Least squares modelling

Learning outcomes

• Understand the difference between correlation and covariance.
• What the objective function of least squares does
• Understand and use an analysis of variance table
• Calculate and interpret the confidence intervals from a least squares model
• Know about the assumptions required to interpret least squares model coefficients
• Use the prediction error range from the model
• Identify outlier points and classify them
• Use the linear model when there are multiple predictor variables (this is what we are building up towards; we will use this extensively in the next topic)

Extended readings/practice

• Run the code below to see how to build and use a linear model in R, but see step 16 and onwards in the R tutorial as well.
• Try some practice problems.
• Describe a linear regression model you have made for a lab report.
  • What was the \(R^2\) value?
  • How did you calculate the regression model values?
  • Use the same data from your report and instead calculate the standard error, \(S_E\). How do you interpret that \(S_E\) value now?
• Do the YouTube challenge: find a video on YouTube that explains the central limit theorem, or the confidence interval, or least squares in a way that is different to explained in class (hopefully you find better explanations than mine). Share the link with a friend in your class.
• Does eating chocolate lead to winning a Nobel prize?

Resources

• Class notes 2015
• Class notes 2014
• Textbook, chapter 4
• Quizzes (with solutions): attempt these after you have watched the videos

Tasks to do first	Quiz	Solution
Watch videos AAA	Quiz	Solution
Watch videos BBB	Quiz	Solution
Watch videos CCC	Quiz	Solution
Watch videos DDD	Quiz	Solution
Watch videos EEE	Quiz	Solution

Class videos from prior years

Videos from 2015

1. Overview of this module [02:59]
2. Covariance and correlation [09:46]
3. Why least squares, and some other alternatives [covered in class]
4. Some of the math behind the LS model [09:44]
5. Understanding the analysis of variance (ANOVA) [11:58]
6. Interpretation of standard error [covered in class]
7. Assumptions to derive LS confidence intervals [05:45]
8. Confidence intervals interpreted and used in 3 LS examples [11:54]
9. Prediction intervals from the least squares model [04:24]
10. Checking for violations of the least squares assumptions (1 of 2) [07:27]
11. Checking for violations of the least squares assumptions (2 of 2) [11:46]
12. Introducing multiple linear regression - why we need to use it [2:59]
13. MLR - the matrix equation form and an example [11:25]
14. Interpreting the MLR model coefficients and confidence intervals [04:58]
15. Integer variables in the multiple linear regression model [09:51]

Covered in class | No video | Script

Covered in class | No video | Script