statisticalrethinkin..

MEDIUM 1695.6. Summaryis chapter introduced multiple regression, a way of constructing descriptive modelsfor how the mean of a measurement is associated with more than one predictor variable.e defining question of multiple regression is: What is the value of knowing each predictor,once we already know the other predictors? Implicit in this question are: (1) the value ofthe predictors for description of the sample, instead of forecasting a future sample, and (2)the assumption that the value of each predictor does depend upon the values of the otherpredictors. In the next two chapters, we confront these two issues.5.7. PracticeEasy5.7.1. e1. Which of the linear models below are multiple linear regressions?(1) µ i = α + βx i(2) µ i = β x x i + β z z i(3) µ i = α + β(x i − z i )(4) µ i = α + β x x i + β z z i5.7.2. e2. Write down a multiple regression to evaluate the claim: Animal diversity is linearlyrelated to latitude, but only aer controlling for plant diversity. You just need to write downthe model definition.5.7.3. e3. Write down a multiple regression to evaluate the claim: Neither amount of fundingnor size of laboratory are by themselves good predictors of time to PhD degree; but together thesevariables are both positively associated with time to degree. Write down the model definitionand indicate which side of zero each slope parameter should be on.5.7.4. e4. Suppose you have a single categorical predictor with 4 levels (unique values), labeledA, B, C and D. Let A i be an indicator variable that is 1 where case i is in category A. Alsosuppose B i , C i , and D i for the other categories. Now which of the following linear modelsare inferentially equivalent ways to include the categorical variable in a regression? Modelsare inferentially equivalent when it’s possible to compute one posterior distribution from theposterior distribution of another model.(1) µ i = α + β A A i + β B B i + β D D i(2) µ i = α + β A A i + β B B i + β C C i + β D D i(3) µ i = α + β B B i + β C C i + β D D i(4) µ i = α A A i + α B B i + α C C i + α D D i(5) µ i = α A (1 − B i − C i − D i ) + α B B i + α C C i + α D D iMedium5.7.5. m1. Invent your own example of a spurious correlation. An outcome variable shouldbe correlated with both predictor variables. But when both predictors are entered in the samemodel, the correlation between the outcome and one of the predictors should mostly vanish(or at least be greatly reduced).5.7.6. m2. Invent your own example of a masked relationship. An outcome variable shouldbe correlated with both predictor variables, but in opposite directions. And the two predictorvariables should be correlated with one another.