Logit Models in Small Samples

Talk given on September 16, 2015, at Texas A&M University.

Transcript

1. Logit Models in
   Small Samples
   Kelly McCaskey and Carlisle Rainey
   

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2. View Slide

3. How likely are the Kuwaitis to form
   an active resistance to the PGFK?
   What factors make this resistance
   more or less likely?
   

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4. View Slide

5. –Weisiger (2014)
   “A good understanding of when resistance is likely after
   conquest should influence leaders’ decisions about whether
   to fight in the first place, whether to accept compromise
   settlements prior to a final military victory, and when it may
   be worthwhile to encourage resistance among friendly
   conquered populations”
   

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6. Variable
   Standardized
   Coefficient Estimate
   90% Confidence
   Interval
   Coordinating Leader 0.44* [0.19, 0.69]
   Polity Score of Conqueror -0.27* [-0.55, 0.00]
   log(Distance) 0.47* [0.21, 0.74]
   Terrain 0.20 [-0.04, 0.45]
   Soldiers per Territory -0.13 [-0.41, 0.15]
   GDP per Capita -0.13 [-0.38, 0.12]
   Constant 0.40* [0.30, 0.50]
   N 35
   Events 14
   R2 0.61
   Linear Probability Model Estimates
   

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7. What is the probability of a Kuwaiti resistance
   in the aftermath of the Iraqi invasion?
   -0.22
   

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8. How many times more likely is a Kuwaiti
   resistance if they have a coordinating leader?
   -0.90
   

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9. a logit model
   What We Need
   

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10. How many times more likely is a Kuwaiti
    resistance if they have a coordinating leader?
    281.9
    

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11. 35 observations
    “It is risky to use ML with samples smaller than 100,
    while samples larger than 500 seem adequate.”
    –J. Scott Long
    

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12. The biostatistics literature recommends 10
    events per explanatory variable.
    2.3 EPEV
    

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13. ML estimates are
    severely biased
    in small samples.
    

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14. Pr(yi = 1) = logit
    1
    ( cons + 0.5x1 +
    k
    X
    j
    =2
    0.2xj)
    

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15. βcons = −1
    25%
    50%
    75%
    100%
    125%
    k = 9
    50 100 150 200
    Sample Size
    Percent Bias
    Percent Bias of the ML Estimator
    

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16. βcons = −1 βcons = −0.5 βcons = 0
    0%
    25%
    50%
    75%
    100%
    125%
    0%
    25%
    50%
    75%
    100%
    125%
    0%
    25%
    50%
    75%
    100%
    125%
    k = 3 k = 6 k = 9
    50 100 150 200 50 100 150 200 50 100 150 200
    Sample Size (N)
    Percent Bias
    Percent Bias of the ML Estimator
    

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17. a logit model
    What We Need
    a good estimator
    

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18. L⇤( |y) = L( |y)
    penalty
    z }| {
    |I( )|1
    2
    

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19. βcons = −1 βcons = −0.5 βcons = 0
    0%
    40%
    80%
    120%
    0%
    40%
    80%
    120%
    0%
    40%
    80%
    120%
    k = 3 k = 6 k = 9
    50 100 150 200 50 100 150 200 50 100 150 200
    Sample Size
    Percent Bias
    Method
    ML
    PML
    Percent Bias of ML and PML Estimators
    30 observations
    about 8 events
    120%
    7%
    

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20. But what about the
    variance?
    

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21. βcons = −1 βcons = −0.5 βcons = 0
    5%
    10%
    20%
    50%
    100%
    200%
    500%
    5%
    10%
    20%
    50%
    100%
    200%
    500%
    5%
    10%
    20%
    50%
    100%
    200%
    500%
    k = 3 k = 6 k = 9
    50 100 150 200 50 100 150 200 50 100 150 200
    Sample Size
    Variance Inflation
    Variance Inflation (%) of ML Relative to PML
    

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22. MSE(ˆ) = E[(ˆ true)2]
    = V ar(ˆ) + [E(ˆ)]2
    

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23. βcons = −1 βcons = −0.5 βcons = 0
    1%
    10%
    100%
    1,000%
    1%
    10%
    100%
    1,000%
    1%
    10%
    100%
    1,000%
    k = 3 k = 6 k = 9
    50 100 150 200 50 100 150 200 50 100 150 200
    Sample Size
    Mean−Squared Error Inflation
    Mean−Squared Error Inflation (%) of ML Relative to PML
    

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24. 0
    50
    100
    150
    50 100 150 200
    Sample Size
    Relative Contribution of Variance Compared to Bias
    Number of Variables
    3
    6
    9
    Intercept
    −1
    −0.5
    0
    The Relative Contribution of the Variance Compared
    to the Bias as the Sample Size Varies
    

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25. ●
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    N = 35 (14 events)
    Intercept
    Conqueror's Polity Score
    log(Intercapital Distance)
    Terrain
    Occupying Force Density
    Per Capita GDP
    Coordinating Leader
    −20 −10 0 10 20
    Logistic Regression Coefficients and 90% Confidence Intervals
    (Variables Standardized)
    Method
    ●
    ●
    ML Estimate
    PML Estimate
    Logistic Regression Model Explaining
    Post−Conflict Guerrilla War
    

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26. 0.0
    0.2
    0.4
    0.6
    Brier Score Log Score
    Score Type
    Score
    Method
    ML
    PML
    Out−of−Sample Prediction Scores
    

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27. How many times more likely is a Kuwaiti
    resistance if they have a coordinating leader?
    ML: 281.9
    PML: 18.5
    

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28. Practical Advice
    

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29. What is a
    “small sample”?
    

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30. ⇠ =
    1
    k
    min
    "
    n
    X
    i=1
    yi,
    n
    X
    i=1
    (1 yi)
    #
    

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    ξ = 51
    ξ = 12
    Slope Coefficient
    0.5%
    1.0%
    2.0%
    2.0%
    3.0%
    5.0%
    10.0%
    20.0%
    50.0%
    2 5 10 20 50 100 200 500
    ξ
    MSE−Inflation
    Sample Size
    ●
    ●
    1000
    2000
    −1
    0
    1
    True Coefficient
    MSE−Inflation of ML Relative to PML for the
    Slope Coefficients as the Information Increases
    

    View Slide

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    ξ = 96
    ξ = 33
    Intercept
    0%
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    200%
    2 5 10 20 50 100 200 500
    ξ
    MSE−Inflation
    Sample Size
    ●
    ●
    1000
    2000
    −2
    0
    2
    True Coefficient
    MSE−Inflation of ML Relative to PML for the
    Intercept as the Information Increases
    

    View Slide

33. Acceptable
    Inaccuracy
    Slope
    Copefficients
    Intercept
    Substantial ξ < 12 ξ < 33
    Noticeable 12 ≤ ξ < 51 33 ≤ ξ < 96
    Negligible ξ ≥ 51 ξ ≥ 96
    Some Thresholds
    

    View Slide

34. Acceptable
    Inaccuracy
    Slope
    Copefficients
    Intercept
    Substantial ξ < 12 ξ < 33
    Noticeable 12 ≤ ξ < 51 33 ≤ ξ < 96
    Negligible ξ ≥ 51 ξ ≥ 96
    Some Thresholds
    

    View Slide

35. Acceptable
    Inaccuracy
    Slope
    Copefficients
    Intercept
    Substantial ξ < 12 ξ < 33
    Noticeable 12 ≤ ξ < 51 33 ≤ ξ < 96
    Negligible ξ ≥ 51 ξ ≥ 96
    Some Thresholds
    

    View Slide

36. PML in Short
    • Always better in theory.
    • Easy to implement.
    • Makes a big difference in small samples.
    • Makes a small, but noticeable, difference in much
    larger samples.
    

    View Slide

37. 1. Choose the number of covariates k randomly from a uniform distribution
    from 3 to 12.
    2. Choose the sample size n randomly from a uniform distribution from 200
    to 3,000.
    3. Choose the intercept cons randomly from a uniform distribution from -4
    to 4.
    4. Choose the slope coefficients 1, ..., k randomly from a normal distribu-
    tion with mean 0 and standard deviation 0.5.
    5. Choose a covariance matrix
    ⌃
    for the explanatory variables randomly
    using the method developed by Joe (2006) such that the variances along
    the diagonal range from from 0.25 to 2.
    6. Choose the explanatory variables x1, x2, ..., xk randomly from a multivari-
    ate normal distribution with mean 0 and covariance matrix
    ⌃
    .
    Generating a Random DGP
    

    View Slide

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    ξ = 51
    ξ = 12
    Sample Size Between 200 and 500 Sample Size Between 501 and 1,000
    Sample Size Between 1,001 and 1,500 Sample Size Between 1,501 and 2,000
    0.5%
    1.0%
    2.0%
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    5.0%
    10.0%
    20.0%
    50.0%
    0.5%
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    50.0%
    2 5 10 20 50 100 200 500 2 5 10 20 50 100 200 500
    ξ
    MSE−Inflation
    Sample Size
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    1000
    2000
    −1
    0
    1
    True Coefficient
    MSE−Inflation of ML Relative to PML for the
    Slope Coefficients as the Information Increases
    

    View Slide

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    ξ = 96
    ξ = 33 ●
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    ξ = 96
    ξ = 33
    Sample Size Between 200 and 500 Sample Size Between 501 and 1,000
    Sample Size Between 1,001 and 1,500 Sample Size Between 1,501 and 2,000
    0%
    1%
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    5%
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    50%
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    200%
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    2 5 10 20 50 100 200 500 2 5 10 20 50 100 200 500
    ξ
    MSE−Inflation
    Sample Size
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    −2
    0
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    True Coefficient
    MSE−Inflation of ML Relative to PML for the
    Intercept as the Information Increases
    

    View Slide