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? \[\begin{align} {μ_i = α + βx_i} \tag{1}\\ μ_i = β_xx_i + β_zz_i \tag{2} \\ μ_i = α + β(x_i − z_i) \tag{3} \\ μ_i = α + β_xx_i + β_zz_i \tag{4} \\ \end{align}\]
# Numbers 2 and 4 are multiple regressions. Number 1 contains only one predictor variable. Number 3, although two variables appear in the model, also only uses the difference of x and z as a single predictor. Only numbers 2 and 4 contain multiple predictor variables5-2. 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.
# We will denote time to PhD degree as T, amount of funding as F, and size of laboratory as L. The model is then defined as:
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# Ti∼Normal(μi,σ)
# μi=α+βFFi+βLLi
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# Both βFand βL should be positive. In order for both funding and lab size to show a positive relationship together but no relationship separately, these two variables would need to be negatively correlated (i.e., large labs have less funding per student and small labs have more funding per student). Both could be positively associated with our outcome, but be negatively associated in the real world. Thus, analyzing the variables in isolation may mask the positive relationship.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 means that there are more people available in the population of single-people that are available to marry. Additionally, people may be getting divorced for the specific purpose of marrying someone else. To evaluate this, we could add marriage number, or a “re-marry” indicator. We would then expect the coefficient for marriage rate to get closer to zero once this predictor is added to the model5-4. 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. \[\begin{align} μ_i = α + β_AA_i + β_BB_i + β_DD_i \tag{1}\\ μ_i = α + β_AA_i + β_BB_i + β_CC_i + β_DD_i \tag{2}\\ μ_i = α + β_BB_i + β_CC_i + β_DD_i \tag{3}\\ μ_i = α_AA_i + α_BB_i + α_CC_i + α_DD_i \tag{4}\\ μ_i = α_A(1 − B_i − C_i − D_i) + α_BB_i + α_CC_i + α_DD_i \tag{5}\\ \end{align}\]
# Numbers 1, 3, 4, and 5 are all inferentially equivalent. Numbers 1 and 3 both use 3 of the 4 indicator variables, meaning that you can always calculate the 4th from the three that are estimated and the intercept. Number 4 is equivalent to an index variable approach, which is inferentially equivalent to the indicator variable approach. Finally, Number 5 is mathematically equivalent to Number 4, assuming that each observation can belong to only 1 of the 4 groups.
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# Number 2 is not a valid model representation, as the model should only contain
# k−1indicator variables with an intercept (when using an indicator variable approach). Because all 4 are included in the definition of Number 2, we should expect to have estimation problems (if the model will estimate at all)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.
# Let’s assume we have four variables: rate of obesity (O), price of gasoline (G), amount of driving (D), amount of exercise (E), and the rate of eating out at restaurants (R). Given these variables, we can outline the first proposed mechanism implies:
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# 1.)D is negatively associated with P
# 2.)E is negatively associated with D
# 3.)O is negatively associated with E
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# This is a chain of causality, where each outcome is the input of the next chain. The second mechanism is similar:
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# 1.)D is negatively associated with P
# 2.)R is positively associated with D
# 3.)O is positively associated with R