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? \[\begin{align} {μ_i = α + βx_i} \tag{1}\\ μ_i = β_xx_i + β_zz_i \tag{2} \\ μ_i = α + β(x_i − z_i) \tag{3} \\ μ_i = α + β_xx_i + β_zz_i \tag{4} \\ \end{align}\]
#2, 3, 4, 5 are the multiple linear regressions5-2. 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.
#μ_i = β_aA_i + β_pP_i
#A is animal diversity and P is plant diversity.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?
#ui=α+βFFi+βSSi
#F= amount of funding S=size of laboratory From the question, we can tell BF and BS are positive5-4. 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. \[\begin{align} μ_i = α + β_AA_i + β_BB_i + β_DD_i \tag{1}\\ μ_i = α + β_AA_i + β_BB_i + β_CC_i + β_DD_i \tag{2}\\ μ_i = α + β_BB_i + β_CC_i + β_DD_i \tag{3}\\ μ_i = α_AA_i + α_BB_i + α_CC_i + α_DD_i \tag{4}\\ μ_i = α_A(1 − B_i − C_i − D_i) + α_BB_i + α_CC_i + α_DD_i \tag{5}\\ \end{align}\]
# The first model consists of a single intercept (for category C) and slopes for categories A, B, and D.
# The second model is unidentifiable due to the fact that it has a slope for every potential category.
# The third model contains one intercept (for category A) and slopes for categories B, C, and D.
# The fourth model employs a distinct intercept for each category based on a unique index (and no slopes).
# The fifth model use the reparameterized technique described on pages 154 and 155 to multiply the intercept for category A by 1 when it falls under category A and by 0 otherwise.
# Hence, models 1, 3, 4, and 5 are inferentially identical since their posterior distributions can be calculated from one another and only number two cannot be computed from others5-5. 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.
# A variable representing the amount of time spent exercising might be an acceptable starting point for the first method. For the second mechanism, a variable related to the frequency of restaurant dining would be an appropriate starting point. We may become even more sophisticated by attempting to monitor things such as calories expended during exercise and calories consumed in restaurants. Consequently, I would suggest the following model of multiple regression:
#μi=α+βGGi+βEEi+βRRi
## When G represents the price of gasoline, E represents a variable linked to exercise, and R represents a variable associated to restaurants. In one version of this model, self-reported frequency of exercise and dining out may be used, but in another version, calories expended via exercise and calories consumed from restaurants may be quantified more precisely.