# Can I probe cross-level interactions without random slope in a hierarchical linear model?

I have a HLM model with significant variance in the level 1 intercepts across groups but no significant variance in the level 1 slopes across groups and find significant cross-level moderation effects. Does it make sense to interpret these or are random slopes a necessary condition for probing cross-level interaction effects?

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

• Thanks! Would you by any chance have any references? Mainstream HLM literature tends to pose significant slope variance as a prerequisite for further analysis. – Bento May 23 '13 at 19:35
• I guess I am inclined to turn the question around and ask you whether you have references which say that such a prerequisite exists. I am confused why you think it should. Nowhere in the definition of a cross-level interaction is the idea of random slopes ever invoked, or random effects at all for that matter; we can even have cross-level interactions in mixed-model ANOVA, as in the traditional analysis of "split-plot" experiments. – Jake Westfall May 23 '13 at 20:25
• One example of a step-by-step discussion that describes significant variance in the level-1 slope as a precondition for testing interactions in slope-as-outcome models would be Hoffman, David A. Journal of Management 1991, Vol. 23, No. 6, 723-744 – Bento May 24 '13 at 8:40
• I looked over the Hofmann paper and I think I see the source of confusion here. I will edit my answer to try to address this. – Jake Westfall May 24 '13 at 19:37
• Thanks a lot Jake. In the meantime I also found a simulation study by LaHuis & Ferguson (2009) in Organizational Research Methods, 12(3): 418-435 which sheds some light on my initial question. – Bento May 25 '13 at 9:34