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Binomial logistic regression is a statistical technique that allows you to model the probability of a binary (yes/no) variable, according to other variables (predictors). Logistic regression is used very frequently in medical research to find predictive factors. For example, if someone wants to predict risk factors for cardiovascular disease in a population, they can use multivariate logistic regression.
A disease is hardly predictable by only one factor. Many variables may influence the occurrence of a disease: genetics, environmental exposure, way of life, tobacco use, etc. If you want to take into account all the parameters concomitantly to explain the disease you are studying, simple statistical analyses will not be sufficient. You will need to perform a multiple regression. In a multiple regression, we study the simultaneous effect of several predictors on one variable we want to predict.
Both terms are often used interchangeably in the medical domain. However, they are truly different and the difference is easy to understand. A multiple (multivariable) regression is the method used to model one variable according to several other variables. For example, modeling the 5-year survival of a patient according to age, BMI, disease stage, etc. Multivariate analysis also models the relation between variables. However, the outcome you want to model is measured for the same individual at multiple time points (repeated measures). So instead of one variable to predict (multivariable analysis or multiple regression), you actually have multiple variables to predict (multivariate analysis).