Midterm Exam 2: Study Guide

STAT3530 Applied Linear Models

Scope

Weeks 6 through 9: the general linear model, model fit and \(F\) tests, simulation-based inference, nonlinear specification, and categorical predictors with interactions. To review, work from:

• homeworks 4 through 7
• the week 6 through 9 slides
• handouts 6 through 9
• in-class activities/scripts, especially those on simulation-based inference and categorical variables and interactions

Format

• in class, open note, calculator allowed, no software
• three questions, each with several short parts
• written to take about eighty minutes, so the period leaves room to think and check
• answers are short: a sentence or two, a small calculation, a line of algebra, a hypothesis written in symbols
• every question is posed in the context of a dataset, and the output you need is printed for you
• ask me during the exam if a prompt is unclear

What you should be able to do

The general linear model

• write a multiple regression model and interpret the parameters in context
• read the standard lm() model summary in R, including the coefficient table, \(\hat\sigma^2\), R squared and adjusted R squared, and the omnibus test
• explain how a coefficient estimate can change in size or in significance when another predictor enters the model

\(F\) tests and contrasts

• recognize nested model comparisons
• write a hypothesis posed in words in terms of model coefficients
• recognize hypotheses expressible as contrasts and write contrast coefficients
• given a hypothesis, set up an extra-sum-of-squares \(F\) test with appropriate numerator and denominator degrees of freedom
• interpret the anova() output for \(F\) tests in R

Categorical predictors and interactions

• express a categorical variable by indicator encoding
• write a linear model with categorical one or more categorical predictors
• write model equations for each level of a categorical variable
• recognize and interpret interactions involving one categorical and one numeric predictor
• interpret regression coefficients when an interaction is in the model

Nonlinear relationships

• interpret a partial residual plot
• specify polynomial models and test for curvature of a given order
• specify a piecewise linear model
• specify a cubic spline model using the b-spline basis
• interpret interactions involving a categorical predictor and a nonlinear specification in a numeric variable

Simulation-based inference

• explain why some quantities of interest have no usable standard error formula
• distinguish randomization from a parametric bootstrap
• identify the assumptions made by each procedure
• compute percentile intervals from a set of simulated estimates and interpret the result appropriately

How much computation to expect

• arithmetic small enough for a calculator
• a confidence interval or two by hand, from an estimate, a standard error, and a quantile that is given to you
• contrast matrices of at most a few rows, written out rather than multiplied
• no matrix inversion, and no \(X^TX\) by hand

How to prepare

• rework homework problems, especially the ones where you read output and write an interpretation
• review the in-class activities and scripts
• prepare a few pages of notes on key concepts/examples
• do the practice set under exam conditions, with a calculator and the clock running, before you look at anything else
• compare answers with a classmate