# Simple slopes analysis

### Introduction

Researchers often need to examine whether one variable moderates the association between two variables (James & Brett, 1984). They might, for instance, want to examine whether feelings of engagement moderate the relationship between workload and dishonesty.

To illustrate, workload might be positively related to dishonesty when individuals do not feel engaged in their tasks. However, workload might not be related to dishonesty when individuals feel engaged in their tasks. Engagement, thus, moderates or changes the relationship between workload and dishonesty.

This pattern of observations appears in the following figure. In this figure, two lines appear in a graph. One line slopes upwards, and represents the relationship between workload and dishonesty when engagement is below average-specifically, when the z value of engagement is -1 or 1 standard deviation below average. The other line is almost horizontal, and represents the relationship between workload and dishonesty when engagement is above average-specifically, when the z value of engagement is +1 or 1 standard deviation above average.

Usually, a technique called moderated regression is applied to ascertain whether or not one variable moderates the association between two other variables. Another procedure, called simple slopes analysis, can then be conducted to determine whether the gradient of one or both these lines differs from 0-that is, departs from the horizontal plane. This article will briefly reiterate moderated regression but will focus on simple slopes analysis.

### Overview of moderated regression

To conduct moderated regression, researchers need to:

• First, ensure the data is entered correctly. That is, each row represents one participant and each column represents one variable, such as workload, dishonesty, and engagement.
• Second, standardize all the variables. That is, in SPSS, select the Descriptives statistics menu, the Descriptives option, then transfer all the variables into the box labeled Variables. Finally, tick the option called Save standardized variables.
• Pressing OK will create standardized variables or z values in the data file, called z_workload, z_dishonesty, and z_engagement respectively. That is, these columns are created by deducting the mean from the original variable and dividing by the standard deviation

You will utilize these variables during the remainder of this analysis. Note that some researchers do not create standardized variables when they conduct moderated regression. Standardized variables-or at least centered variables, in which the mean is deducted-is necessary if you later want to undertake simple slopes analysis. Then, the researcher should:

• Third, construct the interaction term, which is the product of z_workload and z_engagement. That is, select the Transform menu, the Compute option, type any label, such as work_x_engage, in the box labeled Target variable, and finally specify the formula z_workload * z_engagement in the box labeled Numerical expression
• Pressing OK will create a new column in the data file, which represents the interaction
• Fourth, conduct the regression analysis. That is, select the Analyze menu, the Regression and then the Linear option. Designate z_dishonesty as the dependent variable and z_workload, z_engagement, as well as work_x_engage as the independent variables.
• Before pressing OK, press the Statistics button and tick the Covariance matrix option. This phase is necessary only when you need to conduct simple slopes analysis.
• Finally, press Continue and then OK.

The researcher can now interpret the output. A table, called coefficients, should appear, which resembles the following:

 Unstandardized coefficients Standardized coefficients B SE Beta t sig Constant 2.209 2.007 1.101 0.303 z_workload 1.071 0.103 0.830 -2.686 0.020 z_engagement 0.162 0.464 0.119 0.350 0.736 work_x_engage -0.930 0.467 -0.838 2.102 0.004

According to this table, the significance or p value associated with work_x_engage is less than .05. This finding implies the interaction is significant (Aiken & West, 1991). In other words, engagement moderates the relationship between workload and dishonesty.

### Overview of the construction of simple slopes

The next step is to construct the lines that appear on the previous figure. Specifically, the researcher needs to develop two equations, one representing each of these lines. First, they need to construct an equation that represents the relationship between workload and dishonesty when engagement is below average-specifically, when the z value of engagement is -1 or 1 standard deviation below average. Next, they need to construct an equation that represents the relationship between workload and dishonesty when engagement is above average-specifically, when the z value of engagement is +1 or 1 standard deviation above average. To achieve these goals, the researcher should:

• Construct an equation that relates workload to dishonesty, using the unstandardized B values. Researchers often use the standardized B values if they had not earlier standardized or centered the variables.
• In this instance, the equation is: dishonesty = 2.209 + 1.071 x workload + 0.162 x engagement - .930 x workload x engagement.

Although not specified, these labels actually correspond to standardized variables. Next:

• To construct the equation that represents the relationship between workload and dishonesty when engagement is below average, substitute -1 for engagement in the previous formula
• Hence: dishonesty = 2.209 + 1.071 x workload + 0.162 x -1 - .930 x workload x -1.
• After some rearranging, dishonesty = 2.047 + 2.001 x workload
• In other words, when engagement is low, dishonesty is positively related to workload
• To construct the equation that represents the relationship between workload and dishonesty when engagement is below average, substitute +1 for engagement in the previous formula
• Hence: dishonesty = 2.209 + 1.071 x workload + 0.162 x +1 - .930 x workload x +1.
• After some rearranging, dishonesty = 2.371 + .141 x workload
• In other words, when engagement is high, dishonesty is still positively related to workload, but the gradient of .141 is nearly zero.

#### Simple slopes analysis

In the previous section, we developed two equations:

• When engagement is below average: dishonesty = 2.047 + 2.001 x workload
• When engagement is above average: dishonesty = 2.371 + .141 x workload

Simple slopes analysis can be conducted to answer the question as to whether these gradients-in this instance 2.001 or .141-differ from zero. That is, is workload positively related to dishonesty for both low and high levels of engagement.

#### Compute standard error of the slope

To answer this question, the researcher merely needs to calculate the standard error of this gradient. To compute this standard error, the researcher simply needs to apply the following formula:

• Standard error = Square root of [s33 + 2 x Z x s31 + Z x Z x s11]
• To identify these terms, scan the table called coefficient correlations and focus on the bottom half-the half called covariances, as shown below
 work_x_engage z_engagement z_workload work_x_engage .218 .005 -.043 z_engagement .005 .011 -.008 z_workload -.043 -.008 .215
• s33 is the number in the row and column that corresponds to the independent variable: z_workload. This number represents the variance of the B value associated with z_workload. In this instance, s11 is .215
• s31 is the number in the row that corresponds to the independent variable and the column that correspond to the interaction. This number represents the covariance of the B value associated with the independent variable and interaction. In this instance, s31 is -.043
• s11 is the number in the row and column that corresponds to the interaction term: work_x_engage. This number represents the variance of the B value associated with the interaction. In this instance, s33 is .218
• When work engagement is low, Z is set to -1
• Hence, standard error equals the square root of .215 + 2 x -1 x -.043 + -1 x -1 x .18 = .509
• When work engagement is high, Z is set to +1
• Hence, standard error equals the square root of .215 + 2 x 1 x -.043 + 1 x 1 x .18 = .347

#### Compute the t values of the slope

Finally, to ascertain whether the simple slope--that is, the slope of each line--differs from zero, researchers need to:

• Divide the gradient by the standard error, to generate a t value
• When work engagement is low, t = 2.001/ square root of .509 = 2.80
• When work engagement is high, t = .141 / square root of .347 = .24
• To determine the p value when engagement is low, open Excel and type =TDIST(2.80, 280, 2), which is significant at .05. In this example, the first number is the t value, the second number is the number of participants - number of independent variables - 1, and the 2 denotes two tailed
• To determine the p value when engagement is high, open Excel and type =TDIST(.24, 280, 2), which is not significant at .05.

Hence, in this contrived example, one of the two slopes or lines differs significantly from 0 or horizontal.

Simple slopes analysis in practice

Researchers do not have to construct the equations at Z=1 or Z=-1, but can incorporate other z values as well. Indeed, many researchers do not even ascertain whether these simple slopes differ significantly from zero. That is, the overall pattern, uncovered by the moderated regression, is often more important. Whether or not two arbitrary lines differ from the horizontal plane might not be as important.

### References

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Thousand Oaks, CA: Sage.