Covariance Structures
This section provides additional information on covariance structures.
Ante-Dependence: First-Order. This covariance structure has heterogenous variances and heterogenous correlations between adjacent elements. The correlation between two nonadjacent elements is the product of the correlations between the elements that lie between the elements of interest.
(σ_{1} ^{2} | σ_{2}σ_{1}ρ_{1} | σ_{3}σ_{1}ρ_{1}ρ_{2} | σ_{4}σ_{1}ρ_{1}ρ_{2}ρ_{3}) |
(σ_{2}σ_{1}ρ_{1} | σ_{2} ^{2} | σ_{3}σ_{2}ρ_{2} | σ_{4}σ_{2}ρ_{2}ρ_{3}) |
(σ_{3}σ_{1}ρ_{1}ρ_{2} | σ_{3}σ_{2}ρ_{2} | σ_{3} ^{2} | σ_{4}σ_{3}ρ_{3}) |
(σ_{4}σ_{1}ρ_{1}ρ_{2}ρ_{3} | σ_{4}σ_{2}ρ_{2}ρ_{3} | σ_{4}σ_{3}ρ_{3} | σ_{4} ^{2}) |
AR(1). This is a first-order autoregressive structure with homogenous variances. The correlation between any two elements is equal to rho for adjacent elements, rho^{2} for elements that are separated by a third, and so on. is constrained so that –1<<1.
(σ^{2} | σ^{2}ρ | σ^{2}ρ^{2} | σ^{2}ρ^{3}) |
(σ^{2}ρ | σ^{2} | σ^{2}ρ | σ^{2}ρ^{2}) |
(σ^{2}ρ^{2} | σ^{2}ρ | σ^{2} | σ^{2}ρ) |
(σ^{2}ρ^{3} | σ^{2}ρ^{2} | σ^{2}ρ | σ^{2}) |
AR(1): Heterogenous. This is a first-order autoregressive structure with heterogenous variances. The correlation between any two elements is equal to r for adjacent elements, r^{2} for two elements separated by a third, and so on. is constrained to lie between –1 and 1.
(σ_{1} ^{2} | σ_{2}σ_{1}ρ | σ_{3}σ_{1}ρ^{2} | σ_{4}σ_{1}ρ^{3}) |
(σ_{2}σ_{1}ρ | σ_{2} ^{2} | σ_{3}σ_{2}ρ | σ_{4}σ_{2}ρ^{2}) |
(σ_{3}σ_{1}ρ^{2} | σ_{3}σ_{2}ρ | σ_{3} ^{2} | σ_{4}σ_{3}ρ) |
(σ_{4}σ_{1}ρ^{3} | σ_{4}σ_{2}ρ^{2} | σ_{4}σ_{3}ρ | σ_{4} ^{2}) |
ARMA(1,1). This is a first-order autoregressive moving average structure. It has homogenous variances. The correlation between two elements is equal to * for adjacent elements, *(^{2}) for elements separated by a third, and so on. and are the autoregressive and moving average parameters, respectively, and their values are constrained to lie between –1 and 1, inclusive.
(σ^{2} | σ^{2}φρ | σ^{2}φρ^{2} | σ^{2}φρ^{3}) |
(σ^{2}φρ | σ^{2} | σ^{2}φρ | σ^{2}φρ^{2}) |
(σ^{2}φρ^{2} | σ^{2}φρ | σ^{2} | σ^{2}φρ) |
(σ^{2}φρ^{3} | σ^{2}φρ^{2} | σ^{2}φρ | σ^{2}) |
Compound Symmetry. This structure has constant variance and constant covariance.
(σ^{2} + σ_{1} ^{2} | σ_{1} | σ_{1} | σ_{1}) |
(σ_{1} | σ^{2} + σ_{1} ^{2} | σ_{1} | σ_{1}) |
(σ_{1} | σ_{1} | σ^{2} + σ_{1} ^{2} | σ_{1}) |
(σ_{1} | σ_{1} | σ_{1} | σ^{2} + σ_{1} ^{2}) |
Compound Symmetry: Correlation Metric. This covariance structure has homogenous variances and homogenous correlations between elements.
(σ^{2} | σ^{2}ρ | σ^{2}ρ | σ^{2}ρ) |
(σ^{2}ρ | σ^{2} | σ^{2}ρ | σ^{2}ρ) |
(σ^{2}ρ | σ^{2}ρ | σ^{2} | σ^{2}ρ) |
(σ^{2}ρ | σ^{2}ρ | σ^{2}ρ | σ^{2}) |
Compound Symmetry: Heterogenous. This covariance structure has heterogenous variances and constant correlation between elements.
(σ_{1} ^{2} | σ_{2}σ_{1}ρ | σ_{3}σ_{1}ρ | σ_{4}σ_{1}ρ) |
(σ_{2}σ_{1}ρ | σ_{2} ^{2} | σ_{3}σ_{2}ρ | σ_{4}σ_{2}ρ) |
(σ_{3}σ_{1}ρ | σ_{3}σ_{2}ρ | σ_{3} ^{2} | σ_{4}σ_{3}ρ) |
(σ_{4}σ_{1}ρ | σ_{4}σ_{2}ρ | σ_{4}σ_{3}ρ | σ_{4} ^{2}) |
Diagonal. This covariance structure has heterogenous variances and zero correlation between elements.
(σ_{1} ^{2} | 0 | 0 | 0) |
(0 | σ_{2} ^{2} | 0 | 0) |
(0 | 0 | σ_{3} ^{2} | 0) |
(0 | 0 | 0 | σ_{4} ^{2}) |
Direct product AR1 (UN_AR1). Specifies the Kronecker product of one unstructured matrix and the other first-order auto-regression covariance matrix. The first unstructured matrix models the multivariate observation, and the second first-order auto-regression covariance structure models the data covariance across time or another factor.
Direct product unstructured (UN_UN). Specifies the Kronecker product of two unstructured matrices, with the first one modeling the multivariate observation, and second one modeling the data covariance across time or another factor.
Direct product compound symmetry (UN_CS). Specifies the Kronecker product of one unstructured matrix and the other compound-symmetry covariance matrix with constant variance and covariance. The first unstructured matrix models the multivariate observation, and the second compound symmetry covariance structure models the data covariance across time or another factor.
Factor Analytic: First-Order. This covariance structure has heterogenous variances that are composed of a term that is heterogenous across elements and a term that is homogenous across elements. The covariance between any two elements is the square root of the product of their heterogenous variance terms.
(λ_{1} ^{2} + d | λ_{2}λ_{1} | λ_{3}λ_{1} | λ_{4}λ_{1}) |
(λ_{2}λ_{1} | λ_{2} ^{2} + d | λ_{3}λ_{2} | λ_{4}λ_{2}) |
(λ_{3}λ_{1} | λ_{3}λ_{2} | λ_{3} ^{2} + d | λ_{4}λ_{3}) |
(λ_{4}λ_{1} | λ_{4}λ_{2} | λ_{4}λ_{3} | λ_{4} ^{2} + d) |
Factor Analytic: First-Order, Heterogenous. This covariance structure has heterogenous variances that are composed of two terms that are heterogenous across elements. The covariance between any two elements is the square root of the product of the first of their heterogenous variance terms.
(λ_{1} ^{2} + d_{1} | λ_{2}λ_{1} | λ_{3}λ_{1} | λ_{4}λ_{1}) |
(λ_{2}λ_{1} | λ_{2} ^{2} + d_{2} | λ_{3}λ_{2} | λ_{4}λ_{2}) |
(λ_{3}λ_{1} | λ_{3}λ_{2} | λ_{3} ^{2} + d_{3} | λ_{4}λ_{3}) |
(λ_{4}λ_{1} | λ_{4}λ_{2} | λ_{4}λ_{3} | λ_{4} ^{2} + d_{4}) |
Huynh-Feldt. This is a "circular" matrix in which the covariance between any two elements is equal to the average of their variances minus a constant. Neither the variances nor the covariances are constant.
(σ_{1} ^{2} | [σ_{1} ^{2} + σ_{2} ^{2}]/2 - λ | [σ_{1} ^{2} + σ_{3} ^{2}]/2 - λ | [σ_{1} ^{2} + σ_{4} ^{2}]/2 - λ) |
([σ_{1} ^{2} + σ_{2} ^{2}]/2 - λ | σ_{2} ^{2} | [σ_{2} ^{2} + σ_{3} ^{2}]/2 - λ | [σ_{2} ^{2} + σ_{4} ^{2}]/2 - λ) |
([σ_{1} ^{2} + σ_{3} ^{2}]/2 - λ | [σ_{2} ^{2} + σ_{3} ^{2}]/2 - λ | σ_{3} ^{2} | [σ_{3} ^{2} + σ_{4} ^{2}]/2 - λ) |
([σ_{1} ^{2} + σ_{4} ^{2}]/2 - λ | [σ_{2} ^{2} + σ_{4} ^{2}]/2 - λ | [σ_{3} ^{2} + σ_{4} ^{2}]/2 - λ | σ_{4} ^{2}) |
Scaled Identity. This structure has constant variance. There is assumed to be no correlation between any elements.
(σ^{2} | 0 | 0 | 0) |
(0 | σ^{2} | 0 | 0) |
(0 | 0 | σ^{2} | 0) |
(0 | 0 | 0 | σ^{2}) |
Spatial: Power. This covariance structure has homogenous variances and heterogenous correlations between elements. d_{ij} is the estimated Euclidean distance between the i^{th} and j^{th} measurement.
(σ^{2} | σ^{2} ρ^{d12} | σ^{2} ρ^{d13} | σ^{2} ρ^{d14} ) |
(σ^{2} ρ^{d12} | σ^{2} | σ^{2} ρ^{d23} | σ^{2} ρ^{d24} ) |
(σ^{2} ρ^{d13} | σ^{2} ρ^{d23} | σ^{2} | σ^{2} ρ^{d34} ) |
(σ^{2} ρ^{d14} | σ^{2} ρ^{d24} | σ^{2} ρ^{d34} | σ^{2} ) |
Spatial: Exponential. This covariance structure has homogenous variances and heterogenous correlations between elements. d_{ij} is the estimated Euclidean distance between the i^{th} and j^{th} measurement.
(σ^{2} | σ^{2} exp{-d_{12}/θ} | σ^{2}exp{-d_{13}/θ} | σ^{2}exp{-d_{14}/θ} ) |
(σ^{2} exp{-d_{12}/θ} | σ^{2} | σ^{2}exp{-d_{23}/θ} | σ^{2} exp{-d_{24}/θ} ) |
(σ^{2}exp{-d_{13}/θ} | σ^{2} exp{-d_{23}/θ} | σ^{2} | σ^{2} exp{-d_{34}/θ} ) |
(σ^{2}exp{-d_{14}/θ} | σ^{2} exp{-d_{24}/θ} | σ^{2} exp{-d_{34}/θ} | σ^{2} ) |
Spatial: Gaussian. This covariance structure has homogenous variances and heterogenous correlations between elements. d_{ij} is the estimated Euclidean distance between the i^{th} and j^{th} measurement.
(σ^{2} | σ^{2} exp{-d_{12}/ρ^{2}} | σ^{2}exp{-d_{13}/ρ^{2}} | σ^{2}exp{-d_{14}/ρ^{2}} ) |
(σ^{2} exp{-d_{12}/ρ^{2}} | σ^{2} | σ^{2}exp{-d_{23}/ρ^{2}} | σ^{2} exp{-d_{24}/ρ^{2}} ) |
(σ^{2}exp{-d_{13}/ ρ^{2}} | σ^{2} exp{-d_{23}/ρ^{2}} | σ^{2} | σ^{2} exp{-d_{34}/ρ^{2}} ) |
(σ^{2}exp{-d_{14}/ρ^{2}} | σ^{2} exp{-d_{24}/ρ^{2}} | σ^{2} exp{-d_{34}/ρ^{2}} | σ^{2} ) |
Spatial: Linear. This covariance structure has homogenous variances and heterogenous correlations between elements. d_{ij} is the estimated Euclidean distance between the i^{th} and j^{th} measurement, and 1_{ij} is an indicator function which is 1 if ρd_{ij} ≤ 0 and 0 otherwise.
(σ^{2} | σ^{2}(1 - ρd_{12}) 1_{12} | σ^{2}(1 - ρd_{13}) 1_{13} | σ^{2}(1 - ρd_{14}) 1_{14} ) |
(σ^{2}(1 - ρd_{12}) 1_{12} | σ^{2} | σ^{2}(1 - ρd_{23}) 1_{23} | σ^{2}(1 - ρd_{24}) 1_{24} ) |
(σ^{2}(1 - ρd_{13}) 1_{13} | σ^{2}(1 - ρd_{23}) 1_{23} | σ^{2} | σ^{2}(1 - ρd_{34}) 1_{34} ) |
(σ^{2}(1 - ρd_{14}) 1_{14} | σ^{2}(1 - ρd_{24}) 1_{24} | σ^{2}(1 - ρd_{34}) 1_{34} | σ^{2} ) |
Spatial: Linear-log. This covariance structure has homogenous variances and heterogenous correlations between elements. d_{ij} is the estimated Euclidean distance between the i^{th} and j^{th} measurement, and 1_{ij} is an indicator function which is 1 if ρ log(d_{ij}) ≤ 0 and 0 otherwise.
(σ^{2} | σ^{2}(1 - ρ log(d_{12})) 1_{12} | σ^{2}(1 - ρ log(d_{13})) 1_{13} | σ^{2}(1 - ρ log(d_{14})) 1_{14} ) |
(σ^{2}(1 - ρ log(d_{12})) 1_{12} | σ^{2} | σ^{2}(1 - ρ log(d_{23})) 1_{23} | σ^{2}(1 - ρ log(d_{24})) 1_{24} ) |
(σ^{2}(1 - ρ log(d_{13})) 1_{13} | σ^{2}(1 - ρ log(d_{23})) 1_{23} | σ^{2} | σ^{2}(1 - ρ log(d_{34})) 1_{34} ) |
(σ^{2}(1 - ρ log(d_{14})) 1_{14} | σ^{2}(1 - ρ log(d_{24})) 1_{24} | σ^{2}(1 - ρ log(d_{34})) 1_{34} | σ^{2} ) |
Spatial: Spherical. This covariance structure has homogenous variances and heterogenous correlations between elements. r_{ij} = d_{ij}/ρ, where d_{ij} is the estimated Euclidean distance between the i^{th} and j^{th} measurement. 1_{ij} is an indicator function which is 1 if d_{ij} ≤ ρ and 0 otherwise.
(σ^{2} | σ^{2}(1 - 3/2r_{12} + 1/2r^{3}_{12}) 1_{12} | σ^{2}(1 - 3/2r_{13} + 1/2r^{3}_{13}) 1_{13} | σ^{2}(1 - 3/2r_{14} + 1/2r^{3}_{14}) 1_{14} ) |
(σ^{2}(1 - 3/2r_{12} + 1/2r^{3}_{12}) 1_{12} | σ^{2} | σ^{2}(1 - 3/2r_{23} + 1/2r^{3}_{23}) 1_{23} | σ^{2}(1 - 3/2r_{24} + 1/2r^{3}_{24}) 1_{24} ) |
(σ^{2}(1 - 3/2r_{13} + 1/2r^{3}_{13}) 1_{13} | σ^{2}(1 - 3/2r_{23} + 1/2r^{3}_{23}) 1_{23} | σ^{2} | σ^{2}(1 - 3/2r_{34} + 1/2r^{3}_{34}) 1_{34} ) |
(σ^{2}(1 - 3/2r_{14} + 1/2r^{3}_{14}) 1_{14} | σ^{2}(1 - 3/2r_{24} + 1/2r^{3}_{24}) 1_{24} | σ^{2}(1 - 3/2r_{34} + 1/2r^{3}_{34}) 1_{34} | σ^{2} ) |
Toeplitz. This covariance structure has homogenous variances and heterogenous correlations between elements. The correlation between adjacent elements is homogenous across pairs of adjacent elements. The correlation between elements separated by a third is again homogenous, and so on.
(σ^{2} | σ^{2}ρ_{1} | σ^{2}ρ_{2} | σ^{2}ρ_{3}) |
(σ^{2}ρ_{1} | σ^{2} | σ^{2}ρ_{1} | σ^{2}ρ_{2}) |
(σ^{2}ρ_{2} | σ^{2}ρ_{1} | σ^{2} | σ^{2}ρ_{1}) |
(σ^{2}ρ_{3} | σ^{2}ρ_{2} | σ^{2}ρ_{1} | σ^{2}) |
Toeplitz: Heterogenous. This covariance structure has heterogenous variances and heterogenous correlations between elements. The correlation between adjacent elements is homogenous across pairs of adjacent elements. The correlation between elements separated by a third is again homogenous, and so on.
(σ_{1} ^{2} | σ_{2}σ_{1}ρ_{1} | σ_{3}σ_{1}ρ_{2} | σ_{4}σ_{1}ρ_{3}) |
(σ_{2}σ_{1}ρ_{1} | σ_{2} ^{2} | σ_{3}σ_{2}ρ_{1} | σ_{4}σ_{2}ρ_{2}) |
(σ_{3}σ_{1}ρ_{2} | σ_{3}σ_{2}ρ_{1} | σ_{3} ^{2} | σ_{4}σ_{3}ρ_{1}) |
(σ_{4}σ_{1}ρ_{3} | σ_{4}σ_{2}ρ_{2} | σ_{4}σ_{3}ρ_{1} | σ_{4} ^{2}) |
Unstructured. This is a completely general covariance matrix.
(σ_{1} ^{2} | σ_{2} _{1} | σ_{31} | σ_{41}) |
(σ_{2} _{1} | σ_{2} ^{2} | σ_{32} | σ_{4} _{2}) |
(σ_{31} | σ_{32} | σ_{3} ^{2} | σ_{4} _{3}) |
(σ_{41} | σ_{4} _{2} | σ_{4} _{3} | σ_{4} ^{2}) |
Unstructured: Correlation Metric. This covariance structure has heterogenous variances and heterogenous correlations.
(σ_{1} ^{2} | σ_{2}σ_{1}ρ_{21} | σ_{3}σ_{1}ρ_{31} | σ_{4}σ_{1}ρ_{41}) |
(σ_{2}σ_{1}ρ_{21} | σ_{2} ^{2} | σ_{3}σ_{2}ρ_{32} | σ_{4}σ_{2}ρ_{42}) |
(σ_{3}σ_{1}ρ_{31} | σ_{3}σ_{2}ρ_{32} | σ_{3} ^{2} | σ_{4}σ_{3}ρ_{43}) |
(σ_{4}σ_{1}ρ_{41} | σ_{4}σ_{2}ρ_{42} | σ_{4}σ_{3}ρ_{43} | σ_{4} ^{2}) |
Variance Components. This structure assigns a scaled identity (ID) structure to each of the specified random effects.