B&A 448-452; and http://www.youtube.com/watch?v=1b18fO9Vt50 REGRESSION - ANCOVA B&A 448-452; and http://www.youtube.com/watch?v=1b18fO9Vt50

Technique: Statistics of the straight line We have multiple data points (not just two) We need to find a line that best fits them Define “best fit” – Minimum squared deviations slope intercept Independent variable Error Dependent variable Regression coefficients

Technique: Statistics of the straight line The line that “best fits” has the following slope: Standard deviation of y Standard deviation of x Regression coefficient of y on x. Correlation coefficient of x and y The line that “best fits” has the following intercept:

Standard error of prediction Regression does not provide an exact prediction At each value of x, we expect that the value of y will be between two values, most of the time We can estimate a confidence interval for predicted values of y, if we assume that at each value of x, the values of y will be distributed normally with a given standard deviation. Standard deviation of all possible predicted y values obtained from all possible samples. Confidence interval of prediction:

Components of the total variance Standard deviation of all possible predicted y values obtained from all possible samples. Variance that cannot be accounted for by regression Variance that can be accounted for by regression Total variance

Things to know A significant regression coefficient does not “prove” causality A regression can provide “predicted” values Predicted values are “point” estimates Confidence intervals around the point estimates provide very important information

THIS IS A TABLE YOU SHOULD KNOW BEFORE ANALYZING YOUR DATA AND THE FINAL Dependent Variable Independent Variable(s) Method Nominal/Ordinal Chi-Square test Interval/Ratio T-test (2 groups) or ANOVA (3+ groups) Multifactorial ANOVA (2+ IVs) Regression Interval/Ratio AND ANCOVA/GLM

Example Analyses Using all Information

Regression exercise with 2008 SOCI 301 data Attitudes towards marriage will be influenced by gender role attitudes. Gender role attitudes will be influenced by sex and maternal education. Sex Negative attitudes towards marriage Non egalitarian gender role attitudes Maternal education

Are gender role attitudes correlated with the attitudes towards marriage

Sex differences in gender role attitudes

Maternal education differences in gender role attitudes

Sex and maternal education differences in gender role attitudes

Differential effects of maternal education depending on sex?

Profile plot

Do gender role attitudes influence the attitudes towards marriage?

Can I do this in GLM UNIANOVA? Attention: You must ask for parameter estimates from GLM You must specify interval/ratio IV s as “covariates”

Can we do the same in GLM (UNIANOVA)?

Can we include the nominal/ordinal variables too? Attention: You must include nominal/ordinal variables as “fixed factors” You must specify interval/ratio IV s as “covariates”

ANCOVA Model

Assumptions The interval/ratio level independent variable has a LINEAR association with the DV. Error term has FIXED variance (same as regression).

Is the covariate desirable? What if the factor is a predictor of the covariate? Current example What is the factor is independent of the covariate? Covariate adds information to the model