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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
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
1. Scatterplot: Graphs/Scatter
Simple
popgrow
1. Scatterplot: Graphs/Scatter
urb
1. Scatterplot: Graphs/Scatter
URB
120100806040200
POPGROW
8
6
4
2
0
-2
Each dot
represents
a case/country
Chart/Options
To make changes to chart, double click chart in
output window. Chart Editor will appear.
Chart/Options: the least square line
Chart/Axis
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
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.)
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.
2. Simple Linear Regression
Analyze/Regression/Linear...
2. Simple Linear Regression
Dependent: popgrow
Independent(s): urb
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.
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
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
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
popgrow
1. Scatterplot: Graphs/Scatter
fertil
1. Scatterplot: Graphs/Scatter
FERTIL
987654321
POPGROW 7
6
5
4
3
2
1
0
-1
2. Simple Linear Regression
Dependent: popgrow
Independent(s): fertil
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.
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
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
Test alternative regression models for the relationship
between popgrow and fertil using the
Analyze/Regression/Curve Estimation package.
3. Simple Non-Linear Regression
3. Simple Non-linear Regression
Dependent: popgrow
Independent(s): fertil
Linear, Quadratic, Logarithmic
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
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.

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