# lmer fixed effects t-statistic interpretation

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