Dealing with Separation in Logistic Regression Models

Presented on January 9 at the 2016 Annual Meeting of the Southern Political Science Association in San Juan, Puerto Rico.

Transcript

1. 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
   

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2. The prior matters a lot,
   so choose a good one.
   

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3. The prior matters a lot,
   so choose a good one.
   1. in practice
   2. in theory
   3. concepts
   4. software
   

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4. The Prior Matters
   in Practice
   

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5. politics need
   

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6. 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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7. 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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10. Different default priors
    produce different results.
    

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11. The Prior Matters
    in Theory
    

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12. For
    1. a monotonic likelihood
    p(y| )
    decreasing in s,
    2. a proper prior distribution
    p( | )
    , and
    3. a large, negative s,
    the posterior distribution of s is proportional to the prior distribution for s, so
    that
    p( s
    |y) / p( s
    | )
    .
    

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13. For
    1. a monotonic likelihood
    p(y| )
    decreasing in s,
    2. a proper prior distribution
    p( | )
    , and
    3. a large, negative s,
    the posterior distribution of s is proportional to the prior distribution for s, so
    that
    p( s
    |y) / p( s
    | )
    .
    

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14. The prior determines
    crucial parts of the posterior.
    

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15. Key Concepts
    for Choosing a Good Prior
    

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16. Pr
    (
    yi) = ⇤( c + ssi + 1xi1 +
    ...
    + kxik)
    

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17. Transforming the Prior
    Distribution
    ˜ ⇠ p( )
    ˜
    ⇡new = p(ynew
    |˜)
    ˜
    qnew = q(˜
    ⇡new)
    

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18. We Already Know Few Things
    1
    ⇡ ˆmle
    1
    2
    ⇡ ˆmle
    2
    .
    .
    .
    k
    ⇡ ˆmle
    k
    s < 0
    

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19. Partial Prior Distribution
    p⇤( | s < 0, s = ˆmle
    s
    ),
    where ˆmle
    s
    = 1
    

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20. The Pacifying Effects of Nuclear Weapons
    

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21. Software
    for Choosing a Good Prior
    

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22. separation
    (on GitHub)
    

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23. Conclusion
    

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24. The prior matters a lot,
    so choose a good one.
    

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25. 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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