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% File: covariance_matrices.m                                             %
% Last modified: 12/08/2019                                               %
% Author(s): Michael J Grayling (michael.grayling@newcastle.ac.uk)        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%% Initialise required symbolic variables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

e  = sym('e'); % sigma_e
b  = sym('b'); % sigma_b
n  = sym('n');

%%%%% D = 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set up covariance matrix for 'base' case of 9 patients (3 eth, 4 seq)
X             = zeros(27, 11);
% First column is 1's for intercept term
X(:, 1)       = 1;
% Fill in next 2 columns for period effects
per_block     = [0, 0;
                 1, 0;
                 0, 1];
for i = 1:9
    X((1 + 3*(i - 1)):(3*i), 2:3) = per_block;
end
% Fill in next 2 for treatments, using Latin Square allocation
tr_block      = [0, 0;
                 1, 0;
                 0, 1;
                 1, 0;
                 0, 1;
                 0, 0;
                 0, 1;
                 0, 0;
                 1, 0];
X(:, 4:5)     = [tr_block; tr_block; tr_block];
% Fill in next 2 columns for ethnicities
X(10:18, 6)   = 1;
X(19:27, 7)   = 1;
% Fill in final 4 columns for interactions
for i = 10:18
    if X(i,4) == 1
       X(i,8) = 1;
    end
    if X(i,5) == 1
       X(i,9) = 1;
    end
end      
for i = 19:27
    if X(i,4) == 1
       X(i,10) = 1;
    end
    if X(i,5) == 1
       X(i,11) = 1;
    end
end        
% Initialise covariance matrix of the responses        
Sigma_3        = b^2 + e^2*eye(3);
Sigma          = b*zeros(27, 27);
for i = 1:9
  Sigma((1 + 3*(i - 1)):(3*i), (1 + 3*(i - 1)):(3*i)) = Sigma_3;
end
% Compute covariance matrix of fixed effects
cov            = inv(transpose(X)*inv(Sigma)*X);
% Then, in general if there are n patients, the covariance matrix becomes
% 9/n times this
general_cov    = (9/n)*cov;
% This gives 12sigma_e^2/n as the variance of the interaction terms

%%%%% D = 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Set up covariance matrix for 'base' case of 12 patients (3 eth, 4 seq)
X              = zeros(48, 15);
% First column is 1's for intercept term
X(:, 1)        = 1;
% Fill in next 3 columns for period effects
per_block      = [0, 0, 0;
                  1, 0, 0;
                  0, 1, 0;
                  0, 0, 1];
for i = 1:12
    X((1 + 4*(i - 1)):(4*i), 2:4) = per_block;
end
% Fill in next 2 for treatments, using Latin Square allocation
tr_block       = [0, 0, 0;
                  1, 0, 0;
                  0, 1, 0;
                  0, 0, 1;
                  1, 0, 0;
                  0, 1, 0;
                  0, 0, 1;
                  0, 0, 0;
                  0, 1, 0;
                  0, 0, 1;
                  0, 0, 0;
                  1, 0, 0;
                  0, 0, 1;
                  0, 0, 0;
                  1, 0, 0;
                  0, 1, 0];
X(:, 5:7)      = [tr_block; tr_block; tr_block];
% Fill in next 2 columns for ethnicities
X(17:32, 8)    = 1;
X(33:48, 9)    = 1;
% Fill in final 6 columns for interactions
for i = 17:32
    if X(i,5) == 1
       X(i,10) = 1;
    end
    if X(i,6) == 1
       X(i,11) = 1;
    end
    if X(i,7) == 1
       X(i,12) = 1;
    end
end      
for i = 33:48
    if X(i,5) == 1
       X(i,13) = 1;
    end
    if X(i,6) == 1
       X(i,14) = 1;
    end
    if X(i,7) == 1
       X(i,15) = 1;
    end
end        
% Initialise covariance matrix of the responses        
Sigma_4        = b^2 + e^2*eye(4);
Sigma          = b*zeros(48, 48);
for i = 1:12
  Sigma((1 + 4*(i - 1)):(4*i), (1 + 4*(i - 1)):(4*i)) = Sigma_4;
end
% Compute covariance matrix of fixed effects
cov            = inv(transpose(X)*inv(Sigma)*X);
% Then, in general if there are n patients, the covariance matrix becomes
% 12/n times this
general_cov    = (12/n)*cov;
% This gives 12sigma_e^2/n as the variance of the interaction terms