mlds {MLDS}R Documentation

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, ...)

## Default S3 method:
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 contains additional information as attributes. It can also be a one-side formula with parameters p and stimulus variable sx that gives a parametric formula to fit to the data for the formula method.
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’ method is used.
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 number of stimulus levels, deduced from the highest rank in data.
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. Default is “probit”.
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.

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/σ_o = max(Psi_g)

where σ_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.

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 method‘optim’, 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.

Note

The glm method often generates warnings that fitted probabilities are 0 or 1. This does not usually affect the values of the estimated scale. However, it may be wise to check the results with the optim method and obtain standard errors from a bootstrap method (see boot.mlds. The warnings will often disappear if the link is modified or more data are obtained.

Author(s)

Kenneth Knoblauch and Laurence T. Maloney

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.

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)

[Package MLDS version 0.1-7 Index]