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Regression Analysis with SPSS Robert A. Yaffee, Ph.D. Statistics, Mapping and Social Science Group Academic Computing Services Information Technology Services New York University Office: 75 Third Ave Level C3 Tel: 212.998.3402 E-mail: yaffee@nyu.edu February 04 1 Outline 1. Conceptualization 2. Schematic Diagrams of Linear Regression processes 3. Using SPSS, we plot and test relationships for linearity 4. Nonlinear relationships are transformed to linear ones 5. General Linear Model 6. Derivation of Sums of Squares and ANOVA Derivation of intercept and regression coefficients 7. The Prediction Interval and its derivation 8. Model Assumptions 1. Explanation 2. Testing 3. Assessment 9. Alternatives when assumptions are unfulfilled 2 Conceptualization of Regression Analysis • Hypothesis testing • Path Analytical Decomposition of effects 3 Hypothesis Testing • For example: hypothesis 1 : X is statistically significantly related to Y. – The relationship is positive (as X increases, Y increases) or negative (as X decreases, Y increases). – The magnitude of the relationship is small, medium, or large. If the magnitude is small, then a unit change in x is associated with a small change in Y. 4 Regression Analysis Have a clear notion of what you can and cannot do with regression analysis • Conceptualization – A Path Model of a Regression Analysis Path Diagram of A Linear Regression Analysis X1 error YY X2 x3 Yi k b1 x1 b2 x2 b3 x3 ei 5 A Path Analysis Decomposition of Effects into Direct, Indirect, Spurious, and Total Effects Error Error X2 C Y3 A X1 E F B Y2 D Error Y1 Error Direct Effects: Indirect Effects: Total Effects: Paths C, E, F Paths Sum of Direct and Spurious effects are due to AC, BE, DF Indirect Effects common (antecedent) causes In a path analysis, Yi is endogenous. It is the outcome of several paths. Direct effects on Y3: C,E, F Indirect effects on Y3: BF, BDF 6 Total Effects= Direct + Indirect effects Interaction Analysis X1 A C Y B X2 Y= K + aX1 + BX2 + CX1*X2 Interaction coefficient: C X1 and X2 must be in model for interaction to be properly specified. 7 A Precursor to Modeling with Regression • Data Exploration: Run a scatterplot matrix and search for linear relationships with the dependent variable. 8 Click on graphs and then on scatter 9 When the scatterplot dialog box appears, select Matrix 10 A Matrix of Scatterplots will appear Search for distinct linear relationships 11 12 13 Decomposition of the Sums of Squares 14 Graphical Decomposition of Effects Decomposition of Effects Y Yi y a bx ˆ yi yi error ˆ yi y Total Effect Y y y regression effect ˆ X X 15 Decomposition of the sum of squares ˆ Y Y Y Y Y Y ˆ total effect error effects regression (model ) effect ˆ ˆ Yi Y Yi Yi Yi Y per case i ˆ ˆ (Y Y ) 2 (Y Y ) 2 (Y Y ) 2 per case i i i i i n n n (Y Y ) i 1 i 2 (Yˆ Y ) i 1 i i 2 (Yˆ Y ) i 1 i 2 for data set 16 Decomposition of the sum of squares • Total SS = model SS + error SS and if we divide by df n n n (Yi Y ) 2 ˆ (Yi Yi ) 2 ˆ (Yi Y ) 2 i 1 i 1 i 1 n1 nk 1 k • This yields the Variance Decomposition: We have the total variance= model variance + error variance 17 F test for significance and R2 for magnitude of effect • R2 = Model var/total var n ˆ (Yi Y ) 2 i 1 R2 k n ˆ (Y Y ) 2 i 1 i i nk 1 •F test for model significance = Model Var/Error Var 2 R F( k ,n k 1) k 1 R2 nk 1 18 ANOVA tests the significance of the Regression Model 19 The Multiple Regression Equation • We proceed to the derivation of its components: – The intercept: a – The regression parameters, b1 and b2 Yi a b1 x1 b2 x2 ei 20 Derivation of the Intercept y a bx e e y a bx n n n n e i 1 i y a i 1 i i 1 i b xi i 1 n Because by definition ei 0 i 1 n n n 0 y a i 1 i i 1 i b xi i 1 n n n ai yi b xi i 1 i 1 i 1 n n na yi b xi i 1 i 1 a y bx 21 Derivation of the Regression Coefficient Given : yi a b xi ei ei yi a b xi n n e i 1 i (y i 1 i a b xi ) n n ei i 1 2 ( yi a b xi ) 2 i 1 n ei 2 n n i 1 2 xi ( yi ) 2b xi xi b i 1 i 1 n n 0 2 xi ( yi ) 2b xi xi i 1 i 1 n x y i i b i 1 n xi 2 i 1 22 • If we recall that the formula for the correlation coefficient can be expressed as follows: 23 n x i yi r i 1 x y n n 2 2 i i i 1 i 1 where x xi x y yi y n xi yi bj i 1 n i 1 x2 from which it can be seen that the regression coefficient b, is a function of r. sd y bj r * sd x 24 Extending the bivariate case To the Multiple linear regression case 25 ryx1 ryx2 rx1x2 sd y yx . x * (6) 1 2 1 r 2 x1 x2 sd x ryx2 ryx1 rx1x2 sd y yx . x * (7) 2 1 1 r 2 x1 x2 sd x It is also easy to extend the bivariate intercept to the multivariate case as follows. a Y b1 x1 b2 x2 (8) 26 Significance Tests for the Regression Coefficients 1. We find the significance of the parameter estimates by using the F or t test. 2. The R2 is the proportion of variance explained. 2 (n-1) 3.Adjusted R = 1-(1-R ) 2 (n-p-1) where n sample size p number of parameters in model 27 F and T tests for significance for overall model Model variance F error variance R2 / p (1 R 2 ) /(n p 1) where p number of parameters n sample size t F ( n 2) * r 2 1 r 2 28 Significance tests • If we are using a type II sum of squares, we are dealing with the ballantine. DV Variance explained = a + b 29 Significance tests T tests for statistical significance 0 t sea b0 t seb 30 Significance tests Standard Error of intercept SEa (Y Y ) 2 1 * xi 2 n2 n ( n 1) ( xi x ) 2 Standard error of regression coefficient ˆ SEb x 2 where std dev of residual ˆ n e 2 ˆ 2 i 1 31 n2 Programming Protocol After invoking SPSS, procede to File, Open, Data 32 Select a Data Set (we choose employee.sav) and click on open 33 We open the data set 34 To inspect the variable formats, click on variable view on the lower left 35 Because gender is a string variable, we need to recode gender into a numeric format 36 We autorecode gender by clicking on transform and then autorecode 37 We select gender and move it into the variable box on the right 38 Give the variable a new name and click on add new name 39 Click on ok and the numeric variable sex is created It has values 1 for female and 2 for male and those values labels are inserted. 40 To invoke Regression analysis, Click on Analyze 41 Click on Regression and then linear 42 Select the dependent variable: Current Salary 43 Enter it in the dependent variable box 44 Entering independent variables • These variables are entered in blocks. First the potentially confounding covariates that have to entered. • We enter time on job, beginning salary, and previous experience. 45 After entering the covariates, we click on next 46 We now enter the hypotheses we wish to test • We are testing for minority or sex differences in salary after controlling for the time on job, previous experience, and beginning salary. • We enter minority and numeric gender (sex) 47 After entering these variables, click on statistics 48 We select the following statistics from the dialog box and click on continue 49 Click on plots to obtain the plots dialog box 50 We click on OK to run the regression analysis 51 Navigation window (left) and output window(right) This shows that SPSS is reading the variables correctly 52 Variables Entered and Model Summary 53 Omnibus ANOVA Significance Tests for the Model at each stage of the analysis 54 Full Model Coefficients CurSal 12036.3 1.83BeginSal 165.17Jobtime 23.64 Exper 2882.84 gender 1419.7 Minority 55 We omit insignificant variables and rerun the analysis to obtain trimmed model coefficients CurSal 12126.5 1.85BeginSal 163.20Jobtime 24.36 Exper 2694.30 gender 56 Beta weights • These are standardized regression coefficients used to compare the contribution to the explanation of the variance of the dependent variable within the model. 57 T tests and signif. • These are the tests of significance for each parameter estimate. • The significance levels have to be less than .05 for the parameter to be statistically significant. 58 Assumptions of the Linear Regression Model 1. Linear Functional form 2. Fixed independent variables 3. Independent observations 4. Representative sample and proper specification of the model (no omitted variables) 5. Normality of the residuals or errors 6. Equality of variance of the errors (homogeneity of residual variance) 7. No multicollinearity 8. No autocorrelation of the errors 9. No outlier distortion 59 Explanation of the Assumptions 1. 1. Linear Functional form 1. Does not detect curvilinear relationships 2. Independent observations 1. Representative samples 2. Autocorrelation inflates the t and r and f statistics and warps the significance tests 3. Normality of the residuals 1. Permits proper significance testing 4. Equality of variance 1. Heteroskedasticity precludes generalization and external validity 2. This also warps the significance tests 5. Multicollinearity prevents proper parameter estimation. It may also preclude computation of the parameter estimates completely if it is serious enough. 6. Outlier distortion may bias the results: If outliers have high influence and the sample is not large enough, then they may serious bias the parameter estimates 60 Diagnostic Tests for the Regression Assumptions 1. Linearity tests: Regression curve fitting 1. No level shifts: One regime 2. Independence of observations: Runs test 3. Normality of the residuals: Shapiro-Wilks or Kolmogorov-Smirnov Test 4. Homogeneity of variance if the residuals: White’s General Specification test 5. No autocorrelation of residuals: Durbin Watson or ACF or PACF of residuals 6. Multicollinearity: Correlation matrix of independent variables.. Condition index or condition number 7. No serious outlier influence: tests of additive outliers: Pulse dummies. 1. Plot residuals and look for high leverage of residuals 2. Lists of Standardized residuals 3. Lists of Studentized residuals 4. Cook’s distance or leverage statistics 61 Explanation of Diagnostics 1. Plots show linearity or nonlinearity of relationship 2. Correlation matrix shows whether the independent variables are collinear and correlated. 3. Representative sample is done with probability sampling 62 Explanation of Diagnostics Tests for Normality of the residuals. The residuals are saved and then subjected to either of: Kolmogorov-Smirnov Test: Tests the limit of the theoretical cumulative normal distribution against your residual distribution. Nonparametric Tests 1 sample K-S test 63 Collinearity Diagnostics Tolerance 1R 2 small tolerances imply problems Variance Inflation Factor (VIF) 1 Tolerance Small intercorrelations among indep vars means VIF 1 VIF 10 signifies problems 64 More Collinearity Diagnostics condition numbers = maximum eigenvalue/minimum eigenvalue. If condition numbers are between 100 and 1000, there is moderate to strong collinearity condition index k where k condition number If Condition index > 30 then there is strong collinearity 65 Outlier Diagnostics 1. Residuals. 1. The predicted value minus the actual value. This is otherwise known as the error. 2. Studentized Residuals 1. the residuals divided by their standard errors without the ith observation 3. Leverage, called the Hat diag 1. This is the measure of influence of each observation 4. Cook’s Distance: 1. the change in the statistics that results from deleting the observation. Watch this if it is much greater than 1.0. 66 Outlier detection • Outlier detection involves the determination whether the residual (error = predicted – actual) is an extreme negative or positive value. • We may plot the residual versus the fitted plot to determine which errors are large, after running the regression. 67 Create Standardized Residuals • A standardized residual is one divided by its standard deviation. yi yi ˆ resid standardized s where s std dev of residuals 68 Limits of Standardized Residuals If the standardized residuals have values in excess of 3.5 and -3.5, they are outliers. If the absolute values are less than 3.5, as these are, then there are no outliers While outliers by themselves only distort mean prediction when the sample size is small enough, it is important to gauge the influence of outliers. 69 Outlier Influence • Suppose we had a different data set with two outliers. • We tabulate the standardized residuals and obtain the following output: 70 Outlier a does not distort and outlier b does. 71 Studentized Residuals • Alternatively, we could form studentized residuals. These are distributed as a t distribution with df=n-p-1, though they are not quite independent. Therefore, we can approximately determine if they are statistically significant or not. • Belsley et al. (1980) recommended the use of studentized residuals. 72 Studentized Residual ei ei s s 2 (i ) (1 hi ) where ei s studentized residual s(i ) standard deviation where ith obs is deleted hi leverage statistic These are useful in estimating the statistical significance of a particular observation, of which a dummy variable indicator is formed. The t value of the studentized residual will indicate whether or not that observation is a significant outlier. The command to generate studentized residuals, called rstudt is: predict rstudt, rstudent 73 Influence of Outliers 1. Leverage is measured by the diagonal components of the hat matrix. 2. The hat matrix comes from the formula for the regression of Y. ˆ Y X X '( X ' X ) 1 X ' Y where X '( X ' X ) 1 X ' the hat matrix, H Therefore, ˆ Y HY 74 Leverage and the Hat matrix 1. The hat matrix transforms Y into the predicted scores. 2. The diagonals of the hat matrix indicate which values will be outliers or not. 3. The diagonals are therefore measures of leverage. 4. Leverage is bounded by two limits: 1/n and 1. The closer the leverage is to unity, the more leverage the value has. 5. The trace of the hat matrix = the number of variables in the model. 6. When the leverage > 2p/n then there is high leverage according to Belsley et al. (1980) cited in Long, J.F. Modern Methods of Data Analysis (p.262). For smaller samples, Vellman and Welsch (1981) suggested that 3p/n is the criterion. 75 Cook’s D 1. Another measure of influence. 2. This is a popular one. The formula for it is: 1 hi ei 2 Cook ' s Di 2 p 1 hi s (1 hi ) Cook and Weisberg(1982) suggested that values of D that exceeded 50% of the F distribution (df = p, n-p) are large. 76 Using Cook’s D in SPSS • Cook is the option /R • Finding the influential outliers • List cook, if cook > 4/n • Belsley suggests 4/(n-k-1) as a cutoff 77 DFbeta • One can use the DFbetas to ascertain the magnitude of influence that an observation has on a particular parameter estimate if that observation is deleted. b j b(i ) j u j DFbeta j u 2 j (1 h j ) where u j residuals of regression of x on remaining xs. 78 Programming Diagnostic Tests Testing homoskedasiticity Select histogram, normal probability plot, and insert *zresid in Y and *zpred in X Then click on continue 79 Click on Save to obtain the Save dialog box 80 We select the following Then we click on continue, go back to the Main Regression Menu and click on OK 81 Check for linear Functional Form • Run a matrix plot of the dependent variable against each independent variable to be sure that the relationship is linear. 82 Move the variables to be graphed into the box on the upper right, and click on OK 83 Residual Autocorrelation check Durbin Watson d tests first order autocorrelation of residuals d n et et 1 2 i 1 et See significance tables for this statistic 84 Run the autocorrelation function from the Trends Module for a better analysis 85 Testing for Homogeneity of variance 86 Normality of residuals can be visually inspected from the histogram with the superimposed normal curve. Here we check the skewness for symmetry and the kurtosis for peakedness 87 Kolmogorov Smirnov Test: An objective test of normality 88 89 90 Multicollinearity test with the correlation matrix 91 92 93 Alternatives to Violations of Assumptions • 1. Nonlinearity: Transform to linearity if there is nonlinearity or run a nonlinear regression • 2. Nonnormality: Run a least absolute deviations regression or a median regression (available in other packages or generalized linear models [ SPLUS glm, STATA glm, or SAS Proc MODEL or PROC GENMOD)]. • 3. Heteroskedasticity: weighted least squares regression (SPSS) or white estimator (SAS, Stata, SPLUS). One can use a robust regression procedure (SAS, STATA, or SPLUS) to obtain downweighted outlier effect in the estimation. • 4. Autocorrelation: Run AREG in SPSS Trends module or either Prais or Newey-West procedure in STATA. • 4. Multicollinearity: components regression or ridge regression or proxy variables. 2sls in SPSS or ivreg in stata or SAS proc model or proc syslin. 94 Model Building Strategies • Specific to General: Cohen and Cohen • General to Specific: Hendry and Richard • Extreme Bounds analysis: E. Leamer. 95 Nonparametric Alternatives 1. If there is nonlinearity, transform to linearity first. 2. If there is heteroskedasticity, use robust standard errors with STATA or SAS or SPLUS. 3. If there is non-normality, use quantile regression with bootstrapped standard errors in STATA or SPLUS. 4. If there is autocorrelation of residuals, use Newey-West autoregression or First order autocorrelation correction with Areg. If there is higher order autocorrelation, use Box Jenkins ARIMA modeling. 96