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I've used linear mixed models to test if factors genotype and sex influence colon length, while including batch as a random effect. I first ran the testvalue ~ genotype + SEX + (1 | BOX) and got the following results, with sex being significant:

Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: value ~ genotype + SEX + (1 | BOX)
   Data: ColonLength.new

REML criterion at convergence: 94.1

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.69433 -0.55283  0.00537  0.61300  2.01439 

Random effects:
 Groups   Name        Variance Std.Dev.
 BOX      (Intercept) 0.1346   0.3669  
 Residual             0.2819   0.5309  
Number of obs: 49, groups:  BOX, 20

Fixed effects:
              Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)     8.0512     0.1832 16.1116  43.943  < 2e-16 ***
genotypemGlu5   0.0914     0.2281 16.3917   0.401  0.69379    
SEXF           -0.7549     0.2280 16.5801  -3.310  0.00425 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) gntyG5
genotypmGl5 -0.515       
SEXF        -0.518 -0.142

After including a sex-genotype interaction with the formula value ~ genotype*SEX + (1 | BOX) sex is no longer significant

    Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: value ~ genotype * SEX + (1 | BOX)
   Data: ColonLength.new

REML criterion at convergence: 92.7

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.62511 -0.66926  0.05283  0.58236  2.10327 

Random effects:
 Groups   Name        Variance Std.Dev.
 BOX      (Intercept) 0.1362   0.3691  
 Residual             0.2800   0.5291  
Number of obs: 49, groups:  BOX, 20

Fixed effects:
                   Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)          7.9527     0.2055 15.2546  38.707   <2e-16 ***
genotypemGlu5        0.3299     0.3196 14.4285   1.032    0.319    
SEXF                -0.5174     0.3185 18.0472  -1.625    0.122    
genotypemGlu5:SEXF  -0.4888     0.4570 15.6767  -1.070    0.301    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) gntyG5 SEXF  
genotypmGl5 -0.643              
SEXF        -0.645  0.415       
gntyG5:SEXF  0.450 -0.699 -0.697

Should I report both? I.e., something like "the main effect of sex was significant (β = -.75, SE= .23,p= .005), but was no longer significant when the interaction between sex and genotype was included (β = .52, SE= .32, p = .12)"? Is this an appropriate way to report the results? (I know there are also people who recommend reporting the fixed effect estimates, the confidence interval, and the strength of the effect, and still others who somehow report LMM like "F(df,dferror) = F-value, p = p-value". Which is preferable?)

If there is a main effect, is it appropriate to then go and look at the interaction between terms? Or is that something I should primarily be doing if there's not a main effect observed?

Apologies if these questions are inane - I don't have much experience with statistics and have been kind of thrown in the deep end. I'd really appreciate any help.

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2 Answers 2

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In your second model, the effect of sex shown (-0.5174) is the estimate of the effect of sex at the reference level of genotype. The estimate for the sex effect at the mGlu5level is -0.5174-0.448= -0.965. So when the interaction is in the model there is no longer a main effect of sex reported, just different estimates for different genotypes, suggesting a greater effect of sex for the mGlu5 genotype. Yet the p-value for the interaction is 0.301, so there isn't much evidence from the data that those effects are genuinely different in the population.

Now, it probably makes more sense to think about the effect of genotype varying by sex than the effect of sex varying by genotype, (although mathematically they are the same thing). Still, there is little evidence from your data that there is an interaction effect present, so I would probably report the main effects (first model) as your best estimates of the effects of sex and genotype while mentioning that the second model suggests little evidence for an interaction (although it doesn't rule it out, interactions are difficult to detect).

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    $\begingroup$ Thank you very much, this really helped me make sense of things. $\endgroup$
    – oregaymi
    May 4 at 6:27
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"the main effect of sex was significant (β = -.75, SE= .23,p= .005), but was no longer significant when the interaction between sex and genotype was included (β = .52, SE= .32, p = .12)"?

I would give less emphasis on whether results are significant or not, i.e. whether they pass or not a given p-value cutoff. In general, I would report the results of analyses that you consider plausible regardless of significance.

If you think in your study that there is a reasonable main effect of sex and sex-by-genotype interaction is plausible, you could say something like:

the main effect of sex was β = -.75 (SE= .23,p= .005), which was reduced to β = .52 (SE= .32, p = .12) when the interaction between sex and genotype was included.

As noted by @George Savva, the data doesn't support the more complex model with interaction so for further analysis you could work with the simpler one. But again, I would prefer to think in terms of what makes sense rather than in terms of statistical significance.

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