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Dealing with Separation in Logistic Regression Models
Carlisle Rainey Assistant Professor Texas A&M University crainey@tamu.edu
paper, data, and code at crain.co/research
Dealing with Separation in Logistic Regression Models Carlisle - - PowerPoint PPT Presentation
Dealing with Separation in Logistic Regression Models Carlisle - - PowerPoint PPT Presentation
Dealing with Separation in Logistic Regression Models Carlisle Rainey Assistant Professor Texas A&M University crainey@tamu.edu paper, data, and code at crain.co/research The prior matters a lot, so choose a good one. The prior matters
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The prior matters a lot, so choose a good one.
- 1. in practice
- 2. in theory
- 3. concepts
- 4. software
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The Prior Matters
in Practice
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politics need
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Variable Coefficient Confidence Interval Democratic Governor
- 26.35
[-126,979.03; 126,926.33] % Uninsured (Std.) 0.92 [-3.46; 5.30] % Favorable to ACA 0.01 [-0.17; 0.18] GOP Legislature 2.43 [-0.47; 5.33] Fiscal Health 0.00 [-0.02; 0.02] Medicaid Multiplier
- 0.32
[-2.45; 1.80] % Non-white 0.05 [-0.12; 0.21] % Metropolitan
- 0.08
[-0.17; 0.02] Constant 2.58 [-7.02; 12.18]
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Variable Coefficient Confidence Interval Democratic Governor
- 26.35
[-126,979.03; 126,926.33] % Uninsured (Std.) 0.92 [-3.46; 5.30] % Favorable to ACA 0.01 [-0.17; 0.18] GOP Legislature 2.43 [-0.47; 5.33] Fiscal Health 0.00 [-0.02; 0.02] Medicaid Multiplier
- 0.32
[-2.45; 1.80] % Non-white 0.05 [-0.12; 0.21] % Metropolitan
- 0.08
[-0.17; 0.02] Constant 2.58 [-7.02; 12.18]
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Variable Coefficient Confidence Interval Democratic Governor
- 26.35
[-126,979.03; 126,926.33] % Uninsured (Std.) 0.92 [-3.46; 5.30] % Favorable to ACA 0.01 [-0.17; 0.18] GOP Legislature 2.43 [-0.47; 5.33] Fiscal Health 0.00 [-0.02; 0.02] Medicaid Multiplier
- 0.32
[-2.45; 1.80] % Non-white 0.05 [-0.12; 0.21] % Metropolitan
- 0.08
[-0.17; 0.02] Constant 2.58 [-7.02; 12.18]
This is a failure of maximum likelihood.
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Different default priors produce different results.
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The Prior Matters
in Theory
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For
- 1. a monotonic likelihood p(y|β) decreasing in βs,
- 2. a proper prior distribution p(β|σ), and
- 3. a large, negative βs,
the posterior distribution of βs is proportional to the prior distribution for βs, so that p(βs|y) ∝ p(βs|σ).
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For
- 1. a monotonic likelihood p(y|β) decreasing in βs,
- 2. a proper prior distribution p(β|σ), and
- 3. a large, negative βs,
the posterior distribution of βs is proportional to the prior distribution for βs, so that p(βs|y) ∝ p(βs|σ).
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The prior determines crucial parts of the posterior.
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Key Concepts
for Choosing a Good Prior
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Pr(yi) = Λ(βc + βssi + β1xi1 + ... + βkxik)
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Transforming the Prior Distribution
˜ β ∼ p(β) ˜ πnew = p(ynew|˜ β) ˜ qnew = q(˜ πnew)
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We Already Know Few Things
β1 ≈ ˆ βmle
1
β2 ≈ ˆ βmle
2
. . . βk ≈ ˆ βmle
k
βs < 0
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Partial Prior Distribution
p∗(β|βs < 0, β−s = ˆ βmle
−s ),
where ˆ βmle
s
= −∞
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Software
for Choosing a Good Prior
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separation
(on GitHub)
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Stan Project
rstanarm
StataStan
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Conclusion
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The prior matters a lot, so choose a good one.
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What should you do?
- 1. Notice the problem and do something.
- 2. Recognize the the prior affects the inferences
and choose a good one.
- 3. Assess the robustness of your conclusions to a
range of prior distributions.
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