1. Causation does not imply correlation directly
2. Causation implies conditional correlation 

Why Multiple Regression can Cause Spurious Correlation

Spurious correlation is when two variables appear to be directly influencing each other when in reality they are not. Usually this misconception is caused by a confounding variable. DAGs (directed acyclic graphs) can help visualize which variables may influence each other and indicate if there is a possibility for a spurious correlation.  

Spurious Correlation

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

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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. Reveal spurious correlation
2. Uncover hidden association

Benefits of Multiple Regression

Shortcomings of Multiple Regression

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

Example: Marriage and Divorce

1. Nominate the predictor variables 
2. Make a parameter that will measure the association with the outcome for each predictor 
3. Multiply the parameter with variable and add them together

Steps of Applying Multiple Regression Model

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.

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