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.. index:: Examples; Jaws: Repeated Measures Analysis of Variance

Jaws: Repeated Measures Analysis of Variance

An example from OpenBUGS :cite:`openbugs:2014:ex` and Elston and Grizzle :cite:`elston:1962:ETR` concerning jaw bone heights measured repeatedly in a cohort of 20 boys at ages 8, 8.5, 9, and 9.5 years.

Model

Bone heights are modelled as

\bm{y}_i &\sim \text{Normal}(\bm{X} \bm{\beta}, \bm{\Sigma}) \quad\quad i=1,\ldots,20 \\
\bm{X} &= \begin{bmatrix}
  1 & 8 \\
  1 & 8.5 \\
  1 & 9 \\
  1 & 9.5 \\
\end{bmatrix} \quad
\bm{\beta} = \begin{bmatrix}
  \beta_0 \\
  \beta_1 \\
\end{bmatrix} \quad
\bm{\Sigma} = \begin{bmatrix}
  \sigma_{1,1} & \sigma_{1,2} & \sigma_{1,3} & \sigma_{1,4} \\
  \sigma_{2,1} & \sigma_{2,2} & \sigma_{2,3} & \sigma_{2,4} \\
  \sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3} & \sigma_{3,4} \\
  \sigma_{4,1} & \sigma_{4,2} & \sigma_{4,3} & \sigma_{4,4} \\
\end{bmatrix} \\
\beta_0, \beta_1 &\sim \text{Normal}(0, \sqrt{1000}) \\
\bm{\Sigma} &\sim \text{InverseWishart}(4, \bm{I})

where \bm{y}_i is a vector of the four repeated measurements for boy i. In the model specification below, bone heights are arranged into a 1-dimensional vector on which a :ref:`section-Distribution-BDiagNormal` is specified. Also not that since \bm{\Sigma} is a covariance matrix, it is symmetric with M * (M + 1) / 2 unique (upper or lower triangular) parameters, where M is the matrix dimension.

Analysis Program

.. literalinclude:: jaws.jl
    :language: julia

Results

Iterations = 2502:10000
Thinning interval = 2
Chains = 1,2
Samples per chain = 3750

Empirical Posterior Estimates:
              Mean        SD      Naive SE       MCSE        ESS
Sigma[1,1]  6.7915801 2.0232463 0.0233624358 0.1421847433 202.48437
Sigma[1,2]  6.5982624 1.9670001 0.0227129612 0.1469366529 179.20433
Sigma[1,3]  6.1775526 1.9084389 0.0220367541 0.1532226770 155.13523
Sigma[1,4]  5.9477070 1.9358258 0.0223529913 0.1545185214 156.95367
Sigma[2,2]  6.9308723 2.0236387 0.0233669666 0.1531630007 174.56542
Sigma[2,3]  6.6005767 1.9885583 0.0229618936 0.1600864592 154.30055
Sigma[2,4]  6.3803028 2.0196017 0.0233203515 0.1612393116 156.88795
Sigma[3,3]  7.4564163 2.1925641 0.0253175499 0.1705734202 165.22733
Sigma[3,4]  7.4518620 2.2712194 0.0262257824 0.1733769737 171.60713
Sigma[4,4]  8.0594440 2.4746352 0.0285746264 0.1784057891 192.39975
     beta1  1.8742617 0.2272166 0.0026236712 0.0071954415 997.16079
     beta0 33.6379701 1.9912509 0.0229929845 0.0632742554 990.37090

Quantiles:
              2.5%       25.0%      50.0%      75.0%      97.5%
Sigma[1,1]  3.7202164  5.3419070  6.5046777  7.8684049 11.5279247
Sigma[1,2]  3.5674344  5.2009878  6.3564397  7.6419462 11.2720602
Sigma[1,3]  3.2043099  4.8527075  5.9476859  7.1929746 10.8427648
Sigma[1,4]  2.9143241  4.5808041  5.6958961  6.9962164 10.6253935
Sigma[2,2]  3.7936234  5.4940524  6.6730872  8.0151463 11.6796110
Sigma[2,3]  3.4721419  5.2183567  6.3620683  7.6617912 11.4419940
Sigma[2,4]  3.2133129  4.9659531  6.1310937  7.4443619 11.3714037
Sigma[3,3]  4.1213458  5.9139585  7.1780478  8.6551856 12.8617596
Sigma[3,4]  4.0756709  5.8561719  7.1240011  8.7006336 13.0597624
Sigma[4,4]  4.4482953  6.3090779  7.6484712  9.4043857 14.0451233
     beta1  1.4349627  1.7279142  1.8707215  2.0159440  2.3445976
     beta0 29.4960557 32.3922780 33.6327451 34.9577696 37.4067853