Chapter 5 - Many Variables and Spurious Waffles

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. Problems are labeled Easy (E), Medium (M), and Hard(H).

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.

Questions

5E1. 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.

5E2. Write down a multiple regression to evaluate the claim: Animal diversity is linearly related to latitude, but only after controlling for plant diversity. You just need to write down the model definition.

# μ_i = α + β_A A_i + β_P P_i
# where A is animal diversity and P is plant diversity.

5E3. 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 + β_F F_i + β_S S_i
# where F is amount of funding and S is size of laboratory.

5E4. 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 because they each allow the computation of each other's posterior distribution.

5M1. Invent your own example of a spurious correlation. An outcome variable should be correlated with both predictor variables. But when both predictors are entered in the same model, the correlation between the outcome and one of the predictors should mostly vanish (or at least be greatly reduced).

# Invent an example of a spurious correlation between the number of operas seen and score on a standardized achievement test that vanishes when family income is included.
N <- 100
income <- rnorm(n = 100, mean = 0, sd = 1)
operas <- rnorm(n = N, mean = income, sd = 2)
scores <- rnorm(n = N, mean = income, sd = 1)
d <- data.frame(scores, operas, income)
pairs(d)

m <- map(
  alist(
    scores ~ dnorm(mu, sigma),
    mu <- a + bo * operas,
    a ~ dnorm(0, 5),
    bo ~ dnorm(0, 5),
    sigma ~ dunif(0, 5)
  ),
  data = d
)
precis(m)
##            mean         sd       5.5%     94.5%
## a     0.0155318 0.12675067 -0.1870403 0.2181038
## bo    0.2284198 0.05313160  0.1435052 0.3133344
## sigma 1.2674487 0.08962184  1.1242157 1.4106817
m <- map(
  alist(
    scores ~ dnorm(mu, sigma),
    mu <- a + bo * operas + bi * income,
    a ~ dnorm(0, 5),
    bo ~ dnorm(0, 5),
    bi ~ dnorm(0, 5),
    sigma ~ dunif(0, 5)
  ),
  data = d
)
precis(m)
##               mean         sd        5.5%      94.5%
## a     -0.003932419 0.09523315 -0.15613338 0.14826854
## bo     0.026086460 0.04605201 -0.04751354 0.09968646
## bi     1.026332630 0.11677406  0.83970512 1.21296014
## sigma  0.951898721 0.06730884  0.84432619 1.05947125
# So, in this model, family income predicts scores on standardized achievement tests when the number of operas seen is already known, but the number of operas seen does not predict scores on standardized achievement tests when family income is already known. Thus, the bivariate association between test scores and exposure to opera is spurious.

5M2. Invent your own example of a masked relationship. An outcome variable should be correlated with both predictor variables, but in opposite directions. And the two predictor variables should be correlated with one another.

# Invent an example of a masked relationship involving the prediction of happiness ratings from the amount of alcohol one drinks and the amount of time one spends feeling ill.

N <- 100
rho <- 0.6
alcohol <- rnorm(n = N, mean = 0, sd = 1)
illness <- rnorm(n = N, mean = rho * alcohol, sd = sqrt(1 - rho^2))
happiness <- rnorm(n = N, mean = alcohol - illness, sd = 1)
d <- data.frame(happiness, alcohol, illness)
pairs(d)

m <- map(
  alist(
    happiness ~ dnorm(mu, sigma),
    mu <- a + ba * alcohol,
    a ~ dnorm(0, 5),
    ba ~ dnorm(0, 5),
    sigma ~ dunif(0, 5)
  ),
  data = d
)
precis(m)
##            mean         sd        5.5%     94.5%
## a     0.1183827 0.12671446 -0.08413153 0.3208968
## ba    0.5016359 0.12236901  0.30606659 0.6972052
## sigma 1.2643617 0.08940373  1.12147723 1.4072461
m <- map(
  alist(
    happiness ~ dnorm(mu, sigma),
    mu <- a + bi * illness,
    a ~ dnorm(0, 5),
    bi ~ dnorm(0, 5),
    sigma ~ dunif(0, 5)
  ),
  data = d
)
precis(m)
##             mean         sd       5.5%      94.5%
## a      0.0634390 0.13288706 -0.1489402  0.2758182
## bi    -0.3217253 0.13150255 -0.5318918 -0.1115588
## sigma  1.3273024 0.09385035  1.1773114  1.4772934
m <- map(
  alist(
    happiness ~ dnorm(mu, sigma),
    mu <- a + ba * alcohol + bi * illness,
    a ~ dnorm(0, 5),
    ba ~ dnorm(0, 5),
    bi ~ dnorm(0, 5),
    sigma ~ dunif(0, 5)
  ),
  data = d
)
precis(m)
##             mean         sd       5.5%      94.5%
## a      0.1052145 0.10008590 -0.0547421  0.2651711
## ba     1.0439571 0.11922262  0.8534163  1.2344979
## bi    -0.9478041 0.12205035 -1.1428641 -0.7527440
## sigma  0.9983954 0.07059702  0.8855677  1.1112231
# Indeed, the slopes for alcohol and illness became much larger in magnitude in the multivariate model. Thus, in this example (and maybe not in real life), because alcohol increases happiness and feelings of illness, and feelings of illness decrease happiness, the bivariate relationships of alcohol and feelings of illness to happiness are masked.

5M3. 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?

# We might hypothesize that a high divorce rate causes a higher marriage rate by introducing more unmarried individuals (who have a demonstrated willingness to marry) into the dating pool. This possibility could be evaluated using multiple regression by regressing marriage rate on both divorce rate and re-marriage rate (i.e., the rate of non-first marriages or marriages following divorces). If divorce rate no longer predicts marriage rate even when the re-marriage rate is known, this would support our hypothesis.

5M4. In the divorce data, States with high numbers of members of the Church of Jesus Christ of Latter-day Saints (LDS) have much lower divorce rates than the regression models expected. Find a list of LDS population by State and use those numbers as a predictor variable, predicting divorce rate using marriage rate, median age at marriage, and percent LDS population (possibly standardized). You may want to consider transformations of the raw percent LDS variable.

d <- WaffleDivorce
d$LDS <- c(0.0077, 0.0453, 0.0610, 0.0104, 0.0194, 0.0270, 0.0044, 0.0057, 0.0041, 0.0075, 0.0082, 0.0520, 0.2623, 0.0045, 0.0067, 0.0090, 0.0130, 0.0079, 0.0064, 0.0082, 0.0072, 0.0040, 0.0045, 0.0059, 0.0073, 0.0116, 0.0480, 0.0130, 0.0065, 0.0037, 0.0333, 0.0041, 0.0084, 0.0149, 0.0053, 0.0122, 0.0372, 0.0040, 0.0039, 0.0081, 0.0122, 0.0076, 0.0125, 0.6739, 0.0074, 0.0113, 0.0390, 0.0093, 0.0046, 0.1161)
d$logLDS <- log(d$LDS)
d$logLDS.s <- (d$logLDS - mean(d$logLDS)) / sd(d$logLDS)
simplehist(d$LDS)

simplehist(d$logLDS)

simplehist(d$logLDS.s)

m <- map(
  alist(
    Divorce ~ dnorm(mu, sigma),
    mu <- a + bm * Marriage + ba * MedianAgeMarriage + bl * logLDS.s,
    a ~ dnorm(10, 20),
    bm ~ dnorm(0, 10),
    ba ~ dnorm(0, 10),
    bl ~ dnorm(0, 10),
    sigma ~ dunif(0, 5)
  ),
  data = d
)
precis(m)
##              mean         sd        5.5%      94.5%
## a     35.43351219 6.77535513 24.60518610 46.2618383
## bm     0.05341479 0.08261835 -0.07862529  0.1854549
## ba    -1.02948947 0.22469659 -1.38859802 -0.6703809
## bl    -0.60801551 0.29057031 -1.07240298 -0.1436280
## sigma  1.37872427 0.13838892  1.15755204  1.5998965
# In this model, the slope of marriage was widely variable and its interval includes zero. However, the slopes of both median age at marriage and percentage of LDS population were negative and their intervals did not include zero. Thus, states with older median age at marriage or higher percentages of Mormons had lower divorce rates.

5M5. 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.

# μ_i = α + β_G G_i + β_E E_i + β_R R_i
# where 𝐺 represents the price of gasoline, 𝐸 represents one exercise-related variable, and 𝑅 represents one restaurant-related variable. One version of this model might use self-reported frequencies of exercise and eating out, and another version might use more rigorously measured calories burned through exercise and calories ingested from restaurants.