Fit Mixed-Effects Spline Regression

Open Script

This example shows how to fit a mixed-effects linear spline model.

Load the sample data.

load('mespline.mat');

This is simulated data.

Plot $y$ versus sorted $x$.

[x_sorted,I] = sort(x,'ascend');
plot(x_sorted,y(I),'o')

Fit the following mixed-effects linear spline regression model

$$y_i = \beta_1 + \beta_2 x_i + \sum_{j=1}^K b_j \left( x_i-k_j \right)_+
+ \epsilon_i$$

where $k_{j}$ is the $j$ th knot, and $K$ is the total number of knots. Assume that $b_{j} \sim N(0,\sigma^{2}_{b})$ and $\epsilon \sim N(0,\sigma^{2})$.

Define the knots.

k = linspace(0.05,0.95,100);

Define the design matrices.

X = [ones(1000,1),x];
Z = zeros(length(x),length(k));
for j = 1:length(k)
      Z(:,j) = max(X(:,2) - k(j),0);
end

Fit the model with an isotropic covariance structure for the random effects.

lme = fitlmematrix(X,y,Z,[],'CovariancePattern','Isotropic');

Fit a fixed-effects only model.

X = [X Z];
lme_fixed = fitlmematrix(X,y,[],[]);

Compare lme_fixed and lme via a simulated likelihood ratio test.

compare(lme,lme_fixed,'NSim',500,'CheckNesting',true)

ans = 


    SIMULATED LIKELIHOOD RATIO TEST: NSIM = 500, ALPHA = 0.05

    Model        DF     AIC       BIC       LogLik     LRStat    pValue 
    lme            4    170.62    190.25    -81.309                     
    lme_fixed    103    113.38    618.88     46.309    255.24    0.68064


    Lower      Upper  
                      
    0.63784    0.72129

The $p$-value indicates that the fixed-effects only model is not a better fit than the mixed-effects spline regression model.

Plot the fitted values from both models on top of the original response data.

R = response(lme);
figure();
plot(x_sorted,R(I),'o', 'MarkerFaceColor',[0.8,0.8,0.8],...
    'MarkerEdgeColor',[0.8,0.8,0.8],'MarkerSize',4);
hold on
F = fitted(lme);
F_fixed = fitted(lme_fixed);
plot(x_sorted,F(I),'b');
plot(x_sorted,F_fixed(I),'r');
legend('data','mixed effects','fixed effects','Location','NorthWest')
xlabel('sorted x values');
ylabel('y');
hold off

You can also see from the figure that the mixed-effects model provides a better fit to data than the fixed-effects only model.

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