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.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them.
Finally, upon completion, name your final output .html
file as: YourName_ANLY505-Year-Semester.html and publish
the assignment to your R Pubs account and submit the link to Canvas.
Each question is worth 5 points.
5-1. Which of the linear models below are multiple linear regressions? \[\begin{align} {μ_i = α + βx_i} \tag{1}\\ μ_i = β_xx_i + β_zz_i \tag{2} \\ μ_i = α + β(x_i − z_i) \tag{3} \\ μ_i = α + β_xx_i + β_zz_i \tag{4} \\ \end{align}\]
# Model 2, 3, 4 are multiple linear regressions.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.
# μ_i = α + β1*Xi + β2*Yi
# Xi is the amount of funding and Yi is the size of laboratory
# Both β1 and β2 should be positive5-3. 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 higher divorce rate can cause a higher marriage rate because the people who get divorced can remarry later, which increases the marriage rate. We can evaluate this relationship by including the divorce rate and remarriage rate as predictors in a multiple regression with the marriage rate being the dependent variable.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. \[\begin{align} μ_i = α + β_AA_i + β_BB_i + β_DD_i \tag{1}\\ μ_i = α + β_AA_i + β_BB_i + β_CC_i + β_DD_i \tag{2}\\ μ_i = α + β_BB_i + β_CC_i + β_DD_i \tag{3}\\ μ_i = α_AA_i + α_BB_i + α_CC_i + α_DD_i \tag{4}\\ μ_i = α_A(1 − B_i − C_i − D_i) + α_BB_i + α_CC_i + α_DD_i \tag{5}\\ \end{align}\]
# Model 1, 3, 4, 5 are inferentially equivalent ways to include the categorical variable in a regression.
# Model 1 includes a single intercept for category C and slopes for categories A, B, and D.
# Model 2 includes all categories A, B, C, D and an intercept α.
# Model 3 includes a single intercept for category A and slopes for categories B, C, and D.
# Model 4 includes unique intercept for each category.
# Model 5 uses the reparameterized technique to multiply the intercept for category A by 1 when it falls under category A and by 0 otherwise.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.
# μi= α + β1*Xi+ β2*Yi+ β3*Zi
# Xi is the the price of gasoline; Yi is the amount of time spent on exercise; Zi is the number of eating out.
# Yi is used to address the first mechanism related to time spent on exercise. Zi is used to address the second mechanism related to the frequency of eating out at restaurant.