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Dealing with Separation in Logistic Regression Models

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.

Carlisle Rainey

January 09, 2016
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  1. Dealing with Separation in
    Logistic Regression Models
    Carlisle Rainey
    Assistant Professor
    Texas A&M University
    [email protected]
    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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