The Wiener first-passage time family — wiener • frmtmb.eam

A drift-diffusion model for a two-choice decision: a noisy evidence accumulator starts between two boundaries and the response time is the first time it touches one of them. The family models the response time; which boundary was touched is data, supplied through the dec() addition term.

Usage

wiener(
  link = "identity",
  link_bs = "log",
  link_ndt = NULL,
  link_bias = "logit",
  max_ndt = NULL,
  variability = character(0),
  nodes = c(sz = 7L, st = 21L),
  allow_unreachable = FALSE,
  contaminant = FALSE,
  contaminant_range = NULL
)

Arguments

link: Link for the drift rate mu: "identity" (the default), "log", "softplus" or "squareplus", the set brms allows. An unquoted name works too, as in brms: wiener(log). The drift rate is signed, so a link other than the identity is a model in which the accumulator can only drift up.
link_bs: Link for the boundary separation bs: "log" (the default), "identity", "softplus" or "squareplus".
link_ndt: Link for the non-decision time ndt. NULL, the default, is the bounded link of "Non-decision time" above. Give "log", brms's default, or "identity", "softplus" or "squareplus", for brms's own parameterization; see "Links, and the one difference from brms".
link_bias: Link for the start point bias: "logit" (the default) or "identity".
max_ndt: Upper bound for the non-decision time, in the units of the response, applied to every row. NULL, the default, takes the fastest response of each row's ndt_group(), or of the whole data set when the model has no ndt_group(). It cannot be combined with ndt_group(). Give it when a component of a frmtmb::mixture() needs the bound up front.
variability: Which across-trial variability parameters to estimate: any of "sv" (drift rate), "sz" (start point) and "st" (non-decision time). The default estimates none, which is the plain Wiener model.
nodes: Gauss-Legendre node counts for the two quadratures, as a named vector. Only the entries for the variability parameters in use are read. The defaults are measured rather than chosen: sz reaches 1e-10 in the log density by 7 nodes everywhere it was probed, and st needs more because a response time below ndt + st / 2 cuts the range and the integrand turns on sharply at the cut. See vignette("ddm") for the measurement.
allow_unreachable: Permit a max_ndt above the smallest response time. Only correct inside a mixture, where another component carries the rows the Wiener density cannot reach.
contaminant: Mix a uniform contaminant into the density, with its mixing proportion estimated as the distributional parameter lambda on a logit link. FALSE, the default, is the plain model. See Contaminant trials.
contaminant_range: The response times the uniform contaminant spreads over, as c(lower, upper) in the units of the response: the task's response window. NULL, the default, is allowed only when the model declares a deadline with trunc(ub = ), and is then the fastest response to that deadline; otherwise it is refused. See Contaminant trials for why.

Value

A frmtmb_family.

Details

The parameterization is brms's wiener family, name for name:

mu: Drift rate, the mean rate of evidence accumulation (brms and the literature also call it v). Identity link by default, so it is signed: positive drift favors the upper boundary.
bs: Boundary separation, the distance between the two boundaries (a). Log link by default.
ndt: Non-decision time, the part of the response time spent encoding and moving rather than deciding (t0 or tau). Bounded link by default; see Non-decision time below.
bias: Relative start point in (0, 1), the fraction of the boundary separation the accumulator starts at (w). Logit link by default. 0.5 is unbiased.

Each of the four takes the links brms allows for it; see Links, and the one difference from brms.

variability adds Ratcliff's three across-trial variability parameters to that set; see Across-trial variability below.

The decision indicator

The boundary a trial ended at is data, and it reaches the density as an addition term:

frm(bf(rt | dec(response) ~ condition), family = wiener(), data = dat)

dec() is spelled as brms spells it and takes what brms takes: a factor or character vector whose SECOND level is the upper boundary (so "lower"/"upper" and c(FALSE, TRUE) both work as they read), or a numeric 0/1 column. The package contributes the term to frmtmb's addition-term registry when it loads.

vint() also still works, and carries the indicator as a plain 0/1 integer column:

frm(bf(rt | vint(upper) ~ condition), family = wiener(), data = dat)

Supplying neither is refused with a message that says so, because the failure is otherwise silent.

The unit of the response

SECONDS. Every default in this package assumes it: the starting values, the bound the ndt link is scaled onto, and gddm_control(dt = ) and the window it takes from the data. Nothing in any of the likelihoods refuses milliseconds, and a fit to millisecond data converges and reports a boundary separation three orders of magnitude out.

So every family here warns when the fastest response in the data is above 20, names milliseconds as the likely cause and says to divide by 1000. It is a ceiling on the fastest response in the WHOLE data set rather than on any one trial, so one slow trial does not reach it.

It is a warning rather than a refusal because it CAN fire on a correct model: a slow task with a short session. Measured, the rate is zero over 192 designs at the parameters a two-choice task produces, and rises to 0.07 at 20 trials of a task whose median response is 39 seconds, 0.56 at 60 seconds and 1.00 at 100 seconds. A deliberation or matrix-reasoning design is where that lands. The warning carries the class frmtmb_eam_units_warning so that such a design can silence this one condition and keep the rest. NEWS.md carries the full tables.

Non-decision time, and what its coefficients mean

The density is zero for a response time at or below ndt, so the likelihood has a hard edge at the fastest response and an ordinary log link would let the optimizer walk straight over it. The ndt link makes the constraint structural instead, and it does so in one of two ways.

Without ndt_group(), ndt is a TIME, on a logit scaled onto (0, ub) with ub the fastest response in the whole data set, or max_ndt when you give one. This is the parameterization the family has always had. frm_linpred(dpar = "ndt", type = "response") reports seconds, a prior(class = "ndt") is a density on those seconds, and a bf(ndt = 0.2) constant is 0.2 seconds.

With ndt_group(), ndt is a FRACTION of the row's own bound, on a plain logit, and the density multiplies it by that bound. The bound is the fastest response of the row's group. Write

frm(bf(rt | dec(response) + ndt_group(subject) ~ coherence,
       ndt ~ 1 + (1 | subject), bias = 0.5),
    family = wiener(), data = dat)

and each subject's non-decision time is bounded by its own fastest response. That is what a random effect on ndt needs; the next section is what one global bound does to it. The price is that ndt is on a different scale: frm_linpred(dpar = "ndt", type = "response") reports the fraction, and ndt_time() reports the time for either parameterization. A prior(class = "ndt") and a bf(ndt = ) constant are fractions under a grouping too.

Give the grouping as a factor, a character vector, a logical or integer codes. It is keyed on the group's LABEL, so subsetting, droplevels(), relevel() and a prediction grid you build yourself all pair a row with the same bound the fit used. max_ndt and ndt_group() cannot be combined, because they set the same bound to different things, and an ndt_group() no family reads is refused rather than carried into the fit unused.

A max_ndt above the smallest response time is refused, because for this family alone it admits parameter values at which some observed row has no likelihood. Inside a frmtmb::mixture() that is exactly what the other component is for, so allow_unreachable = TRUE lifts the refusal; see Mixtures. A mixture never finalizes its components, so a component needs max_ndt given up front and cannot use ndt_group().

Whichever parameterization a model is in, the bound is fixed when the model frame is assembled and is a property of the FITTED data, so a prediction on new rows is scaled by the bound the fit used rather than by one re-derived from the new rows. A group the fit never saw has no bound and is refused.

Both are the default link. brms's own link, a log link with no bound, is link_ndt = "log"; see Links, and the one difference from brms.

Past a linear predictor of about 37 the logit saturates in double precision. Nothing guards that, and nothing needs to: the density falls off a cliff as the decision time goes to zero, so the log likelihood is already unreachable long before the link runs out of digits.

Why one bound is the wrong constraint under a random effect

A single bound is the GLOBAL fastest response, so a subject whose true ndt is above it cannot be represented at any value of the random effect. At 30 subjects by 400 trials with a between-subject spread of 26 ms on a mean of 250 ms, 20 of the 30 subjects are in that position and NONE is inconsistent with its own data. Without variability such a fit does not converge; with variability = "sv" it converges, reports nothing from diagnose(), and returns a population ndt pinned at the bound and wrong by ten percent with a standard error of 7.2e-06 on it.

What settles it is what happens as data accumulates. On that design, the per-subject root mean squared error of the fitted non-decision times:

trials	with ndt_group(s)	one global bound
100	20.91 ms	27.23 ms
200	14.08 ms	20.51 ms
400	7.67 ms	30.37 ms

The per-group bound converges on the truth and the global bound does not, because more data lowers the global minimum and tightens the ceiling on every subject at once.

What sd(ndt) does and does not tell you

A grouped model estimates ndt as a fraction of each group's own floor, so the fitted per-subject times vary with those floors even when the variance component is exactly zero. On the design above an estimator with NO random effect on ndt returns a between-subject standard deviation of 0.02748 against a truth of 0.02629, where the full model returns 0.02543, and at 100 trials per subject the two are identical in every digit.

The variance component is not empty: at 400 trials it buys 8.44 log-likelihood units and cuts the per-subject error from 11.90 ms to 7.67 ms. But a between-subject standard deviation is the wrong statistic to read that off. Compare the per-subject non-decision times from ndt_time() against what you believe, and compare log-likelihoods, rather than reading VarCorr()'s ndt entry as evidence that the component was estimated.

The bound's own quality is a function of the group's trial count. The group's fastest response overshoots its true non-decision time by about 47 ms at 400 trials and 78 ms at 50, and a group of one trial gets that trial's own response time as its bound. Nothing refuses a small group, because any threshold would fire on correct models; the trial counts are recorded on the fitted family, at family(fit)$ndt_bound$sizes.

Recovery at 30 subjects by 400 trials

Measured on 60 replicate data sets at the design named in the realistic-scale table of dev/extension-gaps-plan.md: 30 subjects, 400 trials each, two conditions, mu ~ cond + (1 | s), bs ~ 1 + (1 | s) and ndt ~ 1 + (1 | s) with ndt_group(s). The truths are a drift of 0.4 and 1.3, a boundary separation of 1.4, a non-decision time of 250 ms, and between-subject standard deviations of 0.35 on the drift, 0.20 on the log boundary and 0.12 on the log non-decision time. Seeds 20260910 to 20260969. dev/eamhier-findings.md holds the tables and dev/eamhier-scripts/ the harness.

All 60 fits converged with a positive definite Hessian, no NaN standard error and nothing from diagnose(), and every subject's fitted non-decision time stayed below its own fastest response.


  quantity              truth    mean     mcse     Wald coverage
  mu intercept          0.4      0.4019   0.0094   54/60   90.0%
  mu condition effect   0.9      0.9004   0.0035   57/60   95.0%
  bs                    1.4      1.3900   0.0067   56/60   93.3%
  sd(mu | s)            0.35     0.3474   0.0059   55/60   91.7%
  sd(log bs | s)        0.20     0.1898   0.0037   54/60   90.0%

What 60 replicates can resolve. The rule applied is that a coverage misses when its Wilson interval excludes 0.95, which at n = 60 rejects on 53 or fewer. Against a nominal rate that is known rather than estimated, the exact binomial power of that rule is 0.82 at a true 85 percent, 0.59 at 88 and 0.39 at 90, with a size of 0.030. So the table can show a ten-point shortfall and cannot show seven. No coefficient's interval excludes 95.

Both variance components come back about as low as maximum likelihood is expected to put them. The shrinkage at 30 groups, including the term for reporting a standard deviation rather than a variance, is sqrt(1 - 1/30) * (1 - 1/116) = 0.9747, that is 2.5 percent low. Against that expectation sd(mu | s) sits 1.06 Monte Carlo standard errors high and sd(log bs | s) 1.39 low: one common shrinkage covers both, and neither is a separate finding. Read a variance component from 30 groups as a slight underestimate.

The non-decision time is read per subject rather than as a spread, for the reason the section above gives. Its root mean squared error is 7.9 ms against a between-subject spread of 30.2 ms, it is smaller than that spread on 60 of 60 replicates, and its correlation with the drawn values is 0.967. The population value comes back about 2 ms high.

The same 60 draws with variability = "sv" recover the same quantities to the same accuracy. What they add is in the next section.

Links, and the one difference from brms

Each parameter takes the links brms 2.23.0 allows for it, through the argument brms uses, in brms's order: link for mu, then link_bs, link_ndt and link_bias. A positional call written for brms means the same thing here. A link outside brms's set is refused.

The defaults are brms's, except for ndt. brms puts ndt on a log link. Without a formula on ndt, brms bounds it in its parameter block, real<lower=0,upper=min_Y> ndt, and gives it a uniform(0, min_Y) prior. With a formula, brms has no bound: each row's density rejects a non-decision time at or above that row's response time. This package has no parameter block, so its default link carries the bound instead.

link_ndt = "log" gives brms's parameterization. The coefficients of ndt ~ ... are then on brms's log scale, and a prior written for brms means the same thing here. A row whose response time is at or below its non-decision time has a log density of minus infinity, and the optimizer treats that as a rejected step. The start is half the fastest response, which every row can reach.

Measured on 100 data sets of 30 trials and 100 of 400 (seeds 20261101 to 20261200, dev/famuni-wiener-ndtrep.R): every fit converged under both links, and on this scalar ndt, where the two links are the same model, the two maxima agreed to 6e-10 in the log-likelihood and to 1.6e-05 standard errors in ndt. Against brms on one data set of 400 trials with ndt ~ 1, the maximum-likelihood estimates sat within 0.15 posterior standard deviations of brms's posterior medians, and the standard errors within 5 percent of the posterior standard deviations.

Under ndt ~ 1 + (1 | s) the per-row rejection is a per-subject bound, which is the constraint ndt_group() gives the default link. Measured on 20 data sets of 20 subjects by 50 trials, with a between-subject standard deviation of 0.12 on the log non-decision time around 250 ms (seeds 20261101 to 20261120, dev/famuni-wiener-ndtrep.R): the log link converged on 20 of 20 with no warning, ndt_group() on 20 of 20, and the one global bound on 12 of 20. The median per-subject root mean squared error of the non-decision time was 13.8 ms under the log link, 16.4 ms under ndt_group() and 24.4 ms under the global bound, and the log link had the smallest error on 20 of the 20 data sets. Its between-subject standard deviation came back at a median of 0.115, where ndt_group() put it at zero for the reason "What sd(ndt) does and does not tell you" gives.

An unbounded ndt link cannot be combined with max_ndt, ndt_group(), variability, contaminant = TRUE or allow_unreachable = TRUE. The first two set the bound that the link does not have. Each of the other three holds a row past the non-decision time at a small positive decision time, and without a bound every row can be held there at the same time, where the density of one point has no upper limit as bs goes to zero. A mixture with a lognormal reached a log-likelihood of +7713.9 on 400 rows that way, with ndt at 1.8e+308 and bs at 7.05e-05 (dev/famuni-wiener-mix2.R).

Under an identity link nothing keeps bs above zero, bias inside (0, 1) or ndt above zero. The density is NaN outside those ranges, on the tape the optimizer evaluates as well as on doubles, and the optimizer treats it as a rejected step. On data with a bias of 0.97 (seeds 20261101 to 20261105, n = 80), link_bias = "identity" fits reach the logit fits' maxima on all five.

A brms family object is read with its links. brms::wiener() and brms::brmsfamily("wiener") carry brms's link_ndt = "log", so family = brms::wiener() fits brms's log link for ndt, while family = "wiener" and this package's wiener() fit the bounded default. With both packages attached, a bare wiener() is the one attached last: brms's gives the log link, this package's the bounded one. On a scalar ndt the two are the same model with the same maximum; under ndt ~ x the coefficients are on different scales, and the log link refuses max_ndt, ndt_group(), variability and contaminant = TRUE. Write frmtmb.eam::wiener() to be sure which you have.

Post-fit quantities against brms

The log-likelihood of a row is the same quantity in both packages: the log density of its response time at the boundary it ended at. On 50 posterior draws of a 400-row brms fit, this density and brms's log_lik() agree to 1.1e-14.

fitted() and simulate() differ from brms on purpose. Both condition on the boundary each row ended at. brms's posterior_predict() draws the boundary as well and returns only the time, or, with negative_rt = TRUE, the time signed by the DRAWN boundary, so a row observed at the upper boundary gets draws at the lower one. predict(negative_rt = TRUE) is refused here. brms's posterior_epred() is the mean response time over both boundaries, from Grasman, Wagenmakers and van der Maas (2009). brms 2.23.0 passes that formula the relative start point bias where it takes the absolute one, bias * bs, so its value is correct only at bs = 1: at bs = 2.5 it is wrong by up to 64 percent, and at a drift of exactly zero it is NaN. The formula with bias * bs agrees with this package's conditional means, weighted by the boundary probabilities, to 1.04e-15 relative (dev/famuni-wiener-fit2.R).

Across-trial variability

Ratcliff's full diffusion model draws three of the four parameters afresh on every trial. variability names which of those to estimate, and each one it names becomes an ordinary distributional parameter that takes its own formula:

frm(bf(rt | dec(response) ~ coherence, bias = 0.5),
    family = wiener(variability = c("sv", "sz", "st")), data = dat)

sv: Standard deviation of a normal drift rate. Log link.
sz: Width of a uniform relative start point, centered on bias, on the same (0, 1) scale as bias. Logit link, so the width is below 1 and the start point stays inside the boundaries whenever bias is 0.5.
st: Width of a uniform non-decision time, centered on ndt, and on whichever scale ndt is on: a duration in the units of the response, on a logit scaled onto (0, 2 * bound), or a FRACTION of twice the row's own bound under ndt_group().

The likelihood is the analytic Wiener density averaged over those distributions, and the three are done three different ways because they are three different integrals. The drift integral is Gaussian against an exponential-quadratic and is evaluated in CLOSED FORM: it is exact, it takes no nodes, and there is nothing to tune. The other two are uniform and are evaluated by fixed-node Gauss-Legendre quadrature, whose node counts are the nodes argument.

The node positions and counts are decided when the family object is built and are constants from then on, because a node count that moved with a parameter would be a branch on a parameter and an automatic-differentiation tape cannot record one. A parameter only rescales the interval the fixed nodes are mapped onto.

fitted() and simulate() follow the variability rather than ignoring it. Both condition on the boundary a row ended at, and conditioning reweights which per-trial parameters that row could have had: the trials that reached a boundary are not a fair sample of the drift rates that could have produced them. So the fitted mean is a ratio of two quadratures and the simulator accepts a drawn drift rate and start point with the boundary probability they imply. The plain closed-form mean is not a usable approximation here: at an unbiased start point it returns the same number for both boundaries, and the full model does not.

Two limits of the parameterization are worth knowing. The uniform start point stays inside the boundaries by construction only when bias is 0.5, which is the usual case and the one the logit link on sz is scaled for; at a strongly biased start a wide sz can push the range past a boundary, where the density is near zero and the likelihood is a barrier rather than a cliff. And nothing holds ndt - st / 2 above zero, so a fit is free to report a range that includes a negative non-decision time. Neither can be made structural from outside frmtmb: both are joint constraints on two distributional parameters, and a link is a property of one.

How well sv is identified, measured. On 60 replicates of one subject with 12,000 trials, no random effects, and a true sv of 0.4, the estimate ranges from 0.194 to 0.599 and its Wald interval covers on 57 of the 59 fits that report a usable one. The estimated sv and an estimated drift contrast are correlated, 0.59 by rank: a draw that returns a low sv returns a low contrast with it, and every replicate that missed on the contrast had an sv below the truth. Read the two together rather than one at a time. In the hierarchical design of "Recovery at 30 subjects by 400 trials" both cover at 57 of 60.

One fit in 60 lost sv entirely and nothing reported it. At one seed the estimate ran to the log link's floor, 4.4e-04, with a Wald interval of (-1163, 1147) on the log scale, convergence code 0, a positive definite Hessian and diagnose() printing "No convergence problems detected". The likelihood is genuinely flat there on that draw: dropping the sv term moves the log-likelihood by 7e-07, while holding sv at its true value costs 4.05 units. So the fit is not wrong, but the interval it prints is not a statement about anything. Read the interval confint() gives sv before you read the estimate. The signature is inside that one fit and needs no replicates: the standard error on log sv was 589.25 where the largest of the other four was 0.0501, a ratio of 11,763. Nothing else in the fit reports it.

Mixtures

A contaminant component covers the trials the diffusion process cannot produce, which is the standard treatment for fast guesses. The Wiener component then wants a non-decision time that some observed rows fall below, so pass the bound and lift the refusal:

frm(bf(rt | dec(response) ~ 1, bias1 = 0.5),
    family = mixture(wiener(max_ndt = 0.4, allow_unreachable = TRUE),
                     lognormal()),
    data = dat)

A row below the non-decision time gets a log density of about -1 / delta where delta is a billionth of the smallest response time: it exponentiates to exactly zero, which is the right likelihood for the component, and it differentiates to exactly zero, which a true -Inf would not.

Censoring and truncation

cens() and trunc() work on the response time, on the plain model. What a censored row means depends on its code, because the boundary is known on some and not on others:

right: The trial had not reached EITHER boundary by the recorded time: a deadline passed with no response. Its probability is the survival over both boundaries, P(T > t). Its dec() value is required, since a declaration cannot depend on a censoring code, and is NOT read: any 0 or 1 gives the same fit, bit for bit.
left and interval: The trial DID reach a boundary, and dec() says which; only its time is coarse. Its probability is that boundary's defective distribution function, P(T <= t, boundary b), or its difference across the interval. dec() is read. A row whose dec() is missing is dropped by frmtmb's na.action with a message, before this family sees it; this family cannot refuse it by name.

A deadline design is

frm(bf(rt | dec(response) + cens(censored) ~ condition, bias = 0.5),
    family = wiener(), data = dat)

with censored set to "right" on the trials with no response and rt set to the deadline there.

trunc() divides by the window's mass over both boundaries. Left or interval censoring cannot be combined with trunc(): frmtmb forms the censored rows and the truncation window from the one distribution-function slot a family has, and the two need different functions here, so the pair is refused by name.

Every quantity is computed as a LOG and never as a complement: the survival from the method of images for small times and from the eigenfunction series for large ones, and each defective function the same way. Against a 700-bit reference over drift -20 to 20, boundary separation 0.3 to 6, relative start 0.001 to 0.999 and normalized time 1e-3 to 10, 3150 points, the worst relative error is 4.3e-14 on the distribution function over both boundaries, 1.2e-12 on either defective one, and on the survival 1.7e-12 where |v| a is at most 1, 7.4e-12 up to 24, 4.6e-11 up to 72 and 8.0e-11 up to 120. These hold on that grid only. The review of 2026-09-24 measured 798 more points at 1200 bits, to |v| a = 250 and a relative start of 0.9999, and there the survival's relative error is 1.7e-11 where |v| a is at most 1, 1.5e-9 at 120 and 1.6e-9 at 250, the worst at relative start 0.9999 and normalized time 0.01. 1.6e-9 is the worst measured anywhere. A survival of exp(-72063) comes back as that log. RWiener::pwiener() agrees to 4.5e-11 absolute on the grid the density is pinned on, and all of that difference is RWiener's own error.

An interval's mass is formed as a difference of logs, never of probabilities: F(t2) (1 - F(t1) / F(t2)), or the same with the mass still to come after each edge, whichever keeps more digits, or a quadrature of the density where the interval holds a tiny share of the mass on both sides. Against 751 intervals at 1200 bits, 511 of them the review's, with the lower edge at normalized time 1e-3 to 3, |v| a up to 80 and masses down to 1e-99 of F(t1), the worst error of the log mass is 4.9e-11. The probability of a left-censored row, or of an interval, is held at 1e-300 or above, because frmtmb takes it on the probability scale: a row whose true probability is smaller than that contributes a flat barrier rather than its value.

What it recovers, at 30 subjects by 400 trials (drift 0.4 and 0.9 by condition with a subject deviation of 0.35, boundary separation 1.4 with a subject deviation of 0.20 on its log, non-decision time 0.25 s). With a 1 s deadline, 16 percent of trials right-censored, 40 replicates: every fixed effect and both variance components covered on 33 to 38 of 38 fits with an interval. With trials faster than 0.45 s recorded as left-censored and every fifth other trial as interval-censored into its 100 ms bin, 29 and 14 percent of trials, 60 replicates: they covered on 47 to 55 of 55, the lowest being the drift intercept at 85.5 percent, Wilson interval 73.8 to 92.4. Every fit converged. The arms draw from the same seeds, so their coverages are not independent of each other.

The plain model only. Under variability the distribution function is the same series averaged over the per-trial parameters, and the drift average has no closed form there, so the pair is refused.

Contaminant trials

contaminant = TRUE mixes a second process into every row: a response time uniform over a window, with a boundary that is a coin flip. Its share is the distributional parameter lambda, on a logit link, which takes a formula like any other:

frm(bf(rt | dec(response) ~ condition, lambda ~ 1, bias = 0.5),
    family = wiener(contaminant = TRUE, contaminant_range = c(0, 3)),
    data = dat)

It needs no frmtmb::mixture(): the uniform has no free parameter, so it is one more term in the density rather than a second family. A right-censored row enters through both parts' survivals; a left- or interval-censored row through the diffusion's defective function at its boundary and half of the uniform's, the coin flip landing there.

The window. Give it as contaminant_range =: the task's response window, such as c(0, deadline). Without it, a model that declares its deadline through trunc(ub = ) uses the fastest response to that deadline, and any other model is refused, with the class frmtmb_eam_contaminant_range_error. A response outside the window is refused with the same class, since the contaminant has density zero there. The refusal of a missing window is the measured reason: without a deadline bounding both processes, the slowest diffusion trial sets the top of the observed range, the window grows with the sample, and lambda collapses. On one subject of 4000 trials, 25 seeds, true share 0.0495, contaminants uniform on 0 to 5 s: the observed range gave lambda 0.0212 and covered on 2 of 25, the true window 0.0498 and 25 of 25. With a 5 s deadline declared by trunc(ub = 5), the window from the fastest response to the deadline gave 0.0508 against a recorded share of 0.0496, and the one to the slowest response 0.0518, both covering on 25 of 25; the deadline is the top used, being the one that does not move with the sample (dev/phase3b-cont-window.R). The review of 2026-09-24 measured the same pattern on its own design: 0.0050 and 5 of 25 without a deadline, 0.0545 and 22 of 25 with one.

Where the practice comes from. Ratcliff and Tuerlinckx (2002) model the contaminant as uniform over the range of the observed response times, and the DMAT toolbox follows them; HDDM uses a uniform of fixed density instead. None of the three was checked against its source or text for this page; the attributions are the review's of 2026-09-24, recorded as such.

What it recovers, at 30 subjects by 400 trials. Drift 0.4 and 0.9 by condition with a subject deviation of 0.35, boundary separation 1.4 with a subject deviation of 0.20 on its log, non-decision time 0.25 s, 5 percent contaminants uniform on 0.1 to 5 s, max_ndt = 0.5. With contaminant_range = c(0.1, 5), over 40 replicates, lambda came back at 0.0499 and its Wald interval covered on 39 of 40; every fixed effect and both variance components covered on 34 to 40 of 40, and every fit converged. The same draws with a 3 s deadline declared by trunc(ub = 3) and the default window, 44 replicates: lambda 0.0306 against a recorded share of 0.0307, covering on 35 of 36. On data with no contaminant and the window given, 30 replicates: see the collapse case below.

A guess faster than the non-decision time. The non-decision time is bounded by the fastest response, so a contaminant faster than the true non-decision time pulls the bound under it and the fit cannot reach the truth. On the design above at the default bound, 20 replicates on the first round's build, the non-decision time came back at 0.111 s against 0.25 and the drift effect covered on 0 of 20. Under the contaminant such a row has a likelihood, so max_ndt above the fastest response is allowed with contaminant = TRUE: give a bound you know the non-decision time is under.

The collapse case. Maximum likelihood has no prior to hold lambda off its edge. When the data give it nothing, it runs down the logit: the estimate is a logit near -20, its standard error is in the thousands, and the log-likelihood is the plain family's. frmtmb::diagnose() reports it as a parameter at the end of its link. On the design above with no contaminant and the window c(0, 5) given, 30 replicates: lambda reached the edge on 25, diagnose() named it on 27, all 25 at the edge among them, and at the edge the log-likelihood was the plain family's to within 8.9e-07 on every one.

At lambda = 0 the density is the plain family's: bit for bit on every row whose diffusion density is at least the contaminant's, and to the rounding of one log-sum anchor on the others.

Accuracy

The density is the Navarro and Fuss (2009) pair of series, both evaluated at a fixed truncation and combined with a smooth weight, because an automatic-differentiation tape cannot choose between them on a parameter. It agrees with RWiener::dwiener() to better than 1e-12 relative on the log scale over normalized times from 1e-3 to 50. See vignette("ddm").

References

Grasman, R. P. P. P., Wagenmakers, E.-J. and van der Maas, H. L. J. (2009). On the mean and variance of response times under the diffusion model with an application to parameter estimation. Journal of Mathematical Psychology, 53(2), 55-68.

Navarro, D. J. and Fuss, I. G. (2009). Fast and accurate calculations for first-passage times in Wiener diffusion models. Journal of Mathematical Psychology, 53(4), 222-230.

Ratcliff, R. and Tuerlinckx, F. (2002). Estimating parameters of the diffusion model: approaches to dealing with contaminant reaction times and parameter variability. Psychonomic Bulletin & Review, 9(3), 438-481.

Examples

set.seed(1)
dat <- ddm_simulate(300, mu = 0.8, bs = 1.4, ndt = 0.3, bias = 0.5)
fit <- frm(bf(rt | dec(upper) ~ 1, bias = 0.5),
           family = wiener(), data = dat)
fixef(fit)
#>            Estimate  Est.Error     Q2.5     Q97.5
#> Intercept 0.6956708 0.08770048 0.523781 0.8675606