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I have a fitted mixed-effects model with a continuous dependent variable and multiple predictors that each vary by multiple random factors. In R, a simplified version of the model has the following form:

lmer(DV ~ A + B + (A + B | X))

A and B (both continuous predictors) are not permitted to interact with each other in the model, and they are strongly correlated with each other (r^2 ≈ .75). (As the actual data set includes >100K observations, though, this collinearity does not seem to pose a problem for significance testing.)

For each predictor A and B, I want to visualize the fixed effect of that predictor -- and that predictor "alone" -- on the DV while reasonably representing the variability in the underlying data. (Showing their joint/simultaneous effect on the DV is likely not a viable option as my actual data set includes 3 continuous variables.) The issue is that I'm not sure how to do this in a statistically fair way given the random effects structure (which makes me unsure of how to apply the advice given to this question). So far, I have considered the following two options for Predictor A, with analogous graphs for Predictor B:

Method 1: Graph the relationship between A and [DV minus the fixed-effect contribution of Predictor B]. Given that A and B are centered, this would show the effect of each variable when the other one is held constant at its mean; and since they're highly correlated/collinear with the DV, I think this would show each of them accounting for the (significant amount of) shared variance.

Method 2: Graph the relationship between A and [residual error plus the overall intercept plus the fixed-effect contribution of Predictor A]. I think this would show each variable accounting only for the fixed-effect variance it uniquely accounts for.

Due (I think) to collinearity and to the random effects structure, these methods do not necessarily generate the same figures; a reproducible example demonstrating this on a subset of the data can be found below (though in fairness, many other subsets of the data did yield remarkably similar plots for the two methods). Of course, other options may well be better than either of these! My question is: What is a fair way to separately visualize the effects of each predictor on the DV?

(Apologies in advance if others think I should have posted this question to StackOverflow instead of CrossValidated, but I feel the crux of the question is statistical in nature.)

# load libraries
library(lme4)
library(ggplot2)
library(gridExtra)

# directly specify data frame
# (sorry for the enormous length -- smaller subsets of data did not seem to do the job)
example.df <- structure(list(RanFact = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 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814L, 834L, 591L, 515L, 629L, 586L, 509L, 777L, 567L, 488L, 752L, 514L, 765L, 789L, 596L, 711L, 743L, 471L, 640L, 815L, 532L, 583L, 550L, 561L, 921L, 656L, 671L, 450L, 552L, 576L, 536L, 769L, 478L, 576L, 700L, 753L, 762L, 538L, 796L, 513L, 899L, 551L, 560L, 562L, 639L, 795L)), .Names = c("RanFact", "FixFact1", "FixFact2", "DV"), class = "data.frame", row.names = c(NA, -1075L))

# run model & extract coefficients
lmer.example <- lmer(DV ~ FixFact1 + FixFact2 + (FixFact1 + FixFact2 | RanFact), data=example.df)
lmer.example.coefs <- coef(summary(lmer.example))[,1]

# for each individual data point, compute the contribution of each fixed-effect predictor to the DV
example.df[["FixFact1_contribution"]] <- example.df[["FixFact1"]] * lmer.example.coefs[["FixFact1"]]
example.df[["FixFact2_contribution"]] <- example.df[["FixFact2"]] * lmer.example.coefs[["FixFact2"]]

# method 1: to graph each predictor, subtract the contributions of the other fixed effects from the DV
example.df[["DV_minus_non_FixFact1_effects"]] <- example.df[["DV"]] - example.df[["FixFact2_contribution"]]
example.df[["DV_minus_non_FixFact2_effects"]] <- example.df[["DV"]] - example.df[["FixFact1_contribution"]]

# method 2: to graph each predictor, add its contribution + the intercept + residual error
example.df[["DV_resids_plus_intercept"]]                  <- lmer.example.coefs[["(Intercept)"]] + resid(lmer.example)
example.df[["FixFact1_effect_plus_resids_and_intercept"]] <- example.df[["DV_resids_plus_intercept"]] + example.df[["FixFact1_contribution"]]
example.df[["FixFact2_effect_plus_resids_and_intercept"]] <- example.df[["DV_resids_plus_intercept"]] + example.df[["FixFact2_contribution"]]

# construct and display plots overlaying methods 1 & 2
fixFact1_plots <- ggplot(example.df, aes(x=FixFact1)) + geom_smooth(method="lm", aes(y=DV_minus_non_FixFact1_effects, colour="Method 1")) + geom_smooth(method="lm", aes(y=FixFact1_effect_plus_resids_and_intercept, colour="Method 2")) + ylab("DV") + theme(legend.title = element_blank())
fixFact2_plots <- ggplot(example.df, aes(x=FixFact2)) + geom_smooth(method="lm", aes(y=DV_minus_non_FixFact2_effects, colour="Method 1")) + geom_smooth(method="lm", aes(y=FixFact2_effect_plus_resids_and_intercept, colour="Method 2")) + ylab("DV") + theme(legend.title = element_blank())
grid.arrange(fixFact1_plots, fixFact2_plots)

enter image description here

$\endgroup$
  • $\begingroup$ "they are strongly correlated with each other (r^2 ≈ .75)" These should not be considered as different independent variables. I would probably just average across them, no? An r^2 of .75 is quite high. Even if you don't think multicollinearity is a problem, it is difficult to interpret a model with predictors that highly correlated, in my opinion. What do the two items represent, conceptually? $\endgroup$ – Mark White Jun 20 '17 at 0:23
  • $\begingroup$ Thanks for your comment, Mark. This data set comes from psychological experiments in which participants named pictures. We were interested in how long they took to name the pictures (DV = reaction time) as a function of several factors. In this analysis, FixFact1 = (log) # of times the participant previously named the same picture & FixFact2 = # of times the participant previously named other pictures. As we did not initially construct the sequence of pictures to dissociate these variables, A and B here are highly correlated; however, we do expect & observe opposing effects of these variables. $\endgroup$ – Dan K. Jun 20 '17 at 0:48
  • $\begingroup$ I'm afraid what you are capturing is just "memory ability" with those two measures. But if you expect them to have different effects, you should really have one IV at level 1 (# times previously named) and an IV at level 2 that is categorical (whether or not it was same or other picture). The cross-level interaction would test that hypothesis that they have opposing effects. If that doesn't make sense, you could post the header of your data, and I can show you how to reshape the data into the correct format. $\endgroup$ – Mark White Jun 20 '17 at 1:36
  • $\begingroup$ Sorry if the length restrictions on comments led me to be imprecise/incomplete. I'm not interested in specifically contrasting the effects of these fixed effects by testing for that interaction; it's enough for me to be able to say that X has a sig. positive effect on the DV and Y has a sig. negative effect on the DV. Plus, as these are both continuous variables, they cannot be obviously recoded to test for such an interaction: On each trial, participants have previously named the same picture m times and have previously named other pictures n times, so both values are needed. ... $\endgroup$ – Dan K. Jun 20 '17 at 2:11
  • $\begingroup$ ... Furthermore, in the full data set, we are examining interactions between these variables and other variables that I have omitted here for simplicity's sake. So while I completely understand your suggestion to combine the variables in some way, I don't think that is the solution here. Given that the variables are all theoretically interpretable in isolation and that we have enough power to detect separate effects for each one, I am comfortable analyzing them simultaneously in a single analysis; my main issue at the moment concerns how to fairly visualize these effects. $\endgroup$ – Dan K. Jun 20 '17 at 2:11

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