Multivariate Data/Statistical Analysis

1. Multivariate Data/Statistical Analysis SC504/HS927 Spring Term 2008 Week 18: Relationships between variables: simple ordinary least squares (OLS) regression

2. Outline • What is regression analysis? • Scatter plots • Linear regression • Terminology and notation • Interpreting a regression equation • Putting it into practice

3. What is regression analysis? A statistical technique for: • analysing the association between variables (e.g. how is alcohol consumption related to income on average ?) • making conditional predictions (e.g. what do we expect to happen to smoking behaviour if tobacco taxes increase?) • testing hypotheses about the nature of conditional relationships (e.g. on average do crime rates vary in proportion to unemployment rates?) • summarizing/describing data on 2+ variables

4. Scatterplot of suicide against unemployment rates

5. How do we summarise the relationship between suicide and unemployment rates? • Assume a straight-line (linear) relationship between suicide rate (y) and unemployment rate (x): y=a + bx • Estimate a and b by applying ordinary least squares regression to the data in the scatter plot: estimate of a = 1.435 estimate of b = 0.324

6. Method of Least Squares • A method of finding the line that best fits the data • The line of ‘best fit’ is found by ascertaining which line, of all possible lines, results in the least amount of difference between observed data points and the line

7. Scatter with fitted line

8. Interpretation y=1.435 + 0.324x • if unemployment (x) is zero, suicide rates are predicted to be 1.435 per 100,000 population • each 1 percentage point increase in unemployment increases the predicted suicide rate by 0.324 • relationship between y and x is not exact so we usually write: y=a + bx + e

9. Terminology and notation yi=a + bxi + e • xi and yi are variables which have different values for each individual/ observation • they vary across cases in dataset (i refers to case (individual) i) • y=dependent variable • x=independent variable • a and b are unknown (not observed) constants • a and b are population parameters • a and b are to be estimated from sample data • e is error/disturbance/residual term

10. a is the y-axis intercept y a 0 x

11. b is the slope or coefficient of x y b 1 a 0 x

12. A note on causality • Just because we write: yi=a + bxi + e • Does not mean x causes y • Suppose y = income, x = whether or not someone is an owner-occupier • would turning renters into homeowners increase their incomes? • or is it that you need a good income to be able to purchase a home? • or that people on low incomes are more likely to be eligible for social rented housing

13. What is the relationship between suicide and unemployment? • Which is your ‘dependent’ variable? • Use Graphs – scatter- simple- define-OK • Double click on chart. Go to: Elements-Fit line at Total. You can also change axes by going to: Edit- Select Y [X] axis • For the values, use Analyse – regression - linear

14. SPSS Output R = .702 (simple correlation between suicide and unemployment) R² = .493 (unemployment rates can account for 49% of the variation in suicide rates)

15. a = intercept (constant) = 1.435 b = gradient (unemployment rate per 100) = .324 In 1997, the unemployment rate was 1 (per 100) therefore…… Suicide rate = 1.435 + .324 x 1 = 1.759 (per 100000)