Multiple Regression

What Techniques Can Tell Us Chi Square- Do groups differ (nominal data)? T Test Do Groups/Variables differ? Gamma/Lambda/Kendall’s Tau etc Are variables related to each other? (nominal data) Correlation Are variables related to each other? (ratio/interval data)

Interpreting Correlations 3 questions we can answer 1.Is there a relationship between 2 variables? 2.What is the direction of the relationship? 3.What is the Strength of a relationship

Interpreting Correlations Are there limitations here? And if so, what? Don’t know amount of effect of one variable on other Don’t know impact of other variables

Strength

Strong Relationships

Perfect Relationship

Basic Equations Let your DV (Y)= total cost of bananas Suppose you buy X lbs of bananas at $.49 a lb How would you express this as an equation to figure out how much your bananas are worth? Y=.49 X Can use for prediction 10lbs=$4.90 2lbs=$.98

Multivariate Equations Suppose you have a phone plan that charges –$5.95 a month –$.10 a minute instate long distance –$.08 a minute interstate long distance –$.01 a minute Local Calls How would you represent? Total=.1x x x

Regression Analysis Lets you work the problem Backwards How much do different IVs contribute to a DV How do different IVs relate to DV Lets you build a model of more complicated relationships In addition to existence, direction, strength, gives you the amount of change

Expressing A regression equation Y=b 1 x 1 +b 2 x 2 +…..b i x i +constant+error Error is part of probabilistic nature of social science Constant- what Y would equal if all Xs=0 Estimation process- fit a line to data that minimizes the distance to all observed data points

Scatter Plots and Regression Lines PID and Ideology Correlation here is.37, not bad, but you can see, there are deviations in some cases

Fitting the Regression Line Goal: Minimize the squared distances (error) between predicted values of Y and observed values. Goal, explain the variance in Y in terms of X Error in prediction is unexplained variance

Party and Ideology Set up PID as DV, Ideology as IV, run analysis Can also do Ideology as DV

Goodness of Fit Measure of how much variance is explained by model you build R 2 = correlation coefficient squared R 2 = proportion of variance explained R 2 is symetrical In previous example R 2 =.256 R 2 ranges from 0-1 Adjusted R 2 takes into account the degrees of freedom, more appropriate measure

Run for the Border Using Multiple Regression Suppose that you and some friends ate at Taco bell every week for a year. For each meal, you know the total amount spent, and the number of each item, but not what each item cost. You could use multiple regression to get parameter estimates of the true values. Data set was constructed by choosing a random number (Between 0 and 4) of Bean Burritos, Tacos, Chalupas, Chicken Tacos, Beef Burritos, 7 Layer Burritos, and Soft drinks Data matrix includes a variable for number of each

Border Model 1 We’ll look at impact of bean burritos on total

Border Model 2 Bean Burritos and Tacos

Border Model 3

Model 4

Model 5

Full Model

Model 4 Revisited Bean Burrito-.69,Taco.79, Chalupa 1.19, Chicken taco 1.39, Beef Burrito 1.59,7 layer 1.89, Drink 1.29

Some Data Requirements for Regression DV must be interval or ratio, and continuous IVs should not be correlated with each other Error should be constant at high and low predicted value (homoschedasticity) Relationship must be linear Errors of subsequent observations should not be correlated (no serial correlation)

For Next time Multicolinearity Heteroskedasticity Interaction terms Pass out Stat Assignment II