A random intercept logistic regression is performed to assess the association between $Y$: Disease (Yes/No) and Standardized Predictor($X_1$) adjusting for control variables ($X_2$, $X_3$) based on clustered survey data. A $X_1^2$ term is considered in the model to explore the nonlinear relationship. Results:
coef p-value
intercept 0.240 <0.001
X1 0.285 <0.01
I(X1)^2 -0.084 <0.01
X2 0.114 <0.05
X3 0.210 0.345
I'm trying to interpret the $X_1$ and $X_1^2$ as follows: "A unit increase in $X_1$ (standardized) is associated with $exp(0.285)$ higher odds of disease when $X_1$ (standardized) is at its mean, each additional level of $X_1$ is associated with $exp(-0.084)$ decrease in the likelihood of disease." Is this appropriate? Does anybody have any thoughts on this?
Thank you.