# 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}) |

**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}) |

**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.