# Fit Difference Scale by Maximum Likelihood 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. ```r 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 . See`glm` 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. ## Returns 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`. ## References Maloney, L. T. and Yang, J. N. (2003). Maximum likelihood difference scaling. **Journal of Vision**, 3(8):5 , 573--585, tools:::Rd_expr_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, tools:::Rd_expr_doi("10.18637/jss.v025.i02") . ## Author(s) Kenneth Knoblauch and Laurence T. Maloney ## 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. ## See Also `glm`, `optim` ## Examples ```r 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) ```

mlds function