mlcm: 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 permits fitting the data with a choice of 3 different models. The formula method permits fitting the data with a parametric model.

Usage

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

Arguments

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.

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.

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 $\sigma$, 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

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

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). This option, now, allows any number of dimensions to be fit.

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.

Examples

Run this code

# NOT RUN {
# 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))
plot(bg.add, pch = 21, bg = c("red", "blue"))
lines(xx, predict(bg.frm, newdata = xx)[seq_along(xx)])
lines(xx, predict(bg.frm, newdata = xx)[-seq_along(xx)])
AIC(bg.frm, bg.add)
par(opar)


#### Analysis of 3-way MLCM data set
# additive model
T.mlcm <- mlcm(Texture)
summary(T.mlcm)
plot(T.mlcm, type = "b")
# independent models
lapply(seq(1, 3), function(wh){ 
	m0 <- mlcm(Texture, model = "ind", which = wh)
	anova(m0, T.mlcm, test = "Chisq")
	})
	
# Deviance differences for 2-way interactions vs 2-way additive models

mlcm(Texture[, -c(4, 5)])$obj$deviance - 
		mlcm(Texture[, -c(4, 5)], model = "full")$obj$deviance
mlcm(Texture[, -c(2, 3)])$obj$deviance - 
		mlcm(Texture[, -c(2, 3)], model = "full")$obj$deviance
mlcm(Texture[, -c(6, 7)])$obj$deviance - 
		mlcm(Texture[, -c(6, 7)], model = "full")$obj$deviance

# deviance differences for 3-way interaction tested against 3 2-way interactions
T3way.mlcm <-  mlcm(Texture, model = "full")  
## construct model matrix from 3 2-way interactions
T3_2way.mf <- cbind(model.frame(mlcm(Texture[, -c(4, 5)], model = "full")$obj), 
		model.matrix(mlcm(Texture[, -c(2, 3)], model = "full")$obj),
		model.matrix(mlcm(Texture[, -c(6, 7)], model = "full")$obj)
	)
T3_2way.mlcm <- glm(Resp ~ . + 0, family = binomial(probit), data = T3_2way.mf) 
Chi2 <- T3_2way.mlcm$deviance -  T3way.mlcm$obj$deviance
degfr <- T3_2way.mlcm$df.residual -  T3way.mlcm$obj$df.residual
pchisq(Chi2, degfr, lower.tail = FALSE)
# }

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