Statistics 359a

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 ) < x for 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. 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 Prediction