This project analyzes the relationship between low Plasma $\beta$-carotene which is categorical variable and other 11 variables using logistic regression models. We first build one logistic regression model between low Plasma $\beta$-carotene and vitamin use and one between Plasma $\beta$-carotene and bmi. Then we build a multiple logistic regression model and make a variable selection. After that we compare the several multiple models we get and find out two best model with criterion function AIC and BIC, then we make residual analysis and perform a goodness-of-fit test. Finally we find the "best" model and make a conclusion with it.

We will investigate whether the plasma $\beta$-carotene concentration is low or not and model the probability of having a low concentration dependent on dietary and/or background factors. To do this, we require an appropriate cut-off value. One that effectively differentiates between those at high and low risk for developing specific medical conditions will be used.

The cut-off value is $0.42 \mu$mol/l (micromoles per liter). Unfortunately, our data is from 1989, when the concentrations were measured in nanograms per milliliter (ng/ml).