Dealing with Separation in Logistic Regression Models Carlisle Rainey Assistant Professor University at Buffalo, SUNY [email protected] paper, data, and code at crain.co/research
“Obamacare is going to be horrible for patients. It’s going to be horrible for taxpayers. It’s probably the biggest job killer ever.” — October 2010 “While the federal government is committed to paying 100 percent of the cost, I cannot, in good conscience, deny Floridians that need it access to healthcare.” — February 2013
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
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 | ) .
1. Choose a prior distribution p( s) . 2. Estimate the model coefficients ˆmle . 3. For i in 1 to nsims, do the following: (a) Simulate ˜[i] s ⇠ p( s) . (b) Replace ˆmle s in ˆmle with ˜[i] s , yielding the vector ˜[i] . (c) Calculate and store the quantity of interest ˜ q[i] = q ⇣ ˜[i] ⌘ . 4. Keep only the simulations in the direction of the separation. 5. Summarize the simulations ˜ q using quantiles, histograms, or density plots. 6. If the prior is inadequate, then update the prior distribution p( s) .
plot(qi, xlim = c(-1, 1), xlab = "First Difference", ylab = "Posterior Density", main = "The Effect of Democratic Partisanship on Opposing the Expansion")
plot(qi, x, xlab = "Percent Uninsured (Std.)", ylab = "Predicted Probability", main = "The Probability of Opposition as the Percent Uninsured (Std.) Varies")
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.
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 | ) .
Theorem 1. For a monotonic likelihood p(y| ) increasing [decreasing] in s, proper prior distribution p( | ) , and large positive [negative] s, the posterior distribution of s is proportional to the prior distribution for s, so that p( s |y) / p( s | ) .
Proof. Due to separation, p(y| ) is monotonic increasing in s to a limit L , so that lim s !1 p(y| s ) = L . By Bayes’ rule, p( |y) = p(y| )p( | ) 1 R 1 p(y| )p( | )d = p(y| )p( | ) p(y| ) | {z } constant w.r.t. . Integrating out the other parameters s = h cons , 1, 2, ..., k i to obtain the posterior distribution of s, p( s |y) = 1 R 1 p(y| )p( | )d s p(y| ) , (1) and the prior distribution of s, p( s | ) = 1 Z 1 p( | )d s . Notice that p( s |y) / p( s | ) iff p( s |y) p( | ) = k , where the constant k 6= 0 .Thus,
p( s | ) = 1 Z 1 p( | )d s . Notice that p( s |y) / p( s | ) iff p( s |y) p( s | ) = k , where the constant k 6= 0 .Thus, Theorem 1 implies that lim s !1 p( s |y) p( s | ) = k Substituting in Equation 1, lim s !1 1 R 1 p ( y | ) p ( | ) d s p ( y | ) p( s | ) = k. Multiplying both sides by p(y| ) , which is constant with respect to , lim s !1 1 R 1 p(y| )p( | )d s p( s | ) = kp(y| ). Setting 1 R p(y| )p( | )d s = p(y| s )p( s | ) ,
s !1 p( s | ) Substituting in Equation 1, lim s !1 1 R 1 p ( y | ) p ( | ) d s p ( y | ) p( s | ) = k. Multiplying both sides by p(y| ) , which is constant with respect to , lim s !1 1 R 1 p(y| )p( | )d s p( s | ) = kp(y| ). Setting 1 R 1 p(y| )p( | )d s = p(y| s )p( s | ) , lim s !1 p(y| s )p( s | ) p( s | ) = kp(y| ). Canceling p( s | ) in the numerator and denominator, lim s !1 p(y| s ) = kp(y| ).
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