mlcm {MLCM}R Documentation

Fit Conjoint Measurement Models by Maximum Likelihood

Description

Generic function mlcm uses different methods to fit the results of a conjoint measurement experiment using glm (Generalized Linear Model). The default method (currently, the only one) permits fitting the data with a choice of 3 different models.

Usage

mlcm(x, ...)

## Default S3 method:
mlcm(x, model = "add", whichdim = NULL, lnk = "probit", 
        control = glm.control(maxit = 50000, epsilon = 1e-14), ...
        )
## S3 method for class 'formula':
mlcm(x, p, data, 
        model = "add", whichdim = NULL,
        lnk = "probit", opt.meth = "BFGS",
        control = list(maxit = 50000, reltol = 1e-14), ...)

Arguments

x a data frame of an odd number of columns (at least 5) or a formula object. In the case of a data frame, the first should be logical or a 2-level factor named Resp indicating the response of the observer. The next columns give the indices in pairs along each dimension for each of the two stimuli being compared.
p numeric indicating initial values of parameters for the formula method.
data data frame of class ‘mlcm.df’ for the formula method.
model character indicating which of three conjoint measurement models to fit to the data: “add”, for additive (default), “ind”, for independence or “full”, for including a dependence with the levels of each dimension with the others. The “full” is not applicable for the formula method.
whichdim integer indicating which dimension of the data set to fit when the independence model is chosen
lnk character indicating the link function to use with the binomial family. Current default is the probit link.
control information to control the fit. See glm and glm.control or optim for the formula method.
opt.meth character indicating optimization method (default: “BFGS”) for optim with the formula method.
... additional arguments passed to glm or optim.

Details

In a conjoint measurement experiment, observers are presented with pairs of stimuli that vary along 2 or more dimensions. The observer's task is to choose which stimulus of the pair is greater along one of the dimensions. Over a large number of trials, mlcm estimates numbers,

psi_1, ..., psi_p, psi'_1, ..., psi'_q, ...

,

by maximum likelihood using glm that best predict the observer's judgments.

The function permits the estimation of 3 different models, independent, additive (the default) and full, by specifying the model argument. The independent model fits the data along only 1 dimension, specified by the whichdim argument. The additive model fits all dimensions with each fixed at 0 at the lowest level on each dimension. Thus, if there are n dimensions each with p_i levels, mlcm estimates sum p_i - n coefficients.

Specifying the full model will fit a saturated model in which an estimate will be made for each combination of the scale values except the lowest (0 on all scales). Currently, this option only allows 2 dimensions to be fit.

Value

a list of class ‘mlcm’ that will include some of the following components depending on whether the default or formual method is used:

pscale a vector or matrix giving the perceptual scale value estimates
stimulus numeric indicating the scale values along each dimension
sigma numeric indicating judgment σ, currently always set to 1
par numeric indicating the fitted parameter values when the formula method is used
logLik log likelihood returned with the formula method
hess Hessian matrix returned with the formual method
method character indicating whether the model was fit by glm or with the formula method
se standard errors returned with the formula method
NumDim numeric indicating number of stimulus dimensions in data set
NumLev numeric indicating the number of levels along both dimensions, currently assumed to be the same
model character indicating which of the 3 models were fit
link character indicating the link used for the binomial family with glm
obj the ‘glm’ object
data the ‘mlcm’ data frame
conv numeric indicating whether convergence was reached in the case of the formula method
formula formula object from argument to formula method
func function constructed from formula object
whichdim numeric indicating which dimension was fit in the case of the “ind” model

Author(s)

Ken Knoblauch

References

Luce, R. D., and Tukey, J. W. (1964). Simultaneous conjoint measurement. Journal of Mathematical Psychology, 1, 1–27.

Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971). Foundations of Measurement, Vol. 1: Additive and Polynomial Representations. New York: Academic Press.

Ho, Y. H., Landy. M. S. and Maloney, L. T. (2008). Conjoint measurement of gloss and surface texture. Psychological Science, 19, 196–204.

See Also

glm

Examples


# Additive model
bg.add <- mlcm(BumpyGlossy)
plot(bg.add, type = "b")

# Independence model for Bumpiness
bg.ind <- mlcm(BumpyGlossy, model = "ind", whichdim = 2)

anova(bg.ind, bg.add, test = "Chisq")

# Full model
bg.full <- mlcm(BumpyGlossy, model = "full")

anova(bg.add, bg.full, test = "Chisq")

opar <- par(mfrow = c(1, 2), pty = "s")
# Compare additive and full model graphically
plot(bg.full, standard.scale = TRUE, type = "b", 
        lty = 2, ylim = c(0, 1.05),
        xlab = "Gloss Level",
        ylab = "Bumpiness Model Estimates")
# additive prediction
bg.pr <- with(bg.add, outer(pscale[, 1], pscale[, 2], "+"))
# predictions are same for arbitrary scaling,
#  so we adjust additive predictions to best fit
#  those from the full model by a scale factor.
cf <- coef(lm(as.vector(bg.full$pscale/bg.full$pscale[5, 5]) ~ 
        as.vector(bg.pr) - 1))
matplot(cf * bg.pr, type = "b", add = TRUE, lty = 1)

#### Now make image of residuals between 2 models
bg.full.sc <- bg.full$pscale/bg.full$pscale[5, 5]
bg.add.adj <- cf * bg.pr
bg.res <- (bg.add.adj - bg.full.sc) + 0.5
image(1:5, 1:5, bg.res, 
        col = grey.colors(100, min(bg.res), max(bg.res)),
        xlab = "Gloss Level", ylab = "Bumpiness Level" 
        )
        
#### Example with formula
# additive model
bg.frm <- mlcm(~ p[1] * (x - 1)^p[2] + p[3] * (y - 1)^p[4], 
   p = c(0.1, 1.3, 1.6, 0.8), data = BumpyGlossy)
summary(bg.frm)
# independence model
bg.frm1 <- mlcm(~ p[1] * (x - 1)^p[2], p = c(1.6, 0.8),
        data = BumpyGlossy, model = "ind", whichdim = 2)
summary(bg.frm1)

### Test additive against independent fits
ddev <- -2 * (logLik(bg.frm1) - logLik(bg.frm))
df <- attr(logLik(bg.frm), "df") - attr(logLik(bg.frm1), "df")
pchisq(as.vector(ddev), df, lower = FALSE)

# Compare additive power law and nonparametric models 
xx <- seq(1, 5, len = 100)
par(mfrow = c(1, 1))
with(bg.frm, plot(xx, func(par, 1, xx), 
        type = "l"))
with(bg.frm, lines(xx, func(par, xx, 1), 
        type = "l"))
plot(bg.add, pch = 21, bg = c("red", "blue"), add = TRUE)
AIC(bg.frm, bg.add)
par(opar)

[Package MLCM version 0.0-6 Index]