1. Reveal spurious correlation
2. Uncover hidden association

Ohio State University, Columbus

Multiple regression model looks like the polynomial model which was introduced before but we will dive into it, and use this method to figure out "correlation". 
Multiple regression have more independent varialbles, or explanatory variables, to determine the output (the dependent variable)

Multiple Regression Model 

McElreath, R. (2016). Statistical rethinking: A Bayesian course with examples in R and Stan. Page 152 


Multiple Regression Model

1. Possible to cause spurious correlation
2. Hide real associations

Shortcomings of Multiple Regression

Q: Does marriage cause divorce?
What are the variables that cause divorce?

Example: Marriage and Divorce

Three good interpretative plot models:
1) Predictor residual plots. These plots show the outcome against residual predictor values. They are useful for understanding the statistical model, but not much else.
2) Posterior prediction plots. These show model-based predictions against raw data, or otherwise display the error in prediction. They are tools for checking fit and assessing predictions. They are not causal tools. 
3) Counterfactual plots. These show the implied predictions for imaginary experiments. These plots allow you to explore the causal implications of manipulating one or more variables.


Plotting multivariate posteriors.

A categorical variable is a type of variable that represents a characteristic or quality, typically assessed by assigning a descriptive category label to each individual rather than a number. Examples include a person's chosen major, nationality, occupation, or whether they are receiving psychotherapy.

Categorical Variable

Applying a multiple regression model involves three main steps:

1. Nominate the predictor variables, represented as $$X_1, X_2, \dots, X_n$$.
2. Define a parameter (slope coefficient) $$\beta_j$$ for each predictor variable $$X_j$$ to measure its association with the outcome variable $$Y$$.
3. Multiply each predictor variable by its corresponding parameter and sum them together (along with an intercept parameter $$\alpha$$) to define the model's linear predictor:
$$\mu_i = \alpha + \sum_{j=1}^n \beta_j X_{j,i}$$

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