This chapter introduced multiple regression, a way of constructing descriptive models for how the mean of a measurement is associated with more than one predictor variable. The defining question of multiple regression is: What is the value of knowing each predictor, once we already know the other predictors? The answer to this question does not by itself provide any causal information. Causal inference requires additional assumptions. Simple directed acyclic graph (DAG) models of causation are one way to represent those assumptions.
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5-1. Which of the linear models below are multiple linear regressions? \[\begin{align} {μ_i = α + βx_i} \tag{1}\\ μ_i = β_xx_i + β_zz_i \tag{2} \\ μ_i = α + β(x_i − z_i) \tag{3} \\ μ_i = α + β_xx_i + β_zz_i \tag{4} \\ \end{align}\]
# Based on the linear models above, Model 2 and Model 4 are multiple linear regressions. This is because both of the models have more than one predictor variables. Model 2 and Model 4 both have β_x and β_z, while Model 1 and Model 3 only have β.5-2. Write down a multiple regression to evaluate the claim: Neither amount of funding nor size of laboratory is by itself a good predictor of time to PhD degree; but together these variables are both positively associated with time to degree. Write down the model definition and indicate which side of zero each slope parameter should be on.
# D_i ~ Normal(μ_i, σ)
# μ_i = α + β_F*F_i + β_L*L_i
# Where D = Time to PhD degree
# F = Funding
# L = Laboratory Size
# Note that both slopes β_F and β_L should both be positive based on the claim above.5-3. It is sometimes observed that the best predictor of fire risk is the presence of firefighters— States and localities with many firefighters also have more fires. Presumably firefighters do not cause fires. Nevertheless, this is not a spurious correlation. Instead fires cause firefighters. Consider the same reversal of causal inference in the context of the divorce and marriage data. How might a high divorce rate cause a higher marriage rate? Can you think of a way to evaluate this relationship, using multiple regression?
# A high divorce rate can also cause a higher marriage rate because presumably people who are divorce are now single and are looking to connect and would likely enter another marriage (since they've done it before). This relationship can be evaluated using multiple regression by looking at marriage rate as a function of divorce rate and re-marriage rate, where re-marriage rate is marriage after divorce rate.5-4. Suppose you have a single categorical predictor with 4 levels (unique values), labeled A, B, C and D. Let Ai be an indicator variable that is 1 where case i is in category A. Also suppose Bi, Ci, and Di for the other categories. Now which of the following linear models are inferentially equivalent ways to include the categorical variable in a regression? Models are inferentially equivalent when it’s possible to compute one posterior distribution from the posterior distribution of another model. \[\begin{align} μ_i = α + β_AA_i + β_BB_i + β_DD_i \tag{1}\\ μ_i = α + β_AA_i + β_BB_i + β_CC_i + β_DD_i \tag{2}\\ μ_i = α + β_BB_i + β_CC_i + β_DD_i \tag{3}\\ μ_i = α_AA_i + α_BB_i + α_CC_i + α_DD_i \tag{4}\\ μ_i = α_A(1 − B_i − C_i − D_i) + α_BB_i + α_CC_i + α_DD_i \tag{5}\\ \end{align}\]
# Assuming that we have a flat priors for all the models, Model 1, Model 3, Model 4, and Model 5 are inferentially equivalent. This is because these models will yield the same predictions, and are convertible amongst each other post model fitting. Additionally, all of these Models also contains the same number of parameters in the multiple regression equation.5-5. One way to reason through multiple causation hypotheses is to imagine detailed mechanisms through which predictor variables may influence outcomes. For example, it is sometimes argued that the price of gasoline (predictor variable) is positively associated with lower obesity rates (outcome variable). However, there are at least two important mechanisms by which the price of gas could reduce obesity. First, it could lead to less driving and therefore more exercise. Second, it could lead to less driving, which leads to less eating out, which leads to less consumption of huge restaurant meals. Can you outline one or more multiple regressions that address these two mechanisms? Assume you can have any predictor data you need.
# Task: Outline how Price of Gasoline (PG) lowers Obesity Rates (OR)
#
# First Mechanism
# PG lowers OR by reducing the Driving Frequency (DF) and increasing Exercise Frequency (EF)
# The following chain of regression could follow:
# DF decreases as PG increases
# EF decreases as DF increases
# OR decreases as EF increases
#
# Each of the predictors above is negatively associated with each outcome
#
# Second Mechanism
# PG lowers OR by reducing Driving Frequency (DF) and reducing Eating Out Frequency (OF)
# The following chain of regression could follow:
# DF decreases as PG increases
# OF decreases as DF decreases
# OR decreases as OF decreases
#
# In terms of a regression model, we could include both mechanism 1 and mechanism 2 into one model.
# OR_i ~ Normal(μ_i, σ)
# μ_i = α + β_PG*PG_i + β_EF*EF_i + β_OF*OF_i
# Where OR = Obesity Rate
# PG = Price of Gasoline
# EF = Exercise Related Predictors (perhaps kcal burn can be use)
# OF = Eating Our Related Predictors (perhaps kcal can intake be use)