# Analyzing the Family of the Averaging Models `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. ```r 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 - `data`: An object of type `matrix`, `data.frame` or `vector` containing the experimental data. Each column corresponds to an experimental design of factorial plan (in order: one-way design, two-way design, ..., full factorial design; see the example for further details). Columns must be sorted combining each level of the first factor with all the levels of the following factors. The first column be used to set an identification code (ID) to label the subjects (see the attribute `subset`). - `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 (`mean = FALSE`) or on the average of columns of the data matrix (`mean = TRUE`). - `lev`: Vector containing the number of levels of each factor. For instance, two factors with respectively 3 and 4 levels require `lev = c(3,4)`. - `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 `NA` is set, the minimum and the maximum values of data matrix will be used. For t-parameters, the values exp(-5) and exp(+5) will be used. This values will be the bounds for parameters in the estimation process when the minimization algorithm is L-BFGS-B. The arguments `s.range` and `w.range` are a simple and quick way to specify the bounds for scale and weight parameters. A more complex but complete way is to use the arguments `lower` and `upper`. If `lower` and `upper` will be specified, `s.range` and `w.range` will be ignored. - `I0`: Logical. If set `FALSE`, the `s0` and `w0` parameters are forced to be zero. If set `TRUE`, the `s0` and `w0` parameters are free to be estimated. - `par.fixed`: This argument allows to constrain one or more parameters to a specified value. Default setting to `NULL` indicates that all the scale and weight parameters will be estimated by the algorithmic procedure. Alternatively, it can be specified the name of the type of parameters to constrain. The argument `par.fixed = "s"` constrains s-parameters, `par.fixed = "w"` constrains w-parameters and `par.fixed = c("s","w")` constrains both s and w parameters. Also, using `"t"` instead of `"w"` constrains directly t-parameters. in these cases a graphical interface is displayed and the values can be specified. - `all`: Logical. If set `TRUE` the information criterion tests all the possible combinations of weights (see details). The default value `FALSE` implies a preselection of a subset of combination based on the results of the previous steps of the algorithm. WARNING: with `all = TRUE` the procedure is generally more time-consuming (depending on the size of the experimental design), but can provide more reliable estimations than the standard procedure. - `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 (`IC.diff = c(2.0, 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 `TRUE` the function prints general informations for every step of the information criterion procedure. - `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 `optim` documentation for further information. - `start`: Vector containing the starting values for respectively scale and weight parameters. For the scale parameters, if the default value `NA` is set, the mean of data is used as starting value. For the weight parameters, the starting default value is 1. - `lower`: Vector containing the lower values for scale and weight parameters when the minimization routine is L-BFGS-B. With the default setting `NULL`, s-parameters are set to the first value specified in `s.range` while w-parameters are set to the first value specified in `w.range`. Values must be specified in the order: `s0,w0,s,w`. For example, for a 3x3 design, in the `lower` vector the positions of parameters must be: `s0,w0, sA1,sA2,sA3, sB1,sB2,sB3, wA1,wA2,wA3, wB1,wB2,wB3`. - `upper`: Vector containing the upper values for scale and weight parameters when the minimization routine is L-BFGS-B. With the default setting `NULL`, s-parameters are set to the second value specified in `s.range` while w-parameters are set to the second value specified in `w.range`. Values must be specified in the order: `s0,w0,s,w`. For example, for a 3x3 design, in the `upper` vector the positions of parameters must be: `s0,w0, sA1,sA2,sA3, sB1,sB2,sB3, wA1,wA2,wA3, wB1,wB2,wB3`. - `control`: A list of control parameters. See the `optim` documentation for further informations. `control` argument can be used to change the maximum iteration number of minimization routine. To increase the number, use: `control=list(maxit=N)`, where `N` is the number of iterations (100 for default). ## 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. ## Returns 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 ```r ## 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) ```

rav function