AGENDA
Linear Regression

LINEAR REGRESSION
Statistical method for determining a linear relationship between two variables assuming normality and independence
Two types of variables
Independent
Dependent
Method of least squares

DATA

WHICH PREDICTION LINE
PREDICTION LINE
DETERMINING PREDICTION LINE
Calculate the total squared error between the
Actual value
Predicted value according to the prediction line

DIFFERENTIATE
Want the minimum of the sum of squared error
Take partial derivatives of the previous equation with respect to a and b
Set to 0

NORMAL EQUATIONS
With some rearrangement
Resulting linear equations are called normal equations

INTERCEPT AND SLOPE
Substitute x, y, xy, and x squared values in normal equations
Solve for a
 Intercept with y axis at x=0
Solve for b
Coefficient of slope for x values

EXAMPLE

NORMAL EQUATIONS

NORMAL EQUATIONS

PREDICTION EQUATION

INFERENCES
R squared…
T value for coefficients…

R SQUARED
What percentage of the variation in the data can be accounted for by the equation
The closer to 1.0, the better the fit
For physical experiments r squared should be > 0.9
For people related experiments r squared should be >0.30
R squared formula…

R SQUARED

INTERCEPT PARAMETER Alpha
Typically tested against the intercept alpha being 0
Hypotheses test
Ho: alpha=0
Ha: alpha < > 0
Two sided critical value based on n-2 df.
Test statistic of…

TEST STATISTIC FOR INTERCEPT
SLOPE PARAMETER Beta
Typically tested against the slope Beta being 0
Hypotheses test
Ho: Beta=0
Ha: Beta < > 0
Two sided critical value based on n-2 df.
Test statistic of…

SLOPE PARAMETER B
IN EXCEL
Data Analysis-Regression
Specify
Input x range
Input y range
Returns
R squared
Coefficients
T stat
P value