SPECIAL TOPICS IN REGRESSION ANALYSIS

Transcription

1 1 SPECIAL TOPICS IN REGRESSION ANALYSIS Representing Nominal Scales in Regression Analysis There are several ways in which a set of G qualitative distinctions on some variable of interest can be represented in regression analysis. All methods share the common feature that (G 1) independent variables are needed to code the information contained in the G different groups. (a) dummy variable coding: use (G 1) dichotomous (0-1) variables to represent membership in the G groups; the group that receives a score of 0 on all the independent variables serves as the reference group: e.g., for G = 4: X1 X2 X3 G G G G the intercept of the regression is the mean for the reference group ( y ref ); the partial regression coefficients bi indicate by how much the mean for group i ( y i ) differs from the mean of the reference group, i.e., b 0 y ref b y y i i ref (b) effects coding: similar to dummy variable coding, except that the reference group is scored as a string of 1's;

2 2 e.g., for G = 4: X1 X2 X3 G G G G the intercept of the regression is the mean of the group means; the partial regression coefficients bi represent contrasts between the mean of group i and the mean of the group means, i.e., y b i i 0 y G b y y i i (c) (orthogonal) contrast coding: any contrast among a set of G means can be tested, i.e., c 1 1 c c G G s.t. c i 0 if possible, use (G 1) orthogonal contrasts to represent the information contained in the G different groups; e.g., for G = 4 two possible sets of contrasts are: (1) X1 X2 X3 G1 ½ ½ 0 G2 ½ ½ 0 G3 ½ 0 ½ G4 ½ 0 ½

3 3 (2) X1 X2 X3 G1 ½ ½ ¼ G2 ½ ½ ¼ G3 ½ ½ ¼ G4 ½ ½ ¼ as in effects coding, the intercept of the regression is the mean of the group means; the partial regression coefficients bi indicate the difference in means for the groups involved in the contrast; An example of testing for group differences using different approaches to scaling nominal variables: The dependent variable of interest is attitude toward using coupons for grocery shopping (AA). Group 1 never or almost never uses coupons, group 2 occasionally uses coupons, and group 3 frequently uses coupons. DATA coupon; [read in the data for AA and membership in one of the three groups] if group=2 then d1=1; else d1=0; if group=3 then d2=1; else d2=0; if group=2 then e1=1; else e1=0; if group=3 then e2=1; else e2=0; if group=1 then e1=-1; if group=1 then e2=-1; if group=1 then c1=1/3; else if group=2 then c1=1/3; else if group=3 then c1=-2/3; if group=1 then c2=.5; else if group=2 then c2=-.5; else if group=3 then c2=0; proc sort; by group; proc means; var AA; proc means; var AA; by group;

4 4 proc reg; model AA = D1 D2; proc reg; model AA = E1 E2; proc reg; model AA = C1 C2; run; N Mean Std Dev Minimum Maximum overall group group group (1) Dummy variable coding: [group 1 serves as the reference group; D1 is coded as 1 for group 2, zero otherwise; D2 as 1 for group 3, zero otherwise] Model: MODEL1 Dependent Variable: AA Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP D D

5 5 (2) Effects coding: [group 1 serves as the reference group; E1 is coded as 1 for group 2, E2 as 1 for group 3] Model: MODEL1 Dependent Variable: AA Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP E E (3) Contrast coding: [ C1 is the contrast between groups 1&2 and 3, coded as 1/3, 1/3, -2/3; C2 is the contrast between groups 1 and 2, coded as.5, -.5, 0 ] Model: MODEL1 Dependent Variable: AA Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP <.0001 C <.0001 C <.0001

6 6 N Mean Std Dev Minimum Maximum overall group group group mean of groups means Dummy variable coding: Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP D D Effects coding: Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP E E Contrast coding: Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP C C

7 7 Multiplicative models Consider a model in which a dependent variable Y is a function of two main effects (X1, X2) and their interaction (X1X2). The model is given by Y = a + b1x1 + b2x2 + b3x1x2 + e where a, b1, b2, and b3 are the estimated regression coefficients. Note that the interpretation of these coefficients is somewhat different from the no-interaction case: b1 the effect of X1 on Y when X2 is zero b2 the effect of X2 on Y when X1 is zero b3 the change in the effect of X1 on Y when X2 changes by one unit (or, equivalently, the change in the effect of X2 on Y when X1 changes by one unit) If the interaction is significant, the exact nature of the interaction can be investigated by testing the significance of the slope of Y on X1 (X2) for selected values of X2 (X1). For example, let's assume we are interested in the effect of X1 on Y for X2=x2. An estimate of this effect is given by b1 + b3x2. To get the standard error of this expression, use the fact that Var(b1 + b3x2) = Var(b1) + x2 2 Var(b3) + 2x2Cov(b1, b3) Issues in testing for interaction effects: (1) Mean-center the main effect variables before forming the multiplicative term to reduce potential problems with multicollinearity. (2) Do not standardize the variables if the invariance of relationships across groups is to be tested. (3) Be aware of the damaging effect of measurement error on tests of interaction effects. (4) Investigate the functional form of the interaction. (5) Make sure the variables are measured on an interval scale. (6) Check the statistical power of the test of interaction.

8 8 An example of testing for interactions: Expectancy-value attitude theory assumes that beliefs (BE) and evaluations (EV) combine multiplicatively to influence attitudes. This hypothesis is tested using beliefs about two positive consequences of using coupons, saving money on the grocery bill and thinking about oneself as a thrifty shopper. [mean-centered regression] DATA coupon; SET coupon; proc standard m=0 data=coupon out=couponmc; var be ev; data couponmc; set couponmc; beev=be*ev; proc corr; var aa be ev beev; proc reg; model aa = be ev beev / stb covb; run; quit; (a) BE and EV coded on 1-7 scales: Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum AA BE EV BEEV Pearson Correlation Coefficients / Prob > R under Ho: Rho=0 / N = 250 AA BE EV BEEV AA BE EV BEEV

9 9 Dependent Variable: AA Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP BE EV BEEV Covariance of Estimates COVB INTERCEP BE EV BEEV INTERCEP BE EV BEEV (b) BE and EV mean-centered: Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum AA BE EV BEEV

10 10 Pearson Correlation Coefficients / Prob > R under Ho: Rho=0 / N = 250 AA BE EV BEEV AA BE EV BEEV Dependent Variable: AA Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP BE EV BEEV Covariance of Estimates COVB INTERCEP BE EV BEEV INTERCEP BE EV BEEV

11 11 AA as a function of BE for different levels of EV AA BE

12 12 Mediation Consider three variables X, M, and Y and assume that they are related in the following way: X a M b c Y Path a is the direct effect of X on M, b is the direct effect of M on Y, and c is the direct effect of X on Y. The total effect of X on Y can be shown to equal c + ab, where c is the direct effect of X on Y and ab is the indirect effect of X on Y (via M). A mediator M is a variable that accounts for the relation between a predictor and a criterion. In the figure, M mediates the influence of X on Y if it channels at least some of the total effect of X on Y. Total mediation occurs if there is no direct effect of X on Y (i.e., the total effect of X on Y is completely accounted for by the indirect effect of X on Y through the mediator M). Partial mediation occurs if the total effect of X on Y is due to both a direct and indirect effect of X on Y. Testing for mediation in the traditional Baron and Kenny (1986) framework: (1) Show that, in a regression of Y on X, Y is significantly related to X. This establishes that there is an effect that can be mediated. The regression coefficient from this regression is an estimate of the total effect of X on Y. [Note: This step is problematic when there is inconsistent or competitive mediation.] (2) Show that, in a regression of M on X, M is significantly related to X. If M is to channel the influence of X on Y, it has to be related to X. The regression coefficient from this regression is an estimate of a. (3) Show that when Y is regressed on both X and M, M affects Y (i.e., the regression coefficient for M, which is an estimate of b, is nonzero) and the direct influence of X on Y (the regression coefficient for X is an estimate of c) is smaller than the total effect. For complete mediation, the direct effect of X on Y should be negligible; for partial mediation, the direct effect of X on Y should be smaller than the total effect.

13 13 Testing for mediation based on the significance of the indirect effect: Since the extent of mediation is defined as the difference between the total effect of X on Y and the direct effect of X on Y and since this difference is equal to the product of a and b, mediation can also be checked by testing whether ab is different from zero. This test (often called the Sobel test) is given by b s 2 2 a ab a s 2 2 b s s 2 2 a b where sa and sb are the standard errors of a and b. This ratio can be treated as a standard normal variate. Because of problems with the normality assumption, a bootstrap test is preferable. Issues in testing for mediation: (1) The mediator M should be neither too close to nor too distant from either the predictor variable X or the criterion variable Y. In particular, although X has to be related to M in order for mediation to occur, the relationship should not be too strong, otherwise the independent variables in the last regression will be collinear. (2) It is assumed that X, M, and Y are related as shown in the Figure. If X is a manipulated variable, it is safe to assume that X influences M and Y and not the other way around. However, M and Y are usually measured variables and Y should not influence M. (3) It is assumed that there is no measurement error in the mediator. If this assumption is incorrect, the estimated effects will be biased. (4) It is assumed that no important influences on M and Y have been omitted from the model specification. (5) If X is manipulated, it is exogenous; however, in general it is not safe to assume that the errors of M and Y are uncorrelated. Note: Recent research has (a) investigated under what conditions mediation analyses can be given a causal interpretation; (b) extended mediation to situations in which there are exposuremediator interactions; and (c) considered the case where the mediator and the outcome variable are not continuous (e.g., binary, counts, etc.). See Valeri and VanderWeele (2013) for details.

14 14 An example of testing for mediation: According to the Theory of Reasoned Action (TRA), behavioral intentions (BI) mediate the effects of attitudes (AA) on behavior (BH). This hypothesis is tested in the context of using coupons for grocery shopping. (1) Regression analysis: %include 'd:\m554\programs\process.sas'; DATA mediation; INFILE 'd:\m554\specreg\sem.dat' PAD; INPUT ID BE1 BE2 BE3 BE4 BE5 BE6 BE7 AA1 AA2 AA3 AA4 BI1 BI2 BH; aa=(aa1+aa2+aa3+aa4)/4; bi=(bi1+bi2)/2; proc corr cov; var aa bi bh; proc reg; model bh=aa / stb covb; proc reg; model bi=aa / stb covb; proc reg; model bh=aa bi / stb covb; %process (data=mediation,vars=bh bi aa, y=bh, x=aa, m=bi, model=4, normal=1,varorder=2,total=1,effsize=1,boot=10000,conf=95); RUN; Pearson Correlation Coefficients / Prob > R under Ho: Rho=0 / N = 250 AA BI BH AA BI BH

15 15 Step 1: Dependent Variable: BH Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP AA Step 2: Dependent Variable: BI Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP AA Step 3: Dependent Variable: BH Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total

16 16 Root MSE R-square Dep Mean Adj R-sq C.V Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP AA BI Output of Process macro: ************************* PROCESS Procedure for SAS Release 2.10 ************************ Model and Variables Model = 4 Y = BH X = AA M = BI Sample size: 250 ***************************************************************************************** Outcome: BI Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI Constant AA ***************************************************************************************** Outcome: BH Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI constant BI AA

17 17 ********************************* TOTAL EFFECT MODEL ********************************* Outcome: BH Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI constant AA *************************** TOTAL, DIRECT AND INDIRECT EFFECTS *************************** Total effect of X on Y Effect SE t p LLCI ULCI Direct effect of X on Y Effect SE t p LLCI ULCI Indirect effect of X on Y Effect Boot SE BootLLCI BootULCI BI Partially standardized indirect effect of X on Y Effect Boot SE BootLLCI BootULCI BI Completely standardized indirect effect of X on Y Effect Boot SE BootLLCI BootULCI BI Ratio of indirect to total effect of X on Y Effect Boot SE BootLLCI BootULCI BI Ratio of indirect to direct effect of X on Y Effect Boot SE BootLLCI BootULCI BI R-squared mediation effect size Effect Boot SE BootLLCI BootULCI BI

18 18 Preacher and Kelley (2011) Kappa-squared Effect Boot SE BootLLCI BootULCI BI Normal theory test for indirect effect Effect se Z p ****************************** ANALYSIS NOTES AND WARNINGS ****************************** Number of bootstrap samples for bias corrected bootstrap confidence intervals: Level of confidence for all confidence intervals in output: (2) Structural equation modeling: proc calis data=mediation EFFPART; path BI <--- AA = ga11, BH <--- BI AA = be21 ga21; pvar BI = psi11, BH = psi22, AA = ph11; run; The CALIS Procedure Covariance Structure Analysis: Maximum Likelihood Estimation PATH List Standard Path Parameter Estimate Error t Value bi <=== aa ga BH <=== bi be BH <=== aa ga Variance Parameters Variance Standard Type Variable Parameter Estimate Error t Value Error bi psi BH psi Exogenous aa ph

19 19 Squared Multiple Correlations Error Total Variable Variance Variance R-Square BH bi Total Effects Effect / Std Error / t Value / p Value bi aa BH <.0001 <.0001 bi <.0001 Direct Effects Effect / Std Error / t Value / p Value bi aa BH < bi <.0001 Indirect Effects Effect / Std Error / t Value / p Value bi aa BH <.0001 bi 0 0

20 20 Moderation A moderator is a variable that affects the direction and/or strength of the relation between a predictor and a criterion. X Y Z Testing for moderation: (1) Both the moderator and the independent variable are categorical: Use ANOVA and check whether the interaction between the moderator and the independent variable is significant. (2) The moderator is categorical, the independent variable is continuous: Use regression analysis with interaction terms between the independent variable and the moderator. If there is differential measurement error in the independent variable across groups, use structural equation modeling. (3) The moderator is continuous, the independent variable is categorical or continuous: The appropriate test depends on the nature of the moderator effect. X Y X Y X Y Z Z Z

21 21 If the effect of X on Y varies linearly as a function of the moderator (case 1), moderation is tested by including the product of X and Z in the regression. If the moderator effect is nonlinear (case 2), higher-order interactions (e.g., XZ 2 ) have to be included in the regression. If the moderator effect takes the form of a stepfunction (case 3), the moderator should be dichotomized at the point where the step occurs. Issues in testing for moderator effects: (1) Multicollinearity may be a problem (esp. when higher-order interactions are included). Mean-center the independent variable and the moderator before forming the interaction term(s). (2) If the interaction is significant, do follow-up tests to investigate the nature of the interaction effect. (3) It is dangerous to compare correlations across groups because (a) the variance of the independent variable may not be constant across groups and (b) measurement error may not be constant across groups. Mediator vs. moderator effects: a. An interest in moderator variables reflects a search for the conditional boundaries of an effect (when question). An interest in mediator variables reflects a concern with the processes underlying an effect (how and why question). b. Finding a moderator of some relationship may stimulate thinking about why or how this occurs (moderation to mediation). Specifying a mediational mechanism between two variables may have implications for when this effect is likely to occur (mediation to moderation). c. Moderated mediation: The mediational effect of some variable varies across levels of the moderator. d. Mediated moderation: A variable mediates the influence of a moderator on another variable. For details of how to test for moderated mediation or mediated moderation, see Edwards and Lambert (2007) and Preacher, Rucker, and Hayes (2007), as well as some of the other papers listed in the syllabus. Also, see the description of the PROCESS macro for an overview of possible models to be tested.

22 22 An example of testing for moderation: According to the theory of action control, people differ in their capacity for action control. People with high self-regulatory capacity are called action-oriented; people with low selfregulatory capacity state-oriented. Action orientation reflects readiness to act; state-orientation inertia to act. Here we test the hypothesis that action-/state-orientation (ASO) moderates the effects of attitudes (AA) and subjective norms (SN) on behavioral intentions (BI). The context is people's usage of coupons for grocery shopping. %include 'd:\m554\programs\process.sas'; OPTIONS LS=100; DATA COUPON; INFILE 'd:\m554\specreg\coupaso.raw' PAD; INPUT obs ASO AA1 AA2 AA3 SN1 SN2 PB BI1 BI2 BH; AA=(AA1+AA2+AA3)/3; SN=(SN1+SN2)/2; BI=(BI1+BI2)/2; PROC STANDARD DATA=COUPON OUT=COUPON M=0; VAR AA SN ASO; proc univariate; var aso; DATA COUPON; SET COUPON; IF ASO<0.208 THEN DASO=0; ELSE DASO=1; AABYASO=AA*ASO; SNBYASO=SN*ASO; AABYDASO=AA*DASO; SNBYDASO=SN*DASO; proc sort; by daso; proc corr; var bi; with aa sn; by daso; PROC REG; MODEL BI = AA SN DASO AABYDASO SNBYDASO / STB COVB; PROC REG; MODEL BI = AA SN ASO AABYASO SNBYASO / STB COVB; run;

23 23 title 'aa as the focal variable, daso as the moderator'; run; %process (data=coupon,vars=bi aa sn daso snbydaso,y=bi,x=aa,m=daso,model=1,plot=1); run; title 'sn as the focal variable, daso as the moderator'; run; %process (data=coupon,vars=bi aa sn daso aabydaso,y=bi,x=sn,m=daso,model=1,plot=1); run; title 'aa as the focal variable, aso as the moderator'; run; %process (data=coupon,vars=bi aa sn aso snbyaso,y=bi,x=aa,m=aso,model=1,center=1,jn=1,plot=1); run; title 'sn as the focal variable, aso as the moderator'; run; %process (data=coupon,vars=bi aa sn aso aabyaso,y=bi,x=sn,m=aso,model=1,center=1,jn=1,plot=1); run; quit; (a) Moderator (ASO) as a dichotomous variable: The UNIVARIATE Procedure Variable: ASO Moments N 149 Sum Weights 149 Mean 0 Sum Observations 0 Std Deviation Variance Skewness Kurtosis Uncorrected SS Corrected SS Coeff Variation. Std Error Mean Basic Statistical Measures Location Variability Mean Std Deviation Median Variance Mode Range Interquartile Range

24 24 Quantile Estimate 100% Max % % % % Q % Median % Q % % % % Min DASO= Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum AA SN BI Pearson Correlation Coefficients, N = 64 Prob > r under H0: Rho=0 BI AA SN < DASO= Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum AA SN BI Pearson Correlation Coefficients, N = 85 Prob > r under H0: Rho=0 BI AA <.0001 SN

25 25 Dependent Variable: BI Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V E17 Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP AA SN DASO AABYDASO SNBYDASO Covariance of Estimates COVB INTERCEP AA SN INTERCEP AA SN DASO AABYDASO SNBYDASO COVB DASO AABYDASO SNBYDASO INTERCEP AA SN DASO AABYDASO SNBYDASO

26 26 Output from the Process macro: ************************* PROCESS Procedure for SAS Release 2.10 ************************ Model and Variables Model = 1 Y = BI X = AA M = DASO Statistical controls: SN SNBYDASO Sample size: 149 ***************************************************************************************** Outcome: BI Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI constant DASO AA INT_ SN SNBYDASO Interactions: INT_1 AA X DASO R-square increase due to interaction(s): R2-chng F df1 df2 p INT_ ***************************************************************************************** Conditional effect of X on Y at values of the moderator(s) DASO Effect se t p LLCI ULCI Values for quantitative moderators are the mean and plus/minus one SD from mean. Values for dichotomous moderators are the two values of the moderator.

27 27 ***************************************************************************************** Data for visualizing conditional effect of X on Y AA DASO yhat Estimates in this table are based on setting covariates to their sample means ****************************** ANALYSIS NOTES AND WARNINGS ****************************** Level of confidence for all confidence intervals in output: ************************* PROCESS Procedure for SAS Release 2.10 ************************ Model and Variables Model = 1 Y = BI X = SN M = DASO Statistical controls: AA AABYDASO Sample size: 149 ***************************************************************************************** Outcome: BI Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI constant DASO SN INT_ AA AABYDASO Interactions: INT_1 SN X DASO

28 28 R-square increase due to interaction(s): R2-chng F df1 df2 p INT_ ***************************************************************************************** Conditional effect of X on Y at values of the moderator(s) DASO Effect se t p LLCI ULCI Values for quantitative moderators are the mean and plus/minus one SD from mean. Values for dichotomous moderators are the two values of the moderator. ***************************************************************************************** Data for visualizing conditional effect of X on Y SN DASO yhat Estimates in this table are based on setting covariates to their sample means ****************************** ANALYSIS NOTES AND WARNINGS ****************************** Level of confidence for all confidence intervals in output: (b) Moderator (ASO) as a continuous variable: Dependent Variable: BI Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Total Root MSE R-square Dep Mean Adj R-sq C.V E17

29 29 Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > T Estimate INTERCEP AA SN ASO AABYASO SNBYASO Covariance of Estimates COVB INTERCEP AA SN INTERCEP AA SN ASO AABYASO SNBYASO COVB ASO AABYASO SNBYASO INTERCEP AA SN ASO AABYASO SNBYASO Output from the Process macro: ************************* PROCESS Procedure for SAS Release 2.10 ************************ Model and Variables Model = 1 Y = BI X = AA M = ASO Statistical controls: SN SNBYASO Sample size: 149

30 30 ***************************************************************************************** Outcome: BI Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI constant ASO AA INT_ SN SNBYASO Interactions: INT_1 AA X ASO R-square increase due to interaction(s): R2-chng F df1 df2 p INT_ ***************************************************************************************** Conditional effect of X on Y at values of the moderator(s) ASO Effect se t p LLCI ULCI Values for quantitative moderators are the mean and plus/minus one SD from mean. Values for dichotomous moderators are the two values of the moderator. ******************************** JOHNSON-NEYMAN TECHNIQUE ******************************** Moderator values(s) defining Johnson-Neyman significance region(s) Value % below % above Conditional effect of X on Y at values of the moderator (M) ASO Effect se t p LLCI ULCI

31 ***************************************************************************************** Data for visualizing conditional effect of X on Y AA ASO yhat Estimates in this table are based on setting covariates to their sample means ****************************** ANALYSIS NOTES AND WARNINGS ****************************** Level of confidence for all confidence intervals in output: NOTE: The following variables were mean centered prior to analysis: AA ASO ************************* PROCESS Procedure for SAS Release 2.10 ************************ Model and Variables Model = 1 Y = BI X = SN M = ASO Statistical controls: AA AABYASO Sample size: 149

32 32 ***************************************************************************************** Outcome: BI Model Summary R R-sq F df1 df2 p Model coeff se t p LLCI ULCI constant ASO SN INT_ AA AABYASO Interactions: INT_1 SN X ASO R-square increase due to interaction(s): R2-chng F df1 df2 p INT_ ***************************************************************************************** Conditional effect of X on Y at values of the moderator(s) ASO Effect se t p LLCI ULCI Values for quantitative moderators are the mean and plus/minus one SD from mean. Values for dichotomous moderators are the two values of the moderator. ******************************** JOHNSON-NEYMAN TECHNIQUE ******************************** Moderator values(s) defining Johnson-Neyman significance region(s) Value % below % above Conditional effect of X on Y at values of the moderator (M) ASO Effect se t p LLCI ULCI

33 ***************************************************************************************** Data for visualizing conditional effect of X on Y SN ASO yhat Estimates in this table are based on setting covariates to their sample means ****************************** ANALYSIS NOTES AND WARNINGS ****************************** Level of confidence for all confidence intervals in output: NOTE: The following variables were mean centered prior to analysis: SN ASO