# How to interpret direction of relationship between continuous predictor and outcome variable in linear mixed model?

Sorry if this is a very simple question. I have run the following mixed model, which tests for the effect of two fixed effects (group and IQ) and participant as an random effect on an outcome (s). Here is the model specification:

m1 <- lmer(s ~ group + IQ + (1|ID), data=data)


Group is a categorical variable, IQ is continuous. Here is the output:


Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: s ~ group + IQ + (1 | ID)
Data: mat

REML criterion at convergence: 229.4

Scaled residuals:
Min      1Q  Median      3Q     Max
-2.6797 -0.5677 -0.1657  0.5448  3.0798

Random effects:
Groups   Name        Variance Std.Dev.
ID       (Intercept) 0.05597  0.2366
Residual             0.06271  0.2504
Number of obs: 499, groups:  ID, 108

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)   0.794787   0.227007 103.852255   3.501 0.000684 ***
group2       -0.056301   0.062010 103.876806  -0.908 0.366017
group3       -0.023709   0.062622 104.590355  -0.379 0.705750
IQ           -0.004828   0.002463 103.531586  -1.960 0.052664 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) gCP/LC gCP/HC
group2      -0.225
group3      -0.322  0.424
IQ          -0.985  0.117  0.216


And here is the output of an Anova:

Analysis of Deviance Table (Type II Wald chisquare tests)

Response: s
Chisq Df Pr(>Chisq)
group 0.8244  2    0.66220
IQ    3.8422  1    0.04998 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


My question is how I can interpret/write up the relationship between IQ and s?

I plotted the relationship, which indicates that as IQ goes up, s goes down. But I'm not sure how I would write this up in a report.

effects_IQ <- effects::effect(term= "IQ", mod= m1)
summary(effects_IQ) #output of what the values are
x_IQ <- as.data.frame(effects_IQ)

IQ_plot <- ggplot() +
#2
geom_point(data=subset(mat), aes(IQ, s)) +
#3
geom_point(data=x_WASI, aes(x=IQ, y=fit), color="blue") +
#4
geom_line(data=x_WASI, aes(x=IQ*, y=fit), color="blue") +
#5
geom_ribbon(data= x_WASI, aes(x=IQ, ymin=lower, ymax=upper), alpha= 0.3, fill="blue") +
#6
labs(x="IQ(centered & scaled)", y="S")

IQ_plot


Here is the plot: https://i.sstatic.net/zsGOi.png

I'd really welcome any advice anyone has, and will happily provide any more information if this is not clear. Thank you!

Your IQ parameter estimate is $$-0.004828$$, which means that for every unit increase in IQ, the dependent variable $$s$$ goes down by $$0.004828$$. (Incidentally, your plot appears to show the standard scaling of IQ to a mean of 100 and a standard deviation of 15, so I would not call it "centered and scaled", which usually refers to a linear transformation to a mean of 0 and a standard deviation of 1.)
Similarly, membership in group 2 is associated with an $$s$$ that is $$0.056301$$ smaller, and in group 3 with an $$s$$ that is $$0.023709$$ smaller than for membership in group 1 (the baseline).