improve beta

.gitignore

@@ -83,3 +83,4 @@ Assets/scraps.cs.meta
 
 content/_book
 Manifest.toml
+activate.jl

Project.toml

@@ -7,4 +7,6 @@ GLMakie = "e9467ef8-e4e7-5192-8a1a-b1aee30e663a"
 Makie = "ee78f7c6-11fb-53f2-987a-cfe4a2b5a57a"
 QuartoNotebookRunner = "4c0109c6-14e9-4c88-93f0-2b974d3468f4"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
-Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"
+SequentialSamplingModels = "0e71a2a6-2b30-4447-8742-d083a85e82d1"
+SubjectiveScalesModels = "da3460ee-0a2f-4d2a-8d76-316f36bde5da"
+Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"

content/.quarto/cites/index.json

@@ -1 +1 @@
-{"3a_scales.qmd":["makowski2023novel"],"5_individual.qmd":[],"references.qmd":[],"3b_choices.qmd":["wagenmakers2008diffusion","heathcote2012linear","theriault2024check"],"4_rt.qmd":["wagenmakers2008diffusion","heathcote2012linear","theriault2024check","lo2015transform","schramm2019reaction","balota2011moving","matzke2009psychological","hohle1965inferred","kieffaber2006switch","matzke2009psychological","schwarz2001ex","heathcote2004fitting","anders2016shifted"],"2_predictors.qmd":[],"4a_rt_descriptive.qmd":["wagenmakers2008diffusion","heathcote2012linear","theriault2024check","wiemann2023using","lo2015transform","schramm2019reaction","wagenmakers2005relation","limpert2001log","balota2011moving","matzke2009psychological","hohle1965inferred","kieffaber2006switch","matzke2009psychological","schwarz2001ex","heathcote2004fitting","anders2016shifted"],"1_introduction.qmd":[],"3_scales.qmd":["makowski2023novel"],"4b_rt_generative.qmd":[],"index.qmd":[],"4_1_Normal.qmd":["wagenmakers2008diffusion","theriault2024check","lo2015transform","schramm2019reaction"]}
+{"2_predictors.qmd":[],"4_rt.qmd":["wagenmakers2008diffusion","heathcote2012linear","theriault2024check","lo2015transform","schramm2019reaction","balota2011moving","matzke2009psychological","hohle1965inferred","kieffaber2006switch","matzke2009psychological","schwarz2001ex","heathcote2004fitting","anders2016shifted"],"3_scales.qmd":["makowski2023novel"],"references.qmd":[],"index.qmd":[],"5_individual.qmd":[],"3b_choices.qmd":["wagenmakers2008diffusion","heathcote2012linear","theriault2024check"],"4_1_Normal.qmd":["wagenmakers2008diffusion","theriault2024check","lo2015transform","schramm2019reaction"],"4b_rt_generative.qmd":[],"3a_scales.qmd":["makowski2023novel"],"1_introduction.qmd":[],"4a_rt_descriptive.qmd":["wagenmakers2008diffusion","heathcote2012linear","theriault2024check","wiemann2023using","lo2015transform","schramm2019reaction","wagenmakers2005relation","limpert2001log","balota2011moving","matzke2009psychological","hohle1965inferred","kieffaber2006switch","matzke2009psychological","schwarz2001ex","heathcote2004fitting","anders2016shifted"]}

content/.quarto/xref/15f266d2

@@ -1 +1 @@
-{"options":{"chapters":true},"headings":["brief-intro-to-julia-and-turing","installing-julia-and-packages","julia-basics","generate-data-from-normal-distribution","recover-distribution-parameters-with-turing","bayesian-linear-models","hierarchical-models"],"entries":[]}
+{"headings":["brief-intro-to-julia-and-turing","installing-julia-and-packages","julia-basics","generate-data-from-normal-distribution","recover-distribution-parameters-with-turing","bayesian-linear-models","hierarchical-models"],"options":{"chapters":true},"entries":[]}

content/.quarto/xref/1a47137c

@@ -1 +1 @@
-{"entries":[],"options":{"chapters":true},"headings":["extracting-individual-parameters-from-mixed-models","population-informed-individual-bayesian-models","task-reliability"]}
+{"entries":[],"headings":["extracting-individual-parameters-from-mixed-models","population-informed-individual-bayesian-models","task-reliability"],"options":{"chapters":true}}

content/.quarto/xref/26afb962

@@ -1 +1 @@
-{"headings":["categorical-predictors-condition-group","interactions","ordered-predictors-likert-scales","non-linear-relationships","polynomials","generalized-additive-models-gams"],"options":{"chapters":true},"entries":[]}
+{"options":{"chapters":true},"entries":[],"headings":["categorical-predictors-condition-group","interactions","ordered-predictors-likert-scales","monotonic-effects","non-linear-relationships","polynomials","generalized-additive-models-gams"]}

content/.quarto/xref/26e6880e

@@ -1 +1 @@
-{"headings":["the-data","gaussian-aka-linear-model","model-specification","posterior-predictive-check","scaled-gaussian-model","solution-1-directional-effect-of-condition","solution-2-avoid-exploring-negative-variance-values","solution-3-use-a-softplus-function","exponential-transformation","softplus-function","the-model","conclusion","the-problem-with-linear-models","shifted-lognormal-model","prior-on-minimum-rt","model-specification-1","interpretation","exgaussian-model","conditional-tau-tau-parameter","interpretation-1","shifted-wald-model","model-specification-2","model-comparison","shifted-lognormal-mixed-model"],"entries":[],"options":{"chapters":true}}
+{"headings":["the-data","gaussian-aka-linear-model","model-specification","posterior-predictive-check","scaled-gaussian-model","solution-1-directional-effect-of-condition","solution-2-avoid-exploring-negative-variance-values","solution-3-use-a-softplus-function","exponential-transformation","softplus-function","the-model","conclusion","the-problem-with-linear-models","shifted-lognormal-model","prior-on-minimum-rt","model-specification-1","interpretation","exgaussian-model","conditional-tau-tau-parameter","interpretation-1","shifted-wald-model","model-specification-2","model-comparison","shifted-lognormal-mixed-model"],"options":{"chapters":true},"entries":[]}

content/.quarto/xref/6cbe9151

@@ -1 +1 @@
-{"entries":[],"options":{"chapters":true},"headings":["preface","why-julia","why-bayesian","looking-for-coauthors"]}
+{"headings":["preface","why-julia","why-bayesian","looking-for-coauthors"],"entries":[],"options":{"chapters":true}}

content/.quarto/xref/a408ff3e

@@ -1 +1 @@
-{"options":{"chapters":true},"entries":[],"headings":["evidence-accumulation","wald-distribution-revisited","drift-diffusion-model-ddm","linear-ballistic-accumulator-lba","other-models-lnr-rdm","including-random-effects","random-intercept","random-slopes","performance-tips","additional-resources"]}
+{"entries":[],"headings":["evidence-accumulation","wald-distribution-revisited","drift-diffusion-model-ddm","linear-ballistic-accumulator-lba","other-models-lnr-rdm","including-random-effects","random-intercept","random-slopes","performance-tips","additional-resources"],"options":{"chapters":true}}

content/2_predictors.qmd

@@ -27,14 +27,119 @@ Likert scales, i.e., ordered multiple *discrete* choices are often used in surve
 
 > The probabilities assigned to discrete probability descriptors are not necessarily equidistant (https://github.com/zonination/perceptions)
 
+### Monotonic Effects
+
 What can we do to better reflect the cognitive process underlying a Likert scale responses? [Monotonic effects](https://cran.r-project.org/web/packages/brms/vignettes/brms_monotonic.html).
 
 - How to do Monotonic effects in Turing?
 
+```
+library(brms)
+
+m <- brm(mpg ~ mo(gear), data = mtcars, refresh=0)
+
+# stancode(m)
+m
+```
+```
+ Family: gaussian 
+  Links: mu = identity; sigma = identity 
+Formula: mpg ~ mo(gear) 
+   Data: mtcars (Number of observations: 32) 
+  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
+         total post-warmup draws = 4000
+
+Regression Coefficients:
+          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
+Intercept    16.51      1.31    14.01    19.19 1.00     2106     2213
+mogear        3.88      1.02     1.81     5.86 1.00     2160     2315
+
+Monotonic Simplex Parameters:
+           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
+mogear1[1]     0.84      0.13     0.52     0.99 1.00     2698     1862
+mogear1[2]     0.16      0.13     0.01     0.48 1.00     2698     1862
+
+Further Distributional Parameters:
+      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
+sigma     5.02      0.68     3.90     6.58 1.00     2850     2485
+
+Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
+and Tail_ESS are effective sample size measures, and Rhat is the potential
+scale reduction factor on split chains (at convergence, Rhat = 1).
+```
+
+
+```{julia}
+#| code-fold: false
+
+# using Downloads, CSV, DataFrames, Random
+# using Turing, Distributions
+# using GLMakie
+# using LinearAlgebra
+
+# # Define the monotonic function
+# function monotonic(scale::Vector{Float64}, i::Int)
+#     if i == 0
+#         return 0.0
+#     else
+#         return length(scale) * sum(scale[1:i])
+#     end
+# end
+
+# # Test
+# using RDatasets
+# data = dataset("datasets", "mtcars")
+
+# # Response variable
+# y = data[!, :MPG]
+# x = data[!, :Gear]
+
+# # Number of observations
+# N = length(y)
+
+# # Length of simplex
+# Jmo = maximum(x)
+
+# # Prior concentration for the simplex (using a uniform prior for simplicity)
+# con_simo_1 = ones(Jmo)
+
+# # Turing Model Specification
+# @model function monotonic_model(y, x, N, Jmo, con_simo_1)
+#     # Parameters
+#     Intercept ~ Normal(19.2, 5.4)
+#     simo_1 ~ Dirichlet(con_simo_1)
+#     sigma ~ truncated(Normal(0, 5.4), 0, Inf)
+#     bsp ~ Normal(0, 1)
+    
+#     # Linear predictor
+#     mu = Vector{Float64}(undef, N)
+#     for n in 1:N
+#         mu[n] = Intercept + bsp * monotonic(simo_1, x[n])
+#     end
+    
+#     # Likelihood
+#     y ~ MvNormal(mu, sigma)
+# end
+
+# # Run the model
+# model = monotonic_model(y, x, N, Jmo, con_simo_1)
+# chain = sample(model, NUTS(), 1000)
+
+# # Display the results
+# display(chain)
+```
+
+
+
 ## Non-linear Relationships
 
 ![](https://img.shields.io/badge/status-good_for_contributing-blue)
 
+It is a common misconception that "linear models" can only model straight relationships between variables.
+In fact, the "linear" part refers to the relationship between the parameters (between the outcome varialbe and the predictors).
+
+
+
 ```{julia}
 #| code-fold: false