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The Use of Dummy Variables • In the examples so far the independent variables are continuous numerical variables. • Suppose that some of the independent variables are categorical. • Dummy variables are artificially defined variables designed to convert a model including categorical independent variables to the standard multiple regression model. Example: Comparison of Slopes of k Regression Lines with Common Intercept Situation: • k treatments or k populations are being compared. • For each of the k treatments we have measured both – Y (the response variable) and – X (an independent variable) • Y is assumed to be linearly related to X with – the slope dependent on treatment (population), while – the intercept is the same for each treatment The Model: Y 1( i ) X for treatm ent i (i 1, 2, ... , k) Graphical Illustration of the above Model 120 Treat k 100 Treat 3 ..... Treat 2 80 Treat 1 y 60 40 Different Slopes 20 Common Intercept 0 0 10 x 20 30 • This model can be artificially put into the form of the Multiple Regression model by the use of dummy variables to handle the categorical independent variable Treatments. • Dummy variables are variables that are artificially defined In this case we define a new variable for each category of the categorical variable. That is we will define Xi for each category of treatments as follows: X if the subject receives treatment i Xi 0 otherwise Then the model can be written as follows: The Complete Model: Y 0 X1 1 (1) 1 ( 2) X 2 1 (k ) Xk where X if the subject receives treatmenti Xi 0 otherwise In this case Dependent Variable: Y Independent Variables: X1, X2, ... , Xk In the above situation we would likely be interested in testing the equality of the slopes. Namely the Null Hypothesis H0 : 1 (1) 1 ( 2) 1 (k ) (q = k – 1) The Reduced Model: Y 0 1 X Dependent Variable: Y Independent Variable: X = X1+ X2+... + Xk Example: In the following example we are measuring – Yield Y as it depends on – the amount (X) of a pesticide. Again we will assume that the dependence of Y on X will be linear. (I should point out that the concepts that are used in this discussion can easily be adapted to the non- linear situation.) • Suppose that the experiment is going to be repeated for three brands of pesticides: • A, B and C. • The quantity, X, of pesticide in this experiment was set at 4 different levels: – 2 units/hectare, – 4 units/hectare and – 8 units per hectare. • Four test plots were randomly assigned to each of the nine combinations of test plot and level of pesticide. • Note that we would expect a common intercept for each brand of pesticide since when the amount of pesticide, X, is zero the four brands of pesticides would be equivalent. The data for this experiment is given in the following table: 2 4 8 A 29.63 28.16 28.45 31.87 33.48 37.21 28.02 28.13 35.06 35.24 28.25 33.99 B 32.95 29.55 44.38 24.74 34.97 38.78 23.38 36.35 34.92 32.08 38.38 27.45 C 28.68 33.79 46.26 28.70 43.95 50.77 22.67 36.89 50.21 30.02 33.56 44.14 60 40 A B C 20 0 0 1 2 3 4 5 6 7 8 Pesticide X (Amount) X1 X2 X3 Y A 2 2 0 0 29.63 A 2 2 0 0 31.87 A 2 2 0 0 28.02 The data as it would appear in a data A 2 2 0 0 35.24 file. B 2 0 2 0 32.95 The variables X1, X2 and X3 are the B 2 0 2 0 24.74 “dummy” variables B 2 0 2 0 23.38 B 2 0 2 0 32.08 C 2 0 0 2 28.68 C 2 0 0 2 28.70 C 2 0 0 2 22.67 C 2 0 0 2 30.02 A 4 4 0 0 28.16 A 4 4 0 0 33.48 A 4 4 0 0 28.13 A 4 4 0 0 28.25 B 4 0 4 0 29.55 B 4 0 4 0 34.97 B 4 0 4 0 36.35 B 4 0 4 0 38.38 C 4 0 0 4 33.79 C 4 0 0 4 43.95 C 4 0 0 4 36.89 C 4 0 0 4 33.56 A 8 8 0 0 28.45 A 8 8 0 0 37.21 A 8 8 0 0 35.06 A 8 8 0 0 33.99 B 8 0 8 0 44.38 B 8 0 8 0 38.78 B 8 0 8 0 34.92 B 8 0 8 0 27.45 C 8 0 0 8 46.26 C 8 0 0 8 50.77 C 8 0 0 8 50.21 C 8 0 0 8 44.14 Fitting the complete model : ANOVA df SS MS F Significance F Regression 3 1095.815813 365.2719378 18.33114788 4.19538E-07 Residual 32 637.6415754 19.92629923 Total 35 1733.457389 Coefficients Intercept 26.24166667 X1 0.981388889 X2 1.422638889 X3 2.602400794 Fitting the reduced model : ANOVA df SS MS F Significance F Regression 1 623.8232508 623.8232508 19.11439978 0.000110172 Residual 34 1109.634138 32.63629818 Total 35 1733.457389 Coefficients Intercept 26.24166667 X 1.668809524 The Anova Table for testing the equality of slopes df SS MS F Significance F common slope 1 623.8232508 623.8232508 31.3065283 3.51448E-06 zero Slope comparison 2 471.9925627 235.9962813 11.84345766 0.000141367 Residual 32 637.6415754 19.92629923 Total 35 1733.457389 Example: Comparison of Intercepts of k Regression Lines with a Common Slope (One-way Analysis of Covariance) Situation: • k treatments or k populations are being compared. • For each of the k treatments we have measured both Y (then response variable) and X (an independent variable) • Y is assumed to be linearly related to X with the intercept dependent on treatment (population), while the slope is the same for each treatment. • Y is called the response variable, while X is called the covariate. The Model: Y 0(i ) 1 X for treatm ent i (i 1, 2, ... , k) Graphical Illustration of the One-way Analysis of Covariance Model 200 Treat k Treat 3 y Treat 2 100 Treat 1 Common Slopes 0 x 0 10 20 30 Equivalent Forms of the Model: 1) Y i 1 X X for treatm ent i i adjusted mean for treatm ent i 2) Y i 1 X X for treatm ent i i overall adjusted mean response i adjusted effect for treatm ent i i i • This model can be artificially put into the form of the Multiple Regression model by the use of dummy variables to handle the categorical independent variable Treatments. In this case we define a new variable for each category of the categorical variable. That is we will define Xi for categories I i = 1, 2, …, (k – 1) of treatments as follows: 1 if the subject receives treatmenti Xi 0 otherwise Then the model can be written as follows: The Complete Model: Y 0 1 X 1 2 X 2 k 1 X k 1 1 X where 1 if the subject receives treatmenti Xi 0 otherwise In this case Dependent Variable: Y Independent Variables: X1, X2, ... , Xk-1, X In the above situation we would likely be interested in testing the equality of the intercepts. Namely the Null Hypothesis H 0 : 1 2 k 1 0 (q = k – 1) The Reduced Model: Y 0 1 X Dependent Variable: Y Independent Variable: X Example: In the following example we are interested in comparing the effects of five workbooks (A, B, C, D, E) on the performance of students in Mathematics. For each workbook, 15 students are selected (Total of n = 15×5 = 75). Each student is given a pretest (pretest score ≡ X) and given a final test (final score ≡ Y). The data is given on the following slide The data Workbook A Workbook B Workbook C Workbook D Workbook E Pre Post Pre Post Pre Post Pre Post Pre Post 43.0 46.4 43.6 52.5 57.5 61.9 59.9 56.1 43.2 46.0 55.3 43.9 45.2 61.8 49.3 57.5 50.5 49.6 60.7 59.7 59.4 59.7 54.2 69.1 48.0 52.5 45.0 46.1 42.7 45.4 51.7 49.6 45.5 61.7 31.3 42.9 55.0 53.2 46.6 44.3 53.0 49.3 43.4 53.3 65.3 74.5 52.6 50.8 42.6 46.5 48.7 47.1 50.1 57.4 47.1 48.9 62.8 60.1 25.6 38.4 45.4 47.4 36.2 48.7 34.8 47.2 41.4 49.5 52.5 57.7 42.1 33.3 55.1 61.9 53.9 59.8 62.1 58.3 51.2 47.1 60.0 53.2 48.9 55.0 42.7 49.6 56.4 58.1 48.8 50.4 32.4 34.1 52.9 63.3 47.6 55.6 54.2 56.8 44.1 52.7 74.4 66.7 51.7 64.7 56.1 62.4 51.6 46.1 73.8 73.6 43.2 43.2 55.3 66.4 39.7 52.1 63.3 56.0 52.6 50.8 44.5 42.5 45.2 59.4 32.3 49.7 37.3 48.8 67.8 66.8 47.1 51.3 37.6 56.9 59.5 67.1 39.2 45.1 42.9 47.2 57.0 48.9 41.7 51.3 46.2 55.2 62.1 58.0 51.7 57.0 The Model: Y 0( i ) 1 X for workbook i (i A, B, C , D, E ) Graphical display of data 80 70 60 50 Final Score 40 Workbook A 30 Workbook B Workbook C 20 Workbook D 10 Workbook E 0 0 20 40 60 80 Pretest Score Some comments 1. The linear relationship between Y (Final Score) and X (Pretest Score), models the differing aptitudes for mathematics. 2. The shifting up and down of this linear relationship measures the effect of workbooks on the final score Y. The Model: Y 0( i ) 1 X for workbook i (i A, B, C , D, E ) Graphical Illustration of the One-way Analysis of Covariance Model 200 A B y C 100 D Common Slopes 0 x 0 10 20 30 Pre Final Workbook 43 46.4 A The data as it would appear in a data 55.3 43.9 A file. 59.4 59.7 A 51.7 49.6 A 53 49.3 A 48.7 47.1 A 45.4 47.4 A 42.1 33.3 A 54.2 56.8 D 60 53.2 A 51.6 46.1 D 32.4 34.1 A 63.3 56 D 74.4 66.7 A 37.3 48.8 D 43.2 43.2 A 39.2 45.1 D 44.5 42.5 A 62.1 58 D 47.1 51.3 A 43.2 46 E 57 48.9 A 43.6 52.5 B 60.7 59.7 E 45.2 61.8 B 42.7 45.4 E 54.2 69.1 B 46.6 44.3 E 45.5 61.7 B 42.6 46.5 E 43.4 53.3 B 25.6 38.4 E 52.5 57.7 E 51.2 47.1 E 48.8 50.4 E 44.1 52.7 E 73.8 73.6 E 52.6 50.8 E 67.8 66.8 E 42.9 47.2 E 51.7 57 E The data as it would appear in a data file with Dummy variables, (X1 , X2, X3, X4 )added Pre Final Workbook X1 X2 X3 X4 43 46.4 A 1 0 0 0 55.3 43.9 A 1 0 0 0 59.4 59.7 A 1 0 0 0 51.7 49.6 A 1 0 0 0 53 49.3 A 1 0 0 0 48.7 47.1 A 1 0 0 0 37.3 48.8 D 0 0 0 1 45.4 47.4 A 1 0 0 0 39.2 45.1 D 0 0 0 1 42.1 33.3 A 1 0 0 0 62.1 58 D 0 0 0 1 60 53.2 A 1 0 0 0 43.2 46 E 0 0 0 0 32.4 34.1 A 1 0 0 0 60.7 59.7 E 0 0 0 0 74.4 66.7 A 1 0 0 0 42.7 45.4 E 0 0 0 0 43.2 43.2 A 1 0 0 0 46.6 44.3 E 0 0 0 0 44.5 42.5 A 1 0 0 0 42.6 46.5 E 0 0 0 0 47.1 51.3 A 1 0 0 0 25.6 38.4 E 0 0 0 0 57 48.9 A 1 0 0 0 52.5 57.7 E 0 0 0 0 43.6 52.5 B 0 1 0 0 51.2 47.1 E 0 0 0 0 45.2 61.8 B 0 1 0 0 48.8 50.4 E 0 0 0 0 44.1 52.7 E 0 0 0 0 73.8 73.6 E 0 0 0 0 52.6 50.8 E 0 0 0 0 67.8 66.8 E 0 0 0 0 42.9 47.2 E 0 0 0 0 51.7 57 E 0 0 0 0 Here is the data file in SPSS with the Dummy variables, (X1 , X2, X3, X4 )added. The can be added within SPSS Fitting the complete model The dependent variable is the final score, Y. The independent variables are the Pre-score X and the four dummy variables X1, X2, X3, X4. The Output b Variables Entered/Removed Variables Variables Model Entered Remov ed Method 1 X4, PRE, a . Enter X3, X1, X2 a. All requested v ariables entered. b. Dependent Variable: FINAL Model Summary Std. Error Adjusted of the Model R R Square R Square Estimate 1 .908a .825 .812 3.594 a. Predictors: (Constant), X4, PRE, X X1, X2 3, The Output - continued ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 4191.378 5 838.276 64.895 .000 a Residual 891.297 69 12.917 Total 5082.675 74 a. Predictors: (Constant), X4, PRE, X3, X1, X2 b. Dependent Variable: FINAL Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 16.954 2.441 6.944 .000 PRE .709 .045 .809 15.626 .000 X1 -4.958 1.313 -.241 -3.777 .000 X2 8.553 1.318 .416 6.489 .000 X3 5.231 1.317 .254 3.972 .000 X4 -1.602 1.320 -.078 -1.214 .229 a. Dependent Variable: FINAL The interpretation of the coefficients Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 16.954 2.441 6.944 .000 PRE .709 .045 .809 15.626 .000 X1 -4.958 1.313 -.241 -3.777 .000 X2 8.553 1.318 .416 6.489 .000 X3 5.231 1.317 .254 3.972 .000 X4 -1.602 1.320 -.078 -1.214 .229 a. Dependent Variable: FINAL The common slope The interpretation of the coefficients Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 16.954 2.441 6.944 .000 PRE .709 .045 .809 15.626 .000 X1 -4.958 1.313 -.241 -3.777 .000 X2 8.553 1.318 .416 6.489 .000 X3 5.231 1.317 .254 3.972 .000 X4 -1.602 1.320 -.078 -1.214 .229 a. Dependent Variable: FINAL The intercept for workbook E The interpretation of the coefficients Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 16.954 2.441 6.944 .000 PRE .709 .045 .809 15.626 .000 X1 -4.958 1.313 -.241 -3.777 .000 X2 8.553 1.318 .416 6.489 .000 X3 5.231 1.317 .254 3.972 .000 X4 -1.602 1.320 -.078 -1.214 .229 a. Dependent Variable: FINAL The changes in the intercept when we change from workbook E to other workbooks. The model can be written as follows: The Complete Model: Y 0 1 X 1 2 X 2 3 X 3 4 X 4 1 X 1. When the workbook is E then X1 = 0,…, X4 = 0 and Y 0 1 X 2. When the workbook is A then X1 = 1,…, X4 = 0 and Y 0 1 1 X hence 1 is the change in the intercept when we change form workbook E to workbook A. Testing for the equality of the intercepts i.e. H 0 : 1 2 3 4 0 The reduced model Y 0 1 X The dependent variable in only X (the pre-score) Fitting the reduced model The dependent variable is the final score, Y. The independent variables is only the Pre-score X. The Output for the reduced model b Variables Entered/Remov ed Variables Variables Model Entered Removed Method 1 PREa . Enter a. All requested variables entered. b. Dependent Variable: FINAL Model Summary Std. Error Adjusted of the Model R R Square R Square Estimate 1 .700 a .490 .483 5.956 a. Predictors: (Constant), PRE Lower R2 The Output - continued ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 2492.779 1 2492.779 70.263 .000 a Residual 2589.896 73 35.478 Total 5082.675 74 a. Predictors: (Constant), PRE b. Dependent Variable: FINAL Increased R.S.S Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 23.105 3.692 6.259 .000 PRE .614 .073 .700 8.382 .000 a. Dependent Variable: FINAL The F Test Reduction in R.S.S q F MSE for complete model The Reduced model ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 2492.779 1 2492.779 70.263 .000 a Residual 2589.896 73 35.478 Total 5082.675 74 a. Predictors: (Constant), PRE b. Dependent Variable: FINAL The Complete model ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 4191.378 5 838.276 64.895 .000 a Residual 891.297 69 12.917 Total 5082.675 74 a. Predictors: (Constant), X4, PRE, X3, X1, X2 b. Dependent Variable: FINAL The F test reduced ANOVA Sum of Squares df Mean Square F Sig. Regression 2492.77885 1 2492.77885 70.2626 4.56272E-13 Residual 2589.89635 73 35.47803219 Total 5082.6752 74 Complete ANOVA Sum of Squares df Mean Square F Sig. Regression 4191.377971 5 838.2755942 64.89532 9.99448E-25 Residual 891.297229 69 12.91735115 Total 5082.6752 74 Test equality of slope Sum of Squares df Mean Square F Sig. slope 2492.77885 1 2492.77885 192.9791 1.13567E-21 equality of int. 1698.599121 4 424.6497803 32.87437 2.46006E-15 Residual 891.297229 69 12.91735115 Total 5082.6752 74 Testing for zero slope i.e. H 0 : 1 0 The reduced model Y 0 1 X 1 2 X 2 3 X 3 4 X 4 The dependent variables are X1, X2, X3, X4 (the dummies) The Reduced model ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 1037.475 4 259.369 4.488 .003 a Residual 4045.200 70 57.789 Total 5082.675 74 a. Predictors: (Constant), X4, X3, X2, X1 b. Dependent Variable: FINAL The Complete model ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 4191.378 5 838.276 64.895 .000 a Residual 891.297 69 12.917 Total 5082.675 74 a. Predictors: (Constant), X4, PRE, X3, X1, X2 b. Dependent Variable: FINAL The F test Reduced Sum of Squares df Mean Square F Sig. Regression 1037.4752 4 259.3688 4.488237 0.002757501 Residual 4045.2 70 57.78857143 Total 5082.6752 74 Complete Sum of Squares df Mean Square F Sig. Regression 4191.377971 5 838.2755942 64.89532 9.99448E-25 Residual 891.297229 69 12.91735115 Total 5082.6752 74 Zero slope Sum of Squares df Mean Square F Sig. Regression 1037.4752 4 259.3688 20.0791 5.30755E-11 zero slope 3153.902771 1 3153.902771 244.1602 2.3422E-24 Residual 891.297229 69 12.91735115 Total 5082.6752 74 The Analysis of Covariance • This analysis can also be performed by using a package that can perform Analysis of Covariance (ANACOVA) • The package sets up the dummy variables automatically Here is the data file in SPSS . The Dummy variables are no longer needed. In SPSS to perform ANACOVA you select from the menu – Analysis->General Linear Model->Univariatee This dialog box will appear You now select: 1. The dependent variable Y (Final Score) 2. The Fixed Factor (the categorical independent variable – workbook) 3. The covariate (the continuous independent variable – pretest score) The output: The ANOVA TABLE Tests of Betw een-Subj ects Effects Dependent Variable: FINAL Type III Sum of Mean Source Squares df Square F Sig. Corrected Model 4191.378 a 5 838.276 64.895 .000 Intercept 837.590 1 837.590 64.842 .000 PRE 3153.903 1 3153.903 244.160 .000 WORKBOOK 1698.599 4 424.650 32.874 .000 Error 891.297 69 12.917 Total 219815.6 75 Corrected Total 5082.675 74 a. R Squared = .825 (Adjusted R Squared = .812) Compare this with the previous computed table Sum of Squares df Mean Square F Sig. slope 2492.77885 1 2492.77885 192.9791 1.13567E-21 equality of int. 1698.599121 4 424.6497803 32.87437 2.46006E-15 Residual 891.297229 69 12.91735115 Total 5082.6752 74 The output: The ANOVA TABLE Tests of Betw een-Subj ects Effects Dependent Variable: FINAL Type III Sum of Mean Source Squares df Square F Sig. Corrected Model 4191.378 a 5 838.276 64.895 .000 Intercept 837.590 1 837.590 64.842 .000 PRE 3153.903 1 3153.903 244.160 .000 WORKBOOK 1698.599 4 424.650 32.874 .000 Error 891.297 69 12.917 Total 219815.6 75 Corrected Total 5082.675 74 a. R Squared = .825 (Adjusted R Squared = .812) This is the sum of squares in the numerator when we attempt to test if the slope is zero (and allow the intercepts to be different) Another application of the use of dummy variables • The dependent variable, Y, is linearly related to X, but the slope changes at one or several known values of X (nodes). Y X nodes Y k The model 1 2 x1 x2 xk X 0 1 X X x1 0 1 x1 2 X x1 x1 X x2 Y 0 1 x1 2 x2 x1 3 X x2 x2 X x3 or 0 1 X X x1 0 1 2 x1 2 X x1 X x2 Y 0 1 2 x1 2 3 x2 3 X x2 X x3 Now define X if X x1 X1 x1 if X x1 0 if X x1 X 2 X x1 if x1 X x2 x x x2 X 2 1 0 if X x2 X 3 X x2 if x2 X x3 x x x3 X 3 3 Etc. Then the model 0 1 X X x1 0 1 x1 2 X x1 x1 X x2 Y 0 1 x1 2 x2 x1 3 X x2 x2 X x3 can be written Y 0 1 X 1 2 X 2 3 X 3 An Example In this example we are measuring Y at time X. Y is growing linearly with time. At time X = 10, an additive is added to the process which may change the rate of growth. The data X 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Y 3.9 5.9 6.4 6.3 7.5 7.9 8.5 X 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Y 10.7 10.0 12.4 11.0 11.5 13.9 17.6 X 14.0 15.0 16.0 17.0 18.0 19.0 20.0 Y 18.2 16.8 21.8 23.1 22.9 26.2 27.7 Graph 30 25 20 15 10 5 0 0 5 10 15 20 Now define the dummy variables X if X 10 X1 10 if X 10 0 if X 10 X2 X 10 if 10 X The data as it appears in SPSS – x1, x2 are the dummy variables We now regress y on x1 and x2. Model Summary The Output Std. Error Adjusted of the Model R R Square R Square Estimate 1 .990 a .980 .978 1.0626 a. Predictors: (Constant), X2, X1 ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 1015.909 2 507.954 449.875 .000 a Residual 20.324 18 1.129 Total 1036.232 20 a. Predictors: (Constant), X2, X1 b. Dependent Variable: Y Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 4.714 .577 8.175 .000 X1 .673 .085 .325 7.886 .000 X2 1.579 .085 .761 18.485 .000 a. Dependent Variable: Y Graph 30 25 20 15 10 5 0 0 5 10 15 20 Testing for no change in slope Here we want to test H0: 1 = 2 vs HA: 1 ≠ 2 The reduced model is Y = 0 + 1 (X1+ X2) + = 0 + 1 X + Fitting the reduced model We now regress y on x. Model Summary The Output Std. Error Adjusted of the Model R R Square R Square Estimate 1 a .971 .942 .939 1.7772 a. Predictors: (Constant), X ANOVAb Sum of Mean Model Squares df Square F Sig. 1 Regression 976.219 1 976.219 309.070 .000 a Residual 60.013 19 3.159 Total 1036.232 20 a. Predictors: (Constant), X b. Dependent Variable: Y Coefficientsa Standardi zed Unstandardized Coefficien Coefficients ts Model B Std. Error Beta t Sig. 1 (Constant) 2.559 .749 3.418 .003 X 1.126 .064 .971 17.580 .000 a. Dependent Variable: Y Graph – fitting a common slope 30 25 20 15 10 5 0 0 5 10 15 20 The test for the equality of slope Reduced Model Sum of Squares df Mean Square F Sig. Regression 976.2194805 1 976.2194805 309.0697 3.27405E-13 Residual 60.01290043 19 3.158573707 Total 1036.232381 20 Complete Model Sum of Squares df Mean Square F Sig. Regression 1015.908579 2 507.9542895 449.8753 0 Residual 20.32380204 18 1.129100113 Total 1036.232381 20 equality of slope Sum of Squares df Mean Square F Sig. slope 976.2194805 1 976.2194805 864.5996 1.14256E-16 equality of slope 39.6890984 1 39.6890984 35.15109 1.30425E-05 Residual 20.32380204 18 1.129100113 Total 1036.232381 20