MLDS (version 0.2-1)

mlds: Fit Difference Scale by Maximum Likelihood

Description

Generic function mlds uses different methods to fit the results of a difference scaling experiment either using glm (Generalized Linear Model), by direct maximization of the likelihood using optim or by maximizing the likelihood with respect to a function of the stimulus dimension specified by a one sided formula.

Usage

mlds(x, ...)
## S3 method for class 'default':
mlds(x, stimulus = NULL, method = "glm", 
	lnk = "probit", opt.meth = "BFGS", opt.init = NULL, 
	control = glm.control(maxit = 50000, epsilon = 1e-14), 
	... )
## S3 method for class 'formula':
mlds(x, p, data, stimulus = NULL, 
	lnk = "probit", opt.meth = "BFGS", 
	control = list(maxit = 50000, reltol = 1e-14), ... )

Arguments

x

When the method is specified as glm or optim a data frame with 5 columns giving the response and the ranks of the stimulus levels for each trial, or an object of class mlds.df which also cont

data

A data. frame with 5 columns giving the response and the ranks of the stimulus levels for each trial, or an object of class mlds.df which also contains additional information as attributes, required when the formula metho

p

numeric vector of parameters of length one greater than the number of parameters in the formula argument that specifies initial values for the parameters. The extra parameter, specified last, is the initial estimate of sigma.

stimulus

A numeric vector that contains the physical stimulus levels used in the experiment. If data is of class mlds.df, this information is included as an attribute. If NULL, a sequence of 1-n is used, where n is the

method

Character, taking the value of glm or optim. Default is glm.

lnk

Character indicating either one of the built-in links for the binomial family or a user defined link of class link-glm. See family and make.link<

opt.meth

If method = optim, the method used by optim can be specified. Defaults to BFGS.

opt.init

Vector of numeric giving initial values which must be provided if you specify the optim method.

control

A list of control values for either glm or optim. Since the method defaults to glm, the default is a glm list but should be changed if the optim method is chosen.

...

Additional arguments passed along to glm or optim.

Value

• A list of class mlds whose components depend on whether the method was specified as glm, optim with the default method, or the formula method was used,
• pscale
  A numeric vector of the estimated difference scale.
• stimulus
  The physical stimulus levels
• sigma
  The scale estimate, always 1.0 for glm
• method
  The fitting method
• link
  The binomial link specified, default probit
• obj
  For method glm, an object of class glm resulting from the fit.
• logLik
  for method optim, the logarithm of likelihood at convergence
• hess
  for method optim, the Hessian matrix at convergence
• data
  For methodoptim, the data.frame or mlds.df entered as an argument.
• conv
  For method optim, a code indicating whether optim converged or not. See optim.
• par
  For formula method, the parameters estimated.
• formula
  The one-sided formula specified with the method.
• func
  For formula method, a function obtained from the one-sided formula.

Details

Observers are presented with pairs of pairs of stimuli, distributed along a physical stimulus axis. For example, for stimuli $a, b, c, d$ with $a < b < c < d$, they see the pairs $(a, b)$ and $(c, d)$. For each trial, they make a judgement as to whether the difference between the elements of pair 1 is greater or not than the difference between the elements of pair 2. From a large number of trials on different quadruples, mlds estimates numbers, $Psi_1,..., Psi_n$, by maximum likelihood such that $(Psi_d - Psi_c) > (Psi_b - Psi_a)$ when the observer chooses pair 2, and pair 1, otherwise. If there are $p$ stimulus levels tested, then $p - 1$ coefficients are estimated. The glm method constrains the lowest estimated value, $Psi_1 = 0$, while the optim method constrains the lowest and highest values to be 0 and 1, respectively. The optim method estimates an additional scale parameter, sigma, whereas this value is fixed at 1.0 for the glm method. In principle, the scales from the two methods are related by $$1/\sigma_o = max(Psi_g)$$ where $\sigma_o$ is sigma estimated with the optim method and $Psi_g$ corresponds to the perceptual scale values estimated with the glm method. The equality may not be exact as the optim method prevents the selection of values outside of the interval [0, 1] whereas the glm method does not.

References

Maloney, L. T. and Yang, J. N. (2003). Maximum likelihood difference scaling. Journal of Vision, 3(8):5, 573--585, http://journalofvision.org/3/8/5/, doi:10.1167/3.8.5.

Knoblauch, K. and Maloney, L. T. (2008) MLDS: Maximum likelihood difference scaling in R. Journal of Statistical Software, 25:2, 1--26, http://www.jstatsoft.org/v25/i02.

See Also

glm, optim

Examples

data(AutumnLab)
#Note the warnings generated by glm method
x.mlds <- mlds(AutumnLab)
summary(x.mlds)
y.mlds <- mlds(AutumnLab, method = "optim", opt.init = c(seq(0, 1, len = 10), 0.16))
summary(y.mlds)
plot(x.mlds)
#How sigma relates the scales obtained by the 2 different methods.
lines(y.mlds$stimulus,  y.mlds$pscale/y.mlds$sigma)
#An example using the formula method
data(kk1)
# with one parameter
kk.frm1 <- mlds(~ sx^p, p = c(3, 0.02), data = kk1)
# with two parameters
kk.frm2 <- mlds(~p[1] * (sx + abs(sx - p[2])) - p[1] * p[2], 
	p = c(0.9, 0.3, 0.2), data = kk1)