Description Usage Arguments Details Value Note Author(s) References See Also Examples

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
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), ... )
``` |

`x` |
For comparisons of two pairs of stimuli, when the |

`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 |

`stimulus` |
A numeric vector that contains the physical stimulus levels used in the experiment. If |

`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 |

`opt.meth` |
If |

`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.meth` |
the method to be used in fitting the model, only when |

`...` |
Additional arguments passed along to |

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/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.

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 |

`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 |

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.

Kenneth Knoblauch and Laurence T. Maloney

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
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)
``` |

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