Table 1 Model equations

β_0i=γ₀₀+ζ_0i

β_1i=γ₁₀+ζ_1i

β_2i=γ₂₀+ζ_2i

i: subject index, j: time index

(t_j−t_KNOT)₊ is a derived variable which becomes (t_j−t_KNOT) only when t_j>t_KNOT. Otherwise it is 0.

γ₀₀ and ζ_0i are fixed and random intercepts

γ₁₀ and ζ_1i are fixed and random first segment slopes

γ₂₀ and ζ_2i are fixed and random incremental slopes

3-segment PW linear

Y_ij=β_0i+β_1i.t_j+β_2i.(t_j−t_KNOT1)₊+β_3i.(t_j−t_KNOT2)₊+ε_ij

β_3i=γ₃₀+ζ_3i

γ₃₀ and ζ_3i are also fixed and random incremental slopes

Cubic

\(Y_{ij} = \beta _{0i} + \beta _{1i}. t_{j} + \beta _{2i}. t_{j}^{2} + \beta _{3i}. t_{j}^{3} + \epsilon _{ij}\)

γ₀₀ and ζ_0i are fixed and random intercepts

γ₁₀ and ζ_1i are fixed and random first order coefficients

γ₂₀ and ζ_2i are fixed and random second order coefficients

γ₃₀ and ζ_3i are fixed and random third order coefficients

2-segment PW quadratic

\(Y_{ij} = \beta _{0i} + \beta _{1i}. (t_{j} - t_{KNOT})_{+} + \beta _{2i}. t_{j}^{2} + \beta _{3i}. (t_{j} - t_{KNOT})_{+}^{2} + \epsilon _{ij}\)

γ₀₀ and ζ_0i are fixed and random intercepts

γ₁₀ and ζ_1i are fixed and random first order coefficients

γ₂₀ and ζ_2i are fixed and random second order coefficients

γ₃₀ and ζ_3i are fixed and random incremental second order coefficients

time as a categorical variable

Y_ij=γ₀₀.D₀+(γ₁₀+ζ_1i).D₁+(γ₂₀+ζ_2i).D₂+(γ₃₀+ζ_2i).D₃+(γ₄₀+ζ_3i).D₄+(γ₅₀+ζ_3i).D₅+ε_ij

γ_{00,10,20,30,40,50} are fixed coefficients

ζ_1i,2i,3i are random coefficients shared between time points

D_0,1,2,3,4,5 are dummy indicator variables for time points