1. Case Replacement for Logistic Regression
Replace cases with null hypothesis cases.
What is a null hypothesis case? One in which probability of success is independent of predictor
 You switch some treatment success cases to treatment failure case

2. Neighborhood Effects: Even Logistic Leverages Product of Associations
Odds ratio relating treatment to outcome
Odds ratio relating omitted variable to outcome
Increments of Γ matter more as Δ increases. Correlation between Γ Δ and odd ratio is -.96
Odds ratio relating omitted variable to treatment
Harding, D. J. (2003). Counterfactual models of neighborhood effects: The effect of neighborhood poverty on dropping out and teenage pregnancy. American Journal of Sociology, 109(3),

3. Replacement of Cases for Logistic: Toy Example
Failure
Success
Total
Control 16 9 25
Treatment 4 21 20 30 50
Odds ratio=16*21/(4*9)=9.3
Odds ratio= ln(odds ratio)= se=.686 =(1/16+1/9+1/4+1/21)
Threshold =t critical*.686=1.96*.686=1.345
% bias to invalidate= /2.2335=40%

4. Replacement of Cases for Logistic = Switching Cases: Toy Example
Failure
Success
Total
Control 16 9 25
Treatment 4 21 20 30 50 7 18
New odds ratio=16*18/(9*7)=4.57 23 27
How many treatment success must you replace with null hypothesis (p=30/50=.6) to invalidate the inference
 How many to switch from treatment success to treatment failure:
% bias to invalidate= /2.2335=40%
Replace 40% of treatment successes (n=21)=8 cases
Replace with null hypothesis cases (p of success=.6).
So switch 8*(1-.6): =about 3.2 cases from treatment success to treatment failure.
# of cases to switch=% bias to invalidate*(treatment success cases)*(1-overall probability of success).

5. Replacement of Cases for Logistic: Toy Example
Failure
Success
Total
Control 16 9 25
Treatment 4 21 20 30 50
Failure
Success
Total
Control
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Treatment
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ

6. Replacement of Cases for Logistic: Toy Example
Failure
Success
Total
Control
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Treatment
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ
Ꙫ Ꙫ Ꙫ
Replace 8 cases = switch 3 cases
Ln(p/(1-p)=β0 +β1 treatment.
H0:β1 =0 Ln(p/(1-p)=β0 p=.6

7. R code brute force
setwd("C:/Users/user/Dropbox (Personal)/sensitivity for logistics") rm(list = ls()) A <- 29 B <- 26 C <- 15 D <- 40 x <- matrix(c(A,B,C,D), byrow = TRUE, 2, 2) # this is the 2 by 2 table we start with p.CD <- p.value <- chisq.test(x,correct = FALSE)$p.value N.CD <- 0 while ( p.value <0.05 ) { C <- C + 1 D <- D - 1 N.CD <- N.CD + 1 print(x) print(chisq.test(x,correct = FALSE)) p.value <- chisq.test(x,correct = FALSE)$p.value p.CD <- c(p.CD, chisq.test(x,correct = FALSE)$p.value) } p.AB <- p.value <- chisq.test(x,correct = FALSE)$p.value N.AB <- 0 A <- A - 1 B <- B + 1 N.AB <- N.AB + 1 p.AB <- c(p.AB, chisq.test(x,correct = FALSE)$p.value) # so p.AB and p.CD record the p values each time we switch one case # N.AB = 5 (5 cases needed to be changed from A to B) # N.CD = 4 (4 cases needed to be changed from D to C)

8. KonFound-it for Logistic Regression

11. Reflection What part if most confusing to you?
Why?
More than one interpretation?
Talk with one other, share
Find new partner and problems and solutions