Having random slopes at level 1 is not a necessary condition for examining cross-level interactions. All that is necessary is that you have 2 predictors that vary at different levels, and their interaction.

**EDIT:** I looked over the Hofmann paper posted in the comments and I think I see the source of confusion here.

Hofmann describes a situation in which one is building a model by starting with the simplest "empty" random-intercept model, and then working up term-by-term to the full HLM, where the very last term added is the predictor representing the cross-level interaction. Under such an approach, it is true that *in the model prior to the cross-level interaction model* (i.e., the model that is identical except that the cross-level interaction term is omitted), there must be variation in the level-1 slopes in order for there to be moderation of these slopes by a level-2 predictor. Intuitively, if every group has the same exact level-1 slope, then it is not possible for us to predict variation in these slopes from another predictor in the dataset, because there is no such variation to predict.

Notice that this is not a statement about the cross-level interaction model itself, but rather a statement about a *different* model which *omits* the cross-level interaction term. In the cross-level interaction model itself, it is entirely possible for there to be no variation in the level-1 slopes. This would essentially mean that all of the seemingly random variation in the level-1 slopes that we observed in the previous model can be accounted for by adding the cross-level interaction term to the model.

I illustrate just such a situation below with some simulated data in R, where we have a cross-level interaction between `x`

varying at level 1, and `z`

varying at level 2:

```
# generate data -----------------------------------------------------------
set.seed(12345)
dat <- merge(data.frame(group=rep(1:30,each=30),
x=runif(900, min=-.5, max=.5),
error=rnorm(900)),
data.frame(group=1:30,
z=runif(30, min=-.5, max=.5),
randInt=rnorm(30)))
dat <- within(dat, y <- randInt + 5*x*z + error)
# model with the x:z interaction ------------------------------------------
library(lme4)
mod1 <- lmer(y ~ x*z + (1|group) + (0+x|group), data=dat)
mod1
# Linear mixed model fit by REML
# Formula: y ~ x * z + (1 | group) + (0 + x | group)
# Data: dat
# AIC BIC logLik deviance REMLdev
# 2658 2692 -1322 2640 2644
# Random effects:
# Groups Name Variance Std.Dev.
# group (Intercept) 8.5326e-01 9.2372e-01
# group x 5.4449e-20 2.3334e-10
# Residual 9.9055e-01 9.9526e-01
# Number of obs: 900, groups: group, 30
#
# Fixed effects:
# Estimate Std. Error t value
# (Intercept) -0.13311 0.17283 -0.770
# x 0.09808 0.11902 0.824
# z -0.24705 0.51424 -0.480
# x:z 5.39969 0.35257 15.315
#
# Correlation of Fixed Effects:
# (Intr) x z
# x -0.010
# z 0.103 0.008
# x:z 0.007 0.137 -0.005
# model without the x:z interaction ---------------------------------------
mod2 <- lmer(y ~ x + z + (1|group) + (0+x|group), data=dat)
mod2
# Linear mixed model fit by REML
# Formula: y ~ x + z + (1 | group) + (0 + x | group)
# Data: dat
# AIC BIC logLik deviance REMLdev
# 2726 2755 -1357 2713 2714
# Random effects:
# Groups Name Variance Std.Dev.
# group (Intercept) 0.85503 0.92468
# group x 3.46811 1.86229
# Residual 0.99607 0.99803
# Number of obs: 900, groups: group, 30
#
# Fixed effects:
# Estimate Std. Error t value
# (Intercept) -0.14148 0.17312 -0.817
# x -0.05178 0.36056 -0.144
# z -0.26570 0.51509 -0.516
#
# Correlation of Fixed Effects:
# (Intr) x
# x -0.004
# z 0.103 0.002
```

1) Please tell me what software you are using.

2) If the interaction is insignificant you should remove it from your model and re-run the analysis.

3) If there is still a significant interaction, then you don't need to interpret the main effects.

4) I usually don't dummy code, and just put variables in as factors. So if you end up having a significant interaction between a categorical variable and continuous variable, then you should plot the groups at different levels of the continuous variable.

5) I found this article to be helpful regarding effect sizes:
http://journal.frontiersin.org/article/10.3389/fpsyg.2013.00863/abstract

And download the Excel sheet that is talked about within the article, it's a handy resource.

## Best Answer

The effect sizes of interactions in a multivariate regression can be assessed in same way as the effect sizes of any other predictor. The common thing is to look at the incremental contribution to R^2 (semi-partial R^2), but there are other possibilities, including Cohen's f^2 for nested models (this is a likelihood ratio test). Chapter 9 of Cohen's Statistical Power Analysis for the Behavioral Sciences has a very good discussion. It is true, too, that the effect sizes of product-term interactions tend to be "small" in terms of incremental addition to R^2. But the

practicaleffect of such an interaction can be very large. This point is especially important to bear in mind when the interaction involves some sort of treatment or intervention -- e.g., in a drug trial, the practical effect of an interaction between the treatment & some individual characteristic of a patient might contribute only a small amount to mode R^2 but have a very appreciable effect on the clinical outcomes. See Rosenthal, R. & Rubin, D.B. A Note on Percent Variance Explained as A Measure of the Importance of Effects. J Appl Soc Psychol 9, 395-396 (1979); Abelson, R.P. A Variance Explanation Paradox: When a Little is a Lot. Psychological Bulletin 97, 129-133 (1985). Both this possibility & the challenge of trying to interpret (or explicate) the simultaneous importance of the coefficients for the predictor, moderator & product-interaction in a regression output, tend to make reporting of the interaction's "effect size" uninformative; better, I'd say, is to illustrate (graphically) the effect size of the interaction in practical terms -- that is, by showing how changes in meaningful levels of the predictor ("high exposure vs. low exposure") and moderator ("being a man vs. being a woman") affect the outcome variable expressed in units that make sense given the context ("additional yrs of life"). I don't have as much experience w/ multilvel modeling, but I do know that the strategy I'm describing is the basic philosophy of Gelman, A. & Hill, J. Data Analysis Using Regression and Multilevel/Hierarchical Models. (Cambridge University Press, Cambridge ; New York; 2007)-- the greatest work on regression, in my opinion, after Cohen & Cohen!