example of a 3x5 design. Original parameters:

fmdata3: example of a 3x2x3 design. Original parameters:

Usage

    data(fmdata1)
    data(fmdata2)
    data(fmdata3)

Format

A matrix object.

Examples

## Not run: 
data(fmdata1)
fm1 <- rav(fmdata1, lev=c(3,3))
data(fmdata2)
fm2 <- rav(fmdata2, lev=c(3,5))
data(fmdata3)
fm3 <- rav(fmdata3, lev=c(3,2,3))

## End(Not run)

Outlier detection and substitution

Description

Starting by a previously estimated averaging model, this function detect outliers according to a Bonferroni method. The outliers can be substituted with a user-defined value.

Usage

outlier.replace(object, whichModel = NULL, alpha = 0.05, value = NA)

Arguments

Value

A data object in which outliers have been removed or replaced.

See Also

rav, rAverage-package,

Examples

data(pasta)
model <- rav(pasta, subset="s04", lev=c(3,3), names=c("Price","Packaging"))
outlier.replace(model, value=mean)
outlier.replace(model, whichModel="IC", value=NA)

Generating random parameters for averaging responses

Description

Generates a random set of parameters that follows an averaging rule.

Usage

pargen(lev, s.range = c(0,20), w.range = exp(c(-5,5)),
        I0 = FALSE, t.par = FALSE, digits = 2)

Arguments

Value

Vector contaning the random-generated parameters in the order s0,w0,s(k,i),w(k,i).

See Also

datgen, rav, rav.indices, rAverage-package

Examples

# Generating random parameters for a 3x4 design:
param <- pargen(lev = c(3,4))

Pasta experiment

Description

Data of four subjects from the Pasta experiment. The table contains data of the one-way sub-designs, folowed by the data of the full-factorial design. Factors: Price (3 levels: 0.89, 0.99, 1.09) and Packaging (3 levels: box with window, box without window, plastic bag).

Usage

data(pasta)

Format

A data.frame object.

References

Massidda D., Polezzi D., Vidotto G. (2011). A Functional Measurement approach to cope the non-linearity of judgments in marketing research. Proceedings of the 10th European Conference on Research Methodology for Business and Management Studies. Normandy Business School, Caen, France, 348-354.

Examples

data(pasta)

Parameter estimation for the averaging model of Information Integration Theory

Description

The R-Average package implements a method to identify the parameters of the Averagingcmodel of Information Integration Theory (Anderson, 1981), following the spirit of the so-called "principle of parsimony".

Name of the parameters:

s0,w0: initial state values of the Averaging Model.
s(k,j): scale value of the j-th level of k-th factor.
w(k,j): weight value of the j-th level of k-th factor.

Details

Functions of the R-Average package:

rav: estimates the parameters for averaging models.
fitted: extracts the predicted values of the best model from a rav object.
residuals: extracts the residuals from a rav object.
coefficients: extracts the parameters from a rav object.
outlier.replace: given an estimated averaging model with the rav function, it detects and replace outliers from the residual matrix. rav.indices: given a set of parameters s and w and a matrix of observed data, it calculates the fit indices for the averaging model.
datgen: returns the responses R for averaging models given the set of parameters s and w.
pargen: generates pseudorandom parameters for the averaging model.
rav.grid: generates an empty matrix in 'rav' format.
rav.single: single subjects analysis over an aggregated data matrix.
rav2file: store the reesults of rav into a text file.

Author(s)

Supervisor: Prof. Giulio Vidotto giulio.vidotto@unipd.it

University of Padova, Department of General Psychology
QPLab: Quantitative Psychology Laboratory

version 0.0:
Marco Vicentini marco.vicentini@gmail.com

version 0.1 and following:
Stefano Noventa stefano.noventa@univr.it
Davide Massidda davide.massidda@gmail.com

References

Akaike, H. (1976). Canonical correlation analysis of time series and the use of an information criterion. In: R. K. Mehra & D. G. Lainotis (Eds.), System identification: Advances and case studies (pp. 52-107). New York: Academic Press. doi: 10.1016/S0076-5392(08)60869-3

Anderson, N. H. (1981). Foundations of Information Integration Theory. New York: Academic Press. doi: 10.2307/1422202

Anderson, N. H. (1982). Methods of Information Integration Theory. New York: Academic Press.

Anderson, N. H. (1991). Contributions to information integration theory: volume 1: cognition. Lawrence Erlbaum Associates, Hillsdale, New Jersey. doi: 10.2307/1422884

Anderson, N. H. (2007). Comment on article of Vidotto and Vicentini. Teorie & Modelli, Vol. 12 (1-2), 223-224.

Byrd, R. H., Lu, P., Nocedal, J., & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. Journal Scientific Computing, 16, 1190-1208. doi: 10.1137/0916069

Kuha, J. (2004). AIC and BIC: Comparisons of Assumptions and Performance. Sociological Methods & Research, 33 (2), 188-229.

Nelder, J. A., & Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7, 308-313. doi: 10.1093/comjnl/7.4.308

Vidotto, G., Massidda, D., & Noventa, S. (2010). Averaging models: parameters estimation with the R-Average procedure. Psicologica, 31, 461-475. URL https://www.uv.es/psicologica/articulos3FM.10/3Vidotto.pdf

Vidotto, G. & Vicentini, M. (2007). A general method for parameter estimation of averaging models. Teorie & Modelli, Vol. 12 (1-2), 211-221.

See Also

rav, datgen, pargen, rav.indices, fmdata1, pasta, optim

Analyzing the Family of the Averaging Models

Description

rav (R-Average for AVeraging models) is a procedure for estimating the parameters of the averaging models of Information Integration Theory (Anderson, 1981). It provides reliable estimations of weights and scale values for a factorial experimental design (with any number of factors and levels) by selecting the most suitable subset of the parameters, according to the overall goodness of fit indices and to the complexity of the design.

Usage

rav( data, subset = NULL, mean = FALSE, lev, s.range = c(NA,NA),
     w.range = exp(c(-5,5)), I0 = FALSE, par.fixed = NULL, all = FALSE,
     IC.diff = c(2,2), Dt = 0.1, IC.break = FALSE, t.par = FALSE,
     verbose = FALSE, title = NULL, names = NULL, method = "BFGS",
     start = c(s=NA,w=exp(0)), lower = NULL, upper = NULL, control = list() )

Arguments

Details

The rav function implements the R-Average method (Vidotto & Vicentini, 2007; Vidotto, Massidda & Noventa, 2010), for the parameter estimation of averaging models. R-Average consists of several procedures which compute different models with a progressive increasing degree of complexity:

1. Null Model (null): identifies a single scale value for all the levels of all factors. It assumes constant weights.

2. Equal scale values model (ESM): makes a distinction between the scale values of different factors, estimating a single s-parameter for each factor. It assumes constant weights.

3. Simple averaging model (SAM): estimates different scale values between factors and within the levels of each factor. It assumes constant weights.

4. Equal-weight averaging model (EAM): differentiates the weighs between factors, but not within the levels of each factor.

5. Differential-weight averaging model (DAM): differentiates the weighs both between factors and within the levels of each factor.

6. Information criteria (IC): the IC procedure starts from the EAM and, step by step, it frees different combinations of weights, checking whether a new estimated model is better than the previous baseline. The Occam razor, applied by means of the Akaike and Bayesian information criteria, is used in order to find a compromise between explanation and parsimony.

Finally, only the best model is shown.

The R-Average procedures estimates both scale values and weight parameters by minimizing the residual sum of squares of the model. The objective function is then the square of the distance between theoretical responses and observed responses (Residual Sum of Squares). For a design with k factors with i levels, theoretical responses are defined as:

R = \sum (s_{ki} w_{ki}) / \sum w_{ki}

where any weight parameter w is defined as:

w = exp(t)

Optimization is performed on t-values, and weights are the exponential transformation of t. See Vidotto (2011) for details.

Value

An object of class "rav". The method summary applied to the rav object prints all the fitted models. The functions fitted.values, residuals and coefficients can be used to extract respectively fitted values (predicted responses), the matrix of residuals and the set of estimated parameters.

Author(s)

Supervisor: Prof. Giulio Vidotto giulio.vidotto@unipd.it

University of Padova, Department of General Psychology
QPLab: Quantitative Psychology Laboratory

version 0.0:
Marco Vicentini marco.vicentini@gmail.com

version 0.1 and following:
Stefano Noventa stefano.noventa@univr.it
Davide Massidda davide.massidda@gmail.com

References

Akaike, H. (1976). Canonical correlation analysis of time series and the use of an information criterion. In: R. K. Mehra & D. G. Lainotis (Eds.), System identification: Advances and case studies (pp. 52-107). New York: Academic Press. doi: 10.1016/S0076-5392(08)60869-3

Anderson, N. H. (1981). Foundations of Information Integration Theory. New York: Academic Press. doi: 10.2307/1422202

Anderson, N. H. (1982). Methods of Information Integration Theory. New York: Academic Press.

Anderson, N. H. (1991). Contributions to information integration theory: volume 1: cognition. Lawrence Erlbaum Associates, Hillsdale, New Jersey. doi: 10.2307/1422884

Anderson, N. H. (2007). Comment on article of Vidotto and Vicentini. Teorie & Modelli, Vol. 12 (1-2), 223-224.

Byrd, R. H., Lu, P., Nocedal, J., & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. Journal Scientific Computing, 16, 1190-1208. doi: 10.1137/0916069

Kuha, J. (2004). AIC and BIC: Comparisons of Assumptions and Performance. Sociological Methods & Research, 33 (2), 188-229.

Nelder, J. A., & Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7, 308-313. doi: 10.1093/comjnl/7.4.308

Vidotto, G., Massidda, D., & Noventa, S. (2010). Averaging models: parameters estimation with the R-Average procedure. Psicologica, 31, 461-475. URL https://www.uv.es/psicologica/articulos3FM.10/3Vidotto.pdf

Vidotto, G. & Vicentini, M. (2007). A general method for parameter estimation of averaging models. Teorie & Modelli, Vol. 12 (1-2), 211-221.

See Also

rAverage-package, rav.single, datgen, pargen, rav.indices, rav2file, outlier.replace, optim

Examples

## Not run: 
# --------------------------------------
# Example 1: 3x3 factorial design
# --------------------------------------
# The first column is filled with a sequence of NA values.
data(fmdata1)
fmdata1
# For a two factors design, the matrix data contains the one-way
# sub-design and the two-ways full factorial design observed data.
# Pay attention to the columns order:
# sub-design: A1, A2, A3, B1, B2, B3
# full factorial: A1B1, A1B2, A1B3, A2B1, A2B2, A2B3, A3B1, A3B2, A3B3
# Start the R-Average procedure:
fm1 <- rav(fmdata1, lev=c(3,3))
# (notice that 'range' argument specifies the range of the response scale)
fm1 # print the best model selected
summary(fm1) # print the fitted models

# To insert the factor names:
fact.names <- c("Name of factor A", "Name of factor B")
fm1 <- rav(fmdata1, lev=c(3,3), names=fact.names)

# To insert a title for the output:
fm1 <- rav(fmdata1, lev=c(3,3), title="Put your title here")

# To supervise the information criterion work flow:
fm1 <- rav(fmdata1, lev=c(3,3), verbose=TRUE)

# To increase the number of iterations of the minimization routine:
fm1 <- rav(fmdata1, lev=c(3,3), control=list(maxit=5000))

# To change the estimation bounds for the scale parameters:
fm1.sMod <- rav(fmdata1, lev=c(3,3), s.range=c(0,20))

# To change the estimation bounds for the weight parameters:
fm1.wMod <- rav(fmdata1, lev=c(3,3), w.range=c(0.01,10))

# To set a fixed value for weights:
fm1.fix <- rav(fmdata1, lev=c(3,3), par.fixed="w")

# rav can work without sub-designs. If any sub-design is not available,
# the corresponding column must be coded with NA values. For example:
fmdata1[,1:3] <- NA
fmdata1
fmdata1 # the A sub-design is empty
fm1.bis <- rav(fmdata1, lev=c(3,3), title="Sub-design A is empty")

# Using a subset of data:
data(pasta)
pasta
# Analyzing "s04" only:
fact.names <- c("Price","Packaging")
fm.subj04 <- rav(pasta, subset="s04", lev=c(3,3), names=fact.names)

# --------------------------------------
# Example 2: 3x5 factorial design
# --------------------------------------
data(fmdata2)
fmdata2 # (pay attention to the columns order)
fm2 <- rav(fmdata2, lev=c(3,5))
# Removing all the one-way sub-design:
fmdata2[,1:8] <- NA
fm2.bis <- rav(fmdata2, lev=c(3,5))

# --------------------------------------
# Example 3: 3x2x3 factorial design
# --------------------------------------
data(fmdata3) # (pay attention to the columns order)
fm3 <- rav(fmdata3, lev=c(3,2,3))
# Removing all the one-way design and the AxC sub-design:
fmdata3[,1:8] <- NA # one-way designs
fmdata3[,15:23] <- NA # AxC design
fm3 <- rav(fmdata3, lev=c(3,2,3))

## End(Not run)

Generating an empty dataset in rav format

Description

rav.grid is a function that generates an empty (NAs filled) dataset according to the 'rAverage' format.

Usage

rav.grid(lev, trials = 1, subset = FALSE, names = NULL)

Arguments

Value

A data.frame object.

See Also

rav, pargen, datgen, rAverage-package

Examples

rav.grid(lev=c(3,2,3), trials=5, names=c("Hk","Fa","Mg"))

Fit indices for averaging models

Description

The function fit.indices returns the fit indices for the averaging model given the parameters s0, w0, s(k,j), and w(k,j).

Usage

rav.indices(param, lev, data, t.par = FALSE, subset = NULL,
    n.pars = NULL, names = NULL, title = NULL)

Arguments

Details

Returns the main fit indices (AIC, BIC, R-squared, Adjusted R-squared), the estimated parameters, the fitted values and the residuals of an averaging model.

Value

An object of class "indices".

See Also

rav, rAverage-package

Examples

## Not run: 
data(fmdata1)
s <- c(12.9, 1.5, 18.3, 5.2, 5.0, 2.3)
w <- c(1.4, 0.3, 0.5, 1.6, 1.7, 1.7)
param <- c(NA,NA, s, w)
# Estimated model by rav:
fit1 <- rav(fmdata1, lev=c(3,3)) ; fit1
# Fitted model by original parameters:
fit2 <- rav.indices(param=param, lev=c(3,3), data=fmdata1) ; fit2

## End(Not run)

Single subject analysis with averaging models

Description

Analyzes averaging models for every single subjects in a data matrix and store the estimated parameters in a list.

Usage

rav.single(data,...)

Arguments

Details

The rav.single function is a wrapper for the rav function. Using rav, rav.sigle analyzes subjects one at time, specifying time by time a different value for subset.

Value

A list object in which each slot contains results of a single subject. The ordering of the subjects is the same as in the input data matrix.

See Also

rav, rAverage-package

Examples

## Not run: 
data(pasta)
model <- rav.single(pasta,lev=c(3,3))
model$s41 # extracts the subject 's41'

## End(Not run)

Export rav results

Description

The function exports to a text file the estimated parameters or the model residuals from a call to rav.

Usage

rav2file(object, what = c("resid","param"), whichModel = NULL,
         file = file.choose(), sep = ",", dec = ".")

Arguments

See Also

rav, rav.single

Examples

## Not run: 
data(pasta)
model <- rav.single(pasta,lev=c(3,3))
rav2file(model, what="resid", file="PastaResid-1.csv")
rav2file(model, what="resid", file="PastaResid-2.csv", sep=";", dec=",")

## End(Not run)

Extract residuals from an averaging model

Description

Function to extract residuals from an object returned by rav.

Usage

residuals(object, ...)

Arguments

Details

Returns the residuals of an averaging model fitted by the rav function. When standard = TRUE, residuals will be transformed in z-scale (mean=0 and sd=1 in each column).

As default, the function extract the residuals of the (first) best model. The optional argument whichModel can be specified to extract the values of another model. Options are:

1. "null": null model

2. "ESM": equal scale values model

3. "SAM": simple averaging model

4. "EAM": equal-weights averaging model

5. "DAM": differential-weight averaging model

6. "IC": information criteria model

Value

A matrix of numeric values.

See Also

rav, rAverage-package

Examples

## Not run: 
data(fmdata1)
fm1 <- rav(fmdata1, lev=c(3,3))
residuals(fm1)
residuals(fm1, whichModel="EAM")
residuals(fm1, whichModel="EAM", standard=TRUE)

## End(Not run)