Review.

Today gave a quick overview of the final examination and distributed a study sheet (downloadable from Blackboard).

ANCOVA and assumptions.

Today we focused on classic ANCOVA and its assumptions. In particular we looked at cases where the covariate and the treatment interact. We also considered the use of adjusted means (sometimes called least squares means) to make inferences such as multiple comparisons involving group means while controlling for the covariate.

Introduction to the analysis of covariance.

The analysis of covariance (ANCOVA) is a design and model that is a nice bridge between analysis of variance and regression. As we saw in our discussion of regression, we can incorporate both categorical and quantitative explanatory variables. A classic ANCOVA is essentially a linear model for a completely randomized design but with a quantitative explanatory variable.

Residual analysis.

Today we focused on residual analysis — i.e., the use of residuals to investigate possible violations of assumptions of regression. I reviewed the four underlying assumptions of regression and illustrated how residuals and plots of residuals can be used to detect violations of the assumptions.

Interpretation of parameters in multiple regression, collinearity, and influence.

Today we focused on some issues that arise when there is more than one explanatory variable in a regression model (i.e., multiple regression). This included the interpretation of the partial slopes/effects and the importance in some cases of controlling for the effects of other variables, and collinearity — what it is and why it can be a problem. I also introduced the concept of influence and leverage — that the influence of an observation depends on its leverage and residual.

 

Simultaneous tests — continued, special cases, and interactions in regression.

We continued looking at simultaneous tests and special cases thereof, and considered an example where we had both a quantitative explanatory variable, and a categorical explanatory variable represented by indicator variables. We also considered how to model an interaction between the quantitative and categorical explanatory variables.

Indicator variables and simultaneous tests.

Today we focused on the use of indicator variables so that we can make the same kinds of inferences with regression that we do with the analysis of variance. I also introduced the idea of the simultaneous test in regression where we test the null hypothesis that a set of beta parameters are equal to zero.

Polynomial regression, and an introduction to indicator variables.

Today I expanded on the idea of using more than one explanatory variable. This is useful of course if there is more than one explanatory variable, but it is also useful for extending regression to handle nonlinear relationships through polynomials, and to handle categorical explanatory variables through the use of indicator/dummy variables.

Homework: 11.4, 11.7, 11.15, 11.22, 11.23, 11.25, 11.26, 11.33, 11.34, 12.5, 12.10.

Inference in simple linear regression — continued.

I continued the introduction of the simple linear regression model. I considered the confidence interval for the mean of the response variable, and the prediction interval for the value of the response variable, both for a given value of the explanatory variable. We also started considering multiple linear regression with an arbitrary number of explanatory variables.

Inference with simple linear regression.

Today we focused on the simple linear regression model, its assumptions, and inferences. For inferences we looked at the estimation of the intercept and slope parameters via least squares, and the estimation of the variance parameter. We also considered the confidence interval and test statistic for the slope parameter. I also explained the ANOVA table in the context of regression. Finally I demonstrated the use of PROC REG and PROC GLM to facilitate these inferences.

Note: Solutions for the most recently set of homework/practice problems are available on Blackboard.