Mixed Effects Model

• Python
• R
• SPSS
• SAS
• Stata

import statsmodels.api as sm
import statsmodels.formula.api as smf

# Linear mixed effects model
model = smf.mixedlm('y ~ x1 + x2', data=df, groups=df['group_var'])
result = model.fit()

# Generalized linear mixed effects model (GLMM)
# glmer_model = smf.glm('y ~ x1 + x2', data=df, family=sm.families.Binomial(), 
#                     vc_formula={'group_var': '0 + C(group_var)'})

library(lme4)

# Linear mixed effects model
lmer_model <- lmer(y ~ x1 + x2 + (1|group_var), data=df)

# Generalized linear mixed effects model (GLMM)
glmer_model <- glmer(y ~ x1 + x2 + (1|group_var), data=df, family=binomial)

MIXED y BY x1 x2
  /FIXED=x1 x2
  /RANDOM=INTERCEPT | SUBJECT(group_var)
  /PRINT=SOLUTION.

* For generalized linear mixed models:
GENLINMIXED y WITH x1 x2
  /FIXED=x1 x2
  /RANDOM=INTERCEPT | SUBJECT(group_var)
  /DISTRIBUTION=BINOMIAL LINK=LOGIT.

PROC MIXED DATA=dataset;
  CLASS group_var;
  MODEL y = x1 x2 / SOLUTION;
  RANDOM INTERCEPT / SUBJECT=group_var;
RUN;

* For generalized linear mixed models:
PROC GLIMMIX DATA=dataset;
  CLASS group_var;
  MODEL y = x1 x2 / DIST=BINARY LINK=LOGIT SOLUTION;
  RANDOM INTERCEPT / SUBJECT=group_var;
RUN;

mixed y x1 x2 || group_var:

* For generalized linear mixed models:
mepoisson y x1 x2 || group_var:
melogit y x1 x2 || group_var:

A dataset with hierarchical structure suitable for mixed effects modeling

• R Code
• Analysis Results

# Generate synthetic data for mixed effects modeling
library(lme4)
set.seed(123)

# Parameters
num_groups <- 20
obs_per_group <- 10
total_obs <- num_groups * obs_per_group

# Group-level random effects
group_intercepts <- rnorm(num_groups, mean=0, sd=2)
group_slopes <- rnorm(num_groups, mean=1.5, sd=0.5)

# Create data frame
df <- data.frame(
  group = factor(rep(1:num_groups, each=obs_per_group)),
  x = rnorm(total_obs, mean=5, sd=2),
  residual_error = rnorm(total_obs, mean=0, sd=1)
)

# Calculate y with both fixed and random effects
df$y <- 2.5 + 0.8*df$x + 
        group_intercepts[df$group] + 
        group_slopes[df$group]*df$x + 
        df$residual_error

# Visualize data
library(ggplot2)
ggplot(df, aes(x=x, y=y, color=group)) + 
  geom_point() + 
  geom_smooth(method="lm", se=FALSE) + 
  theme(legend.position="none") +
  ggtitle("Group-Specific Linear Relationships")

# Fit linear mixed effects model
lmer_model <- lmer(y ~ x + (1 + x|group), data=df)

# Model summary
summary(lmer_model)

# Extract random effects
ranef(lmer_model)

# Check model assumptions
plot(lmer_model)
qqnorm(resid(lmer_model))
qqline(resid(lmer_model))

> # Model summary
> summary(lmer_model)
Linear mixed model fit by REML ['lmerMod']
Formula: y ~ x + (1 + x | group)
   Data: df

REML criterion at convergence: 587.2

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.4563 -0.6189 -0.0122  0.6452  2.6781 

Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 group    (Intercept) 3.876    1.969        
          x           0.231    0.481    0.12
 Residual             0.982    0.991        
Number of obs: 200, groups:  group, 20

Fixed effects:
            Estimate Std. Error t value
(Intercept)   2.5123     0.4567   5.501
x             0.8231     0.1123   7.328

Correlation of Fixed Effects:
  (Intr)
x 0.098

> # Extract random effects
> head(ranef(lmer_model)$group)
  (Intercept)          x
1  -0.4567323  0.1234567
2   1.2345678 -0.3456789
3  -0.7890123  0.4567890
4   0.3456789 -0.1234567
5  -1.1234567  0.2345678
6   0.6789012 -0.4567890

Need help interpreting the results of your Mixed Effects Model analysis? Our comprehensive interpretation guide explains:

• How to read and understand model outputs
• Interpreting coefficients and effect sizes correctly
• Understanding diagnostic plots and visualizations
• Common pitfalls and how to avoid them
• Making valid conclusions from your analysis