CONDUCTING HIERARCHICAL MULTIPLE REGRESSION Hierarchical multiple regression is provided automatically for every regression analysis that is performed. If you are not familiar with hierarchical models, see Chapters 3 & 4 in the text by Cohen, J. & Cohen, P. 'Applied Multiple Regression/ Correlation Analysis for the Behavioral Sciences', (2nd ed). Hillsdale, NJ: Lawrence Erlbaum, 1983. In order to illustrate, suppose you choose to analyze the Cohen & Cohen data (p. 99) using an hierarchical model which asks, 'To what extent is salary explained by years of teaching, number of publications, and number of citations -- under the condition that any effect of gender has been eliminated?' In order to address this question, it will be necessary to order the variables as Order Code: 0 2 3 1 4 Variable: SALARY YEARS PUBLICATIONS GENDER CITATIONS GROUPING SETS OF VARIABLES Since you are primarily interested in the COMBINED effect of three independent variables (after eliminating the effect of gender), you should specify the number of variables in each 'set' of variables. The program will allow you to do that after you have reordered your variables. For this example, you would indicate that the first set contains one variable (gender) and the second set contains three variables. The important point here is that you can have any number of sets of variables as long as the number of sets is less than the number of independent variables that are available. The ability to group variables to examine 'set effects' is an essential requirement in many hiearchical regression problems. If you are not familiar with the analysis of variables according to organized sets you will find invaluable information in Chapter 4 of the text by Cohen & Cohen (1983). Partial results of the hierarchical analysis for the above example are shown in the following screens. SIMULTANEOUS MULTIPLE REGRESSION RESULTS Standardized Raw Score b- Beta Variable Name Coefficients Coefficients t-ratio p <= ------------- ------------ ------------ ----------- --------- Intercept 20589.87389 GENDER -2945.23932 -0.21363 -0.91096 0.61297 YEARS 293.14251 0.34842 1.00966 0.33814 PUBLICATIONS 175.82097 0.14315 0.45220 0.66360 CITATIONS 1145.35457 0.30267 1.18643 0.26228 ANALYSIS OF VARIANCE Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= ------------ ---- ----------- ---------- ---------- -------- GENDER 1 3.833E+07 3.833E+07 1.350 0.2718 YEARS 1 1.908E+08 1.908E+08 6.722 0.0257 PUBLICATIONS 1 4.548E+06 4.548E+06 0.160 0.6982 CITATIONS 1 3.996E+07 3.996E+07 1.408 0.2622 For this set: 3 2.353E+08 7.844E+07 2.763 0.0969 Error 10 2.839E+08 2.839E+07 Total 14 5.575E+08 SUMMARY STATISTICS Dependent Variable = SALARY Omnibus F-ratio = 2.410040 Significance Level, p <= 0.118022 Multiple R = 0.700599 Squared Multiple R = 0.490839 Shrunken R = 0.535886 Shrunken Squared R = 0.287174 Determinant of Rxx = 0.388804 Regression Sum of Squares = 2.736528E+08 Error Sum of Squares = 2.838675E+08 Standard Deviation of y' = 5231.183000 Standard Error of Regression = 5327.922000 CONDUCTING POLYNOMIAL (CURVILINEAR) REGRESSION It is rather easy to conduct polynomial or non-linear regression. An excellent discussion of the use of powered polynomials can be found in Chapter 6 of the text by Cohen, J. & Cohen, P. 'Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences', (2nd ed). Hillsdale, NJ: Lawrence Erlbaum, 1983. In order to use polynomial models you must construct your own powered terms in a raw data matrix. Suppose, for example, that you wish to determine whether salary (using the Cohen & Cohen data, p. 99) is a non-linear function of number of publications. Moreover, you may wish to fit a cubic polynomial to the data. In such a case you would build the raw data file as shown below in which you manually construct the values of PUB_SQR and PUB_CUBE. Order Code: 0 1 2 3 Variable: SALARY PUBLICATIONS PUB_SQR PUB_CUBE 18000 2 4 8 19961 4 16 64 19828 5 25 125 17030 12 144 1728 19925 5 25 125 19041 9 81 729 27132 3 9 27 27268 1 1 1 32483 8 64 512 27029 12 144 1728 25362 9 81 729 28463 4 16 64 32931 8 64 512 28270 11 121 1331 38362 21 441 9261 In order to analyze these data it is important to examine the hierarchical solution to determine whether the quadratic term makes an additionally significant contribution above that made by the linear component of the model. If so, the question then becomes one of determining whether the cubic term makes an additionally significant contribution (above that which is explained by the linear and quadratic terms). If you wish to analyze these data, choose the summary data option and use the summary data file on your SPPC Disk 1 that has the name POLYS.DAT. Or, choose the raw data option and use the data file named POLYR.DAT. Partial results of the analysis are shown as follows: SIMULTANEOUS MULTIPLE REGRESSION RESULTS Raw Score b- Standardized Variable Name Coefficients Beta t-ratio p <= -------------- ------------ ------------ --------- ------ Intercept 21566.50000 PUBLICATIONS 1226.55000 0.99866 0.39229 0.7030 PUB_SQR -151.83500 -2.66629 -0.42540 0.6808 PUB_CUBE 6.22221 2.29727 0.57327 0.5834 ANALYSIS OF VARIANCE Hierarchical VARIANCE SUM OF MEAN Step-down SOURCE df SQUARES SQUARES F-ratio p <= ------------- -- ----------- ----------- ----------- ------ PUBLICATIONS 1 1.18295E+08 1.18295E+08 3.41094 0.0891 PUB_SQR 1 4.63345E+07 4.63345E+07 1.33601 0.2718 PUB_CUBE 1 1.13976E+07 1.13976E+07 0.32863 0.5834 Error 11 3.81493E+08 3.46812E+07 Total 14 5.57520E+08