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Correlation & Regression_

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Correlation & Regression_

Correlation & Regression_

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Correlation & Regression_ Correlation & Regression_ Presentation Transcript

    • 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 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 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
  • Regression Results r 2 = 0.06 About 6% of the variation in popgrow can be explained by variation in urb.
  • Regression Results < 0.05 Reject H 0 : 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 H 0 .) > 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 (r 2 =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
  • 2. Simple Linear Regression Dependent: popgrow Independent(s): fertil
  • Regression Results r 2 = 0.547 About 55% of the variation in popgrow can be explained by variation in fertil.
  • Regression Results < 0.05 Reject H 0 : 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 H 0 .) 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 (r 2 =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 r 2 = .547   LOGARITHMIC: popgrow = b0 + b1 * log(fertil) popgrow = .045 + 1.67 * log(fertil) r 2 = .635   QUADRATIC: popgrow = b0 + b1 * fertil + b2 * fertil 2 popgrow = -1.337 + 1.508 * fertil - .132 * fertil 2 r 2 = .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
  • 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.