Data Analysis: Relationships Continued Regression Research Methods Dr. Gail Johnson

Simple Regression Enables us to estimate the: Strength of relationship Expressed as the percent of variance explained How much change you can expect in the dependent variable based on a one unit change in the independent variable Enables you to make predictive estimates

Relationships Correlation is not causation. Statistical measurement includes the measurement of relationships. There are 2 ways to measure the strength of a relationship: 1. How great a difference the independent variable makes on the dependent variable (sometimes called the effect-description). This allows you to predict the effect of the IV on the DV but you have to have interval data.

Relationships 2. How completely the dependent variable is explained by the independent variable. (Correlational). (R squared).

Simple Regression Assumes a linear relationship Interval level (or dichotomous: which means coded 0 or 1) data Independent Variable: interval level Random or census

Simple Regression Y = a + bX + error Where: a = the constant or Y intercept b = the regression coefficient, or slope Y = predicted value of the dependant variable X = the independent variable.

Simple Regression Estimate car repair costs for motor pool Y= car repair costs X = miles driven Collect data and crunch it. You get these results: Y = -267 and .018X

Simple Regression Estimate car repair costs Y = -267 and .018X Interpretation: for every mile driven, the repair costs goes up by 1.8 cents. For every 100 miles driven, costs go up by $1.80

Simple Regression Y = -267 and .018X If you expect the cars to be driven a total of 100,000 miles, how much will car repair costs likely be? 100,000 x .018 = $1,800 Solve equation: Y = -267 + 1,800 = $1,763

Simple Regression r= correlation coefficient (overall fit) (measure of association but non-directional; zero-order correlational coefficient). r2 = proportion of explained variation 1-r2 = proportion of unexplained variation

Life is more complex Rarely will any one single variable cause something to happen Life is inherently multivariate What are the possible causes for urban decay?

What are the possible causes for urban decay? lack of jobs high % of absentee landlords low % of homeowners poor quality of schools increased concentration of poor increase in drugs, crime aging housing stock flight of middle class to suburbs corruption aging infrastructure business flight to suburbs

What caused drop in crime? Changing demographics? Better policing? Strong economy? Gun control laws? Concealed weapons laws? Increased use of death penalty? Increase in number of police? Rising prison population? Waning Crack epidemic? Legalization of abortion?

Multiple Regression Multiple Regression lets you do four things: test your hypothesis predict the dependent variable if you know the values for independent variables Predicts the independent effect of each independent variable while controlling for the others tells you the relative strength of each of the independent variable using the beta weights

Multiple Regression Y = a1 + bX1 + bX2 + bX3 + b X4 + e. Y = dependent variable X1 = independent variable 1, controlling for X2, X3, X4 X2 = independent variable 2 controlling for X1, X3, X4 X3 = independent variable 3 controlling for X1, X2, X4 X4= independent variable 4 controlling for X1, X2, X3

Multiple Regression Income as a function of education and seniority? Y = Income (dep. Var.) Y (Income) = a + education + seniority Y= 6000 + 400X1 + 200X2 based on Lewis-Beck example

Multiple Regression Y= 6000 + 400X1 + 200X2 R square. = .67 67% of the variation in income is explained by these two variables. Excellent! For every year of education, holding seniority constant, income increases by $400. For every year of seniority, holding education constant, income increases by $200.

Multiple Regression Y= 6000 + 400X1 + 200X2 Example: Estimate the income of someone who has 10 years of education and 5 years of seniority: Y=6000 + 400(10) + 200(5) Y= $ 11,000

Multiple Regression Relationship between contributions to political campaigns as a function of age and income? Y= campaign contribution (dollars) x1 = age (years) X2 = income (dollars)

Multiple Regression Relationship between contributions to political campaigns as a function of age and income. Y = 8 + 2X1 + .010X2 (age) (income) For every increase in age, contributions go up by $2. For every increase in income, contributions go up .01 dollars

Multiple Regression Y = 8 + 2X1 + .010X2 Y= campaign contribution (dollars) But which is stronger? Need to look at the Beta weights Age = .15 Income = .45

Beta Weights Workbook: Table 10.15

Quick Analysis Whenever you are dealing with a correlation (regression analysis) First check the R squared value. A good study will have this If it is low, then you know that it is not a strong model and they shouldn’t be making grand conclusions Make sure they meet 4 conditions necessary for causality

Example Study tried to determine what explained why some cities introduced reinvention. Sent out a survey, respectable response rate Tested 13 factors they thought would explain reinvention R squared was .05 What do you conclude?

Example They ran a second model Included managers’ attitudes about innovation R squared was .22 What do you conclude?

The Levitt Article What data does he show? What kind of question is he asking? Does he show correlations? Does he build a multivariate model? Did anyone see an R squared in this article?