In short
A moderation analysis is reported with four pieces and none of them is optional. The interaction term with its unstandardised coefficient, standard error, t with degrees of freedom and p value: b3 = −0.15, SE = 0.05, t(210) = −3.00, p = .003. The R2 change the product contributes over the model without it: ΔR2 = .03, F(1, 210) = 9.00, p = .003. The simple slopes at −1 SD, at the mean and at +1 SD of the moderator, each with its confidence interval. And the Johnson-Neyman region of significance, which tells you the moderator value beyond which the effect stops being distinguishable from zero. Both variables in the product are mean-centered before you multiply them: centering moves neither the interaction coefficient nor its p value, but it is the only thing that makes the first two rows of the table interpretable.
The mistake I correct most often in a moderation analysis is not in the model, it is in the first row of the table. Someone multiplies two variables without centering them, the coefficient on the predictor comes out small and strange, and the Results paragraph ends up describing the effect of stress in people who score zero on a scale that starts at one. The interaction, meanwhile, is fine: centering never touches it. Here is what APA 7 asks for in a moderation analysis — the interaction term, the R2 change, the simple slopes, Johnson-Neyman and the figure — with a full worked example to copy. The general guide is in how to report results in APA 7, and if your model also has a mediator in the middle, what you have is moderated mediation, which takes a different route.
What a moderation tests, and how it differs from moderated mediation
A moderation analysis does not claim that X influences Y. It claims that the strength of that relationship depends on a third variable. The whole hypothesis lives in a single coefficient, the one on the product X·W: if that coefficient is zero, the slope of X on Y is the same for everyone and there is nothing left to probe. That is why the order of the report is not negotiable — the interaction first, and only if the interaction holds, the simple slopes.
The model is symmetric, and it is worth knowing that before you write the discussion. The product does not know which of the two variables you call the moderator: swap X and W and the numbers come out identical, only the sentence changes. What decides who moderates whom is your theory and your design — what was manipulated, what is a trait and what is a state, what came first in time — never the software output. If in your paper cognitive reappraisal moderates stress rather than the other way round, that has to be argued in the introduction.
Do not confuse it with mediation. A mediation answers “how?”, by placing a mechanism M between X and Y; a moderation answers “for whom?” or “when?”. Moderated mediation combines the two: the indirect effect through M depends on W, and it is tested with the index of moderated mediation and its bootstrap interval, not with the ΔR2 of a product term. If you have no mediator there is no index to compute; if you do have one, a simple moderation falls short of your own hypothesis and a reviewer will say so.
Center your variables before you multiply
Centering means subtracting the mean from X and from W before computing the product. The first thing to understand is what it does not change: the interaction coefficient, its standard error, its t, its p value, the ΔR2, the R2 of the full model, the predicted values and the simple slopes all come out exactly the same with centered and uncentred variables. This is an algebraic result, not a matter of luck (Aiken & West, 1991). If you were taught that centering “fixes multicollinearity” and thereby rescues an interaction that would not come out, you were taught badly: an interaction that is not significant uncentred is not significant centered either.
What does change is the meaning of the two lower-order coefficients. With the product in the model, b1 is no longer “the effect of X”: it is the effect of X when the moderator equals zero. Uncentred, with a moderator scored from 1 to 7, zero does not exist in your sample and that coefficient extrapolates to nobody. Centered, zero is the mean, and b1 becomes the effect for the average participant, which is what you actually wanted to report. The first two rows of your table change in both number and story depending on whether you center; the bottom row, the product, never notices.
The collinearity part deserves an honest sentence. Uncentred, the correlation between X and the product X·W can exceed .90 through the pure arithmetic of where the scale origin sits: that is non-essential collinearity, an artifact that centering removes completely. What remains afterwards is essential collinearity, which comes from the skewness of X and W, and that does not go away. If the variance inflation factor of the product term is still high after centering, it is not a coding problem: it is information about your data, and it is discussed as such.
Three practical rules to keep you out of trouble. Do not center dummy variables: in a dummy, zero means something, and centering it turns your coefficient into the effect at “0.43 of a group”. Do not copy the standardized beta of the product term that SPSS prints: what has been standardized there is the raw product, and the product of two z scores is not the z of the product; if you need standardized coefficients, standardize X, W and Y first and re-estimate the model, letting the software build the product from the already standardized variables. And state in the Method that you centered, and on which mean — that of the analytic sample, after exclusions — because centering on a different N moves b1.
The interaction term and the R² change
The product coefficient is reported with all its pieces: unstandardised value, standard error, t with degrees of freedom, p value and 95% confidence interval. In the example below, b3 = −0.15, SE = 0.05, t(210) = −3.00, p = .003, 95% CI [−0.25, −0.05]. And it is interpreted in the units of the two scales, not in the abstract: each additional point of cognitive reappraisal reduces by 0.15 points the effect that one point of perceived stress has on anxiety symptoms. That sentence, with both units inside it, is what separates a result you have reported from a result you have copied.
The R2 change comes from comparing two models: step 1 with X and W, step 2 adding the product. You report the R2 at each step, the ΔR2 and its F with two degrees of freedom, the numerator (1, because you added a single term) and the denominator (N − k − 1, where k is the number of predictors in the full model). One check that will save you an embarrassment: with a single added product term, the F for the change is exactly the square of the t on the interaction coefficient, 9.00 and (−3.00)². If those two numbers do not match in your table, one of them has been copied from the wrong block of output.
The effect size of an interaction is given as f2 = ΔR2 / (1 − R2); here, .03 / .68 = 0.04, small by Cohen’s (1988) conventions, which are conventions rather than laws. Interactions explain little variance almost always, and there is a technical reason for it: the product carries the measurement error of both of its factors, so its reliability is lower than either one and power suffers (Aiken & West, 1991). Two symmetric warnings follow. A ΔR2 of .03 is not “trivial” if the slope moves from 0.47 to 0.14 within the observed range, because that is what happens to a real participant. And a non-significant interaction with N = 90 does not say the moderator is irrelevant: it says you had nothing to see it with.
Probing the interaction: simple slopes and Johnson-Neyman
A simple slope is the effect of X on Y evaluated at a specific value of the moderator, computed as b1 + b3w. Watch its standard error: it is not the standard error of b1, because it combines the variances of both coefficients and their covariance, which is why the slopes at the extremes have wider intervals than the one in the middle. The conventional probe values are the moderator mean and ±1 SD (Aiken & West, 1991), and each slope is reported with its SE, its t with degrees of freedom, its p value and its interval.
The ±1 SD convention breaks down in two situations that turn up constantly in psychology. If the moderator is skewed or has a floor or ceiling effect, the mean minus one standard deviation can fall below the minimum of the scale, and you are then describing people who do not exist; the alternative is the 16th, 50th and 84th percentiles, which is exactly what PROCESS uses by default: the mean and ±1 SD have to be asked for explicitly with moments=1. And if the moderator is categorical, the probe values are the categories, not ±1 SD of an arbitrary code. Whichever you choose, say so in the Results: “slopes were probed at the mean and at ±1 SD” is half a line that heads off a reviewer question.
The underlying problem with simple slopes is that you choose the three points yourself. Change the convention and the number of asterisks changes, without the model having moved an inch. The Johnson-Neyman technique removes that arbitrariness: instead of fixing values and inspecting significance, it solves the other way round and finds the moderator values at which the t statistic for the simple slope equals the critical value. What it returns is a region of significance — the stretch of the moderator where the effect of X on Y is distinguishable from zero, and the stretch where it is not.
It is reported with the boundary in raw moderator units, which side carries the significance, and what proportion of the sample falls in each stretch. Plus one check nearly everybody skips: that the boundary lies inside the observed range. The Johnson-Neyman equation is quadratic and usually has two solutions; in my example the second one lands at 10.73 on a scale that stops at 7, so it is a mathematical solution rather than a finding, and it is mentioned only to note that it falls out of range. With a dichotomous moderator the technique makes no sense, since there is no continuum to travel along. In SPSS, PROCESS produces it for model 1 with the jn=1 option; in R, the interactions package with johnson_neyman(). How the model is set up step by step is covered in the guide to mediation and moderation with PROCESS.
Worked example in the text, the table and the figure
“We estimated a hierarchical regression in a sample of N = 214 university students. Perceived stress (PSS-10) and cognitive reappraisal (ERQ), both mean-centered, explained 29% of the variance in anxiety symptoms (GAD-7), R2 = .29, F(2, 211) = 43.09, p < .001. Adding their product increased the explained variance, ΔR2 = .03, F(1, 210) = 9.00, p = .003, f2 = 0.04, for a final model of R2 = .32, F(3, 210) = 32.94, p < .001. Perceived stress was associated with anxiety symptoms, b1 = 0.30, SE = 0.07, t(210) = 4.29, p < .001, cognitive reappraisal was associated in the opposite direction, b2 = −0.86, SE = 0.24, t(210) = −3.58, p < .001, and the interaction term was significant, b3 = −0.15, SE = 0.05, t(210) = −3.00, p = .003, 95% CI [−0.25, −0.05].”
“Simple slopes were probed at the moderator mean and at ±1 SD (M = 4.60, SD = 1.10). Perceived stress predicted anxiety symptoms at low reappraisal (−1 SD, 3.50), b = 0.47, SE = 0.09, t(210) = 5.22, p < .001, 95% CI [0.29, 0.65], and at the mean (4.60), b = 0.30, SE = 0.07, t(210) = 4.29, p < .001, 95% CI [0.16, 0.44], but not at high reappraisal (+1 SD, 5.70), b = 0.14, SE = 0.09, t(210) = 1.56, p = .120, 95% CI [−0.04, 0.32]. The Johnson-Neyman procedure placed the boundary of significance at 5.50 points of cognitive reappraisal: above that value, where 21.5% of the sample sits (n = 46), the effect of stress was no longer distinguishable from zero.” If what you have in front of you is raw PROCESS or R output and you cannot tell which block is which, paste the output into the interpreter and it breaks it down in this format.
The table is a two-step hierarchical regression, one row per term, with columns for b, SE, the 95% confidence interval and p. At the foot of each step, the R2; at the foot of the second, the ΔR2 with its F and degrees of freedom as well. Below that, in a block of its own with its own heading, the simple slopes: one row per probe value, with that value in raw units and not merely labeled “low”, “medium” and “high”. The note carries three things: that the predictors are mean-centered, what the moderator values represent, and which covariates are in the model if there are any. Horizontal rules at the top, under the headings and at the foot; no vertical rules anywhere. If you would rather have it built already formatted, the APA 7 regression table generator will hand it to you ready to paste into Word.
The figure puts X on the horizontal axis, Y on the vertical axis and one line per moderator value. Four details nearly everybody skips: the lines are drawn only within the observed range of X, not from edge to edge of the canvas; the axes are labeled with the instrument name and its range, not with the variable name in the data file; the legend gives the moderator values (“reappraisal = 3.50 / 4.60 / 5.70”), because “high” and “low” do not say how much; and the lines are distinguished by line type as well as by color, since the journal may print in grayscale. With a continuous moderator there is a better figure than three straight lines: the Johnson-Neyman plot, which puts the moderator on the horizontal axis and the slope of X on Y on the vertical axis with its confidence band, so the boundary is where the band crosses zero and you see the whole function instead of three cross-sections. Number, title and note follow the format for figures in APA 7, and the note states that predicted values were computed with covariates held at their means.
The mistakes a reviewer flags
Reading the coefficient on X as “the effect of X”. With the product in the model that coefficient is conditional, and if you have not centered it describes a moderator value that may not exist. It is the mistake I see most often, and it cannot be patched in the discussion: it is fixed by centering and rewriting the paragraph.
Concluding moderation from the simple slopes. “Stress predicted anxiety in the low-reappraisal group but not in the high-reappraisal group” does not show that the two slopes differ; b3 shows that, which is why it comes first. If the interaction term is not significant you do not probe it, and you certainly do not present the difference between two p values as a finding.
Splitting the moderator at the median. Cutting a continuous variable in two throws away power, moves the boundary with every new sample, and puts two people scoring 4.59 and 4.61 into different categories. If the moderator is continuous, analyze it as continuous — the simple slopes already give you the portrait of “the high scorers” and “the low scorers” without breaking anything.
Extrapolating beyond the observed range. This applies to the figure, to the probe values and to Johnson-Neyman alike: if the mean minus one standard deviation falls below the minimum of the scale, or the boundary appears at a value nobody reaches, it is not describing your participants. Check the minimum and the maximum before choosing values, and say so if any of them falls outside.
Leaving the procedure unstated. If the paper does not say that you centered, which probe values you used, which software and version estimated the model and which covariates are inside it, the analysis is not reproducible. It is two sentences in the Method, and it saves the review round whose only purpose is to ask for them.
Frequently asked questions
How do you report a moderation analysis in APA 7?
With four elements. The interaction term with its unstandardised coefficient, standard error, t, degrees of freedom, p value and confidence interval (b3 = −0.15, SE = 0.05, t(210) = −3.00, p = .003). The R2 change relative to the model without the product (ΔR2 = .03, F(1, 210) = 9.00, p = .003). The simple slopes at the moderator mean and at ±1 SD, each with its interval. And the Johnson-Neyman region of significance when the moderator is continuous. The Method section states that the variables were centered and which software was used.
Why do you need to center variables in a moderation analysis?
Because in a model containing a product term, the coefficient on each lower-order variable is the effect of that variable when the other one equals zero. If the moderator is scored from 1 to 7, zero does not exist in your sample and that coefficient extrapolates to nobody. Subtracting the mean makes zero the sample average, and the coefficient recovers the interpretation you wanted for it: the effect of X for the average participant. Centering also removes the artificial correlation between X and the product, which uncentred can exceed .90.
Does centering change the interaction result?
No. The product coefficient, its standard error, its t, its p value, the ΔR2, the R2 of the full model, the predicted values and the simple slopes all come out identical with centered and uncentred variables (Aiken & West, 1991). The only things that change are the coefficients on X and on the moderator, and they change because the point at which they are evaluated changes. An interaction that is not significant uncentred will not become significant once centered: centering is an interpretation decision, not a rescue.
What are simple slopes and how do you interpret them?
They are the effect of X on Y evaluated at a specific value of the moderator, computed as b1 + b3w. The convention is to report them at the mean and at ±1 SD, each with its standard error — which is not the standard error of b1 — its t, its p value and its interval. They are interpreted like any regression slope, but bounded: “at low cognitive reappraisal, each point of stress was associated with 0.47 more points of anxiety”. They only make sense once the interaction term is significant; probing an interaction that is not significant amounts to describing differences the model has not detected.
What is the Johnson-Neyman region of significance?
It is the stretch of moderator values over which the effect of X on Y is distinguishable from zero. Rather than picking three probe points and inspecting their significance, the procedure solves the problem backwards: it finds the moderator values at which the t statistic for the simple slope equals the critical value. It is reported with the boundary in raw units, the side that carries the significance and the percentage of the sample that falls there (“above 5.50, where 21.5% of the sample sits”). You have to check that the boundary lies inside the observed range, and it makes no sense with a dichotomous moderator.
What is the difference between moderation and moderated mediation?
A moderation analysis tests whether the relationship between X and Y depends on a third variable, and it is carried by the interaction term and its ΔR2. Moderated mediation tests whether the indirect effect of X on Y through a mediator M depends on that third variable, and it is carried by the index of moderated mediation and its bootstrap interval. The practical difference is whether there is a mediator: without one there is no index to compute, and with one a simple moderation does not answer the hypothesis announced in your title.
Is your moderation carried by the interaction term, or by two slopes compared by eye?
It is the first thing an experienced reviewer checks in a model with a product term, and it cannot be fixed in the response letter: it has to be fixed before you submit.
Run it through Previum →With the interaction term, the R2 change, the simple slopes, the Johnson-Neyman boundary and a figure drawn inside the observed range, your moderation is reported the way APA 7 asks. If the model gets heavier — two moderators, a three-way interaction, repeated measures, or data nested within sites — we can work through it together in my statistical consulting.