Donkin et al LBA code and example of fitting the LBA to data

1_lba-math.R

fptcdf = function(z,x0max,chi,driftrate,sddrift) {
    zs = z*sddrift
    zu = z*driftrate
    chiminuszu = chi-zu
    xx = chiminuszu-x0max
    chizu = chiminuszu/zs
    chizumax = xx/zs
    tmp1 = zs*(dnorm(chizumax)-dnorm(chizu))
    tmp2 = xx*pnorm(chizumax)-chiminuszu*pnorm(chizu)
    1+(tmp1+tmp2)/x0max
}

fptpdf = function(z,x0max,chi,driftrate,sddrift) {
    zs = z*sddrift
    zu = z*driftrate
    chiminuszu = chi-zu
    chizu = chiminuszu/zs
    chizumax = (chiminuszu-x0max)/zs
    (driftrate*(pnorm(chizu)-pnorm(chizumax)) + sddrift*(dnorm(chizumax)-dnorm(chizu)))/x0max
}

allrtCDF = function(t,x0max,chi,drift,sdI) {
    # Generates CDF for all RTs irrespective of response.
    N = length(drift) # Number of responses.
    tmp = array(dim = c(length(t),N))
    for(i in 1:N){
        tmp[,i] = fptcdf(
            z = t
            , x0max = x0max
            , chi = chi
            , driftrate = drift[i]
            , sddrift = sdI
        )
    }
    1-apply(1-tmp,1,prod)
}

n1PDF = function(t,x0max,chi,drift,sdI) {
    # Generates defective PDF for responses on node #1. 
    N = length(drift) # Number of responses.
    if(N>2){
        tmp = array(dim = c(length(t),N-1))
        for(i in 2:N){
            tmp[,i-1] = fptcdf(
                z = t
                , x0max = x0max
                , chi = chi
                , driftrate = drift[i]
                , sddrift = sdI
            )
        }
        G = apply(1-tmp,1,prod)
    }else{
        G = 1-fptcdf(
            z = t
            , x0max = x0max
            , chi = chi
            , driftrate = drift[2]
            , sddrift = sdI
        )
    }
    G*fptpdf(
        z = t
        , x0max = x0max
        , chi = chi
        , driftrate = drift[1]
        , sddrift = sdI
    )
}

n1CDF = function(t,x0max,chi,drift,sdI) {
    # Generates defective CDF for responses on node #1. 
    outs = numeric(length(t))
    bounds = c(0,t)
    for(i in 1:length(t)){
        tmp = "error"
        repeat{
            if(bounds[i]> = bounds[i+1]){
                outs[i] = 0
                break
            }
            tmp = try(
                integrate(
                    f = n1PDF
                    , lower = bounds[i]
                    , upper = bounds[i+1]
                    , x0max = x0max
                    , chi = chi
                    , drift = drift
                    , sdI = sdI
                )$value
                , silent = T
            )
            if(is.numeric(tmp)){
                outs[i] = tmp
                break
            }
            # Try smart lower bound.
            if(bounds[i]< = 0){
                bounds[i] = max(c((chi-0.98*x0max)/(max(mean(drift),drift[1])+2*sdI),0))
                next
            }
            # Try smart upper bound.
            if(bounds[i+1] == Inf){
                bounds[i+1] = 0.02*chi/(mean(drift)-2*sdI)
                next
            }
            stop("Error in n1CDF that I could not catch.")
        }
    }
    cumsum(outs)
}

n1mean = function(x0max,chi,drift,sdI) {
    # Generates mean RT for responses on node #1. 
    pc = n1CDF(Inf,x0max,chi,drift,sdI)
    fn = function(t,x0max,chi,drift,sdI,pc){
        t*n1PDF(t,x0max,chi,drift,sdI)/pc
    }
    tmp = integrate(
        f = fn
        , lower = 0
        , upper = 100*chi
        , x0max = x0max
        , chi = chi
        , pc = pc
        , drift = drift
        , sdI = sdI
    )$value
    list(mean = tmp,p = pc)
}

actCDF = function(z,t,x0max,chi,drift,sdI) {
    # CDF for the distribution (over trials) of activation values at time t.
    zat = (z-x0max)/t
    zt = z/t
    sdi2 = 2*(sdI^2)
    exp1 = exp(-((zt-drift)^2)/sdi2)
    exp2 = exp(-((zat-drift)^2)/sdi2)
    tmp1 = t*sdI*(exp1-exp2)/sqrt(2*pi)
    tmp2 = pnorm(zat,mean = drift,sd = sdI)
    tmp3 = pnorm(zt,mean = drift,sd = sdI)
    (tmp1+(x0max-z+drift*t)*tmp2+(z-drift*t)*tmp3)/x0max
}

actPDF = function(z,t,x0max,chi,drift,sdI) {
    # CDF for the distribution (over trials) of activation values at time t.
    tmp1 = pnorm((z-x0max)/t,mean = drift,sd = sdI)
    tmp2 = pnorm(z/t,mean = drift,sd = sdI)
    (-tmp1+tmp2)/x0max
}    
    
lbameans = function(Is,sdI,x0max,Ter,chi) {
    # Ter should be a vector of length ncond, the others atomic, 
    # except Is which is ncond x 2.
    ncond = length(Is)/2
    outm<-outp<-array(dim = c(ncond,2))
    for(i in 1:ncond){
        for(j in 1:2){
            tmp = n1mean(x0max,chi,drift = Is[i,switch(j,1:2,2:1)],sdI)
            outm[i,j] = tmp$mean+Ter[i]
            outp[i,] = tmp$p
        }
    }
    list(
        mns = c(outm[1:ncond,2],outm[ncond:1,1])
        , ps = c(outp[1:ncond,2],outp[ncond:1,1])
    )
}

deadlineaccuracy = function(t,x0max,chi,drift,sdI,guess = .5,meth = "noboundary") {
    # Works out deadline accuracy, using one of three 
    # methods:
    #     - noboundary    =    no implicity boundaries.
    #     - partial    =     uses implicit boundaries, and partial information.
    #     - nopartial    =    uses implicit boundaries and guesses otherwise.
    meth = match.arg(meth,c("noboundary","partial","nopartial"))
    noboundaries = function(t,x0max,chi,drift,sdI,ulimit = Inf){
        # Probability of a correct response in a deadline experiment
        # at times t, with no implicit boundaries.
        N = length(drift)
        tmpf = function(x,t,x0max,chi,drift,sdI){
            if(N>2){
                tmp = array(dim = c(length(x),N-1))
                for(i in 2:N){
                    tmp[,i-1] = actCDF(x,t,x0max,chi,drift[i],sdI)
                }
                G = apply(tmp,1,prod)*actPDF(x,t,x0max,chi,drift[1],sdI)
            }else{
                G = actCDF(x,t,x0max,chi,drift[2],sdI)*actPDF(x,t,x0max,chi,drift[1],sdI)
            }
        }
        outs = numeric(length(t))
        for(i in 1:length(t)){
            if(t[i]< = 0){
                outs[i] = .5
            }else{
                outs[i] = integrate(
                    f = tmpf
                    , lower = -Inf
                    , upper = ulimit
                    , t = t[i]
                    , x0max = x0max
                    , drift = drift
                    , sdI = sdI
                )$value
            }
        }
        outs
    }
    if(meth == "noboundary"){
        noboundaries(t,x0max,chi,drift,sdI,ulimit = Inf)
    }else{
        pt = n1CDF(t = t,x0max = x0max,chi = chi,drift = drift,sdI = sdI)
        pa = allrtCDF(t = t,x0max = x0max,chi = chi,drift = drift,sdI = sdI)
        pguess = switch(
            meth
            , "nopartial" = guess*(1-pa)
            , "partial" = noboundaries(
                t = t
                , x0max = x0max
                , chi = chi
                , drift = drift
                , sdI = sdI
                , ulimit = chi
            )
        )
        pt+pguess    
    }
}

2_pq-lba.R

# Like pnorm,  punif etc evaluates the CDF of the linear BA model at a 
# pre-specified set of quantiles. If input "qs" is null instead
# returns predicted quantiles given by qps.
source("1_lba-math.r")
pqlba = function(Is, sdI, x0max, Ter, chi, qs, qps = seq(.1, .9, .2)){
    # Input checks.
    if(length(Is)! = 2) stop("Not two drifts in pqlba.")
    dop = (!is.null(qs)) # Switch for return p-values or q-values.
    if(dop){
        nq = dim(qs)[1] # qs should be quants x responses.
        if(length(dim(qs))! = 2) stop("Crazy q-values in pqba.")
        if((dim(qs)[2])! = 2) stop("Wrong number of response choices in q-values for pqlba.")
    }
    # First get probability of each response. 
    out = list(p = numeric(2))
    out$pfail = prod(pnorm(-Is/sdI))
    out$p[1] = n1CDF(t = Inf, x0max = x0max, chi = chi, drift = Is, sdI = sdI)
    out$p[2] = n1CDF(t = Inf, x0max = x0max, chi = chi, drift = Is[2:1], sdI = sdI)
#   out$p[2] = 1-out$p[1]
    if(dop){
        # Calculate probability masses in inter-quantile ranges.
        out$predp = array(dim = c(nq+1, 2))
        for(i in 1:2){
            tmpI = switch(i, Is, Is[2:1]) 
            tmp = n1CDF(t = qs[, i]-Ter, x0max = x0max, chi = chi, drift = tmpI, sdI = sdI)
            out$predp[, i] = diff(c(0, tmp, out$p[i]))
        }
    }else{
        # Calculate predicted quantile values.
        out$predq = array(dim = c(length(qps), 2))
        tmpf = function(t, drift, sdI, x0max, chi, p){
            n1CDF(t = t, x0max = x0max, chi = chi, drift = drift, sdI = sdI)-p
        }
        for(i in 1:2){
            tmpI = switch(i, Is, Is[2:1]) 
            for(j in 1:length(qps)){
                interval = switch((Ter<5)+1, c(1, 1e4), c(1e-3, 10)) # Sec or MSEC.
	            tmp = uniroot(
	                f = tmpf
	                , interval = interval
	                , x0max = x0max
	                , chi = chi
	                , drift = tmpI
	                , sdI = sdI
	                , p = qps[j]*out$p[i]
	            )
                out$predq[j, i] = tmp$root+Ter
            }
            out$p[i] = n1CDF(t = Inf, x0max = x0max, chi = chi, drift = tmpI, sdI = sdI)
        }
    }
    out$p = out$p+out$pfail/2
    out
}

getpreds = function(I1, I2, sdI, x0max, Ter, chi, pred = F, dat, qps = seq(.1, .9, .2)){
    # If pred = F,  returns chi squared. If pred = T,  returns a list
    # like "dat" with predicted quantiles and probabilities.
    # This function expects the parameters(I1, I2, x0max, chi, Ter, sdI)
    # to be 3 arrays(stim1/stim2 x accuracy/neutral/speed).
    mod = list()
    if(pred){
        mod = list(p = array(dim = c(3)), q = array(dim = c(length(qps), 2, 3)))
    } else{
        mod = list(p = array(dim = c(3)), q = array(dim = c(length(qps)+1, 2, 3)))
    }
    for(j in 1:3){
        tmp = pqlba(
            Is = c(I1[j], I2[j])
            , sdI = sdI[j]
            , x0max = x0max[j]
            , Ter = Ter[j]
            , chi = chi[j]
            , qs = switch(pred+1, dat$q[, , j], NULL)
        )
        mod$p[j] = tmp$p[1]
        mod$q[, , j] = switch(pred+1, tmp$predp, tmp$predq)
    }
    mod
}

obj = function(x, dat, pred = F, qps = seq(.1, .9, .2), trace = 0){
    numits <<- numits+1
    sdI = rep(exp(x[1]), 3)
    x0max = rep(exp(x[2]), 3)
    Ter = rep(exp(x[3]), 3)
    chi = rep(exp(x[4]), 3)+x0max
    I1 =(x[5:7])
    I2 = 1-I1
    preds = getpreds(
        I1 = I1
        , I2 = I2
        , sdI = sdI
        , x0max = x0max
        , Ter = Ter
        , chi = chi
        , pred = pred
        , dat = dat
        , qps = qps
    )
    if(pred == F){
        tmp = -sum(dat$pb*log(pmax(preds$q, 1e-10)))
        if(trace>0){
            if((numits%%trace) = = 0){
                names(x) = NULL
                print(c(exp(x[1:4]), x[5:7], tmp), 2)
            }
        }
        return(tmp)
    }else{
        return(preds)
    }
}

fitter = function(dat, maxit = seq(1000, 9000, 1000), qps = seq(.1, .9, .2), trace = 0){
    fit = length(maxit)
    I1 = qnorm((1-dat$p), sd = .3*sqrt(2))*2
    I1 = .5+.5*I1
    tmp = sum1 = .5
    Ter = min(dat$q)*.9
    x0max = mean(dat$q[4, , ]-dat$q[2, , ])*(4*tmp)
    chi =(Ter/4)
    sdI = 0.3
    par = c(log(c(sdI, x0max, Ter, chi)), I1)
    attr(par, "obj") = NA
    for(fitnum in 1:fit){ # Do the fitting or not.
        numits <<- 0
        #return(par)
        tmp = optim(
            fn = obj
            , par = par
            , control = list(
                maxit = maxit[fitnum]
                , parscale = par
            ) 
            , dat = dat
            , qps = qps
            , trace = trace
        )
        out <- par <- tmp$par
        attr(out, "obj") = tmp$value
    }
    out
}

3_fit-example.R

#remove everything previously loaded in the R workspace
rm(list=ls())
#load all functions required to fit the LBA
source("2_pq-lba.r")
#setting this as T minimises the output printed to the R workspace
quiet=T

#set the quantiles used in the QMPE fitting
qps=seq(.1,.9,.2)
#which RT values in the data that should be discarded (smaller than the first element, larger than the second)
trim=c(180,10000)

#read in the data with column names "difficulty", "correct" and "rt"
rawdata=read.table("4_exampledata.txt",col.names=c("difficulty","correct","rt"))

#Format the data for QMPE fitting
#creates a vector indicating which data will be used based on values given to the trim variable defined earlier
use=(rawdata$rt>trim[1])&(rawdata$rt<trim[2])
#set up a list of names for variables for later use
nms=list(c("err","crct"),c("easy","medium","hard"))
dims=unlist(lapply(nms,length))
#q gets the values of the quantile values (set by qps above) for correct and error responses for each of the three difficulties
q=tapply(rawdata$rt[use],list(rawdata$correct[use],rawdata$difficulty[use]),quantile,probs=qps)
q=array(unlist(q),dim=c(length(qps),dims),dimnames=c(list(qps),nms))
#n gets the number of correct and error responses in each difficulty condition
n=tapply(rawdata$rt[use],list(rawdata$correct[use],rawdata$difficulty[use]),length)
#p gets the proportion of correct responses for each difficulty level
p=tapply(rawdata$correct[use],list(rawdata$difficulty[use]),mean)
#gives names to previous variables
dimnames(n)=nms ; dimnames(p)=nms[-1]
#calculates how many observations make up each quantile value for correct and error responses in each of the three difficulties
pb=array(rep(n,each=length(qps)+1),dim=c(length(qps)+1,dim(n)),
  dimnames=c(list(NULL),dimnames(n)))*c(.1,.2,.2,.2,.2,.1)
#use p, n, q and pb to make a list
data=list(p=p,n=n,q=q,pb=pb)

#here is the call to fit the LBA to the model. It requires two arguments - dat, the data, and maxit, the number of iterations, or steps, the fitting algorithm should take when trying to fit the model to data
fit=fitter(dat=data,maxit=c(100,200,500,1000,2000,5000,10000))
#put the results of the fit into a variable called pars
pars=c(exp(fit[1:4]),fit[5:7])
#the fitting function gives back (b-A) instead of b. To make the output look like that given in Brown and Heathcote (2008) we add A to b-A to get b
pars[4]=pars[4]+pars[2]
#the fitting algorithm returns 1 minus the drift rates for the correct responses. Change these so that they are the actual drift rates
pars[5:7]=1-pars[5:7]
fit[5:7]=1-fit[5:7]
#give names to the parameters
names(pars)=c("s","A","ter","b","v1","v2","v3")
#print the parameters onscreen
print(pars,3)

4_exampledata.txt

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