LEVEL (Logical Explanations & Visualizations of Estimates in Linear mixed models): recommendations for reporting multilevel data and analyses

Monsalves, Maria Jose; Bangdiwala, Ananta Shrikant; Thabane, Alex; Bangdiwala, Shrikant Ishver

doi:10.1186/s12874-019-0876-8

Table 1 Example simple linear mixed models in 2-level and 3-level study designs

From: LEVEL (Logical Explanations & Visualizations of Estimates in Linear mixed models): recommendations for reporting multilevel data and analyses

	Nature of design	Random intercept effects only	Random intercept effects and random slope effects
2-level	• Clustered	Y_ij = β₀ + β_0i + β₁X_ij + ε_ij	Y_ij = β₀ + β_0i + (β₁ + β_1i)X_ij + ε_ij
2-level	• No repeated measurements	\( {\beta}_{0i}\sim N\left(0,{\sigma}_I^2\right) \) \( {\varepsilon}_{ij}\sim N\left(0,{\sigma}_E^2\right) \)	\( {\beta}_{0i}\sim N\left(0,{\sigma}_{Iint.}^2\right) \) \( {\beta}_{1i}\sim N\left(0,{\sigma}_{Islope}^2\right) \) \( {\varepsilon}_{ij}\sim N\left(0,{\sigma}_E^2\right) \)
2-level	• Not clustered	Y_jt = β₀ + β_0j + β₁X_jt + ε_jt	Y_jt = β₀ + β_0j + (β₁ + β_1j)X_ij + ε_jt
2-level	• Repeated measurements	\( {\beta}_{0j}\sim N\left(0,{\sigma}_J^2\right) \) \( {\varepsilon}_{jt}\sim N\left(0,{\sigma}_E^2\right) \)	\( {\beta}_{0j}\sim N\left(0,{\sigma}_{Jint.}^2\right) \) \( {\beta}_{1j}\sim N\left(0,{\sigma}_{Jslope}^2\right) \) \( {\varepsilon}_{jt}\sim N\left(0,{\sigma}_E^2\right) \)
3-level	• Clustered	Y_ijt = β₀ + β_0i + β_0ij + β₁X_ijt + ε_ijt	Y_ijt = β₀ + β_0i + β_0ij + (β₁ + β_1i + β_1ij)X_ijt + ε_ijt
3-level	• Repeated measurements	\( {\beta}_{0i}\sim N\left(0,{\sigma}_I^2\right) \) \( {\beta}_{0 ij}\sim N\left(0,{\sigma}_J^2\right) \) \( {\varepsilon}_{ijt}\sim N\left(0,{\sigma}_E^2\right) \)	\( {\beta}_{0i}\sim N\left(0,{\sigma}_{Iint.}^2\right) \) \( {\beta}_{0 ij}\sim N\left(0,{\sigma}_{Jint.}^2\right) \) \( {\beta}_{1i}\sim N\left(0,{\sigma}_{Islope}^2\right) \) \( {\beta}_{1 ij}\sim N\left(0,{\sigma}_{Jslope}^2\right) \) \( {\varepsilon}_{ijt}\sim N\left(0,{\sigma}_E^2\right) \)

Note: Clusters are indexed by i, Subjects are indexed by j, and Time points are indexed by t

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