Comparing coefficients in between linear and linear mixed effects models

Question

I have a naive question, but it will help me to get a better conceptual understanding of how the mixed-effects model works.

Question: If I do linear regression and linear mixed-effects regression on the same data (as summarized below), then how can I compare/interpret the estimates from the linear regression to the estimates of the fixed effects from the mixed model? Can I say that estimates of the fixed effects are independent of the estimates of the random effect in a mixed model, which is included in a simple linear model? Or it's the other way around.

Any explanation would be helpful. Thanks.

Linear Model

Call:
lm(formula = pSHP2 ~ condition, data = dat_stim_unstim)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.6408 -0.5963 -0.1943  0.5046  6.9230 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.583864   0.001941  300.83   <2e-16 ***
conditionstim -0.106348   0.002609  -40.77   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7409 on 326360 degrees of freedom
Multiple R-squared:  0.005066,  Adjusted R-squared:  0.005063 
F-statistic:  1662 on 1 and 326360 DF,  p-value: < 2.2e-16

Linear Mixed Effects Model

Linear mixed model fit by REML ['lmerMod']
Formula: pSHP2 ~ condition + (1 | cluster)
   Data: dat_stim_unstim

REML criterion at convergence: 668294.9

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.9819 -0.7061 -0.2411  0.6576 10.0387 

Random effects:
 Groups   Name        Variance Std.Dev.
 cluster  (Intercept) 0.09619  0.3101  
 Residual             0.45352  0.6734  
Number of obs: 326362, groups:  cluster, 24

Fixed effects:
               Estimate Std. Error t value
(Intercept)    0.719139   0.063388   11.35
conditionstim -0.097682   0.002375  -41.14

Correlation of Fixed Effects:
            (Intr)
conditinstm -0.022
```

Answer 1

The models use different ways to get the fixed-effect coefficient estimates, and it might help to think a bit differently about the interpretations of the coefficients in the two models. In general the estimates of fixed-effect coefficients won't agree in the two types of models.

The simple linear regression model ignores all of the information about the clusters within the data and correlations of results within clusters. For estimating the intercept and slope, each data point is just equally weighted regardless of its associated cluster in a formula that has a closed form.

Your mixed model estimates an intercept along with a Gaussian distribution of cluster-specific intercepts around it. So you might think of the intercept as a sort of average among clusters* rather than among the individual data points as in the simple linear regression. Your model specified a single slope with respect to condition for all clusters, but that now represents the condition-related change from the corresponding cluster-specific intercept. A restricted maximum likelihood (REML) method was used to fit the model with these necessarily interdependent estimates, for which there is no closed-form solution.

Your results show that the fixed-effect coefficients are similar but not identical, even in this simple model with 1 fixed and 1 random effect.

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*The distribution of cluster intercepts is forced to be Gaussian, so this "type of average" is not necessarily the same as the arithmetic average intercept among clusters.