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I'm struggling to explain some output from a linear mixed effects model. I've done a lot of reading and searching of previous questions but haven't been able to find what I'm looking for. I have 3 conditions (HH, IH, and PH). For each of these I've calculated the change from baseline. The dependent variable is s1_d1_hbo (a measure of deoxygenated blood in the brain, loosely). I'm expecting this to increase with each condition but am also interested in any between condition differences; this is why I set the equation as s1_d1_hbo ~ 0 + Condition + (1 | subject_ID) rather than s1_d1_hbo ~ Condition + (1 | subject_ID).

From the fixed effects results, I understand that the pre-post difference for HH is 63.9 – am I correct that the probability this is the same as zero has a t-value of 4.78 and the associated p-value? The confidence interval from emmeans shows that the 95 % CI is way above zero, which means that this difference is statistically significant from zero. Post hoc tests (Tukey's HSD) shows no between-condition differences.

The main point I'm confused about is how to write and explain the fixed effects – since the model has obviously done some test to get those test statistics do I need to correct for multiple comparisons? Can I just state that the fixed effects were significantly different from zero? Code is below. Any help appreciated!

Formula: s1_d1_hbo ~ 0 + Condition + (1 | subject_ID)
   Data: hbo_summary.df

REML criterion at convergence: 350.9

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.08725 -0.42395 -0.02953  0.59511  2.03162 

Random effects:
 Groups     Name        Variance Std.Dev.
 subject_ID (Intercept) 1177     34.30   
 Residual               1148     33.88   
Number of obs: 36, groups:  subject_ID, 16

Fixed effects:
                              Estimate Std. Error    df t value Pr(>|t|)    
ConditionHH hypoxia - HH room    63.90      13.37 28.80   4.780 4.75e-05 ***
ConditionIH hypoxia - IH room    54.52      13.83 29.91   3.943 0.000448 ***
ConditionPH hypoxia - PH room    55.66      12.98 27.60   4.286 0.000199 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            CHh-Hr CIh-Ir
CndtIHh-IHr 0.393        
CndtPHh-PHr 0.406  0.412 
$emmeans
 Condition            emmean   SE   df lower.CL upper.CL
 HH hypoxia - HH room   63.9 13.4 28.8     36.5     91.3
 IH hypoxia - IH room   54.5 13.8 29.9     26.3     82.8
 PH hypoxia - PH room   55.7 13.0 27.6     29.0     82.3

Degrees-of-freedom method: satterthwaite 
Confidence level used: 0.95 

$contrasts
 contrast                                        estimate   SE   df t.ratio p.value
 (HH hypoxia - HH room) - (IH hypoxia - IH room)     9.38 15.0 20.1  0.626  0.8076 
 (HH hypoxia - HH room) - (PH hypoxia - PH room)     8.24 14.4 20.3  0.574  0.8353 
 (IH hypoxia - IH room) - (PH hypoxia - PH room)    -1.14 14.5 19.6 -0.078  0.9966 

Degrees-of-freedom method: satterthwaite 
P value adjustment: tukey method for comparing a family of 3 estimates 
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From your mixed model formulation, the unconditional mean is strictly a function of the fixed effects term "Condition" whereas the conditional mean (with respect to subject ID) is the sum of the unconditional mean and a subject-specific intercept term. The unconditional mean may also be interpreted as the mean (with respect to subject ID) of the conditional mean. Hence, an interpretation is: The estimated mean 's1_d1_hbo', marginalized over all subject IDs, for a subject with condition 'HH hypoxia - HH room' is 63.90.

When testing for the inclusion of a categorical variable, I do not look at the t-test outputs, which are based on asymptotic Wald statistics, but I calculate the likelihood ratio test statistic myself by fitting a Null model (intercept only) and the model with the categorical variable (your output given). However, make sure to use the MLE fitting algorithm, not REML. This overall test answers the question: does my categorical variable explain an appreciable amount of variation in the response variable? This overall test does not require controlling for multiple comparisons.

After the above overall test (in your case it will have 2 degrees of freedom) shows significance, then you can test the differences within the levels of the categorical variable using the 'contrast' statements. These should be controlled for multiple comparisons, which your R output does automatically for you. From your output it is clear that given the presence of any of these 3 conditions the mean 's1_d1_hbo' is significantly different than the mean 's1_d1_hbo' for a subject without any of the conditions; however, no statistically significant difference in mean 's1_d1_hbo' is detected between subjects with two differing conditions at the .05 significance level.

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