The BayesDiffIRT package provides functions to sample posterior
distributions and posterior predictive distributions of item and subject
parameters of diffusion item response theory models for responses and
reaction times Kang, De Boeck, and Ratcliff (2022). BayesDiffIRT also
provides functions to visualize posterior distributions of Diffusion
item response theory model parameters and construct credible intervals.
Under the hood, the package relies on NUTS sampling with STAN (Carpenter
et al. 2017). Up to know, the following diffusion item response theory
models have been implemented:
- D-diffusion model (Tuerlinckx and Boeck 2005),
- Q-diffusion model (Van Der Maas et al. 2011),
- D-diffusion model with random variability (Kang, De Boeck, and Ratcliff 2022),
- Q-diffusion model with random variability (Kang, De Boeck, and Ratcliff 2022).
The two versions of the D-diffusion model are appropriate for survey items where persons decide whether to accept or reject an item. The two flavours of the Q-diffusion model were designed to model ability tests.
Important: BayesDiffIRT uses Stan through the cmdstanr interface. Fitting models therefore requires a C++ toolchain and a separate installation of CmdStan. Installing the BayesDiffIRT R package alone is not sufficient.
Diffusion item response theory combines item response theory with the
drift diffusion model of decision making. According to the drift
diffusion decision model (Stone 1960; Link and Heath 1975; Ratcliff et
al. 2016), the sensory system repeatedly generates momentary evidence
about which of two possible choice options is correct. This momentary
evidence is drawn from a Gaussian distribution and accumulated over
time. The newly acquired evidence is therefore continuously added to the
evidence collected up to that moment.The accumulation process is bounded
by an upper and a lower threshold, where each threshold represents one
of the two possible choice options. When the accumulated evidence
reaches one of the thresholds, a choice is made for the corresponding
option. The quality of information favouring one response option over
the other is reflected in the drift rate
In diffusion item response theory models, two traditional parameters of
the drift diffusion model, boundary separation and drift rate, are
decomposed into person and item parameters (Tuerlinckx and Boeck 2005).
When person
where
The D-diffusion and Q-diffusion models differ in how the drift rate is decomposed. In the D-diffusion model (Tuerlinckx and Boeck 2005), which is applicable to survey items, the drift rate is given by
According to the Q-diffusion model (Van Der Maas et al. 2011), which is applicable to ability tests, the drift rate is given by
In both the D-diffusion and Q-diffusion models, the accumulation process starts midway between the two response alternatives. Thus, there is assumed to be no a priori bias toward either choice alternative.
Kang, De Boeck, and Ratcliff (2022) proposed extensions that include
random trial-to-trial variability in both the starting point
The drift rate
Random variability in starting points and drift rates accounts for the conditional dependency between accuracy and reaction times (Kang, De Boeck, and Ratcliff 2022), but note that sampling is considerably slower for these models.
The BayesDiffIRT package relies on STAN via cmdstanr. Stan models are
translated into C++ and compiled before they are run. Consequently, a
working C++ compiler and GNU Make are required. Please install the
appropriate toolchain for your operating system before installing
CmdStan:
- Windows: Install the version of Rtools appropriate for your version of R.
- macOS: Install the Xcode Command Line Tools by entering the following command in the Terminal:
xcode-select --install- Debian/Ubuntu Linux: Install g++ and make:
sudo apt update
sudo apt install g++ makeFurther operating-system-specific information is available in the CmdStan installation guide.
If cmdstanr is not yet installed on oyur system, install it from the
Stan R-universe repository:
install.packages("cmdstanr", repos = c("https://mc-stan.org/r-packages/", getOption("repos")))First check whether the C++ toolchain is correctly configured:
cmdstanr::check_cmdstan_toolchain()Once the toolchain check succeeds, install CmdStan:
cmdstanr::install_cmdstan()Verify the installation with:
cmdstanr::cmdstan_version()CmdStan is installed separately from the R package, normally in the .cmdstan directory in the user’s home directory. The initial compilation of CmdStan may take several minutes and require substantial memory.
The development version of BayesDiffIRT is available from GitHub. The
easiest way to install it is using the devtools package:
devtools::install_github("ManuelRausch/BayesDiffIRT")
The function fitBayesDiffIRT fits Bayesian diffusion item-response
theory models by sampling from the posterior distributions of item and
subject parameters using the No-U-Turn Sampler (NUTS) as implemented in
Stan (Carpenter et al. 2017). The data should be a dataframe with
columns identifying the subject, item, response, and response time.
Response times should be numeric and measured in seconds. For ability
tests, binary responses should be coded as 0 for incorrect responses and
1 for correct responses. For questionnaire items, binary responses
should be coded as 0 for rejected items and 1 for accepted items. Here,
we prepare a dataset contained in the diffIRT package as example.
library(tidyverse)# Example for preparing the data set.
data(extraversion, package = "diffIRT")
Extra <- as.data.frame(extraversion)
names(Extra)[1:10] <- paste0("Item", 1:10, "_resp")
names(Extra)[11:20] <- paste0("Item", 1:10, "_rt")
Extra$sbj <- 1:nrow(Extra)
Extra <- tidyr::pivot_longer(
Extra,
cols = tidyselect::matches("^Item\\d+_(resp|rt)$"),
names_to = c("item", ".value"),
names_pattern = "^(Item\\d+)_(resp|rt)$"
)
Extra$item <- factor(Extra$item)
Extra$item <- factor(Extra$item)
head(Extra)## # A tibble: 6 × 4
## sbj item resp rt
## <int> <fct> <dbl> <dbl>
## 1 1 Item1 0 2.73
## 2 1 Item2 1 0.915
## 3 1 Item3 1 3.48
## 4 1 Item4 1 1.02
## 5 1 Item5 1 1.11
## 6 1 Item6 0 2.38
Priors distributions for parameter classes are specified using the
function prior(). The function creates objects of class
BayesDiffIRTPrior, which can be passed to the priors argument of
fitBayesDiffIRT. Priors are specified using Stan distribution syntax,
for example normal(0, 1) or lognormal(0, 0.5). Priors on the
following parameters can be specified:
| Parameter | Description |
|---|---|
"omega_theta" |
Standard deviation of theta, the latent trait. |
"omega_gamma" |
Standard deviation of gamma, the item response tendency parameter. |
"nu" |
Item difficulty parameter. |
"a" |
Item boundary separation parameter. |
"tnd" |
Person-specific non-decision time. |
"s_delta" |
Standard deviation of Gaussian trial-to-trial variability in drift rate. Only relevant for models "dRV" and "qRV". |
"s_beta" |
Range of uniform trial-to-trial variability in starting point. Only relevant for models "dRV" and "qRV". |
If priors are supplied for only some parameter classes, the remaining parameter classes are filled in with their model-specific defaults.
# Example for prior specification
library(BayesDiffIRT)## BayesDiffIRT: Bayesian diffusion-IRT models with RTs.
## Backend: cmdstanr.
myPrior <- list(prior(normal(0, 2.5), class = "omega_theta"),
prior(normal(0, 0.5), class = "omega_gamma"),
prior(lognormal(0, 0.75), class = "nu"),
prior(lognormal(0, 0.5), class = "a"),
prior(lognormal(-1.25, 0.3), class = "tnd"))Finally, a string identifying the chosen model, the prepared data and a
list of priors should be passed to fitBayesDiffIRT to fit a Bayesian
diffusion item-response theory model. The following model names are
recognized:
- “d” for the D-diffusion model (for survey items, default),
- “dRV” for the D-diffusion model with random variability (for survey items),
- “q” for the Q-diffusion model (for ability tests),
- “qRV” for the Q-diffusion model with random variability (for ability tests).
samples <-
fitBayesDiffIRT(Extra,
rt = "rt", resp = "resp", sbj = "sbj",
item = "item", model = "d")## Running MCMC with 4 parallel chains...
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## Mean chain execution time: 212.4 seconds.
## Total execution time: 217.2 seconds.
The results of a fitted Bayesian diffusion item-response theory model
can be inspected using the summary method.
summary(samples)## Summary of BayesDiffIRT model fit
## ---------------------------------
## Model: D-Diffusion model
##
## Data:
## Observations: 1429
## Persons: 143
## Items: 10
##
## Call:
## fitBayesDiffIRT(data = Extra, rt = "rt", resp = "resp", sbj = "sbj",
## item = "item", model = "d")
##
## Posterior summaries:
##
## Hyperparameters:
## # A tibble: 2 × 9
## variable mean median sd q5 q95 rhat ess_bulk ess_tail
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 omega_theta 0.69 0.69 0.06 0.6 0.8 1 1109. 1589.
## 2 omega_gamma 0.2 0.2 0.03 0.15 0.25 1 725. 1464.
##
## Item parameters:
## # A tibble: 20 × 9
## variable mean median sd q5 q95 rhat ess_bulk ess_tail
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 nu[1] -0.65 -0.65 0.11 -0.82 -0.48 1 1486. 2497.
## 2 nu[2] -0.15 -0.15 0.11 -0.32 0.03 1 1605. 2279.
## 3 nu[3] -1.23 -1.23 0.13 -1.44 -1.03 1 1783. 2943.
## 4 nu[4] -1.7 -1.7 0.15 -1.94 -1.46 1 1705. 2632.
## 5 nu[5] -0.21 -0.21 0.11 -0.39 -0.03 1 1633. 2463.
## 6 nu[6] -1.3 -1.3 0.12 -1.51 -1.09 1 1733. 2594.
## 7 nu[7] -1.69 -1.69 0.15 -1.93 -1.45 1 1999. 2621.
## 8 nu[8] -1.91 -1.91 0.15 -2.16 -1.67 1 2038. 2553.
## 9 nu[9] -0.83 -0.83 0.11 -1 -0.66 1 1306. 2561.
## 10 nu[10] -1.42 -1.42 0.14 -1.65 -1.2 1 2113. 2612.
## 11 a[1] 0.44 0.44 0.02 0.41 0.47 1 1891. 2533.
## 12 a[2] 0.49 0.49 0.02 0.46 0.53 1 2143. 2732.
## 13 a[3] 0.5 0.49 0.02 0.46 0.54 1 1682. 3054.
## 14 a[4] 0.51 0.51 0.03 0.47 0.56 1 1508. 2357.
## 15 a[5] 0.51 0.51 0.02 0.48 0.55 1 1921. 2418.
## 16 a[6] 0.43 0.43 0.02 0.4 0.47 1 2117. 2981.
## 17 a[7] 0.4 0.4 0.02 0.37 0.45 1 2080. 2672.
## 18 a[8] 0.42 0.42 0.02 0.38 0.46 1 1905. 2425.
## 19 a[9] 0.35 0.35 0.02 0.32 0.38 1 2569. 2313.
## 20 a[10] 0.55 0.55 0.03 0.5 0.6 1 1779. 2337.
##
## Subject parameters:
## # A tibble: 429 × 9
## variable mean median sd q5 q95 rhat ess_bulk ess_tail
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 tnd[1] 0.37 0.36 0.1 0.22 0.54 1 4895. 3073.
## 2 tnd[2] 0.4 0.4 0.09 0.24 0.54 1 3672. 2885.
## 3 tnd[3] 0.46 0.47 0.12 0.27 0.65 1 3868. 3010.
## 4 tnd[4] 0.27 0.27 0.06 0.18 0.37 1 4644. 2926.
## 5 tnd[5] 0.39 0.39 0.08 0.26 0.51 1 4105. 3447.
## 6 tnd[6] 0.31 0.32 0.06 0.21 0.41 1 4428. 3376.
## 7 tnd[7] 0.39 0.39 0.09 0.24 0.53 1 4004. 2712.
## 8 tnd[8] 0.52 0.52 0.15 0.27 0.78 1 3288. 2695.
## 9 tnd[9] 0.34 0.34 0.07 0.21 0.46 1 4880. 2951.
## 10 tnd[10] 0.43 0.42 0.13 0.24 0.65 1 4587. 2806.
## # ℹ 419 more rows
##
## Other parameters:
## # A tibble: 429 × 9
## variable mean median sd q5 q95 rhat ess_bulk ess_tail
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 z_theta[1] -0.82 -0.82 0.36 -1.4 -0.24 1 3458. 2830.
## 2 z_theta[2] -0.3 -0.31 0.54 -1.17 0.62 1 5767. 2674.
## 3 z_theta[3] 0.4 0.39 0.54 -0.45 1.32 1 5866. 3116.
## 4 z_theta[4] 0.49 0.48 0.46 -0.26 1.26 1 4103. 3000.
## 5 z_theta[5] 1.47 1.48 0.67 0.4 2.59 1 5223. 3058.
## 6 z_theta[6] -0.07 -0.07 0.53 -0.96 0.8 1 5785. 3105.
## 7 z_theta[7] -0.07 -0.07 0.52 -0.91 0.8 1 5294. 2862.
## 8 z_theta[8] 0.33 0.32 0.56 -0.59 1.28 1 4870. 3072.
## 9 z_theta[9] 0.27 0.27 0.49 -0.53 1.07 1 5256. 2935.
## 10 z_theta[10] 0.37 0.36 0.46 -0.35 1.13 1 4306. 2974.
## # ℹ 419 more rows
The method checkDiagnostics provides common Stan diagnostics such as
number of divergeant transitions, R hat, effective sample size, maximum
treedepth hits, and E-BFMI.
checkDiagnostics(samples)## Stan diagnostics
## ----------------
## Chains: 4
## Divergences: 0
## Max treedepth hits: 0
## R-hat warnings: 0
## Low ESS warnings: 0
## E-BFMI warnings: 0
##
## Overall status: OK
Marcov chains of selected parameters can be visualized using the plot
method:
plot(samples, parameter = "omega_theta", type = "trace")
There is also the possibility to plot posterior means with 50% and 95% credible intervals as well as marginal posterior densities.
plot(samples, parameter = "theta", type = "interval")
plot(samples, parameter = "omega_theta",
type = "density")
The function fitBayesDiffIRT returns a BayesDiffIRTfit-object. The
samples from a BayesDiffIRTfit can be extracted using the
extractSamples method.
samplesDf <- extractSamples(samples)
head(samplesDf)## # A draws_df: 6 iterations, 1 chains, and 881 variables
## lp__ z_theta[1] z_theta[2] z_theta[3] z_theta[4] z_theta[5] z_theta[6]
## 1 -1729 -0.81 -0.37 0.84 0.309 0.51 0.994
## 2 -1740 -1.05 -0.25 0.34 0.444 2.60 -1.189
## 3 -1733 -0.57 -0.49 0.31 0.035 0.76 0.702
## 4 -1711 -0.76 0.18 1.17 0.665 1.80 -0.443
## 5 -1713 -1.06 -1.03 -0.29 0.071 2.16 -0.249
## 6 -1706 -0.96 -1.03 -0.16 0.735 2.37 -0.026
## z_theta[7]
## 1 -0.20
## 2 0.60
## 3 -0.50
## 4 0.61
## 5 -0.30
## 6 -0.71
## # ... with 873 more variables
## # ... hidden reserved variables {'.chain', '.iteration', '.draw'}
Should you really prefer working with point estimates, you can extract them conveniently with the coef method:
pointEstim <- coef(samples, parameter = "theta")
pointEstim ## theta[1] theta[2] theta[3] theta[4] theta[5] theta[6]
## -0.569520019 -0.209285584 0.277636153 0.336519997 1.023910705 -0.049829556
## theta[7] theta[8] theta[9] theta[10] theta[11] theta[12]
## -0.051630431 0.230456798 0.184561586 0.255965047 -0.035412350 0.347768629
## theta[13] theta[14] theta[15] theta[16] theta[17] theta[18]
## 0.291585559 -0.048192310 0.502675284 0.317207223 -0.112446303 -0.050642397
## theta[19] theta[20] theta[21] theta[22] theta[23] theta[24]
## -0.248443253 0.666722577 -1.109748113 1.226674588 0.454476878 0.089180505
## theta[25] theta[26] theta[27] theta[28] theta[29] theta[30]
## 0.129068258 -0.925067323 1.103761641 0.233593194 0.527437541 -0.780084583
## theta[31] theta[32] theta[33] theta[34] theta[35] theta[36]
## 0.428069266 -0.864653524 0.717329926 0.041588960 0.084318169 0.153425651
## theta[37] theta[38] theta[39] theta[40] theta[41] theta[42]
## -0.139378609 0.263625318 -0.037596780 0.336340478 -1.107169604 -0.371515273
## theta[43] theta[44] theta[45] theta[46] theta[47] theta[48]
## 0.736954869 0.403726323 0.638236034 -0.042707669 0.033432663 -0.283141711
## theta[49] theta[50] theta[51] theta[52] theta[53] theta[54]
## -0.625372419 0.411520380 0.407479429 -0.207486155 1.190459440 -0.046561946
## theta[55] theta[56] theta[57] theta[58] theta[59] theta[60]
## 0.005897704 1.159979454 0.406145197 0.423256696 -1.159252943 -1.048539675
## theta[61] theta[62] theta[63] theta[64] theta[65] theta[66]
## -1.100214038 -0.078644509 -0.482871591 1.120388319 -0.129883298 -0.695972741
## theta[67] theta[68] theta[69] theta[70] theta[71] theta[72]
## -0.711958675 0.163137250 0.892144985 0.792333848 -0.392710605 0.501243717
## theta[73] theta[74] theta[75] theta[76] theta[77] theta[78]
## -0.393733304 0.991091822 -0.138492906 -1.681032655 0.270416812 -0.684742756
## theta[79] theta[80] theta[81] theta[82] theta[83] theta[84]
## -0.266418176 -0.189490327 -0.313077470 -0.498725094 0.412683318 0.132127167
## theta[85] theta[86] theta[87] theta[88] theta[89] theta[90]
## 0.362474891 -0.088304637 0.042290239 -0.398056927 -0.194056580 0.283639296
## theta[91] theta[92] theta[93] theta[94] theta[95] theta[96]
## -0.025972026 -1.347483643 0.314232349 -0.374185652 -0.210352702 -0.494349631
## theta[97] theta[98] theta[99] theta[100] theta[101] theta[102]
## -0.136139282 -0.338163725 -0.941078150 -0.649550482 0.134793934 0.905775589
## theta[103] theta[104] theta[105] theta[106] theta[107] theta[108]
## -0.212478751 -0.322475715 0.008521914 0.088084597 0.379291563 -0.147629974
## theta[109] theta[110] theta[111] theta[112] theta[113] theta[114]
## 0.226490632 -0.788087726 -0.191515119 0.534028985 -0.578334130 1.058881166
## theta[115] theta[116] theta[117] theta[118] theta[119] theta[120]
## 0.041381601 0.042342203 -0.395949337 -0.422913056 0.029548343 0.171584846
## theta[121] theta[122] theta[123] theta[124] theta[125] theta[126]
## -1.015805697 0.802383888 0.523905988 -1.383991773 0.074167739 0.782436970
## theta[127] theta[128] theta[129] theta[130] theta[131] theta[132]
## 0.457817822 0.064473234 0.008655518 0.612048040 0.153635861 0.480023626
## theta[133] theta[134] theta[135] theta[136] theta[137] theta[138]
## 0.956621014 -0.669607382 0.173875118 0.374055617 -0.857826497 1.235726873
## theta[139] theta[140] theta[141] theta[142] theta[143]
## -0.334005857 -1.332447561 0.618582328 -0.187328022 -0.394417040
The posterior predictive distributions can be visualized using
ppCheck. Set type = “response” to visualize the predicted probability
of a correct response / item acceptance as a function of item or person.
yrep <- posteriorPredict(samples, ndraws=20)
ppCheck(samples, type = "response", yrep=yrep)
ppCheck(samples, type = "response", group = "item", yrep=yrep )
ppCheck(samples, type = "response", group = "person",
index=1:10, yrep=yrep)
Set type = “rtQuantile” to compares observed and posterior-predictive reaction-time quantiles:
ppCheck(samples, type = "rtQuantile")
According to drift diffusion item response theory, whether a person
solves a test item or accepts a survey item depends on two latent
variables, person ability
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The package is under active development. Please feel free to contact us to suggest diffusion item response theory models that we might have not yet implemented, or to volunteer adding additional features.
For comments, bug reports, and feature suggestions please feel free to either write to manuel.rausch@aau.at or submit an issue.
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Kang, Inhan, Paul De Boeck, and Roger Ratcliff. 2022. “Modeling Conditional Dependence of Response Accuracy and Response Time with the Diffusion Item Response Theory Model.” Psychometrika 87 (2): 725–48. https://doi.org/10.1007/s11336-021-09819-5.
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Ratcliff, Roger, Philip L Smith, Scott D Brown, and Gail McKoon. 2016. “Diffusion Decision Model : Current Issues and History.” Trends in Cognitive Sciences 20 (4): 260–81. https://doi.org/10.1016/j.tics.2016.01.007.
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Van Der Maas, Han L. J., Dylan Molenaar, Gunter Maris, Rogier A. Kievit, and Denny Borsboom. 2011. “Cognitive Psychology Meets Psychometric Theory: On the Relation Between Process Models for Decision Making and Latent Variable Models for Individual Differences.” Psychological Review 118 (2): 339–56. https://doi.org/10.1037/a0022749.