Logistic Regression – Interpreting Model with Negative Coefficients and Odds Ratios <1

likelihoodlogisticodds-ratio

I'm doing logistic regression where X has four factors (1-4) and Y has two factors (0-1). I did:

model=glm(formula=y~x,family=binomial(link=logit),data=data) #x=='1' is reference level
summary(model)


I get:

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)                -1.6740     0.3632  -4.608 4.06e-06 ***
2                          -1.0916     0.5000  -2.183    0.029 *
3                          -1.9369     1.0766  -1.799    0.072 .
4                          -16.8921  1058.1118  -0.016    0.987


Then for my odds ratios, I get:

                          ORs 2.5 %       97.5 %
(Intercept)              0.19  0.09 3.600000e-01
2                        0.34  0.12 9.100000e-01
3                        0.14  0.01 8.200000e-01
4                        0.00    NA 1.057042e+21


I'm most interested in variable 2 and I'm confused how I would interpret its odds ratio. I would think that an OR<1 would mean a lesser likelihood of outcome Y=1, but seeing how the log-likelihood coefficient for the intercept/variable 1 is less than variable 2. I'm confused if the OR for var2 would still be considered 'lower' than var1.

Any help is really appreciated. Thank you!

I think you have a desired result in mind (var2 lowers the probability of $$Y=1$$ than var1) and are perplexed by the numbers. The intercept $$\beta_0$$ in logistic regression should be interpreted carefully since $$e^{\beta_0}$$ is not an odds ratio. Observe this: \begin{align} \beta_0 &= \mathrm{logit}(P(Y=1\mid \text{var2}=0, \text{var3}=0))\\ e^{\beta_0} &= \frac{P(Y=1\mid \text{var2}=0, \text{var3}=0)}{1-P(Y=1\mid \text{var2}=0, \text{var3}=0)}. \end{align} But the slopes are different: \begin{align} \beta_1 &= \mathrm{logit}(P(Y=1\mid \text{var2}=1,\text{var3}=0)) - \mathrm{logit}(P(Y=1\mid \text{var2}=0,\text{var3}=0))\\ &= \mathrm{logit}(P(Y=1\mid \text{var2}=1,\text{var3}=0)) - \beta_0\\ e^{\beta_1} &= \frac{P(Y=1\mid \text{var2}=1,\text{var3}=0)/P(Y=0\mid \text{var2}=1,\text{var3}=0)}{P(Y=1\mid \text{var2}=0,\text{var3}=0)/P(Y=0\mid \text{var2}=0,\text{var3}=0)}. \end{align} The interpretation of slopes is always relative to the baseline (i.e., the intercept).