1. YOU NEED TO KNOW WHAT THIS MEANS
REGRESSION ANALYSIS
In the OUTCOME that you will commence second week back, you might be given data and asked to perform a REGRESSION ANALYSIS
YOU NEED TO KNOW WHAT THIS MEANS

2. REGRESSION ANALYSIS is the process of fitting a linear model to a data set.
The aim is to determine the best linear model possible and to use it to make predictions.

3. What do we mean by the “best possible linear model”?
The best possible linear model is the one in which:
a. The data is linear or has been linearized by a data transformation:

4. and we also want b. the linear model which has the greatest possible value of r2
REMEMBER: the value of the coefficient of determination measures the predictive power of our regression model.
PREDICTIVE POWER R2

5. If r2 > 30%, then our model will have
Predictive power

6. STEP 1:
Construct a scatterplot of the RAW (Original ) Data and note:
Its shape
The value of the coefficient of determination
FIRST: We must decide
Which is the INDEPENDENT (x) VARIABLE:
Which is the DEPENDENT (y ) VARIABLE
We are predicting LIFE EXPECTANCY from GDP, so: X
GDP Y
LIFE EXPECTANCY

7. CONCLUSION: Data is NON-LINEAR List A = gdp List B = le
lifeex
950 58
1670 65
4250 68
11520 74
12280 73
4170
14300 75
5540 71
9830 72
1680 61
320 67
22260 66
550 50
930
940 64
2670
11220
1420 48
150 41
330 44
520 49
350
180
Life expectancy
CONCLUSION:
Data is NON-LINEAR

8. From the Home screen determine the value of r2. Value of r2 = 0.3665.

9. CHECK THE CIRCLE OF TRANSFORMATIONS!!
STEP 2: We seek a Transformation to linearize the data.
CHECK THE CIRCLE OF TRANSFORMATIONS!!
Our scatterplot most closely resembles Quadrant 2!
POTENTIALLY SUITABLE TRANSFORMATIONS are: Y2
Logx 1 x
Quadrant 2
Quadrant 1
Quadrant 3
Quadrant 4

10. Step 3
Try each of these transformations to determine which one effectively linearizes the data and gives the highest value for r2.
In each case, obtain a RESIDUAL PLOT to confirm that the transformed data is linear.

11. R2 = 38.3% List A gdp ( x variable) Y SQUARED TRANSFORMATION List B
lesqu
950 58
3364
1670 65
4225
4250 68
4624
11520 74
5476
12280 73
5329
4170
14300 75
5625
5540 71
5041
9830 72
5184
1680 61
3721
320 67
4489
22260 66
4356
550 50
2500
930
940 64
4096
2670
11220
1420 48
2304
150 41
1681
330 44
1936
520 49
2401
350
180
List B
le (y variable)
List C
lesqu
(y transformed variable )
R2 = 38.3%
TRANSFORMED DATA APPEARS NON-LINEAR STILL

12. Establish the value of r2 in HOMESCREEN:

13. CONFIRM WITH RESIDUAL PLOT
Remember: to get the correct residual plot use the split screen view. Make sure that the scatterplot at the top has the correct transformed variable.
CONCLUSION: The residual plot shows a definite curved pattern, indicating that the transformed data is still not linear.
The y2 transformation has NOT succeeded in producing an effective linear model.

14. NEXT STEP….
You guessed it!!
Now we try the next potential candidate transformation.
It was the log x transformation!

15. (Delete the y2 column, as we have discarded this transformation.)
GDP
lifeex
logGDP
950 58
2.98
1670 65
3.22
4250 68
3.63
11520 74
4.06
12280 73
4.09
4170
3.62
14300 75
4.16
5540 71
3.74
9830 72
3.99
1680 61
3.23
320 67
2.51
22260 66
4.35
550 50
2.74
930
2.97
940 64
2670
3.43
11220
4.05
1420 48
3.15
150 41
2.18
330 44
2.52
520
2.72 49
350
2.54
List A= gdp List B= le List C= loggdp
R2 = 66.0%
CONCLUSION: It appears that the log(GDP) transformation has successfully linearized the data! Scatterplot appears linear, and R2 has increased.

16. NOTE THE VARIABLES ARE LISTED HERE SO YOU CAN CHECK

17. The value of r2 has now increased to 66.0%.
Now confirm this by creating a RESIDUAL PLOT for the log(x) transformation. Open a new graphing screen!!
CONCLUSION: The residual plot shows a random scattering of points with no pattern, indicating that the transformed data is linear.
The value of r2 has now increased to 66.0%.
The logx transformation has succeeded in producing an effective linear model for the data with significant predictive power.

18. And now……
Yes you guessed it!
We need to check out the reciprocal x transformation, because …..
maybe it will give a higher coefficient of determination than the logx!
(here we go again)

19. Don’t delete log x column because we think this model was effective!
List A = gdp List B = le List C = loggdp List D = recgdp
Life expectancy
list A
listB
list C
list D
log(list1)
1/list1
950 58
2.98
1670 65
3.22
4250 68
3.63
11520 74
4.06
12280 73
4.09
4170
3.62
14300 75
4.16
5540 71
3.74
9830 72
3.99
1680 61
3.23
320 67
2.51
22260 66
4.35
550 50
2.74
930
2.97
940 64
2670
3.43
11220
4.05
1420 48
3.15
150 41
2.18
330 44
2.52
520
2.72 49
350
2.54
180
2.26
R2 = 51.5%
1/GNP
CONCLUSION: The transformed data appears to be linear, but the value of the coefficient of determination is 51.5%, lower than for the loggdp transformation.

20. Coefficient of determination

21. Remember to create a new graphing screen for the new transformation!!
CONCLUSION: The residual plot shows a random scattering of points with no pattern, indicating that the 1/x transformation has made the data linear.

22. Y squared transformation: Ineffective (did not linearize the data)
OVERALL CONCLUSIONS
We have tested three transformations:
Y squared transformation: Ineffective (did not linearize the data)
Log (x ) transformation: Effective in linearizing data with r2 = 66.0%
1/x transformation: Effective in linearizing data with r2 = 51.5%
Based on this regression analysis, we conclude that the log(GDP) transformation provides the best model for making predictions from this data.

23. MAKING A PREDICTION
Use your linear regression model to predict the Life Expectancy in a country where the GNP is $8000
gnp le
List3
loggnp
950 58
2.98
1670 65
3.22
4250 68
3.63
11520 74
4.06
12280 73
4.09
4170
3.62
14300 75
4.16
5540 71
3.74
9830 72
3.99
1680 61
3.23
320 67
2.51
22260 66
4.35
550 50
2.74
930
2.97
940 64
2670
3.43
11220
4.05
1420 48
3.15
150 41
2.18
330 44
2.52
520
2.72 49
350
2.54
180
2.26
Find the equation of the LEAST SQUARES REGRESSION line for the Log transformation
Regression(a+bx)
Xlist = log(GNP)
Ylist=le
a = 14.3
b = 14.5
Life Expectancy =  log(GNP)
Life Expectancy = × log(8000)
= 70.9
Predicted Life Expectancy = 70.9 years