Description Usage Arguments Details Value Author(s) References See Also Examples

`rav`

(**R**-Average for **AV**eraging 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.

1 2 3 4 5 | ```
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() )
``` |

`data` |
An object of type |

`subset` |
Character, numeric or factor attribute that selects a subset of experimental data for the analysis (see the examples). |

`mean` |
Logical value wich specifies if the analysis must be performed on raw data ( |

`lev` |
Vector containing the number of levels of each factor. For instance, two factors
with respectively 3 and 4 levels require |

`s.range,w.range` |
The range of s and w parameters. Each vector must contains, respectively, the minimum and the
maximum value. For s-parameters, if the default value |

`I0` |
Logical. If set |

`par.fixed` |
This argument allows to constrain one or more parameters to a specified value. Default
setting to |

`all` |
Logical. If set |

`IC.diff` |
Vector containing the cut-off values (of both BIC and AIC indices) at which different
models are considered equivalent. Default setting: BIC difference = 2.0, AIC difference = 2.0
( |

`Dt` |
Numeric attribute that set the cut-off value at which different t-parameters must be considered equal (see details). |

`IC.break` |
Logical argument which specifies if to run the Information Criteria Procedure. |

`t.par` |
Logical. Specifies if the output must shows the t-parameters instead of the w-parameters. |

`verbose` |
Logical. If set |

`title` |
Character. Label to use as title for output. |

`names` |
Vector of character strings containing the names of the factors. |

`method` |
The minimization algorithm that has to be used. Options are: "L-BFGS-B", "BFGS", "Nelder-Mead",
"SANN" and "CG". See |

`start` |
Vector containing the starting values for respectively scale and weight parameters. For the scale
parameters, if the default value |

`lower` |
Vector containing the lower values for scale and weight parameters when the minimization
routine is L-BFGS-B. With the default setting |

`upper` |
Vector containing the upper values for scale and weight parameters when the minimization
routine is L-BFGS-B. With the default setting |

`control` |
A list of control parameters. See the |

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:

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

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.

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

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

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

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 = ∑ (s_{ki} w_{ki}) / ∑ 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.

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.

**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

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.

`rAverage-package`

,
`rav.single`

,
`datgen`

,
`pargen`

,
`rav.indices`

,
`rav2file`

,
`outlier.replace`

,
`optim`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | ```
## 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)
``` |

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