1. Statistics 359a Regression Analysis

2. Necessary Background Knowledge - Statistics expectations of sums variances of sums distributions of sums of normal random variables t distribution – assumptions and use calculation of confidence intervals simple tests of hypotheses and p-values

3. Necessary Background Knowledge – Linear Algebra multiplication of conformable matrices transpose of a matrix determinant of a square matrix inverse of a square matrix eigenvalues of a square matrix quadratic forms

4. Origin of Least Squares Introduction of the metric system and the length of a meter 1790 – French National Assembly commissions the French Academy of Sciences to design a simple decimal-based system of weights and measures 1791 – French Academy defines the meter to be 10 -7 or one ten-millionth of the length of the meridian through Paris from the north pole to the equator.

6. Adrien-Marie Legendre Legendre on the French commission in 1792 to determine the length of the meridian quadrant measurements of latitude made in 1795 complex calculations made from the measurements in 1799 Legendre proposes the method of least squares in 1805 to determine the length of a meter

7. Data old French units of measurement: 1 module = 2 toises old French to imperial English: 1 toise = 6.395 feet metric to imperial: 1 meter = 3.2808 feet

8. From Spherical Geometry

9. Including measurement errors, the data and model reduce to:

10. Solution is: D = 28497.78 modules 90D = 2564800.2 modules = length of the meridian quadrant Therefore 1 meter = 0.256480 modules = 0.512960 toises = 3.280 feet modern meter = 3.2808 feet

11. Origin of the Term “Regression” Francis Galton, 1886, ‘Regression towards mediocrity in hereditary stature.’ Journal of the Anthropological Institute, 15: 246 – 263 See JSTOR under UWO library databases

12. Data on Heights of Children and Parents

13. ‘Regression Line’

14. Theoretical Basis For X and Y bivariate normal with equal means variances For  > 0 E(Y |X )  and E(Y |X ) > x for x < 

15. Example in Data Analysis Through Regression Relationship between the price of a violin bow and its attributes such as age, shape and ornamentation on the bow

17. Price and Date of Sale 1995 seems to be a more expensive year Is the effect confounded with some other attribute common to 1995?

18. Price and Year of Manufacture Is there anything special about 1920? Is there a quadratic trend in the data?

19. Price and Weight of the Bow Is there any trend with respect to the weight?

20. Octagonal vs. Round Bows No apparent trend

21. The Gold Standard? The presence of gold on a bow generally makes it more expensive

22. Tortoise Shell Frogs Some evidence of added expense for tortoise shell

23. Price and Pearl Accessories No apparent effect

24. Prediction Can we use the model built with the current data to predict the future price of a bow Example: some 1999 data from auctions 1920 bow, 60.5 g., round with gold and pearl accessories - $4098 1933 bow, 61 g., octagonal with pearl accessories only - $2421