# Odds ratio with given third covariate

Call:
glm(formula = cbind(yes, no) ~ Defendant + Victim, family = binomial())

Deviance Residuals:
1        2        3        4
-0.1269   0.6778   0.1752  -0.2136

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)     -2.6943     0.3913  -6.885 5.76e-12 ***
DefendantWhite  -0.2955     0.4018  -0.735   0.4621
VictimWhite      1.0824     0.4976   2.175   0.0296 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 6.01225  on 3  degrees of freedom
Residual deviance: 0.55178  on 1  degrees of freedom
AIC: 20.762

Number of Fisher Scoring iterations: 4


The response describes wheather an offender receives a death penalty (yes or no).

I now want to calculate the relative "chance" for a black defendant receiving the death penalty when the victim was white vs. black just by using results of the R-output. Thanks for helping me.

• Just to note something: with 2 covariates, an intercept and only 4 points any inference from this sample is spurious; I would not trust it at all. You have next to zero degrees of freedom. As an extremely coarse rule of thumb I would suggest you have at least 10 times more data-points than covariates. – usεr11852 says Reinstate Monic Feb 4 '17 at 20:48

From your model, the odds ratio for a defendant receiving the death penalty, whatever he's race is, when the victim was white vs. black is $\exp(1.0824)$, which is statistically significant ($\alpha = 0.05$). This means that, based on your model, when the victim is a white, the defendant of receiving death penalty is 8% more possible than when the victim is a black.

I think your model cannot answer the question of "relative chance" for a black defendant receiving the death penalty when the victim is a white of a black. You should include an interaction term between defendant race and victim race by running following command in R,

glm(formula = cbind(yes, no) ~ Defendant + Victim + Defendant:Victim, family = binomial())


Then you can calculate the odd radio of a black defendant receiving the death penalty when the victim was white vs. black AND a white defendant receiving the death penalty when the victim was white vs. black. Then the radio of these two odds ratios can answer your question.

• Thank you for your help. But I am not sure if you got my concern, maybe I reformulate it: I want to calculate the odds ratio of response and Victim under the condition that the defendant is black. Is this possible with the output of this model or do I have to proceed like you said above? – lemontree Feb 4 '17 at 21:30
• I still think you need to include interaction term. An interaction model is a model where the interpretation of the effect of X1 depends on the value of X2 and vice versa. – WCMC Feb 4 '17 at 22:16
• How does the odds ratio look like with an interaction term? – lemontree Feb 4 '17 at 23:40
• For example, logit(Y) ~ a + bX1 + cX2 + dX1*X2. X1 is defendant race, 0 being white and 1 being black, while X2 is victim, 0 being white and 1 being black. Thus when the defendant is black, the odds ratio of victim is $\exp(c+d)$, while when the defendant is white, the odds ratio of victim is $\exp(c)$. Then there is a difference of odds ratios between defendant race, if the interaction term is significant. – WCMC Feb 5 '17 at 0:14