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Research Methods I Lecture 10: Regression Analysis on SPSS Introduction • Lecture 8: looked at descriptive statistics; relationships between variables: correlation; cross-tabulation • This lecture: regression • Focus on how regression can be done on SPSS • Focus on OLS, although SPSS is capable of WLS and logistic regression Regression: fundamentals • Relationship between two variables lead to prediction of values of a dependent variable • Prediction of Yi based on Y(bar), model (effect of X on Y) and residual • Regression equation: Y^ = a^ + b^X + e^ Regression: fundamentals • Actual values of X will predict values of Y^ based upon intercept term (a^), slope coefficient (b^) and residual (e^) • Ordinary Least Squares finds values of a^ and b^ which minimise sum of squared residuals from the regression • Under certain assumptions, OLS is BLUE Evaluating the regression • SPSS allows various ways to evaluate the generated regression equation: • Goodness of fit • Individual significance of regressors • Tests of classical assumptions (necessary for inference) Goodness of fit • R2=ESS/TSS • F-test: F=ESS/TSS • Check outliers via standardised residuals or studentised residuals • Can check for significant outliers, i.e., ones which would affect the value of the regressors Significant outliers • • • • Cook’s distance (critical>1) Leverage values (critical: > 2(k+1)/n Mahalanobis distance Covariance ratio: if CVRi>/<1 + [3(k+1)/n], deleting case damages/improves parameter precision • Casewise diagnostics • CI of regression coefficients Methods of regression • Hierarchical: when there is a good statistical or theoretical reason for including one variable first • Forward (specific to general) or Backward (general to specific): based on statistical criteria • Stepwise: forward + removal test • Use Enter command to do usual regression Regression equation • Individual coefficients: unstandardised b and standardised b • SPSS will give t-statistics, s.e.(b), and p-value for significance of b • If p<0.05, coefficient significant at 5% • This is all OK for description but for inference, require specific assumptions: need to test these “Diagnostic” tests • No direct specification test (no equivalent of RESET in Microfit) • Variation in ind. variables (2≠0): test by prior observation • SPSS provides many tests for multicollinearity: e.g. VIF (critical value=1), tolerance (=1/VIF), covariance matrix, partial correlations between regressors “Diagnostic” tests • Autocorrelation: DW test (DW(2) not directly available) • Linearity and homoscedasticity can be tested via scatter plots of standardised residuals versus standardised predicted residuals (from the assump) • Normality: ask SPSS for histogram and P-P plot of the regression residuals (can also do K-S and S-W tests) Conclusions • SPSS is capable of doing regression analysis of various types • SPSS offers a range of tests of specification, fit and underlying assumptions of regression • Tests are more extensive but less userfriendly than a package like Microfit