The goal of this assignment is to allow you to get more experience writing the equations that define your model. Writing these equations correctly (including all the appropriate subscripts) is critical to properly and succinctly communicate your ideas to reviewers and other scientists. As a result, for all problems below, please remember to define each variable and each subscript that you use.

1. We are interested in understanding the risk factors of malaria (equal to 1 if infected, zero otherwise). We collected data using a clustered survey, where we first randomly select a village and then randomly select 10 people to be tested for malaria. I am particularly interested in understanding how malaria prevalence (i.e., the probability of being infected) is influenced by distance to the nearest health facility. To understand this relationship, I will rely on a logistic regression. Please write the equations (likelihood and priors) that describe this model, assuming that:

a - we just have a fixed-effect intercept and a fixed-effect slope

b - we have a village-level random-effect intercept and a fixed-effect slope

2. We are interested in understanding the effect of an educational intervention on stats anxiety (measured as a questionnaire score) on students. To this end, we conduct an experiment in which some students receive the intervention (i.e., treatment group), given in the middle of the semester, and others do not (i.e., control group). Furthermore, we survey the students in the beginning and at the end of the semester in 10 different schools. We will model the end-of-semester anxiety score using a Gaussian regression, in which the covariates consist of the educational intervention (a binary variable equal to zero and one for the control and treatment groups), the beginning-of-semester anxiety score, and an interaction term between these two variables. Please write the equations (likelihood and priors) that describe this model, assuming that:

a - we just have a fixed-effect intercept and fixed-effect slopes

b - we have a fixed-effect intercept that varies from school to school and fixed-effect slopes

3. We are interested in understanding the movement patterns of birds. Our landscape contains 30 habitat patches. The number of movements between each pair of sites is modeled using a Poisson regression, where the covariates are the distance between these sites, density of individuals at the originating site, and area of the destiny site. Please write the equations (likelihood and priors) that describe this model, assuming fixed-effects for all parameters.