Density and random generation functions for the EZ-Diffusion Model. The model operates on aggregated data: mean reaction time, variance of reaction time, and number of responses to the upper boundary.
Arguments
- mean_rt
Observed mean reaction time(s) in seconds. For version "3par", a numeric vector or single value. For version "4par", either a vector of length 2 (c(mean_rt_upper, mean_rt_lower)) for single observation, or a matrix with 2 columns for multiple observations.
- var_rt
Observed variance of reaction times in seconds^2. For version "3par", a numeric vector or single value. For version "4par", either a vector of length 2 (c(var_rt_upper, var_rt_lower)) for single observation, or a matrix with 2 columns for multiple observations.
- n_upper
Number of responses to the upper boundary
- n_trials
Total number of trials
- drift
Drift rate (evidence accumulation rate; can be positive or negative for below-chance performance).
- bound
Boundary separation (distance between decision thresholds).
- ndt
Non-decision time (seconds).
- zr
Relative starting point (0 to 1). Only used for version "4par".
- s
Diffusion constant (standard deviation of noise), default = 1.
- version
Character; either "3par" (default) or "4par"
- log
Logical; if
TRUE, values are returned on the log scale.- n
Number of samples to generate
Value
dezdm gives the log-density of the observed summary statistics
under the EZDM, and rezdm generates random summary statistics from the
implied sampling distributions.
Details
The number of upper-boundary responses is binomial. The two RT
summaries follow the joint sampling distribution of the mean and the
variance of n independent decision times, matched to the exact first four
cumulants of the first-passage time. Writing \(\mathrm{MDT}\) and
\(\mathrm{VRT}\) for its mean and variance and \(\kappa_3\),
\(\kappa_4\) for its third and fourth cumulants, and
\(W = \kappa_4 / n + 2\,\mathrm{VRT}^2 / (n - 1)\) for the exact variance
of the sample variance,
$$\mathrm{var\_rt} \sim \mathrm{Gamma}(\mathrm{VRT}^2 / W, \mathrm{VRT} / W),$$
$$\mathrm{mean\_rt} \mid \mathrm{var\_rt} \sim N\left(\mathrm{ndt} + \mathrm{MDT} + \frac{\kappa_3 / n}{W}(\mathrm{var\_rt} - \mathrm{VRT}),\ \sqrt{\mathrm{VRT} / n - (\kappa_3 / n)^2 / W}\right),$$
so that \(\mathrm{Var}(\mathrm{mean\_rt}) = \mathrm{VRT} / n\) and
\(\mathrm{Cov}(\mathrm{mean\_rt}, \mathrm{var\_rt}) = \kappa_3 / n\) are
exact. Decision times are right-skewed (with a symmetric start point their
kurtosis is 8.8 at zero drift and falls towards 3 as drift grows), so the
older form that assumes normal reaction times — independent normal and
scaled chi-square
\(\mathrm{Gamma}((n - 1)/2, (n - 1)/(2\,\mathrm{VRT}))\) terms —
understates the sampling variance of var_rt by a factor of
\(1 + (\mathrm{kurtosis} - 3)(n - 1)/(2n)\): about 3.8 with 100 trials
per cell and 3.5 to 3.6 with 10, less at accuracies above .95. With an
asymmetric start point in version "4par" it is more at the boundary
nearer the start point (up to 5.7 for zr between .3 and .7) and less at
the other. It also ignores a correlation of about 0.7 between the
two statistics, making posteriors too narrow. The terms above reduce to it when \(\kappa_3 = \kappa_4 = 0\).
For version "3par" the start point is symmetric, so the decision time is
independent of which boundary is hit and all n_trials responses inform
one set of cumulants. For version "4par" the two boundaries have
different decision-time distributions, so each is given its own summaries
and its own response count. A boundary reached fewer than twice has no
sample variance, and one whose summaries are NA has nothing to evaluate;
dezdm() lets either contribute only through the binomial term.
rezdm() and ezdm_summary_stats() code such summaries as NA, and
bmm() keeps these cells: it replaces the summaries of such a boundary
with a placeholder that the likelihood never reads, so the response
counts still inform the fit. The per-boundary formulas condition on the
realised counts, which are themselves random.
The two additional cumulants cost sampling time. In two simulated designs
(30 subjects with 200 or 250 trials, 3 seeds each, one machine) a gradient
of the whole model took 1.3 times as long as with the older form for
version "3par" and 1.9 times for version "4par". The likelihood alone
took 1.5 to 2.0 and 1.7 to 2.5 times as long, most where accuracy is near
chance, so the ratio grows with the number of cells.
Simulated mean_rt is not truncated at ndt. When a summary rests on few
responses (a handful of trials in version "3par", or a rarely reached
boundary in version "4par", whatever n_trials is), rezdm() can
return mean_rt below ndt or even mean_rt <= 0, which bmm()
rejects; drop those rows before fitting.
References
Wagenmakers, E.-J., Van Der Maas, H. L. J., & Grasman, R. P. P. P. (2007). An EZ-diffusion model for response time and accuracy. Psychonomic Bulletin & Review, 14(1), 3-22.
Chávez De la Peña, A. F., & Vandekerckhove, J. (2025). An EZ Bayesian hierarchical drift diffusion model for response time and accuracy. Psychonomic Bulletin & Review.
Examples
# 3-parameter version (single observation)
dezdm(
mean_rt = 0.5, var_rt = 0.02, n_upper = 80, n_trials = 100,
drift = 2, bound = 1.5, ndt = 0.3
)
#> [1] -30.99022
# 3-parameter version (vectorized)
dezdm(
mean_rt = c(0.5, 0.55), var_rt = c(0.02, 0.025),
n_upper = c(80, 75), n_trials = c(100, 100),
drift = 2, bound = 1.5, ndt = 0.3
)
#> [1] -30.99022 -27.30733
# 4-parameter version (single observation)
dezdm(
mean_rt = c(0.45, 0.55), var_rt = c(0.018, 0.025),
n_upper = 80, n_trials = 100,
drift = 2, bound = 1.5, ndt = 0.3, zr = 0.55, version = "4par"
)
#> [1] -37.10626
# generate random summary statistics
rezdm(n = 100, n_trials = 100, drift = 2, bound = 1.5, ndt = 0.3)
#> mean_rt var_rt n_upper n_trials
#> 1 0.6111635 0.05418579 95 100
#> 2 0.6086589 0.04968604 91 100
#> 3 0.6043066 0.06557410 92 100
#> 4 0.6337035 0.04187681 98 100
#> 5 0.6628371 0.06905988 98 100
#> 6 0.6306668 0.06789205 92 100
#> 7 0.6683561 0.09017106 97 100
#> 8 0.6390170 0.05048177 92 100
#> 9 0.6594366 0.06443427 97 100
#> 10 0.6388977 0.05186053 96 100
#> 11 0.6466595 0.05898557 99 100
#> 12 0.6441727 0.06067506 96 100
#> 13 0.5960057 0.03501029 95 100
#> 14 0.6566931 0.07312851 97 100
#> 15 0.6480473 0.05061371 99 100
#> 16 0.6338738 0.06588902 97 100
#> 17 0.6749657 0.07643939 93 100
#> 18 0.6315851 0.04292302 94 100
#> 19 0.6299139 0.04702333 95 100
#> 20 0.6415729 0.06164678 97 100
#> 21 0.6569962 0.04665327 94 100
#> 22 0.6475104 0.06588247 96 100
#> 23 0.6607518 0.09228922 95 100
#> 24 0.6486437 0.06127875 96 100
#> 25 0.6397690 0.06979176 92 100
#> 26 0.6414065 0.07433980 96 100
#> 27 0.6021627 0.05596795 95 100
#> 28 0.6385743 0.05330900 95 100
#> 29 0.6524296 0.07905578 94 100
#> 30 0.6554120 0.06261139 92 100
#> 31 0.6469188 0.07973757 99 100
#> 32 0.6369412 0.05716800 96 100
#> 33 0.6245654 0.04790737 93 100
#> 34 0.6647938 0.07865629 97 100
#> 35 0.6356469 0.06488684 96 100
#> 36 0.6579946 0.06010943 94 100
#> 37 0.6288020 0.04890998 99 100
#> 38 0.6955779 0.08011255 95 100
#> 39 0.6633387 0.03779677 95 100
#> 40 0.6497766 0.05726640 95 100
#> 41 0.6020050 0.03912827 95 100
#> 42 0.6512410 0.07179975 94 100
#> 43 0.6187174 0.04709756 94 100
#> 44 0.6831167 0.07639139 98 100
#> 45 0.6561083 0.05187717 94 100
#> 46 0.6456383 0.07429988 93 100
#> 47 0.6413068 0.06202381 94 100
#> 48 0.6439319 0.08259685 94 100
#> 49 0.6545809 0.05948838 91 100
#> 50 0.6902880 0.09919671 93 100
#> 51 0.6490158 0.04714586 92 100
#> 52 0.6391610 0.07818092 97 100
#> 53 0.6669382 0.07987881 95 100
#> 54 0.6470221 0.07570689 99 100
#> 55 0.6697892 0.07322692 96 100
#> 56 0.6629982 0.05985320 96 100
#> 57 0.6085790 0.04388921 97 100
#> 58 0.6733613 0.08550546 97 100
#> 59 0.6025789 0.04238616 93 100
#> 60 0.6782462 0.08774496 90 100
#> 61 0.5853326 0.03055388 97 100
#> 62 0.6808352 0.06220229 96 100
#> 63 0.6804366 0.06035126 96 100
#> 64 0.6473923 0.05182805 96 100
#> 65 0.6464125 0.06720487 93 100
#> 66 0.6808136 0.08828020 93 100
#> 67 0.6593359 0.09232922 96 100
#> 68 0.6183544 0.06604877 94 100
#> 69 0.6550676 0.06929830 94 100
#> 70 0.6021303 0.06651822 94 100
#> 71 0.6212957 0.05692183 91 100
#> 72 0.6120274 0.03727352 89 100
#> 73 0.6525679 0.05112530 90 100
#> 74 0.6318721 0.03595612 95 100
#> 75 0.6268748 0.03405663 92 100
#> 76 0.6122607 0.04591200 91 100
#> 77 0.7305996 0.12676027 91 100
#> 78 0.6501535 0.07278958 95 100
#> 79 0.6840221 0.07520687 97 100
#> 80 0.6258505 0.07007770 98 100
#> 81 0.6213556 0.04796828 96 100
#> 82 0.6270388 0.04742668 99 100
#> 83 0.6663180 0.07532801 90 100
#> 84 0.6386077 0.06691333 94 100
#> 85 0.6302745 0.04252136 93 100
#> 86 0.5907167 0.04764884 97 100
#> 87 0.6222910 0.05049253 93 100
#> 88 0.6280339 0.05476211 95 100
#> 89 0.6579823 0.07378763 93 100
#> 90 0.6320513 0.06339803 91 100
#> 91 0.6010266 0.04239233 94 100
#> 92 0.6229369 0.04027990 95 100
#> 93 0.6783554 0.09407671 91 100
#> 94 0.6453377 0.08696251 96 100
#> 95 0.6168118 0.05827374 97 100
#> 96 0.6156420 0.04072509 99 100
#> 97 0.6278741 0.04883660 96 100
#> 98 0.6728986 0.07497164 97 100
#> 99 0.6355790 0.04843782 96 100
#> 100 0.6425947 0.03357066 97 100
rezdm(
n = 100, n_trials = 100, drift = 2, bound = 1.5, ndt = 0.3,
zr = 0.55, version = "4par"
)
#> mean_rt_upper mean_rt_lower var_rt_upper var_rt_lower n_upper n_trials
#> 1 0.6253226 0.5823772 0.07911455 5.282973e-03 98 100
#> 2 0.6171880 0.5839276 0.05528031 5.966834e-02 95 100
#> 3 0.6485196 0.7139917 0.08391141 2.858171e-01 97 100
#> 4 0.5829925 1.0441456 0.04721922 1.909124e-01 96 100
#> 5 0.6273172 0.5092828 0.07401407 2.321978e-02 93 100
#> 6 0.6249412 0.6171310 0.07956924 2.244483e-03 97 100
#> 7 0.6114124 0.8940065 0.05081401 2.970588e-03 98 100
#> 8 0.5880939 0.8036448 0.06025104 1.227541e-01 96 100
#> 9 0.6447928 0.6301377 0.06424441 4.105989e-02 96 100
#> 10 0.6082380 0.6723502 0.03238539 1.181740e-03 97 100
#> 11 0.5904479 0.7559707 0.05169755 2.350302e-02 93 100
#> 12 0.6196726 0.6583712 0.07667082 2.837501e-02 97 100
#> 13 0.5899915 NA 0.05189302 NA 99 100
#> 14 0.6181806 0.5681570 0.06644203 1.374754e-02 96 100
#> 15 0.6175320 0.7707452 0.07191115 4.200276e-02 95 100
#> 16 0.5931011 0.7873282 0.03816982 1.337478e-04 96 100
#> 17 0.5673028 0.5018691 0.04601163 2.907999e-02 98 100
#> 18 0.6278858 0.6996673 0.07084437 4.308046e-02 93 100
#> 19 0.5999634 0.6190288 0.04638203 1.800411e-03 94 100
#> 20 0.5716147 NA 0.04551777 NA 99 100
#> 21 0.5933489 0.6675141 0.03567301 1.414062e-04 95 100
#> 22 0.6270457 0.6874140 0.07178717 4.104850e-02 94 100
#> 23 0.6327607 0.6135700 0.08203474 6.503511e-03 96 100
#> 24 0.6026259 0.6489808 0.04942633 4.013743e-03 95 100
#> 25 0.6261162 0.4786807 0.06545772 1.936777e-03 97 100
#> 26 0.6402308 NA 0.07972958 NA 99 100
#> 27 0.5963508 NA 0.05022043 NA 99 100
#> 28 0.6202917 0.6139906 0.04721892 1.773589e-01 97 100
#> 29 0.6292854 0.8031477 0.07546246 1.045227e-01 98 100
#> 30 0.6201752 0.6355989 0.06832560 6.812979e-03 98 100
#> 31 0.6184427 0.5993869 0.04891331 1.497810e-10 98 100
#> 32 0.6200807 0.3899048 0.04971070 1.130956e-02 98 100
#> 33 0.5821060 0.6999998 0.04120795 1.410072e-01 94 100
#> 34 0.5808450 NA 0.04444016 NA 99 100
#> 35 0.5996229 0.7369012 0.05796532 1.158442e-01 97 100
#> 36 0.5896469 0.8406242 0.05602702 3.661818e-01 98 100
#> 37 0.5749020 0.6998572 0.05193127 8.825138e-02 97 100
#> 38 0.6089534 0.7608306 0.05016780 1.185885e-01 97 100
#> 39 0.6091953 0.6913986 0.05791185 1.166646e-01 98 100
#> 40 0.6566303 0.7872351 0.09057143 1.427873e-01 97 100
#> 41 0.6259888 0.5853079 0.07171781 6.039863e-04 97 100
#> 42 0.6524525 NA 0.09050340 NA 100 100
#> 43 0.5883644 NA 0.04354750 NA 100 100
#> 44 0.5955397 0.5841933 0.05831974 5.377575e-03 96 100
#> 45 0.6346774 0.5960504 0.06707157 9.774911e-03 95 100
#> 46 0.6210293 0.7869111 0.05421985 1.308940e-01 95 100
#> 47 0.5643313 0.7206490 0.04127384 4.529126e-02 95 100
#> 48 0.6290337 0.8056297 0.05998544 3.975849e-04 95 100
#> 49 0.5864586 0.5344239 0.06338933 2.497840e-03 94 100
#> 50 0.6017033 0.5552666 0.04908010 7.124369e-03 95 100
#> 51 0.6382134 0.6836493 0.08997426 6.404295e-06 97 100
#> 52 0.5878631 0.7653400 0.03712318 1.468800e-04 96 100
#> 53 0.6099902 0.9212601 0.03800446 2.197637e-01 95 100
#> 54 0.6127636 0.5738889 0.08001620 2.442924e-03 94 100
#> 55 0.5898540 NA 0.04125372 NA 99 100
#> 56 0.6565393 0.5712908 0.05915474 6.186089e-02 98 100
#> 57 0.6036652 0.7524118 0.07486943 1.142665e-03 95 100
#> 58 0.6205222 NA 0.07666748 NA 99 100
#> 59 0.6186504 0.7601851 0.06530945 4.286781e-03 96 100
#> 60 0.5846995 0.6870436 0.02863498 6.128926e-02 96 100
#> 61 0.6007758 0.7520200 0.05632263 3.525548e-04 95 100
#> 62 0.6146287 0.6095731 0.06517262 9.433624e-04 96 100
#> 63 0.6079881 0.7482211 0.03627917 2.192291e-01 95 100
#> 64 0.5717904 0.6237479 0.02249992 1.621140e-02 98 100
#> 65 0.6201225 NA 0.06193363 NA 99 100
#> 66 0.6113292 0.7473014 0.05148123 1.414600e-01 97 100
#> 67 0.6075502 0.4635556 0.04719018 2.600340e-03 96 100
#> 68 0.6120317 0.5996071 0.05008479 6.928488e-03 96 100
#> 69 0.6371032 0.6414161 0.08146846 6.582110e-02 93 100
#> 70 0.6095704 0.6700533 0.06765039 1.487197e-02 96 100
#> 71 0.5794014 0.5437786 0.02781296 1.159084e-03 98 100
#> 72 0.5805540 0.8679745 0.04403614 2.297599e-01 95 100
#> 73 0.5882489 0.6572810 0.03381068 4.042184e-03 95 100
#> 74 0.6337591 0.6219328 0.08497394 3.914663e-02 95 100
#> 75 0.6519035 0.6074190 0.09859535 2.255194e-03 96 100
#> 76 0.6020498 NA 0.06652119 NA 100 100
#> 77 0.5973022 1.0329858 0.04875611 1.740835e-01 97 100
#> 78 0.5637603 0.7237134 0.03017398 7.604240e-03 93 100
#> 79 0.6162830 0.7907933 0.06911791 5.392265e-02 95 100
#> 80 0.6616871 0.6805740 0.09730298 7.029844e-02 96 100
#> 81 0.6310367 0.5784882 0.07253849 4.318314e-02 95 100
#> 82 0.6056327 0.6992810 0.03734506 6.048835e-04 98 100
#> 83 0.6501273 0.8648440 0.08690550 2.984268e-01 98 100
#> 84 0.5939566 NA 0.03903370 NA 99 100
#> 85 0.6073185 0.9779838 0.03715447 2.959167e-02 96 100
#> 86 0.6047717 0.4530562 0.03138902 8.292242e-03 95 100
#> 87 0.5719962 0.6500034 0.04307318 6.181304e-02 95 100
#> 88 0.6053661 0.7520025 0.03703732 1.544416e-03 96 100
#> 89 0.6325180 NA 0.06526068 NA 99 100
#> 90 0.6139106 0.4433208 0.04435218 1.282726e-02 95 100
#> 91 0.6117540 0.6004376 0.07323417 1.341352e-02 94 100
#> 92 0.6574545 0.6501949 0.06220291 1.239173e-03 97 100
#> 93 0.5754018 0.5888569 0.06561530 2.414883e-03 98 100
#> 94 0.5895091 0.6046400 0.03646934 1.080198e-01 97 100
#> 95 0.6030274 0.6675923 0.04760016 7.117333e-02 93 100
#> 96 0.6019021 0.7496666 0.05104677 5.595125e-02 92 100
#> 97 0.6342686 0.4714468 0.06731189 1.101831e-02 97 100
#> 98 0.6026688 NA 0.04723144 NA 100 100
#> 99 0.6025780 0.8252478 0.06193917 8.912872e-03 98 100
#> 100 0.6258862 0.7871865 0.04819153 9.731447e-02 96 100