# LBA-race: LBA race functions: Likelihood for first accumulator to win. In rtdists: Response Time Distributions

## Description

n1PDF and n1CDF take RTs, the distribution functions of the LBA, and corresponding parameter values and put them throughout the race equations and return the likelihood for the first accumulator winning (hence n1) in a set of accumulators.

## Usage

 ```1 2 3 4 5``` ```n1PDF(rt, A, b, t0, ..., st0 = 0, distribution = c("norm", "gamma", "frechet", "lnorm"), args.dist = list(), silent = FALSE) n1CDF(rt, A, b, t0, ..., st0 = 0, distribution = c("norm", "gamma", "frechet", "lnorm"), args.dist = list(), silent = FALSE) ```

## Arguments

 `rt` a vector of RTs. `A, b, t0` LBA parameters, see `LBA`. Can either be a single numeric vector (which will be recycled to reach `length(rt)` for trialwise parameters) or a `list` of such vectors in which each list element corresponds to the parameters for this accumulator (i.e., the list needs to be of the same length as there are accumulators). Each list will also be recycled to reach `length(rt)` for trialwise parameters per accumulator. `...` two named drift rate parameters depending on `distribution` (e.g., `mean_v` and `sd_v` for `distribution=="norm"`). The parameters can either be given as a numeric vector or a list. If a numeric vector is passed each element of the vector corresponds to one accumulator. If a list is passed each list element corresponds to one accumulator allowing again trialwise driftrates. The shorter parameter will be recycled as necessary (and also the elements of the list to match the length of `rt`). See examples. `st0` parameter specifying the variability of `t0` (which varies uniformly from `t0` to `t0` + `st0`). Can be trialwise, and will be recycled to length of `rt`. `distribution` character specifying the distribution of the drift rate. Possible values are `c("norm", "gamma", "frechet", "lnorm")`, default is `"norm"`. `args.dist` list of optional further arguments to the distribution functions (i.e., `posdrift` or `robust` for `distribution=="norm"`). `silent` logical. Should the number of accumulators used be suppressed? Default is `FALSE` which prints the number of accumulators.

## Details

For a set of N independent accumulators i = 1...N, the race likelihood for a given accumulator i is given by

L(unit i wins) = f_i(t) * prod_j<>i [ S_j(t) ]

where f(t) is the PDF (`dlba_...`) and S_j(t) = 1 - F_j(t) is the survivor function, that is the complement of the CDF F(t) (`plba_...`) at time t.

In other words, this is just the PDF/CDF for the winning accumulator at time t times the probability that no other accumulators have finished at time t.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84``` ```## check random generated values against race functions: ## 1. Without st0: r_lba <- rLBA(1e4, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1), sd_v=0.2) x <- seq(0.5, 4, length.out = 100) # for plotting # PDF y <- n1PDF(x, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2) # PDF hist(r_lba\$rt[r_lba\$response==1],probability = TRUE, breaks = "FD") lines(x=x,y=y/mean(r_lba\$response == 1)) # CDF plot(ecdf(r_lba\$rt[r_lba\$response==1])) y <- n1CDF(x, A=0.5, b=1, t0 = 0.5, st0 = 0, mean_v=c(1.2, 1.0), sd_v=0.2) lines(x=x,y=y/mean(r_lba\$response == 1), col = "red", lwd = 4.5, lty = 2) # KS test ## Not run: normalised_n1CDF = function(rt,...) n1CDF(rt,...)/n1CDF(rt=Inf,...) ks.test(r_lba\$rt[r_lba\$response==1], normalised_n1CDF, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2) ## End(Not run) ## Not run: ## Other examples (don't run to save time): ## 2. With st0 = 0.2: r_lba <- rLBA(1e4, A=0.5, b=1, t0 = 0.5, st0 = 0.2, mean_v=c(1.2, 1), sd_v=0.2) x <- seq(0.5, 4, length.out = 100) # for plotting # PDF y <- n1PDF(x, A=0.5, b=1, t0 = 0.5, st0 = 0.2, mean_v=c(1.2, 1.0), sd_v=0.2) # PDF hist(r_lba\$rt[r_lba\$response==1],probability = TRUE, breaks = "FD") lines(x=x,y=y/mean(r_lba\$response == 1)) # CDF plot(ecdf(r_lba\$rt[r_lba\$response==1])) y <- n1CDF(x, A=0.5, b=1, t0 = 0.5, st0 = 0.2, mean_v=c(1.2, 1.0), sd_v=0.2) lines(x=x,y=y/mean(r_lba\$response == 1), col = "red", lwd = 4.5, lty = 2) # KS test normalised_n1CDF = function(rt,...) n1CDF(rt,...)/n1CDF(rt=Inf,...) ks.test(r_lba\$rt[r_lba\$response==1], normalised_n1CDF, A=0.5, b=1, t0 = 0.5, st0 = 0.2, mean_v=c(1.2, 1.0), sd_v=0.2) xx <- rLBA(10, A=0.5, b=1, t0 = 0.5, mean_v=1.2, sd_v=0.2) # default uses normal distribution for drift rate: n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2) # other distributions: n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, shape_v=c(1.2, 1), scale_v=c(0.2,0.3), distribution = "gamma") n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, shape_v=c(1.2, 1), scale_v=c(0.2,0.3), distribution = "frechet") n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, meanlog_v = c(0.5, 0.8), sdlog_v = 0.5, distribution = "lnorm") # add st0: n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2, st0 = 0.4) # use different A parameters for each RT: n1PDF(xx\$rt, A=runif(10, 0.4, 0.6), b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2) # use different A parameters for each RT and each accumulator: n1PDF(xx\$rt, A=list(runif(10, 0.4, 0.6), runif(10, 0.2, 0.4)), b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2) ### vectorize drift rates: # vector versus list: v1 <- n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1.0), sd_v=0.2) v2 <- n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, mean_v=list(1.2, 1.0), sd_v=0.2) identical(v1, v2) # TRUE # drift rate per trial: n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, mean_v=list(rnorm(10, 1.2), rnorm(10, 1)), sd_v=0.2) # combine list with vector: n1PDF(xx\$rt, A=0.5, b=1, t0 = 0.5, mean_v=list(rnorm(10, 1.2), rnorm(10, 1)), sd_v=c(0.2, 0.1)) # t0 per trial and accumulator: n1PDF(xx\$rt, A=0.5, b=1, t0 = c(0.5), mean_v=c(1.2, 1.0), sd_v=0.2) n1PDF(xx\$rt, A=0.5, b=1, t0 = c(0.5, 0.6), mean_v=c(1.2, 1.0), sd_v=0.2) # per trial only n1PDF(xx\$rt, A=0.5, b=1, t0 = list(0.5, 0.6), mean_v=c(1.2, 1.0), sd_v=0.2) # per drift rate only n1PDF(xx\$rt, A=0.5, b=1, t0 = list(c(0.4, 0.5), c(0.5, 0.6)), mean_v=c(1.2, 1.0), sd_v=0.2) ## End(Not run) ```