mlds {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 'mlds.df'
mlds(x, stimulus = NULL, method = "glm", 
	lnk = "probit", opt.meth = "BFGS", glm.meth = "glm.fit",
	opt.init = NULL, control = glm.control(maxit = 50000, epsilon = 1e-14), 
	... )
	
## S3 method for class 'mlbs.df'
mlds(x, stimulus = NULL, method = "glm", 
	lnk = "probit",
	control = glm.control(maxit = 50000, epsilon = 1e-14), 
	glm.meth = "glm.fit", 
	... )

## S3 method for class 'data.frame'
mlds(x, ... )

## 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

For comparisons of two pairs of stimuli, 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. For comparisons of triples of stimuli, only the method ‘glm’ is currently defined. The object can be a data frame of 4 columns with the first specifying the response and the other 3 the stimulus level ranks, or an object of class ‘mlbs.df’, which contains additional 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 4 or 5 columns giving the response and the ranks of the stimulus levels for each trial, or an object of class ‘mlbs.df’ or ‘mlds.df’, respectively, 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.

glm.meth

the method to be used in fitting the model, only when method = glm. The default value is “glm.fit”. Seeglm for further details.

...

Additional arguments passed along to glm or optim.

Details

Observers are presented with either triples or pairs of pairs of stimuli, distributed along a physical stimulus axis. For example, for stimuli a, b, c with a < b < c, they see the triple a, b, c, or 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 respectivily as to whether the difference between stimuli 1 and 2 is greater or not that between stimuli 2 and 3 or 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.

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, 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, doi:10.18637/jss.v025.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)

#Example with triads
data(kktriad)
kkt.mlds <- mlds(kktriad)
plot(kkt.mlds, type = "b")

#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)