Correlation & Regression_

1. 1. Scatterplot
2. Simple (Linear) Regression
3. Simple Non-Linear Regression
Data: Country database website
– information on national characteristics of 160 countries
– measures of quality of life of the population (e.g. life
expectancy and infant mortality)
– measures of wealth (e.g. GNP)
Simple Regression/Correlation

2. Research Question:
Is there a relationship between population growth
rate and level of urbanization (measured as the
percent of population living in urban areas)?
A question of
“whether or not there is,”
as well as
“how much.”
Scatterplot &
Linear Simple Regression
T-tests and chi-
square tests
Correlation and
regression analysis

3. 1. Scatterplot: Graphs/Scatter
Simple

4. popgrow
1. Scatterplot: Graphs/Scatter
urb

5. 1. Scatterplot: Graphs/Scatter
URB
120100806040200
POPGROW
8
6
4
2
0
-2
Each dot
represents
a case/country

6. Chart/Options
To make changes to chart, double click chart in
output window. Chart Editor will appear.

7. Chart/Options: the least square line

8. Chart/Axis

9. 1. Scatterplot: Graphs/Scatter
URB
100806040200
GROWTH
6
4
2
0
-2
-4
File/Print,
Export Chart to
export as a Graphics
Figure (e.g. .jpg) to
Word,
Copy/Paste as
picture to Word
The least
square line

10. 1. Scatterplot: Graphs/Scatter
Identify cases using the
Point ID tool in the
Chart Editor window.
Select the button in the
menu, and use the
pointer to select the
point in the upper right
corner with high
urbanization and high
growth rate. (Case
number 46: United Arab
Emirates.)

11. Format/Color or Format/Marker
While in the Chart Editor, click on a feature (e.g. the least square line or
the dots), you can change the color (on the button menu above) or right
click and select “properties window” to change the line or marker/dot
types.

12. 2. Simple Linear Regression
Analyze/Regression/Linear...

13. 2. Simple Linear Regression
Dependent: popgrow
Independent(s): urb

14. Model Summary
.249a .062 .057 1.1537
Model
1
R R Square
Adjusted
R Square
Std. Error of
the Estimate
Predictors: (Constant), URBa.
Regression Results
r2
= 0.06
Variables Entered/Removedb
URBa . Enter
Model
1
Variables
Entered
Variables
Removed Method
All requested variables entered.a.
Dependent Variable: POPGROWb.
About 6% of the variation in popgrow can be explained by
variation in urb.

15. Coefficientsa
2.511 .220 11.389 .000
-1.25E-02 .004 -.249 -3.357 .001
(Constant)
URB
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardized
Coefficients
t Sig.
Dependent Variable: POPGROWa.
ANOVAb
15.002 1 15.002 11.271 .001a
226.272 170 1.331
241.274 171
Regression
Residual
Total
Model
1
Sum of
Squares df Mean Square F Sig.
Predictors: (Constant), URBa.
Dependent Variable: POPGROWb.
Regression Results
< 0.05
Reject H0: No linear relationship between
growth and urbanization
Confidence interval on the slope for URB, b:
[-0.0125 + 1.96 * 0.004] = [-.01984 ≤ b ≤ -.00416 ]
--> does not contain 0
There is a significant linear relationship. (reject H0.)
> 1.96

16. There is a weak negative relationship between
POPGROW and URB.
Only 6% of the variation in growth rate is explained
by variation in the level of urbanization (r2
=0.062)
The Linear Model (refer to B in coefficients):
POPGROW = 2.511 - 0.0125 * URB
The predicted growth rate for the US (URB=77.2%) is 1.58%
(2.511- 0.0125*77.2), in comparison to the actual rate of 1%.
Analysis of Regression Results

17. Research Question:
Perhaps the average number of live births per
female (FERTIL) will also explain variation in
growth rate?
1. Scatterplot &
2. Linear Simple Regression

18. popgrow
1. Scatterplot: Graphs/Scatter
fertil

19. 1. Scatterplot: Graphs/Scatter
FERTIL
987654321
POPGROW 7
6
5
4
3
2
1
0
-1

20. 2. Simple Linear Regression
Dependent: popgrow
Independent(s): fertil

21. Model Summary
.740a .547 .544 .7892
Model
1
R R Square
Adjusted
R Square
Std. Error of
the Estimate
Predictors: (Constant), FERTILa.
Regression Results
r2
= 0.547
Variables Entered/Removedb
FERTILa . Enter
Model
1
Variables
Entered
Variables
Removed Method
All requested variables entered.a.
Dependent Variable: POPGROWb.
About 55% of the variation in popgrow can be explained by
variation in fertil.

22. Coefficientsa
.245 .132 1.851 .066
.469 .033 .740 14.027 .000
(Constant)
FERTIL
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardized
Coefficients
t Sig.
Dependent Variable: POPGROWa.
ANOVAb
122.537 1 122.537 196.759 .000a
101.513 163 .623
224.051 164
Regression
Residual
Total
Model
1
Sum of
Squares df Mean Square F Sig.
Predictors: (Constant), FERTILa.
Dependent Variable: POPGROWb.
Regression Results
< 0.05
Reject H0: No linear relationship between
growth and fertility
Confidence interval on the slope for FERTIL, b:
[0.469 + 1.96 * 0.033] = [0.404 < b < 0.533] --> not containing 0
There is a significant linear relationship. (reject H0.)
t > 1.96

23. There is a moderate positive relationship between POPGROW
and FERTIL.
About 55% of the variation in growth rate is explained by
variation in fertility rate (r2
=0.547)
The Linear Model (refer to B in coefficients):
POPGROW = 0.245 + 0.469 * FERTIL
T-scores and the significance levels indicates the constant and coefficient
ON FERTIL are significantly different from 0.
However, the relationship does not look linear on the scatterplot.
Analysis of Regression Results

24. Test alternative regression models for the relationship
between popgrow and fertil using the
Analyze/Regression/Curve Estimation package.
3. Simple Non-Linear Regression

25. 3. Simple Non-linear Regression
Dependent: popgrow
Independent(s): fertil
Linear, Quadratic, Logarithmic

26. Analysis Results
LINEAR: popgrow = b0 + b1 * fertil
popgrow = .245 + .469 * fertil r2
= .547
LOGARITHMIC: popgrow = b0 + b1 * log(fertil)
popgrow = .045 + 1.67 * log(fertil) r2
= .635
QUADRATIC: popgrow = b0 + b1 * fertil + b2 * fertil2
popgrow = -1.337 + 1.508 * fertil - .132 * fertil2
r2
= .655
About 65% of the variation in growth rates can be explained
by variation in fertility using logarithmic or quadratic as
compared to only 55% using the linear model.
Independent: FERTIL
Dependent Mth Rsq d.f. F Sigf b0 b1 b2
POPGROW LIN .547 163 196.76 .000 .2447 .4686
POPGROW LOG .635 163 284.08 .000 .0445 1.6661
POPGROW QUA .655 162 154.03 .000 -1.3371 1.5078 -.1315

27. POPGROW
FERTIL
987654321
7
6
5
4
3
2
1
0
-1
Observed
Linear
Logarithmic
Quadratic
Analysis Results
Both the logarithmic
and quadratic curves
seem to better
represent the
relationship between
growth rate and
fertility: growth rates
increase at a decreasing
rate with fertility.