fddm

fddm provides the function dfddm(), which evaluates the density function (or probability density function, PDF) for the Ratcliff diffusion decision model (DDM) using different methods for approximating the full PDF, which contains an infinite sum. fddm also provides the family of functions d*_dfddm(), which evaluate the first-order partial derivatives of the DDM density function with respect to the parameter indicated by the * in the function name; the available parameters are listed below. Similarly, fddm provides the family of functions d*2_dfddm(), which evaluate the second-order partial derivatives of the DDM density function for the same parameters. Based on the density function and its partial derivatives, fddm provides the function ddm(), which fits the DDM to provided data. fddm also provides the function pfddm(), which evaluates the distribution function (or cumulative distribution function, CDF) for the DDM using two different methods for approximating the CDF.

Our implementation of the DDM has the following parameters: a ϵ (0, ∞) (threshold separation), v ϵ (-∞, ∞) (drift rate), t₀ ϵ [0, ∞) (non-decision time/response time constant), w ϵ (0, 1) (relative starting point), sv ϵ (0, ∞) (inter-trial-variability of drift), and sigma ϵ (0, ∞) (diffusion coefficient of the underlying Wiener Process).

Installation

You can install the released version of fddm from CRAN with:

install.packages("fddm")

And the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("rtdists/fddm")

Example

As a preliminary example, we will fit the DDM to the data from one participant in the med_dec data that comes with fddm. This dataset contains the accuracy condition reported in Trueblood et al. (2018), which investigates medical decision making among medical professionals (pathologists) and novices (i.e., undergraduate students). The task of participants was to judge whether pictures of blood cells show cancerous cells (i.e., blast cells) or non-cancerous cells (i.e., non-blast cells). The dataset contains 200 decisions per participant, based on pictures of 100 true cancerous cells and pictures of 100 true non-cancerous cells. Here we use the data collected from the trials of one experienced medical professional (pathologist). First, we load the fddm package, remove any invalid responses from the data, and select the individual whose data we will use for fitting.

library("fddm")
data(med_dec, package = "fddm")
med_dec <- med_dec[which(med_dec[["rt"]] >= 0), ]
onep <- med_dec[ med_dec[["id"]] == "2" & med_dec[["group"]] == "experienced", ]
str(onep)
#> 'data.frame':    200 obs. of  9 variables:
#>  $ id            : Factor w/ 37 levels "1","2","3","4",..: 2 2 2 2 2 2 2 2 2 2 ...
#>  $ group         : Factor w/ 3 levels "experienced",..: 1 1 1 1 1 1 1 1 1 1 ...
#>  $ block         : int  3 3 3 3 3 3 3 3 3 3 ...
#>  $ trial         : int  1 2 3 4 5 6 7 8 9 10 ...
#>  $ classification: Factor w/ 2 levels "blast","non-blast": 1 2 2 2 1 1 1 1 2 1 ...
#>  $ difficulty    : Factor w/ 2 levels "easy","hard": 1 1 2 2 1 1 2 2 1 2 ...
#>  $ response      : Factor w/ 2 levels "blast","non-blast": 1 2 1 2 1 1 1 1 2 1 ...
#>  $ rt            : num  0.853 0.575 1.136 0.875 0.748 ...
#>  $ stimulus      : Factor w/ 312 levels "blastEasy/AuerRod.jpg",..: 7 167 246 273 46 31 132 98 217 85 ...

Easy Fitting with Built-in ddm()

The ddm() function fits the 5-parameter DDM to the user-supplied data via maximum likelihood estimation. Each DDM parameter can be modeled using R’s formula interface; the model parameters can either be fixed or estimated, except for the drift rate which is always estimated.

We will demonstrate a simple example of how to fit the DDM to the onep dataset from the above code chunks.

Because we use an ANOVA approach, we set orthogonal sum-to-zero contrasts.

op <- options(contrasts = c('contr.sum', 'contr.poly'))

Now we can use the ddm() function to fit the DDM to the data. Note that we are using formula notation to indicate the interaction between variables for the drift rate. The first argument of the ddm() function is the formula indicating how the drift rate should be modeled. By default, the boundary separation and non-decision time are estimated, and the initial bias and inter-trial variability in the drift rate are held fixed.

fit0 <- ddm(rt + response ~ classification*difficulty, data = onep)
summary(fit0)
#> 
#> Call:
#> ddm(drift = rt + response ~ classification * difficulty, data = onep)
#> 
#> DDM fit with 3 estimated and 2 fixed distributional parameters.
#> Fixed: bias = 0.5, sv = 0 
#> 
#> drift coefficients (identity link):
#>                             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)                  -0.5924     0.1168  -5.073 3.91e-07 ***
#> classification1              -2.6447     0.1168 -22.647  < 2e-16 ***
#> difficulty1                   0.2890     0.1168   2.475   0.0133 *  
#> classification1:difficulty1  -1.4987     0.1168 -12.834  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> boundary coefficients (identity link):
#>             Estimate Std. Error
#> (Intercept)    2.064      0.058
#> 
#> ndt coefficients (identity link):
#>             Estimate Std. Error
#> (Intercept)   0.3938      0.007

This output first shows the input to the function call and which parameters are held fixed; this information is useful to verify that the formula inputs to the ddm() function call were correct. For the model of the drift rate that we input, we can see the estimates and summary statistics for each coefficient. Below this, we can see the simple estimates for the boundary separation and non-decision time (default behavior).

We can reset the contrasts after fitting.

options(op) # reset contrasts

Alternative Fitting Method with nlminb()

Although we strongly recommend using the ddm() function for fitting the DDM to data because it is faster and more convenient, we will also show how to use the probability density function in a manual optimization setup. We further prepare the data by defining upper and lower responses and the correct response bounds.

onep[["resp"]] <- ifelse(onep[["response"]] == "blast", "upper", "lower")
onep[["truth"]] <- ifelse(onep[["classification"]] == "blast", "upper", "lower")

For fitting, we need a simple likelihood function; here we will use a straightforward log of sum of densities of the study responses and associated response times. This log-likelihood function will fit the standard parameters in the DDM, but it will fit two versions of the drift rate v: one for when the correct response is "blast" (v_u), and another for when the correct response is "non-blast" (v_l). A detailed explanation of the log-likelihood function is provided in the Example Vignette (vignette("example", package = "fddm")). Note that this likelihood function returns the negative log-likelihood as we can simply minimize this function to get the maximum likelihood estimate.

ll_fun <- function(pars, rt, resp, truth) {
  v <- numeric(length(rt))

  # the truth is "upper" so use vu
  v[truth == "upper"] <- pars[[1]]
  # the truth is "lower" so use vl
  v[truth == "lower"] <- pars[[2]]

  dens <- dfddm(rt = rt, response = resp, a = pars[[3]], v = v,
                t0 = pars[[4]], w = pars[[5]], sv = pars[[6]], log = TRUE)

  return( ifelse(any(!is.finite(dens)), 1e6, -sum(dens)) )
}

We then pass the data and log-likelihood function to an optimization function with the necessary additional arguments. As we are using the optimization function nlminb for this example, the first argument must be the initial values of our DDM parameters that we want optimized. These are input in the order: v_u, v_l, a, t₀, w, and sv; we also need to define upper and lower bounds for each of the parameters. Fitting the DDM to this dataset is basically instantaneous using this setup.

fit <- nlminb(c(0, 0, 1, 0, 0.5, 0), objective = ll_fun,
              control = list(iter.max = 300, eval.max = 300),
              rt = onep[["rt"]], resp = onep[["resp"]], truth = onep[["truth"]],
              # limits:   vu,   vl,   a,                t0, w,  sv
              lower = c(-Inf, -Inf, .01,                 0, 0,   0),
              upper = c( Inf,  Inf, Inf, min(onep[["rt"]]), 1, Inf))
fit
#> $par
#> [1]  5.6813063 -2.1886615  2.7909130  0.3764465  0.4010115  2.2813001
#> 
#> $objective
#> [1] 42.47181
#> 
#> $convergence
#> [1] 0
#> 
#> $iterations
#> [1] 231
#> 
#> $evaluations
#> function gradient 
#>      251     1656 
#> 
#> $message
#> [1] "relative convergence (4)"