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Statistics 359a

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Statistics 359a

Regression Analysis

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

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

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.

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

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

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

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

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 <

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

Price and Date of Sale

- 1995 seems to be a more expensive year
- Is the effect confounded with some other attribute common to 1995?

Price and Year of Manufacture

- Is there anything special about 1920?
- Is there a quadratic trend in the data?

Price and Weight of the Bow

- Is there any trend with respect to the weight?

Octagonal vs. Round Bows

- No apparent trend

The Gold Standard?

- The presence of gold on a bow generally makes it more expensive

Tortoise Shell Frogs

- Some evidence of added expense for tortoise shell

Price and Pearl Accessories

- No apparent effect

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