Regression

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Regression

• K300 Class 27
• April 21, 2009

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Overview

• Regression versus correlation
• Sample park use problem
• Correlation analysis
• Idea of regression
• Regression analysis
• Coefficient of determination
• Prediction
• SPSS analysis

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Regression versus correlation

• Correlation
• Is there a linear relationship between two
  continuous variables?
• What is the strength of that relationship?
• Regression
• What is the best estimate of the mathematical
  form of the linear relationship?

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Simple park use problem

• You are a park planner for a city
• You hypothesize that there is a relationship
  between the use made of parks and the number of
  people living within 1.5 miles of the parks
• You collect data on number of users per day and
  populations for six parks

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Dependent and independent variables

• In regression, we refer to y as the dependent
  variable
• and x as the independent variable
• Based on the assumption that the value of y
  depends on the value of x
• or, put another way, that changes in x cause
  changes in y

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Park use data
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Park use data scattergram
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Correlation analysis

• We begin by looking at the correlation
• Gives us a measure of the strength of the
  relationship
• Test for significance indicates whether
  relationship could have or could not have
  occurred by chance
• No value in doing regression if no relationship,
  if it could have occurred by chance

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Calculating correlation coefficient
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Correlation between users and persons
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Hypothesis test

• Step 1 state hypothesis
• H0 ? 0, H1 ? ltgt 0
• Step 2 critical value
• a 0.05, d.f. n 2 4, CV r 0.811
• Step 3 test statistic
• r 0.835
• Step 4 make decision
• Reject null hypothesis
• Step 5 summarize the results
• Relationship between users and population, could
  not have occurred by chance

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Idea of regression

• Object is to find the mathematical formula for
  the straight line representing the relationship
• Formula for the regression equation straight
  line
• Need to estimate values for a and b
• a is the y intercept y value where line crosses
  y axis
• b is the slope change in y divided by change in x

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Regression line through data points
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Determining the regression line

• Want to find line that is closest to the data
  points
• Minimize the sum of squared errors
• Take the vertical distance from each point to the
  line (these are the errors)
• Square the errors and sum them
• Find the line the minimizes this sum

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Errors in regression
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Regression analysis

• Formulas for a (intercept) and b (slope) make use
  of the same sums used in calculating the
  correlation coefficient

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Calculating a, intercept
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Calculating b, slope
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Estimated regression equation
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Variation and goodness of fit

• Total variation is the variation of the values of
  y around its mean
• Explained variation is the variation of the
  predicted values of y from the mean
• Unexplained variation is the error, difference in
  actual and predicted values of y

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Variation
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Calculating variation
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Coefficient of determination

• Ratio of explained variation to total
  variation
• Proportion of the total variation explained
  (accounted for) by the regression
• Coefficient of determination is correlation
  coefficient r, squared

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Coefficient of determination

• Coefficient of determination provides an
  alternative way of describing the strength of the
  relationship associated with the estimated
  regression line
• Coefficient of determination r2 0.697 says that
  69.7 of the variation in the dependent variable
  y is accounted for by the independent variable x

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Interpreting the correlation and regression
analysis

• Significance of correlation coefficient says
  there is a linear relationship between the two
  variables in the population that is not likely to
  have occurred in the sample by chance
• If correlation were not significant, it would not
  make sense to proceed with the regression analysis

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What the regression tells you about the
relationship

• Have the estimated regression equation
• Value of the slope b 8.41 tells us that for
  each additional thousand population within 1.5
  miles we would expect an additional 8.41 park
  users

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Using the estimated regression for prediction

• The final application of regression is using the
  estimated regression equation for prediction of y
  for other values of x
• Suppose you are planning to build a new park
• 30 thousand people live within 1.5 miles of the
  new park site
• How many persons are likely to use the new park
  each day?

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Prediction
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SPSS analysis

• Analysis of rent versus family income
• Interactive scatterplot
• Graphs, Chart Builder, Scatter/Dot
• Add regression line in Chart Editor
• Regression analysis
• Analyze, Regression, Linear

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Scatterplot with regression line
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Regression output goodness of fit
Correlation
Coefficient of Determination
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Regression output significance of model
Significance Test of null hypothesis ? 0 or
Population R2 0
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Regression output regression coefficients
b - slope
a y intercept