Jason W. Miller a, William R. Stromeyer b & Matthew A. Schwieterman a a Department of Marketing & Logistics, The Ohio

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1 This article was downloaded by: [New York University] On: 16 July 2014, At: 12:57 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Multivariate Behavioral Research Publication details, including instructions for authors and subscription information: Extensions of the Johnson- Neyman Technique to Linear Models With Curvilinear Effects: Derivations and Analytical Tools Jason W. Miller a, William R. Stromeyer b & Matthew A. Schwieterman a a Department of Marketing & Logistics, The Ohio State University b Department of Management & Human Resources, The Ohio State University Published online: 15 Apr To cite this article: Jason W. Miller, William R. Stromeyer & Matthew A. Schwieterman (2013) Extensions of the Johnson-Neyman Technique to Linear Models With Curvilinear Effects: Derivations and Analytical Tools, Multivariate Behavioral Research, 48:2, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the

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3 Multivariate Behavioral Research, 48: , 2013 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / Extensions of the Johnson-Neyman Technique to Linear Models With Curvilinear Effects: Derivations and Analytical Tools Jason W. Miller Department of Marketing & Logistics The Ohio State University William R. Stromeyer Department of Management & Human Resources The Ohio State University Matthew A. Schwieterman Department of Marketing & Logistics The Ohio State University The past decade has witnessed renewed interest in the use of the Johnson-Neyman (J-N) technique for calculating the regions of significance for the simple slope of a focal predictor on an outcome variable across the range of a second, continuous independent variable. Although tools have been developed to apply this technique to probe 2- and 3-way interactions in several types of linear models, this method has not been extended to include quadratic terms or more complicated models involving Correspondence concerning this article should be addressed to Jason W. Miller, Department of Marketing & Logistics, Fisher Hall 500, Fisher College of Business, The Ohio State University, 2100 Neil Avenue, Columbus, OH Miller_5350@fisher.osu.edu 267

4 268 MILLER, STROMEYER, SCHWIETERMAN quadratic terms and interactions. Curvilinear relations of this type are incorporated in several theories in the social sciences. This article extends the J-N method to such linear models along with presenting freely available online tools that implement this technique as well as the traditional pick-a-point approach. Algebraic and graphical representations of the proposed J-N extension are provided. An example is presented to illustrate the use of these tools and the interpretation of findings. Issues of reliability as well as spurious moderator effects are discussed along with recommendations for future research. Research in the social sciences commonly features hypotheses involving moderated and nonlinear relations. According to Aiken and West (1991), These more complex relationships often take the form of a monotonically increasing (or decreasing) curvilinear relationship or a U-shaped or inverted U-shaped function (p. 62). It is further possible to incorporate curvilinear and moderated specifications within the same model to determine if a curvilinear effect between a focal predictor and an outcome is contingent on a moderator (Aiken & West, 1991; Cohen, Cohen, West, & Aiken, 2003). Interest in curvilinear effects and moderation of curvilinear effects has been on the rise as evidenced by a number of recent publications that address these concepts. Examples of such studies include Villena, Revilla, and Choi s (2011) examination of a negative curvilinear relation between social capital and operational and strategic performance and Le et al. s (2011) study of the moderated curvilinear relation between personality traits and performance with job complexity serving as a moderator. A range of other studies have also examined various types of curvilinear relations including Baer and Oldham (2006); Barber et al. (2008); Adams and Laursen (2007); Agustin and Singh (2005); Janssen (2001); and Zhou, Shin, Brass, Choi, and Zhang (2009). Currently, methods to probe significant linear models with curvilinear terms and/or linear-by-linear or quadratic-by-linear interactions rely on the picka-point approach (Rogosa, 1980) in which the researcher selects arbitrary values, commonly one standard deviation from the mean, and tests the significance of the simple slope(s) at these values (Aiken & West, 1991; Jaccard & Turrisi, 2003). Focusing on a particular example helps to illustrate the specifics of this style of analysis. Zhou et al. s (2009) study of the relations between social networks, creativity, and conformity value hypothesized and supported the presence of a curvilinear relation between the number of weak ties of an employee and the exhibition of creativity. This work revealed that employees with a moderate level of weak ties exhibited greater creativity than employees with low or high levels of weak ties. They further demonstrated that this relation was moderated by the degree that the employee valued conformity (split between low and high levels). In order to examine this moderation effect they relied on the approach of utilizing the mean of weak ties and plus/minus

5 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 269 one standard deviation to estimate these three simple slopes at each level of conformity. Within the past decade, there has been renewed interest in utilizing the Johnson-Neyman (J-N) technique (Johnson & Fay, 1950; Johnson & Neyman, 1936; Potthof, 1964) for probing significant interactions when the moderator is continuous (Bauer & Curran, 2005; Hayes & Matthes, 2009; Preacher, Curran, & Bauer, 2006). Unlike the pick-a-point method, this technique allows an analyst to calculate a region of significance for the simple slope of a focal predictor conditioned on the value of the continuous moderator. Bauer and Curran (2005) denote two distinct advantages of this method: (a) it yields exact values of the moderator where the simple slope of the focal predictor is statistically different from zero rather than selecting arbitrary values and (b) the corresponding confidence bands indicate the precision of the point estimate of the simple slope. Returning to the Zhou et al. (2009) example, by utilizing the J-N techniques presented in this article the authors could extend their analysis to precisely determine at what level of weak ties the simple slope transitions from being significant to nonsignificant. Further such techniques permit the examination of the moderating effect of conformity treated as a continuous variable rather than needing to split the variable into high and low levels. Despite these benefits, to the best of the authors knowledge, this method has not been extended to analyze (a) quadratic effects, (b) linear-by-linear interactions (i.e., an X M ) where there is a quadratic term for the focal predictor (i.e., X 2 ) in the model, or (c) models that contain a quadratic-bylinear interaction term (X 2 M ). Although multiple theories in the social sciences imply such curvilinear effects, it has been recognized that simple linear moderated relations have received substantially more attention than tests of nonlinear relations (Cortina, 1993; Ganzach, 1997, 1998). In this article the J-N technique is extended to these three scenarios. To complement this extension, Excel-based tools are presented. These tools automatically calculate and graph these relations for ordinary least squares (OLS) regression, logistic regression, fixed-effect hierarchical linear models, and structural equation models. The value of using these tools is demonstrated by comparing the information provided using this new extension of the J-N technique relative to the pick-a-point approach presented in Aiken and West (1991). The remainder of this article is structured as follows: first the testing of quadratic effects is presented, along with the technique that was utilized to derive the formulae to calculate the regions of significance with the J-N method. Next the Excel-based tools developed by the authors to allow researchers to test such effects are presented. These tools are then utilized to probe the quadratic effects presented in Aiken and West (1991) to demonstrate the utility of the method. Finally, this article provides a brief discussion, an overview of limitations, and directions for future research.

6 270 MILLER, STROMEYER, SCHWIETERMAN ADVANTAGES OF THE JOHNSON-NEYMAN TECHNIQUE RELATIVE TO THE PICK-A-POINT METHOD Given that the underlying principles of post-hoc probing of a quadratic relation, a quadratic relation with a linear-by-linear interaction, and a quadratic-by-linear interaction are conceptually the same as probing a significant two-way or threeway interaction, the advantages of using the J-N approach are transferred to these settings. Bauer and Curran (2005) state that the two principal advantages of the J-N approach are that it (a) provides an exact region of significance for the conditional effect of the focal predictor and (b) the incorporation of the confidence bands quantifies the precision of the point estimate of the conditional effect. To clarify the first point, using the pick-a-point method when the moderator is continuous, the analyst typically selects arbitrary values of the moderator (often 1.0 standard deviation from the mean) in order to capture at what values of the moderator the conditional effect of the focal predictor significantly changes. However, there are two disadvantages to this approach. First, the selected arbitrary values may not be representative of the underlying distribution of the moderator; for example, if the moderator is positively skewed, selecting a value of minus 1.0 standard deviation below the mean could result in probing the interaction at a value of the moderator outside of the range of the data. The second disadvantage of this approach is that the selection of the moderator is arbitrary and can fail to capture when the relation between the focal predictor and outcome variable meaningfully changes. The J-N technique remedies the two disadvantages of the pick-a-point method highlighted earlier. First, the method accurately calculates the exact values of the moderator where the confidence bands for the conditional effect of the focal predictor either begin to contain or no longer contain zero, thus providing a region of significance (Bauer & Curran, 2005). Second, because the analyst can use the J-N method to calculate the region of significance across all values of the moderator, the arbitrary selection of moderator values in the pick-a-point technique is addressed (Hayes & Matthes, 2009; Preacher et al., 2006). Furthermore, in the following settings where the focal predictor has a quadratic relation with the outcome variable along with possible interactions with a moderator, it is argued that the J-N technique provides a far more interpretable method for examining such relations. Following the pick-a-point strategy highlighted in Aiken and West (1991) to determine the simple slope, the analyst must visualize taking the first (partial) derivative of the outcome variable with respect to the focal predictor and then calculate whether this simple slope differs from zero. The proposed extension of the J-N technique and corresponding tools directly plot the simple slope, provide confidence bands to calculate the region of significance, and calculate the points of transition.

7 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 271 TESTING QUADRATIC RELATIONS AND QUADRATIC RELATIONS WITH MODERATED RELATIONS Simple Quadratic Relations The most basic form of a polynomial relation (Cohen et al., 2003) in a regression model incorporates the quadratic term of an independent variable as follows: OY D 0 C 1 X C 2 X 2 : (1) An analyst wishing to calculate the simple slope of the effect of X on Y at a specific value of the independent variable calculates this by taking the first (partial) derivative of the regression equation with respect to X to determine the instantaneous rate of change (Hayes & Preacher, D 1 C 2 2 X: (2) To determine the significance of a simple slope at a given value of X, the analyst can conduct a hypothesis test or construct a confidence interval. The standard error for the simple slope with a quadratic term is as follows, with values of s ii and s ij coming from the variance-covariance matrix of the predictor variables (Aiken & West, 1991): SE D p s 11 C 4Xs 12 C 4X 2 s 22 : (3) To extend the J-N technique to this setting, we solve for the value of X where the confidence bands of the simple slope formula cross the X-axis (Bauer & Curran, 2005; Hayes & Matthes, 2009) using the quadratic formula. This is calculated as follows: t crit D 1 C 2 2 X p s11 C 4Xs 12 C 4X 2 s 22 (4) t 2 crit D 2 1 C X C X C X2 s 11 C 4Xs 12 C 4X 2 s 22 (5) t 2 crit s 11 C 4t 2 crit Xs 12 C 4t 2 crit X2 s 22 D 2 1 C X C X2 (6) t 2 crit s 11 C 4t 2 crit Xs 12 C 4t 2 crit X2 s X X2 D 0 (7).4tcrit 2 s /X2 C.4tcrit 2 s /X C.tcrit 2 s /1 D 0 (8)

8 272 MILLER, STROMEYER, SCHWIETERMAN a D 4t 2 crit s (9) b D 4t 2 crit s (10) X D c D t 2 crit s (11) v.4tcrit 2 s u / t.4t2 crit s / 2 4.4tcrit 2 s /.t2 crit s / 2.4tcrit 2 s / : (12) The values of X that are calculated indicate the exact values where the simple slope crosses the threshold for significance. The researcher then examines the calculated values of X to determine if they are within the interval covered by the data. If both values of X fall outside the range of the data, it indicates that the simple slope is either significant across all values of X or the simple slope is not significant across any value of X within the range of the data (Hayes & Matthes, 2009). One value within the interval of the data indicates that the simple slope is nonsignificant over some portion of X but at some point becomes significant and remains significant. If both values are within the range of X, then the researcher knows that the simple slope either (a) is significant, becomes nonsignificant, and then becomes significant again or (b) is nonsignificant, becomes significant, and then returns to being nonsignificant. As subsequently shown, plotting this function is a critical step that aids in the interpretability of the results. Quadratic Relations With a Linear Interaction Between X and a Moderator, M A more complicated linear model occurs when a moderator.m/ influences the strength and/or direction of the linear term of a focal predictor that also has a quadratic term in a regression equation. Following Aiken and West (1991), this equation can be denoted as follows: OY D 0 C 1 X C 2 X 2 C 3 M C 4 XM: (13) In this setting, when the simple slope is calculated as the first partial derivative of the regression equation with respect to X, the resulting simple slope D 1 C 2 2 X C 4 M: (14)

9 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 273 The significance of this simple slope at an arbitrary value of both X and M is calculated as usual by dividing this value by the associated standard error, which from Aiken and West (1991) is SE D p s 11 C 4Xs 12 C 4X 2 s 22 C 2Ms 14 C 4XMs 24 C M 2 s 44 : (15) In this setting, to apply the J-N approach, one solves for the values of the focal predictor and moderator to determine the region of significance for the simple slope. As such, the value of either X or M is held constant in order to solve the quadratic formula. Here we present the component solutions (a, b, and c) for the quadratic equation given that either X or M is fixed: t crit D 1 C 2 2 X C 4 M p s11 C 4Xs 12 C 4X 2 s 22 C 2Ms 14 C 4XMs 24 C M 2 s 44 (16) t 2 crit D 2 1 C X C M C XM C X2 C 2 4 M 2 s 11 C 4Xs 12 C 4X 2 s 22 C 2Ms 14 C 4XMs 24 C M 2 s 44 (17) t 2 crit s 11 C 4t 2 crit Xs 12 C 4t 2 crit X2 s 22 C 2t 2 crit Ms 14 C 4t 2 crit XMs 24 C t 2 crit M 2 s 44 D 2 1 C X C M C XM C X2 C 2 4 M 2 (18) t 2 crit s 11 C 4t 2 crit Xs 12 C 4t 2 crit X2 s 22 C 2t 2 crit Ms 14 C 4t 2 crit XMs 24 C t 2 crit M 2 s X M XM X2 2 4 M 2 D 0 (19) When X is fixed, a D t 2 crit s (20) b D 2t 2 crit s 14 C 4t 2 crit Xs X (21) c D t 2 crit s 11 C 4t 2 crit Xs 12 C 4t 2 crit X2 s X X2 : (22) When M is fixed, a D 4t 2 crit s (23) b D 4t 2 crit s 12 C 4t 2 crit Ms M (24) c D t 2 crit s 11 C 2t 2 crit Ms 14 C t 2 crit M 2 s M 2 4 M 2 : (25) As noted earlier, one challenge in this setting is given that both X and M are included in the simple slope equation, the analyst must fix a value of either

10 274 MILLER, STROMEYER, SCHWIETERMAN X or M in order to solve for only one unknown. Although this involves the specification of arbitrary values that were previously noted as a limitation of the pick-a-point approach, three caveats need to be mentioned. First, by fixing the value of one variable, the analysis is shifted to a two-dimensional plot that improves interpretability. Second, upon examining histograms of X and M, the analyst can determine a few values (3 5) at which to conduct the analysis that are representative of the data. Alternatively, values of the focal predictor could be selected that correspond to theory. Third, the pick-a-point method requires both X and M to be arbitrary. Apart from probing the conditional effect with respect to the focal predictor, the analyst may want to examine the conditional effect of the moderator depending on the value of X. This effect is a traditional two-way interaction, and several works (Bauer & Curran, 2005; Hayes & Matthes, 2009; Preacher et al., 2006) have already presented applications of the J-N technique in this setting, and as such it will not be discussed here. Quadratic Relations With a Quadratic Interaction Between X and a Moderator, M The most complicated model used here extends the J-N technique to when there is a quadratic-by-linear interaction between the focal predictor and the moderator (Aiken & West, 1991). Such an equation can be specified as follows: OY D 0 C 1 X C 2 X 2 C 3 M C 4 XM C 5 X 2 M: (26) As before, the simple slope is calculated as the first partial derivative of the regression equation with respect to D 1 C 2 2 X C 4 M C 2 5 XM: (27) The standard error for testing the significance of such a simple slope is given as v u SE D t s 11 C 4Xs 12 C 2Ms 14 C 4XMs 15 C 4X 2 s 22 C 4XMs 24 C8X 2 Ms 25 C M 2 s 44 C 4XZ 2 s 45 C 4X 2 Z 2 s 55 : (28) To conserve space, the solution to determine a, b, and c from the quadratic equation is provided as an appendix. If a significant quadratic by linear interaction is present, the analyst will likely wish to probe such a relation while interchanging the roles of the focal predictor

11 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 275 and moderator. In this setting, the simple slope is calculated as the first (partial) derivative with respect to M as D 3 C 4 X C 5 X 2 : (29) From Aiken and West (1991), the standard error for this simple slope is SE D p s 33 C 2Xs 34 C 2X 2 s 35 C X 2 s 44 C 2X 3 s 45 C X 4 s 55 : (30) The challenge of directly implementing a strategy for finding the values of X where the simple slope of M becomes significant/nonsignificant is clearly compounded in this setting given that it would be necessary to solve a fourthorder polynomial. As such, only the graphical procedure is implemented. This procedure shown in the analysis section provides the analyst with a means of ascertaining approximate values for the region of significance for the simple slope of M. SUMMARY OF THE ONLINE TOOLS AVAILABLE FOR USE Given that the methods presented here are not available in commercial software packages, nor to the authors knowledge do working macros exist, a set of Excel-based tools that allow the implementation of these aforementioned extensions has been made freely available online at nqw1w40nujty38u/bceknmkfy8. A detailed PDF illustrating the exact instructions for the use of each Excel tool is provided along with a description of the mathematical underpinnings for each technique. The researcher is required to specify minimal information into the spreadsheet tab for the J-N technique as well as an optional second tab that uses the traditional pick-a-point approach. It should be noted that these two approaches can be operated independently. To use the J-N tools, the researcher only needs to enter necessary information concerning (a) the critical value for the significance tests; (b) the beta coefficients; (c) the variances and covariances of these coefficients; and (d) in the case of models involving a quadratic focal predictor and linear or quadratic interaction, the researcher must fix the value of the focal predictor or moderator. In this special case, given the ease by which the fixed focal predictor or moderator value can be changed, it is recommend that either the convention from Aiken and West (1991) and Cohen et al. (2003) of using 1.0 standard deviations and the mean of the focal predictor be utilized or the 10th, 25th, 50th, 75th, and 90th percentiles of the focal predictor be utilized as these specified

12 276 MILLER, STROMEYER, SCHWIETERMAN values. It is further recommended that researchers mean-center predictors in such models to aid in the interpretability of the first-order coefficients. The last task the researcher must do is to define the range of the focal predictor or moderator to allow the tool to generate a graph of the simple slope and confidence bands. The second portion of the tool uses the pick-a-point method for testing simple slopes at specified levels of the moderator; this tool allows researchers to test the simple slope of the focal predictor when the moderator is dichotomous, and thus the J-N technique cannot be utilized. The researcher needs to provide (a) the critical value for the significance tests; (b) the beta coefficients; (c) the variances and covariances of the predictors; (d) the mean and standard deviation of the moderator as well as the number of standard deviations above and below the mean to test the simple slopes; (e) the range of the focal predictor to graph the interaction; and (f) in more complex models, the researcher must specify a fixed value of the focal predictor as in the J-N tools. These requirements are similar to the online tools developed in Preacher et al. (2006) for probing two- and three-way interactions in OLS regression, multilevel, and latent curve models. These tools test the significance of the simple slopes using both hypothesis tests as well as providing confidence intervals, and upon specification, plot the interaction along with the regression equation. EXAMPLE SCENARIOS To demonstrate the value of the proposed J-N tools, a reanalysis of the data published in Chapter 5 of Aiken and West (1991) is presented. This strategy was selected for several reasons: (a) it provided a means to calibrate the tools, (b) reanalyzing the provided data allows direct illustration of the additional information provided by the extended J-N approach, and (c) Aiken and West have already provided a context for the simulated data, thus allowing an easier interpretation of the findings. To provide background for this study, Aiken and West used a sample of N D 400 simulated pairs of moderately correlated, univariate normal data to generate the regression equations used for their analysis. In this setting, the criterion variable.y / represents the individual s level of self-disclosure, the focal predictor.x/ represents an individual s self-concept, and the moderator.m/ represents the amount of alcohol consumed in a social situation in which an individual has an opportunity to self-disclose (Aiken & West, 1991, p. 67). Quadratic Effect of Self-Concept Presented here is the plot of the curvilinear effect of self-concept along with a traditional plot of the simple slopes using the pick-a-point method.

13 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 277 Figure 1 is the plot of the predicted value of the criterion variable on the Y- axis versus the value of the focal predictor on the X-axis. To calculate the simple slope, the analyst takes the first partial derivative of the criterion with respect to the focal predictor, resulting in a line tangent to the curve. A significance test is then completed to determine whether the slope of this tangent line is significantly different from zero. Given the symmetric distribution of the selfconcept predictor, the analyst would likely select 1.0 standard deviation ( units) as well as the mean to probe the relation. This is shown in Figure 2 and in the table of significance tests (Table 1); note, these results are consistent with results presented on page 75 of Aiken and West (1991). Although the pick-a-point procedure works effectively in this setting, one thing that is apparent is that at some point the simple slope for regressing self-concept on self-disclosure must be nonsignificant because the simple slope moves from negative and significantly different from zero to positive and significantly different from zero. Application of the J-N technique, presented in Figure 3, to this same relation captures this missing information. The important difference between the J-N plot and the pick-a-point plot of the simple slopes is that the Y -axis of the J-N plot is the simple slope of self-concept FIGURE 1 self-concept. Plot of the regression equation for regressing self-disclosure on the value of

14 278 MILLER, STROMEYER, SCHWIETERMAN FIGURE 2 Plot of the predicted value of self-disclosure at self-concept at low, average, and high levels of self-concept. The slope of each plotted line is the simple slope of selfconcept at the fixed value of self-concept (color figure available online). conditional on the level of self-concept. As before, this plot indicates that when self-concept is low there is a significant, negative simple slope, indicating that when self-concept is low, increasing self-concept will decrease self-disclosure. The confidence bands indicate the precision of the point estimate and indicate whether the simple slope is expected to differ significantly from zero. As can be seen, from approximately 0.75 to 0.15 units, the simple slope of selfconcept does not differ significantly from zero. The exact values calculated from the Excel-based tool are and 0.154, which is consistent with the graph of the regression equation. TABLE 1 Table Containing Significance Tests for the Simple Slope of Self-Concept at Low, Average, and High Self-Concept Value of Self-Concept Simple Slope Standard Error t-test Lower 95% CI Upper 95% CI Low self-concept Average self-concept High self-concept

15 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 279 FIGURE 3 Johnson-Neyman plot of the region of significance for the simple slope of self-concept on self-disclosure (color figure available online). To summarize, in this scenario, the J-N plot provided more information than the pick-a-point plot of the simple slopes in that it provided the regions of significance and nonsignificance for the simple slope of self-concept. Based on the J-N plot, it is seen that when self-concept is less than units, the simple slope of self-concept is negative and significantly different from zero, indicating that an increase in self-concept results in a statistically significant decrease in self-disclosure. When self-concept ranges from to 0.154, an increase in self-concept is not expected to have a significant effect on selfdisclosure. When self-concept is above units, an increase in self-concept is expected to result in a significant, positive increase in self-disclosure. Quadratic Self-Concept Effect and a Linear-by-Linear Self-Concept Alcohol Consumption Interaction As shown in the plot of the regression equation returned by the tool (Figure 4), when one adds an interaction term with a second predictor to a model that already includes a quadratic term for the focal predictor, interpretational challenges greatly increase. Figure 4 shows a plot of a regression model defined by coefficients from Equation 13.

16 280 MILLER, STROMEYER, SCHWIETERMAN FIGURE 4 Plot of the predicted value of self-disclosure when there is a significant quadratic effect of self-concept and a significant Self-Concept Alcohol Consumption interaction (color figure available online). In this scenario, the pick-a-point approach (Figure 5) would involve probing this complex effect by graphing the predicted value of the outcome variable across the values of the focal predictor contingent upon the moderator similar to what is shown here. To calculate the simple slopes, the analyst would fix a value of the focal predictor (self-concept) and test the simple slope of self-concept at this fixed value at the specified values of the moderator (alcohol consumption). The clear disadvantage of this approach is that two sets of arbitrary values must be selected, both for the focal predictor and moderator. The proposed implementation of the J-N approach requires only the selection of arbitrary values for the focal predictor, which removes the chance of selecting arbitrary values of the moderator that may miss valuable information. Here is presented a series of three J-N plots when self-concept has been fixed at 1.0 standard deviations, its mean, and C1.0 standard deviations, thus corresponding to approximately the 16th, 50th, and 84th percentiles of self-concept given that the data are normally distributed. The interpretation is begun by examining the J-N plot of the simple slope of self-concept when self-concept is held at 1.0 standard deviations ( units) and alcohol consumption is allowed to vary across all values (Figure 6A). The plot indicates that for individuals who have a low self-concept, if these individuals have not consumed much alcohol, then an increase in self-concept

17 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 281 FIGURE 5 Plot of the predicted value of self-disclosure at low, average, and high levels of alcohol consumption across the range of self-concept (color figure available online). will decrease their level of self-disclosure. However, as alcohol consumption increases, the negative impact that an increase in self-concept has on selfdisclosure becomes nonsignificant. The calculated values of alcohol consumption for the region of significance are and 6.07, which, based on the plot, allows us to conclude that when self-concept is fixed at 1.0 standard deviations and when alcohol consumption is < units, then an increase in self-concept is expected to have a significant, negative impact on self-disclosure. When an individual s self-concept is low and his or her alcohol consumption is units, an increase in self-concept is not expected to have a significant impact on self-disclosure. Figure 6B is the J-N plot of the simple slope of self-concept when self-concept is held at its mean and alcohol consumption is allowed to vary. It indicates that when alcohol consumption is low (specifically < 1.79 units), an increase in self-concept is expected to result in significantly lower levels of self-disclosure. The simple slope of self-concept does not differ significantly from zero from 1.79 to 0.67 units of alcohol consumption, but when alcohol consumption is > 0.67 units, one would expect that an increase in self-concept would result in a significant, positive increase in self-disclosure. A similar finding occurs when self-concept is fixed at C1.0 standard deviations (0.945 units) above the mean except that in this scenario (Figure 6C), regardless of the level of alcohol consumption, the simple slope of self-concept is not negative. Given that a

18 282 MILLER, STROMEYER, SCHWIETERMAN FIGURE 6A Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at a low value ( 1 standard deviation) of self-concept across the range of alcohol consumption (color figure available online). participant already has a high level of self-concept, we would not expect an increase in self-concept to result in a significant decrease in self-disclosure. An alternative method for examining this complex nonlinear effect is to utilize the J-N tool but in this setting fix the value of the moderator and solve the quadratic equation for X. Previously, the interpretation of the conditional effects was given a current value of the focal predictor (self-concept), at what values of the moderator (alcohol consumption) does an increase in the focal predictor (self-concept) result in a significant increase or decrease in the criterion (self-disclosure)? When the analyst fixes the value of M and solves for X, the interpretation changes to given a current value of the moderator (alcohol consumption), at what values of the focal predictor (self-concept) does an increase in the focal predictor (self-concept) result in a significant increase or decrease in the criterion (self-disclosure)? Whereas the former method is designed to analyze the interaction effect, the later method is designed to better understand the quadratic effect. Here we report a series of three J-N plots when the moderator has been fixed to examine the simple slope across values of the focal predictor.

19 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 283 FIGURE 6B Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at the average value (0) of self-concept across the range of alcohol consumption (color figure available online). Figure 7A indicates that given a low level of alcohol consumption, if participants have a low level of self-concept, then an increase in self-concept is expected to reduce their level of self-disclosure, but if self-concept is greater than 0.08 units above the mean an increase in self-concept is not expected to influence self-disclosure. Figure 7B indicates that given an average level of alcohol consumption, for participants who have a low self-concept (< 0.92 units below the mean), an increase in self-concept is expected to reduce selfdisclosure. Furthermore, for participants whose self-concept is currently above 0.32 units, we expect an increase in self-concept to result in a significant increase in self-disclosure. Figure 7C indicates that given a high level of alcohol consumption, we no longer expect participants who have a low self-concept to reduce their self-disclosure given an increase in self-concept, whereas the increase in self-disclosure for participants with a high self-concept is expected to increase rapidly due to an increase in self-concept. Along with probing this relation, the analyst would likely be interested in probing the simple slope of alcohol consumption to understand how alcohol consumption impacts self-disclosure at various values of self-concept. Figure 8

20 284 MILLER, STROMEYER, SCHWIETERMAN FIGURE 6C Johnson-Neyman plot of the simple slope on self-disclosure of self-concept at a high value (C1 standard deviation) of self-concept across the range of alcohol consumption (color figure available online). shows a standard two-way J-N interaction plot of this conditional effect in the vein of Bauer and Curran (2005). As can be seen, the impact that alcohol consumption has on self-disclosure is conditional on the level of self-concept. When self-concept is above 0.68 units, an increase in alcohol consumption is expected to have a significant positive impact on self-disclosure; this finding indicates that increasing alcohol consumption when participants have a higher level of self-concept will increase self-disclosure. However, when individuals have low self-concept, an increase in alcohol consumption does not increase self-disclosure. A researcher would thus conclude that alcohol consumption has the greatest impact on self-disclosure when individuals have higher levels of self-concept. Quadratic Self-Concept Effect and a Quadratic-by-Linear Self-Concept Alcohol Consumption Interaction Figure 9 shows the three-dimensional plot of the predicted value of self-disclosure in the given scenario, indicating a very complicated relation is present.

21 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 285 FIGURE 7A Johnson-Neyman plot of the simple slope of self-concept on self-disclosure when alcohol consumption is low ( 1 standard deviation) across the range of self-concept (color figure available online). In this example, Equation 26 can be represented by a plot of the regression equation. The feature that distinguishes this model from the previous model is that the shape of the quadratic effect for self-concept changes depending on the level of alcohol consumption. This can be seen in a plot of the simple slopes shown in Figure 10. Following the aforementioned procedure, three J-N plots are provided when self-concept has been fixed at 1.0 standard deviations, its mean, and C1.0 standard deviations, or the 16th, 50th, and 84th percentiles. Given that these plots are interpreted in the same manner as in the previous example, we do not focus on the interpretation here other than to note that the simple slope of self-concept across alcohol consumption differs in direction in this example. When self-concept is low (Figure 11A), given that an individual has higher levels of alcohol consumption, we would expect a decrease in selfdisclosure if there was an increase in self-concept. However, given an average (Figure 11B) or high (Figure 11C) level of self-concept, we would expect that at higher levels of alcohol consumption an increase in self-concept results in an

22 286 MILLER, STROMEYER, SCHWIETERMAN FIGURE 7B Johnson-Neyman plot of the simple slope of self-concept on self-disclosure when alcohol consumption is average (0) across the range of self-concept (color figure available online). increase in self-disclosure. The J-N approach presented here allows the analyst to easily identify this distinct pattern. Additionally, as before, to better understand the change of the quadratic component, the analyst would want to fix the value of the moderator (alcohol consumption) and examine the simple slope of the focal predictor across values of itself. Shown here are three J-N plots when alcohol consumption has been fixed at 1, 0, and C1 standard deviations from the mean, or the 16th, 50th, and 84th percentiles. These three J-N plots capture how the quadratic effect of self-concept changes due to different levels of alcohol consumption. When alcohol consumption is low, Figure 12A reveals that we expect an increase in self-concept to reduce self-disclosure only for those individuals in the middle of the pack of self-concept. However, when alcohol consumption is held at its mean (Figure 12B), we see that an increase in self-concept is expected to reduce selfdisclosure for individuals at a lower level of self-concept (those with selfconcepts < 0.22 units), whereas an increase in self-concept does not impact self-disclosure for those above this threshold. Finally, when alcohol consumption is high (Figure 12C), we expect that an increase in self-concept will reduce

23 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 287 FIGURE 7C Johnson-Neyman plot of the simple slope of self-concept on self-disclosure when alcohol consumption is high (C1 standard deviation) across the range of self-concept (color figure available online). self-disclosure for individuals with low self-concept, whereas an increase in self-concept will increase self-disclosure for individuals with a higher level of self-concept. As before, in this regression model, the analyst will likely want to probe the effects using alcohol consumption as the focal predictor. It is important that, as previously noted, the simple slope of alcohol consumption includes a squared term for self-concept; this implies that the J-N plot will in fact be quadratic. This plot and an interpretation of the findings are presented in Figure 13A and Figure 13B. The aforementioned plot indicates that when self-concept is low, an increase in alcohol consumption is not expected to have a significant impact on selfdisclosure for individuals with self-concept scores from to 0.25 units. Thus one could say, based on this plot, that one would expect an increase in alcohol consumption to significantly increase the self-disclosure of individuals with the lowest level of self-disclosure along with significantly increasing self-disclosure for participants who have self-concepts above 1/4th standard deviations below the mean level of self-concept. The flexibility of the Excel-

24 288 MILLER, STROMEYER, SCHWIETERMAN FIGURE 8 Johnson-Neyman plot of the simple slope of alcohol consumption on selfdisclosure across the range of self-concept (color figure available online). based tool allows for the easy respecification of the chart area to gain a more detailed look at the region of significance if desired as shown here. DISCUSSION In this article the Johnson-Neyman (J-N) technique was extended to (a) quadratic effects, (b) linear-by-linear interactions where there is a quadratic term for the focal predictor (i.e., X 2 ) in the model, and (c) models that contain a quadraticby-linear interaction term.x 2 M/. Results were presented from a series of tools freely available online that calculate regions of significance as well as graph the simple slope and confidence bands. These tools were utilized to reanalyze the data provided in Chapter 5 of Aiken and West (1991) to demonstrate the value of using the J-N technique in these settings. Simply stated, when the moderator is continuous, the J-N approach is more effective than the pick-a-point method in that it (a) provides a region of significance for the simple slope of the focal predictor, (b) quantifies the precision of the estimate of the simple slope, and

25 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 289 FIGURE 9 Plot of the predicted value of self-disclosure when there is a significant Self- Concept 2 Alcohol Consumption interaction (color figure available online). FIGURE 10 Plot of the predicted value of self-disclosure at low, average, and high levels of alcohol consumption across the range of self-concept (color figure available online).

26 290 MILLER, STROMEYER, SCHWIETERMAN FIGURE 11A Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at a low value ( 1 standard deviation) of self-concept across the range of alcohol consumption (color figure available online). (c) when there are linear-by-linear or quadratic-by-linear interactions, the analyst must pick arbitrary values of only the focal predictor or moderator rather than both the focal predictor and moderator as is done in the pick-a-point approach. It is important to note that accurate conclusions can be drawn from these tools only if the underlying assumptions of the linear models and estimation methods are met in the analyst s data. For example, in ordinary least squares (OLS) regression, critical assumptions are that residuals exhibit constant variance (i.e., homoscedasticity) and are normally distributed. In the case of OLS, if the assumption of homoscedasticity is violated, then the use of HC3 or HC4 heteroscedastic-consistent standard errors (Hayes & Cai, 2007; Long & Ervin, 2000) or the use of a weighted least squares estimator (Cohen et al., 2003) is recommended. Likewise, nonnormal residuals can often be addressed through appropriate transformations of the dependent variable (Cohen et al., 2003). In the analysis of the data in Aiken and West (1991), the predictors are assumed to be measured without error. However, in applied research in the social sciences, this is rarely the case (Cohen et al., 2003). As has been well reported,

27 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 291 FIGURE 11B Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at the average value (0) of self-concept across the range of alcohol consumption (color figure available online). when predictors are measured with error the resulting regression coefficients are attenuated compared with those derived from a structural model measured without error, and thus the probability of a Type II error is increased (Busemeyer & Jones, 1983). Bohrnstedt and Marwell (1978) showed that the reliability of a product term is a function of the reliability of the individual variables as well as the correlation between the true scores of the variables. When the true scores have a correlation of 0, the reliability of the product term is the product of the reliabilities of the two components; if X and M have a reliability of 0.70 and 0 true score correlation, the reliability of XM will equal As the correlation increases between the true scores, the reliability of XM will approach that of the individual components (MacCallum & Mar, 1995). Given that X 2 is also a product term (Shepperd, 1991), as demonstrated by MacCallum and Mar (1995), whenever the true scores of X and M are correlated, then the reliability of X 2 will be less than XM, and as a result, the Type II error rate for tests of quadratic effects will be higher than tests of linear-by-linear interactions. Consequently, when researchers test models such as those discussed

28 292 MILLER, STROMEYER, SCHWIETERMAN FIGURE 11C Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at a high value (C1 standard deviation) of self-concept across the range of alcohol consumption (color figure available online). in this article, special attention should be placed on using measurement instruments with high reliability. Fortunately, methodological advancements using structural equation modeling including the double-mean-centered approach (Lin, Wen, Marsh, & Lin, 2010) as well as the latent moderated structural model (Klein & Moosebrugger, 2000) and more robust quasi-maximum likelihood estimation (Klein & Muthén, 2007) should help address these issues of measurement error. A second topic concerns spurious moderator effects (Busemeyer & Jones, 1983; Lubinski & Humphreys, 1990). As noted by these authors and others (Cortina, 1993; Ganzach, 1997, 1998), when the correlation between the focal predictor and moderator is large, the quadratic term.x 2 / and linear-by-linear (XM) terms will have considerable overlap in the variance that they explain in the criterion variable. Lubinski and Humphreys (1990) proposed a stepwise procedure where the additive model with X and M is first tested, then a stepwise procedure is used to determine whether X 2, M 2, or XM should be included based on the increase in explained variance. Cortina (1993) proposed

29 EXTENSIONS OF THE JOHNSON-NEYMAN TECHNIQUE 293 FIGURE 12A Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at a low value ( 1 standard deviation) of alcohol consumption across the range of selfconcept (color figure available online). a more stringent requirement by arguing that linear-by-linear interaction models should be estimated with both X 2 and M 2. Ganzach (1998) concurs with this recommendation, noting that although inclusion of these quadratic terms does reduce the power to detect a significant XM term if the quadratic components are not present in the true structural model, in most of the situations encountered by researchers in management, adding quadratic terms does not result in a considerable increase in the probability of Type II error in detecting interaction if the true model does not include the quadratic terms (p. 621). Shepperd (1991), Aiken and West (1991), MacCallum and Mar (1995), and Cohen et al. (2003) cautioned against the automatic inclusion of quadratic and interaction terms but instead argued that theory should drive the inclusion of such terms. First, assuming that X and M are correlated, including X 2, M 2, and XM in the same model will result in higher levels of multicollinearity, less stable beta weights, and consequently less power to detect significant effects (Aiken & West, 1991; MacCallum & Mar, 1995). Second, in many instances, these

30 294 MILLER, STROMEYER, SCHWIETERMAN FIGURE 12B Johnson-Neyman plot of the simple slope of self-concept on self-disclosure at the average value (0) of alcohol consumption across the range of self-concept (color figure available online). terms may have little substantive or theoretically meaning. Third, increasing the number of coefficients in the model increases the probability of committing a Type I error (Shepperd, 1991). Fourth, given that one of the stated goals of science is parsimony, Cohen (1990) summarized the argument succinctly by stating, In any given investigation that isn t explicitly exploratory, we should be studying few independent variables and fewer dependent variables, for a variety of reasons (p. 1304). Although both of these perspectives have merit, we support the argument that theoretical justification should exist before including curvilinear and interaction terms in linear models. Limitations As with all research, this article has its limitations. One limitation is that when testing more complicated models involving a third variable as a moderator, it was necessary to fix an arbitrary value of the focal predictor X or moderator M. Sources such as Aiken and West (1991) and Jaccard and Turrisi (2003) utilize a