Algorithm 1:

Nonparametric Bootstrap for fixed effects inference

Data: {Y(sl), l = 1,…,L}, X, Z

Result: Var(β(sl)), l = 1,…,L.

for b = 1,…,B do

1. Re-sample / subject indices from {1,…,I} with replacement. Denote the vector of re-sampled indices as M(b);

2. For the i′th element of M(b),i′ = 1,…,I, include all observations of the corresponding subject in the bootstrap sample. Denote the bth bootstrap sample as $\left\{\left\{Y_{M_{(b)}}\left(s_l\right),l=1,\dots ,L\right\},X_{M_{(b)}},Z_{M_{(b)}}\right\}$;

3. Fit the model in Section 2 using the bth bootstrap sample. Derive the fixed effects estimates {β(sl)(b), l = 1,…,L};

end

4. For l = 1,…,L, derive Var(β(sl)) from B bootstrap estimates {β(sl)(1),…,β(sl)(B)}. In practice we calculate the sample variance and use it as the estimator.