return error covariance constraints for basic ANOVA designs
FORMAT [xVi] = pr_spm_non_sphericity(xVi)
required fields:
xVi.I - n x 4 matrix of factor level indicators
I(n,i) is the level of factor i for observation n
xVi.var - 1 x 4 vector of flags
var(i) = 1; different variance among levels of factor i
xVi.dep - 1 x 4 vector of flags
dep(i) = 1; dependencies within levels of factor i
Output:
xVi.Vi - cell of covariance components
or
xVi.V - speye(n,n)
See also; pr_spm_Ce.m & pr_spm_ui.m
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Non-sphericity specification
=========================
In some instances the i.i.d. assumptions about the errors do not hold:
Identity assumption:
The identity assumption, of equal error variance (homoscedasticity), can
be violated if the levels of a factor do not have the same error variance.
For example, in a 2nd-level analysis of variance, one contrast may be scaled
differently from another. Another example would be the comparison of
qualitatively different dependant variables (e.g. normals vs. patients). If
You say no to identity assumptions, you will be asked whether the error
variance is the same over levels of each factor. Different variances
(heteroscedasticy) induce different error covariance components that
are estimated using restricted maximum likelihood (see below).
Independence assumption.
In some situations, certain factors may contain random effects. These induce
dependencies or covariance components in the error terms. If you say no
to independence assumptions, you will be asked whether random effects
should be modelled for each factor. A simple example of this would be
modelling the random effects of subject. These cause correlations among the
error terms of observation from the same subject. For simplicity, it is
assumed that the random effects of each factor are i.i.d.
ReML
The ensuing covariance components will be estimated using ReML in spm_spm
(assuming the same for all responsive voxels) and used to adjust the
statistics and degrees of freedom during inference. By default spm_spm
will use weighted least squares to produce Gauss-Markov or Maximum
likelihood estimators using the non-sphericity structure specified at this
stage. The components will be found in xX.xVi and enter the estimation
procedure exactly as the serial correlations in fMRI models.
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Copyright (C) 2005 Wellcome Department of Imaging Neuroscience
0001 function [xVi] = pr_spm_non_sphericity(xVi) 0002 % return error covariance constraints for basic ANOVA designs 0003 % FORMAT [xVi] = pr_spm_non_sphericity(xVi) 0004 % 0005 % required fields: 0006 % xVi.I - n x 4 matrix of factor level indicators 0007 % I(n,i) is the level of factor i for observation n 0008 % xVi.var - 1 x 4 vector of flags 0009 % var(i) = 1; different variance among levels of factor i 0010 % xVi.dep - 1 x 4 vector of flags 0011 % dep(i) = 1; dependencies within levels of factor i 0012 % 0013 % Output: 0014 % xVi.Vi - cell of covariance components 0015 % or 0016 % xVi.V - speye(n,n) 0017 % 0018 % See also; pr_spm_Ce.m & pr_spm_ui.m 0019 %___________________________________________________________________________ 0020 % Non-sphericity specification 0021 % ========================= 0022 % 0023 % In some instances the i.i.d. assumptions about the errors do not hold: 0024 % 0025 % Identity assumption: 0026 % The identity assumption, of equal error variance (homoscedasticity), can 0027 % be violated if the levels of a factor do not have the same error variance. 0028 % For example, in a 2nd-level analysis of variance, one contrast may be scaled 0029 % differently from another. Another example would be the comparison of 0030 % qualitatively different dependant variables (e.g. normals vs. patients). If 0031 % You say no to identity assumptions, you will be asked whether the error 0032 % variance is the same over levels of each factor. Different variances 0033 % (heteroscedasticy) induce different error covariance components that 0034 % are estimated using restricted maximum likelihood (see below). 0035 % 0036 % Independence assumption. 0037 % In some situations, certain factors may contain random effects. These induce 0038 % dependencies or covariance components in the error terms. If you say no 0039 % to independence assumptions, you will be asked whether random effects 0040 % should be modelled for each factor. A simple example of this would be 0041 % modelling the random effects of subject. These cause correlations among the 0042 % error terms of observation from the same subject. For simplicity, it is 0043 % assumed that the random effects of each factor are i.i.d. 0044 % 0045 % ReML 0046 % The ensuing covariance components will be estimated using ReML in spm_spm 0047 % (assuming the same for all responsive voxels) and used to adjust the 0048 % statistics and degrees of freedom during inference. By default spm_spm 0049 % will use weighted least squares to produce Gauss-Markov or Maximum 0050 % likelihood estimators using the non-sphericity structure specified at this 0051 % stage. The components will be found in xX.xVi and enter the estimation 0052 % procedure exactly as the serial correlations in fMRI models. 0053 % 0054 %___________________________________________________________________________ 0055 % Copyright (C) 2005 Wellcome Department of Imaging Neuroscience 0056 0057 % Karl Friston 0058 % $Id: spm_non_sphericity.m 112 2005-05-04 18:20:52Z john $ 0059 0060 0061 % create covariance components Q{:} 0062 %=========================================================================== 0063 [n f] = size(xVi.I); % # observations, % # Factors 0064 l = max(xVi.I); % levels 0065 0066 % if var(i): add variance component for each level of factor i, 0067 %--------------------------------------------------------------------------- 0068 Q = {}; 0069 for i = find(xVi.var) 0070 for j = 1:l(i) 0071 u = xVi.I(:,i) == j; 0072 q = spdiags(u,0,n,n); 0073 Q{end + 1} = q; 0074 end 0075 end 0076 0077 % effects (discounting factors with dependencies) as defined by interactions 0078 %--------------------------------------------------------------------------- 0079 X = ones(n,1); 0080 for i = find(~xVi.dep & (l > 1)) 0081 Xi = sparse(1:n,xVi.I(:,i),1,n,l(i)); 0082 Xj = X; 0083 X = sparse(n,0); 0084 for j = 1:size(Xi,2) 0085 for k = 1:size(Xj,2) 0086 X(:,end + 1) = Xi(:,j) & Xj(:,k); 0087 end 0088 end 0089 end 0090 0091 % dependencies among repeated measures created by the hadamrad product %--------------------------------------------------------------------------- 0092 for i = find(xVi.dep) 0093 q = sparse(1:n,xVi.I(:,i),1,n,l(i)); 0094 P = q*q'; 0095 for j = 1:size(X,2) 0096 for k = (j + 1):size(X,2) 0097 Q{end + 1} = (X(:,j)*X(:,k)' + X(:,k)*X(:,j)').*P; 0098 end 0099 end 0100 end 0101 0102 % set Q in non-sphericity structure 0103 %--------------------------------------------------------------------------- 0104 0105 0106 % if i.i.d nonsphericity (V) is known otherwise there are components {Vi} 0107 %--------------------------------------------------------------------------- 0108 if length(Q) > 1 0109 xVi.Vi = Q; 0110 else 0111 xVi.V = speye(n,n); 0112 end