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GLM submission for the Humans of Julia
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#= | |
Motivation: I want to model psychometric data to some kind of sigmoid function, but the nature | |
of psychometric data is that it is forced choice. This means that the output is binary. | |
GLM doesn't support forced choice models yet——however, with a little bit of work and some | |
type piracy, the results are attractive. | |
=# | |
using GLM, DataFrames, CSV, FreqTables, StatsPlots, Distributions, Statistics | |
const Dist = Distributions | |
# imports for overloading | |
import GLM: Link, Link01, linkfun, linkinv, mueta, inverselink | |
# obtain dataset | |
df_raw = DataFrame(CSV.File(download("https://gist.githubusercontent.com/jakewilliami/7a84208f33de8076c7b17abb52a9b242/raw/b27d63700f4072830251eb4063e790c27c3154d0/mcd.csv"))) | |
# statistical notations | |
Φ(x::Real) = Dist.cdf(Dist.Normal(), x) | |
Φ⁻¹(x::Real) = Dist.quantile(Dist.Normal(), x) | |
φ(x::Real) = Dist.pdf(Dist.Normal(), x) | |
# define a new link function for 2-alternative forced choice | |
struct Probit2AFCLink <: Link end | |
# overloading | |
linkfun(::Probit2AFCLink, μ::Real) = Φ⁻¹(2 * max(μ, nextfloat(0.5)) - 1) | |
linkinv(::Probit2AFCLink, η::Real) = (1 + Φ(η)) / 2 | |
mueta(::Probit2AFCLink, η::Real) = φ(η) / 2 | |
function inverselink(::Probit2AFCLink, η::Real) | |
μ = (1 + Φ(η)) / 2 | |
d = φ(η) / 2 | |
return μ, d, oftype(μ, NaN) | |
end | |
# define models | |
logit(p) = log(p / (1 - p)) | |
logit⁻¹(α) = 1 / (1 + exp(-α)) | |
logit⁻¹(α) = logit⁻¹(α) * 0.5 + 0.5 # shift and squish for 2-AFC | |
logit2afc⁻¹(α) = 0.5 + 0.5 / (1 + exp(-α)) | |
probit(p) = Dist.quantile(Dist.Normal(), p) | |
probit⁻¹(x) = Dist.cdf(Dist.Normal(), x) | |
Φ⁻¹(z) = probit(z) # statistical notation | |
Φ(α) = probit⁻¹(α) # statistical notation | |
probit2afc⁻¹(x) = probit⁻¹(x) * 0.5 + 0.5 # shift and squish for 2-AFC | |
df_pivot = combine(groupby(df_raw, :condition1)) do df | |
μ = mean(df.correct) | |
n = nrow(df) | |
( | |
correct_mean = μ, | |
n = n, | |
k = μ * n | |
) | |
end | |
model = glm(@formula(correct ~ condition1) , df_raw, Binomial(), Probit2AFCLink()) | |
a, b = coef(model) | |
theme(:solarized) | |
plot = @df df_pivot scatter( | |
:condition1, | |
:correct_mean, | |
title = "Psychometric Curve of Motion Coherence", | |
label = false, | |
xaxis = "Coherence", | |
yaxis = "Accuracy", | |
fontfamily = font("Times") | |
) | |
# add models to graph | |
plot!(plot, x -> probit2afc⁻¹(a + b*x), 0, 0.32, label = "Probit Link") | |
plot!(plot, x -> logit2afc⁻¹(a + b*x), 0, 0.32, label = "Logit Link") |
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