Regression Analysis

1
Regression Analysis

• Deterministic model
• No chance of an error in calculating y for a
  given x
• Probabilistic model
• chance of an error
• First order linear probabilistic model
• ?0 ?x ?

2
Least Square Method

• Minimizes the sum of squared differences between
  observed and values from the regression line.
• ? slope of the line and
• Ssxy? SSx
• Look for the short cut formula on page 731
• ?o y intercept y- - ?x-
• Residual yi - Y

3
Regression continued

• R2 SSR/SST
• Proportion of the total variation in Y explained
  by the regression line
• R2 1, all scatter points on the regression line
• R2 0 no scatter point on the regression line
• Square root of R2 is called coefficient of
  correlation

4
Regression continued

• When Coefficient of correlation is positive,
  there is direct relationship between variables
• When Coefficient of correlation is negative, y
  value increases when x decrease and vice versa
• When Coefficient of correlation is zero, there is
  no linear relationship.

5
Regression

• SSTSSRSSE
• Formulae for predicted interval and expected
  interval are on page 756
• To infer on the population coefficient of
  correlation, use t -test, formula on page 761
• To find t-value from t-table, you must know
• degree of freedom
• the level of significance
• For two tailed test, divide the level of
  significance by 2.

6
Assessing the Model

• Standard error of the estimate
• Divide the standard error of the estimate by the
  average of y
• Smaller its value, better the fit
• Coefficient of determination
• Closer its value to 1, better the fit
• R2 1, all scatter points fit on the
  regression/least square line
• R2 0, non of the scatter points lie on the
  regression line.
• Explains the proportion of the total deviation
  explained by the regression line.

7
Inference on Slope

• Apply t-test because
• standard deviation of the population is unknown
• H0? 0
• Ha ?? 0
• t (? - ?)/S?
• S? is the standard deviation of the slope and
  Se /?SSx
• use level of significance and the degree of
  freedom (n-2) to derive a conclusion based on
  the data.

• Predicting a value of Y for a given value of x
• use the formula on page 756 or use the Excel
  print-out under PI
• Estimating expected value
• Use the formula on page 756 or use the computer
  print out under confidence interval.