# Interpretation of coefficients from a mixed-effect linear model with an interaction

I have the following output from a mixed-effects linear regression model with an interaction. This model comprises:

1. A continuous outcome (ranging from 590 to 1401).
2. A group variable (binary; control vs. treatment).
3. A continuous covariate (ranging from 0 to 1).
4. An interaction term between a group variable and a covariate.
> summary(test)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: outcome ~ group * covariate + (1 | subject)
Data: ex.dat

REML criterion at convergence: 1942.7

Scaled residuals:
Min      1Q  Median      3Q     Max
-2.8484 -0.4493 -0.0084  0.3499  3.8192

Random effects:
Groups   Name        Variance Std.Dev.
subject  (Intercept) 20772    144.12
Residual              4511     67.16
Number of obs: 164, groups:  subject, 48

Fixed effects:
Estimate Std. Error     df t value Pr(>|t|)
(Intercept)                833.73      31.85  55.31  26.175  < 2e-16 ***
groupTreatment              20.95      45.08  55.47   0.465 0.643903
covariate                   66.99      18.68 114.30   3.586 0.000496 ***
groupTreatment:covariate    14.14      26.49 114.39   0.534 0.594531
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


In this case: I believe I can say (+ please let me know if there are any issues with these interpretations.):

1. The covariate coefficient of 66.99 represents the average increase in the outcome for each unit increase in the covariate within the (baseline) control group. (Significant)
2. With each unit increase in the covariate, the outcome is, on average, 14.14 units higher for the treatment group compared to the control group. (Not significant)

However, considering the coefficient and its corresponding p-value of the group variable, I am unsure whether it would be appropriate to conclude that there is no significant difference between the groups.

If I cannot draw such a conclusion based on this model, which model or which number should I examine to determine if there is a significant difference between the control and treatment groups?

"However, considering the coefficient and its corresponding p-value of the group variable, I am unsure whether it would be appropriate to conclude that there is no significant difference between the groups." No, you do not know that for sure yet. The difference 20.95 holds for the covariate being 0, but if the covariate is 1, the difference would be 20.95 + 14.14 = 35.09. In order to see if this difference of 35.09 would be significant, you could subtract 1 from the covariate (newcov <- covariate - 1) and run the model again with newcov as covariate. This works fine, since the effect of groupTreatment is the effect for newcov=0, meaning covariate=1.

Suppose the effect of groupTreatment would be significant in that new model, then there must be a particular value of the covariate, somewhere between 0 and 1, for which the group effect gets significant. Suppose you would like to know if that value is 0.6, then you could subtract value 0.6 from your original covariate (newcov <- covariate - 0.6) and run the model again with newcov as covariate. The idea is: the effect of "groupTreatment" shown in the summary is the effect for value 0 of "newcov", whichever original value of "covariate" that may be. There may be an R package to find the precise value for the "turning point" of the covariate, i.e. the value for which the group effect gets significant. Below is an R script which show generates data and shows the estimates of the group (g) effect, the covariate (x2) effect and the interaction effect (g:x2) for all values of 0 through 9 of the covariate. Within the for loop, each consecutive value from 0 through 9 is subtracted from the original covariate x, to obtain x2:

set.seed(245)
g <- c(rep(0,100), rep(1,100))
x <- ifelse(g==0, runif(100,-0.5,7.5), runif(100,2.5,9.5))
x <- round(x)
table(x)

y <- 0.1*g + 0.2*x + 0.2*g*x + rnorm(10,0,3)

for (i in 0:9){
print(i)
x2 <- x - i
model <- lm(y ~ g*x2)
print(summary(model))
}

[1] 0

Estimate Std. Error t value Pr(>|t|)
(Intercept)  -1.2471     0.4336  -2.876  0.00447 **
g             0.4980     0.8590   0.580  0.56275
x2            0.2570     0.1072   2.397  0.01747 *
g:x2          0.1084     0.1587   0.683  0.49523
---

[1] 1

Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.9901     0.3499  -2.829  0.00515 **
g             0.6064     0.7225   0.839  0.40234
x2            0.2570     0.1072   2.397  0.01747 *
g:x2          0.1084     0.1587   0.683  0.49523
---

[1] 2

Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.7330     0.2827  -2.593   0.0102 *
g             0.7148     0.5972   1.197   0.2328
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---

[1] 3

Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.4760     0.2457  -1.937   0.0541 .
g             0.8233     0.4916   1.675   0.0956 .
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---

[1] 4

Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.2190     0.2526  -0.867   0.3871
g             0.9317     0.4207   2.215   0.0279 *
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---

[1] 5

Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.03806    0.30044   0.127   0.8993
g            1.04010    0.40328   2.579   0.0106 *
x2           0.25703    0.10723   2.397   0.0175 *
g:x2         0.10842    0.15868   0.683   0.4952
---

[1] 6

Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.2951     0.3738   0.789   0.4308
g             1.1485     0.4457   2.577   0.0107 *
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---

[1] 7

Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.5521     0.4606   1.199   0.2321
g             1.2570     0.5339   2.354   0.0196 *
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---

[1] 8

Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.8091     0.5546   1.459   0.1462
g             1.3654     0.6495   2.102   0.0368 *
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---

[1] 9

Estimate Std. Error t value Pr(>|t|)
(Intercept)   1.0662     0.6527   1.633   0.1040
g             1.4738     0.7803   1.889   0.0604 .
x2            0.2570     0.1072   2.397   0.0175 *
g:x2          0.1084     0.1587   0.683   0.4952
---


The significance of the group (g) effect decreases to values below 0.05 and then increases again. So, for some values of the covariate there is a significant difference between the two groups of g (alpha=0.05). (The decrease and increase of the p-value of the group (g) effect over the range of values of covariate x2 is caused by the fact that the squared std. error of the group (g) effect is a quadratic function of the covariate value).

A final remark about your last question: "If I cannot draw such a conclusion based on this model, which model or which number should I examine to determine if there is a significant difference between the control and treatment groups?". If your data are OK as far as the assumptions for regression are met, it's best to stop at this point and not look for further analysis techniques. Your question sounds like "I definitely want to show that my treatment has a significant effect no matter what statistical method I have to use". Sometimes, an effect simply does not exist contrary to your expectations!

• Thank you so much for your detailed explanation! It helps me a lot. I have one quick follow-up question. Would it then be appropriate to conclude that there is no significant difference between the groups 'after adjusting for' the covariate and the interaction between the covariate and group variables? Or should I conclude that there is no significant difference between the groups when the covariate is equal to 0?
– KLee
Commented Mar 7 at 15:32
• The last formulation the one that makes sense. Your first way of formulating is only "a list of your model terms" and does not provide a real interpretation.
– BenP
Commented Mar 7 at 16:32

The covariate coefficient of 66.99 represents the average increase in the outcome for each unit increase in the covariate within the (baseline) control group. (Significant)

Mathematically, when the data is not transformed, the coefficient represents the associated increase in the outcome with every unit increase in the predictor after controlling for the other predictors. But you are right that this represents the slope for the control group, whereas the interaction shows the slope for the treatment group.

With each unit increase in the covariate, the outcome is, on average, 14.14 units higher for the treatment group compared to the control group. (Not significant)

As noted above, this is the increase in the slope for the treatment group for this covariate (it would be negative if the slope decreased). This seems to indicate that for the treatment group, the covariate has a stronger magnitude in increasing the outcome variable.

However, considering the coefficient and its corresponding p-value of the group variable, I am unsure whether it would be appropriate to conclude that there is no significant difference between the groups.

The $$p$$ value should not really weight interpretation of the raw effects. It only states the probability of the point estimate having a null effect on the outcome. This doesn't actually say if or how much the estimate actually increases or decreases the outcome variable. Its still clear to me that there is a $$\beta = 20.95$$ increase in the outcome variable on average for the treatment group over and above the control group.

If I cannot draw such a conclusion based on this model, which model or which number should I examine to determine if there is a significant difference between the control and treatment groups?

Do not determine the model post-hoc because of statistical significance. Your model and your data speak plainly about what is going on. Simply report what you have found here and move on. Statistical significance on it's own is rubbish anyway.

• Thank you so much for your comment! I have one follow-up question regarding the coefficient of the groupTreatment (β = 20.95). Would it be appropriate to conclude that the treatment group, on average, has a 20.95-unit higher outcome compared to the control group, after adjusting for the covariate and the interaction between the covariate and the group variables?
– KLee
Commented Mar 7 at 15:36
• Yes this is correct. The coefficient for treatment represents the adjustment to the conditional mean as compared to the reference group, here the control. Commented Mar 7 at 20:03