---
tags: stat340, learning-targets
---

# Stat 340 Learning Target Quiz 4 Study Guide

Learning Target Quiz #4 will include questions on learning targets 16-20. In addition, you can reattempt targets that appeared for the first time on Quiz 3.

#### 16. Given a fitted hiearchical model, check whether the fitted model is appropriate/adequate.

- State the appropriate/available diagnostic tools for assessing the adequacy of a fitted model
- Given the diagnostics for your model, comment on whether the model assumptions are reasonable, and how wrong the model is.
- Explore the posterior predictive densities and comment on how reasonable the predictions are

*Example question:*

Below is an overlayed density plot with 100 posterior simulated data sets for the hierarchical model.


What does this plot indicate about the adequacy of the hierarchical model?

#### 17. Given your prior belief, specify an appropriate prior distribution for a regression model.

- Interpret appropriate prior models for the regression parameters
- Understand how to specify weakly informative prior distributions
- Understand three methods to specify informative prior distributions: based on a centered predictor, based on standardized variables, and based on conditional means

*Example question:*

In this problem, you'll consider a simple linear regression model of rides (Y) by humidity (X). Based on past bikeshare analyses, suppose we have the following prior understanding of this relationship:

- On an average humidity day, there are typically around 5000 riders, though this average could be somewhere between 1000 and 9000.
- Ridership tends to decrease as humidity increases. Specifically, for every one percentage point increase in humidity level, ridership tends to decrease by 10 rides, though this average decrease could be anywhere between 0 and 20.
- Ridership is only weakly related to humidity. At any given humidity, ridership will tend to vary with a large standard deviation of 2000 rides.

Specify what prior distributions you would use for $\beta_0$, $\beta_1$, and $\sigma$ (or for functions of these parameters) to reflect this prior belief.

#### 18. Given a research question and your prior belief, write out a regression model in statistical notation.

- Be able to specify the likelihood and link function
- Understand what assumptions/conditions are expressed by this model equation

*Example question:*

Given the prior distributions defined in the previous problem, write out the entire Bayesian regression model of rides (Y) vs. humidity (X) using proper notation.

#### 19. Given draws from the (approximate) posterior distribution of your regression model, draw inferences about the appropriate parameters in the context of the research question.

- Calculate and interpret a point estimate for a parameter or function of parameters
- Calculate and interpret a credible interval for a parameter or function of parameters
- Perform and interpret a Bayesian hypothesis test
- Utilize simulation results to build a posterior understanding of the relationship between Y and X and to build posterior predictive models of Y

*Example question:*

Let $Y_i$ denote the winning time in seconds for the women's 100 meter butterfly race for the Olympics from 1964 through 2016, and let $x_i$ denote the year for the $i$th Olympics. Suppose we fit the following simple linear regression model:

##           1%   2.5%    25%    50%    75%  97.5%    99%
## beta0 62.235 62.518 63.506 63.924 64.317 65.154 65.410
## beta1 -0.221 -0.212 -0.185 -0.172 -0.159 -0.128 -0.119
## sigma  0.758  0.810  1.011  1.162  1.352  1.879  2.060

Based on the posterior draws, is there substantial evidence that the winning time is decreas- ing? Briefly justify your answer.

#### 20. Given a fitted regression model, check whether the fitted model is appropriate/adequate.

- State the appropriate/available diagnostic tools for assessing the adequacy of a fitted model
- Given the diagnostics for your model, comment on whether the model assumptions are reasonable, and how wrong the model is.
- Explore the posterior predictive densities and comment on how reasonable the predictions are

*Example question:* Suppose that you fit a simple linear regression model predicting the divorce rate in a state based on the median age of first marriage. Below is a plot of the predicted divorce rate vs. the observed divorce rate. What does this plot suggest about the adequacy of your fitted model?