Estimating Distributional Parameters in Hierarchical Models - - PowerPoint PPT Presentation

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Estimating Distributional Parameters in Hierarchical Models - - PowerPoint PPT Presentation

Jan 14, 2024 105 likes β€’577 views

Estimating Distributional Parameters in Hierarchical Models Introduction: Variability in Hierarchical Models Linear Models = 0 + 1 + (0, 2 ) Modelling central

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Estimating Distributional Parameters in Hierarchical Models

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Introduction: Variability in Hierarchical Models

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Linear Models

  • Modelling central tendency
  • Response (π‘§π‘—π‘˜) is a sum of intercept (𝛾0), slopes (𝛾1, 𝛾2, …), and error (π‘“π‘—π‘˜)
  • Error is assumed to be normally distributed around zero

π‘§π‘—π‘˜ = 𝛾0 + 𝛾1π‘Œπ‘—π‘˜ + π‘“π‘—π‘˜ π‘“π‘—π‘˜ ∼ 𝑂(0, Οƒ2)

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Linear Models

  • Modelling central tendency
  • Response (y) is a sum of intercept (implicit), slopes (pred), and error

(implicit)

  • Error is assumed to be normally distributed around zero

lm(y ~ pred)

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Linear Mixed Effects Models

  • Modelling central tendency
  • Response (π‘§π‘—π‘˜) is a sum of intercept (𝛾0), slopes (𝛾1, 𝛾2, …), random unit

intercepts (ΞΌ0𝑗), random unit slopes (ΞΌ1𝑗), and error (π‘“π‘—π‘˜)

  • Error, random intercepts, and random slopes are assumed to be normally

distributed around zero

π‘§π‘—π‘˜ = 𝛾0 + ΞΌ0𝑗 + (𝛾1 + ΞΌ1𝑗)π‘Œπ‘—π‘˜ + π‘“π‘—π‘˜ ΞΌ0𝑗 ∼ 𝑂(0, Οƒ2) ΞΌ1𝑗 ∼ 𝑂(0, Οƒ2) π‘“π‘—π‘˜ ∼ 𝑂(0, Οƒ2)

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Linear Mixed Effects Models

  • Modelling central tendency
  • Response (y) is a sum of intercept (implicit), slopes (pred), random unit

intercepts (pred || rand_unit), random unit slopes (pred | rand_unit), and error (implicit)

  • Error, random intercepts, and random slopes are assumed to be normally

distributed around zero

lmer(y ~ pred + (pred | rand_unit))

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Example Non-Gaussian Data: RT

  • 2AFC: does the word match the picture?
  • Congruency (2) x Predictability (12% – 100%)
  • 35 Subjects, 200 trials

+ bandage sardine

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Gamma Family GLMM

m_glmer <- glmer( rt ~ cong * pred + (cong * pred | subj) + (cong | image) + (1 | word), family = Gamma(identity), control = glmerControl(

  • ptimizer = β€œbobyqa”,
  • ptCtrl = list(maxfun = 2e5)

) )

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GLMM Results

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GLMM Results – Random Effects

summary(m_glmer)

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GLMM Results – Random Effects

ranef(m_glmer)

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GLMM Results – Random Effects

m_glmer %>% ranef() %>% as.data.frame()

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GLMM Results – Random Effects

ranef(m_glmer) %>% as_tibble() %>% filter(grpvar == β€œsubj") %>% mutate(grp = fct_reorder2(grp, term, condval)) %>% ggplot(aes( x = grp, y = condval, ymin = condval - condsd, ymax = condval + condsd )) + geom_pointrange(size=0.25) + facet_wrap(vars(term), scales="free", nrow=2)

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GLMM Results – Random Effects – Subject

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GLMM Results – Random Effects – Image

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GLMM Results – Random Effects – Word

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Estimating Distributional Parameters in Hierarchical Models

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What if Meaningful Effects on Variance?

  • All glm variants model single parameters

(i.e. central tendency)

  • What if your effect looks like this?
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What if Meaningful Effects on Variance?

  • Mu is higher F(1, 1998) = 3237, p<.001
  • Sigma is higher Levene’s F(1, 1998) = 550, p<.001
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Assumption-free Distribution Comparison

  • Within a single model?
  • Assumption free distribution comparison (e.g.

Kolmogorov–Smirnov) could be one approach!

  • Overlapping index (Pastore & Calcagni, 2019)

from 0 (no overlap) to 1 (identical distribution)

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Assumption-free Distribution Comparison

x <- rnorm(1000, 10, 1), y <- rnorm(1000, 10.5, 1.5)

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Assumption-free Distribution Comparison

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Overlap Index Mu * Sigma Parameter Space

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Overlap Index Mu * Sigma Parameter Space

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Overlap Index Mu * Sigma Parameter Space

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Weirder Distribution Example

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Weirder Distribution Example

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Summary so far

  • Assumption-free approaches are flexible but don’t allow

us to test/make any specific predictions

  • Equivalent of shrugging and saying β€œyeah idk probs

something going on there” (though useful for very weird distributions)

  • Explicitly modelling multiple parameters of an assumed

distribution can give us more meaningful info

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Distributional Parameters in brms

brm( bf( dv ~ Intercept + iv + (iv | rand_unit), sigma ~ Intercept + iv + (iv | rand_unit) ), control = list( adapt_delta = 0.999, max_treedepth = 12 ), sample_all_pars = TRUE )

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Shifted Log-Normal Distribution

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Shifted Log-Normal Distribution

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Bayesian Shifted Log-Normal Mixed Effects Model with Distributional Parameters

brms::bf( rt ~ Intercept + cong * pred + (cong * pred | subj) + (cong | image) + (1 | word), sigma ~ rt ~ Intercept + cong * pred + (cong * pred | subj) + (cong | image) + (1 | word), ndt ~ rt ~ Intercept + cong * pred + (cong * pred | subj) + (cong | image) + (1 | word) )

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ranef(m_bme)

Bayesian Results – Random Effects

ID (e.g. subj_01, subj_02…) * value (est, err, Q2.5, Q97.5) * fixed parameter

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Caveats

  • Computationally intensive if using non-

informative priors for complex hierarchical formulae

  • Have to avoid temptation to try over-infer about

mechanisms unless using more cognitively informed models (e.g. drift diffusion)

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Summary

Hierarchical models with maximal structures for distributional parameters are a robust and appropriate way of looking at or accounting for subject/item/etc variability in fixed effects when you’re interested in more than central tendency. But, if you can assume no systematic differences in distributional parameters, GLMMs will suffice (and save you a lot of time and effort)!