ASSESSING THE STRENGTH

1

• ASSESSING THE STRENGTH
• OF THE
• REGRESSION MODEL

2
Assessing the Models Strength

• Although the best straight line through a set of
  points may have been found and the assumptions
  for ? may appear valid, is the resulting
  regression line useful in predicting y?

3
STEP 4HOW GOOD IS THE MODEL?

• Can we conclude that there is a linear relation
  between y and x?
• This is a hypothesis test (t-test)
• What proportion of the overall variability in y
  (from its mean) can be explained by changes in x?
• This is a performance measure called -- the
  coefficient of determination (denoted by r2)

4
Can we conclude a linear relation exists between
y and x?

• We are hypothesizing that y changes linearly with
  x y ?0 ?1x. That is, if x goes up by 1, y
  will change by ?1.
• But if no linear relation exists, then that means
  if x goes up by 1, y will not change, i.e. ?1
  0.

5
The Hypothesis Test

• To test whether or not a linear relation exists
• H0 ?1 0 (No linear relation exists)
• HA ?1 ? 0 (A linear relation does exists)
• ? the significance level
• Reject H0 (Accept HA) if t gt t?/2 or if t lt
  -t?/2
• with Degrees of Freedom n- ( betas) n-2

6
The t statistic for the test of ?1 0

7
HAND CALCULATIONS
Test Reject H0 if t gt t.025,8 2.306 or t lt
-t.025,8 -2.306

5.123 gt 2.306 Can conclude ß1 ?0, i.e. a linear
relation exists.
8
95 Confidence Interval for ?1

• (Point Estimate) ? t.025,n-2(Appropriate std
  dev.)

9
Coefficient of Determination -- r2
The proportion of the total change in y that can
be explained by changes in the x values is called
the coefficient of determination, denoted r2.

10
Hand Calculation of SSR, SSE, SST
1 1200 101000 109567.57 186802403.21 73403214.02 26010000
2 800 92000 88540.54 54161643.54 11967859.75 15210000
3 1000 110000 99054.05 9948056.98 119813732.7 198810000
4 1300 120000 114824.32 358130051.13 26787618.7 580810000
5 700 90000 83283.78 159168911.61 45107560.26 34810000
6 800 82000 88540.54 54161643.54 42778670.56 193210000
7 1000 93000 99054.05 9948056.98 36651570.49 8410000
8 600 75000 78027.03 319443162.89 9162892.622 436810000
9 900 91000 93797.30 4421358.66 7824872.169 24010000
10 1100 105000 104310.81 70741738.50 474981.7385 82810000
SUM 1226927027.03 373972972.97 1600900000
11
Hand Calculation of r2
12
Interpretation of r2

• r2 1 -- perfect (positive or negative) relation
• i.e. points fit exactly along the regression
  line
• r2 close to 0 -- very little relation
• The higher the value of r2 the better the model
  fits the data

13
Pearson Correlation Coefficient, r

• r ?r2, which can also be calculated by
  cov(x,y)/sxsy is called the Pearson correlation
  coefficient.
• This is also used to measure the strength of the
  relation between y and x.
• r -1 means perfect negative correlation (i.e.
  all points fit exactly on a line with negative
  slope).
• r 1 means perfect positive correlation (i.e.
  all points fit exactly on a line with positive
  slope).
• r 0 means no correlation.
• Other values give relative strength, but have no
  exact meaning like r2 so we usually use r2
• When we take the square root of r2 to get r, the
  sign in front of r is the sign of b1 positive
  or negative slope

14
EXCEL
15
Steps Using Excel

• Determine regression equation
• Equation y 46486.49 52.56757x
• Can you conclude a linear relation exists between
  y and x?
• The p-value for the test is .000904 lt ?.05YES
• What proportion of the overall variation in y is
  explained by changes to x?
• This is r2 .766398 -- a high r2
• CONCLUSION Overall a good model!

16
Review

• Can we conclude a linear relation exists?
• Two-tailed t-test of ?1? 0
• Look at p-value for the x-variable on Excel
• Computation of a confidence interval for the
  amount y will change per unit increase in x (i.e.
  for ?1)
• By hand
• Printed on Excel Output
• What proportion of the overall variation in y is
  explained by changes in x? r2
• By hand
• Printed on Excel
• Pearson correlation coefficient r
• Square root of r2
• Sign is same as b1