Notes:
p represents different profiles (or mixture components).
each covariance parameterization is represented by a 4 x 4 covariance matrix and therefore would represent the parameterization for a four-profile solution.
two options for easily carrying out Latent Profile Analysis are MCLUST and Rmixmod
see ?mclust::mclustModelNames() for the list of MCLUST model names
see Rmixmod::?mixmodGaussianModel() for the list of Rmixmod model names
A. Varying means, equal variances, covariances fixed to 0
corresponds to the MCLUST model “EEI”, “diagonal, equal volume and shape”
corresponds to the Rmixmod model “Gaussian_pk_L_B”
\[ \left[ \begin{matrix} { \sigma }_{ 1 }^{ 2 } & 0 & 0 & 0 \\ 0 & { \sigma }_{ 2 }^{ 2 } & 0 & 0 \\ 0 & 0 & { \sigma }_{ 3 }^{ 2 } & 0 \\ 0 & 0 & 0 & { \sigma }_{ 4 }^{ 2 } \end{matrix} \right] \] B. Varying means, equal variances and covariances
corresponds to the MCLUST model “EEE”, “ellipsoidal, equal volume, shape, and orientation”
corresponds to the Rmixmod model “Gaussian_pk_L_C”
\[ \left[ \begin{matrix} { \sigma }_{ 1 }^{ 2 } & { \sigma }_{ 21 } & { \sigma }_{ 31 } & { \sigma }_{ 41 } \\ { \sigma }_{ 12 } & { \sigma }_{ 2 }^{ 2 } & { \sigma }_{ 23 } & { \sigma }_{ 24 } \\ { \sigma }_{ 13 } & { \sigma }_{ 12 } & { \sigma }_{ 3 }^{ 2 } & { \sigma }_{ 33 } \\ { \sigma }_{ 14 } & { \sigma }_{ 12 } & { \sigma }_{ 12 } & { \sigma }_{ 4 }^{ 2 } \end{matrix} \right] \]
C. Varying means and variances, covariances fixed to 0
corresponds to the MCLUST model “VVI”, “diagonal, varying volume and shape”
corresponds to the Rmixmod model “Gaussian_pk_Lk_Bk”
\[ \left[ \begin{matrix} { \sigma }_{ 1p }^{ 2 } & 0 & 0 & 0 \\ 0 & { \sigma }_{ 2p }^{ 2 } & 0 & 0 \\ 0 & 0 & { \sigma }_{ 3p }^{ 2 } & 0 \\ 0 & 0 & 0 & { \sigma }_{ 4p }^{ 2 } \end{matrix} \right] \]
D. Varying means and variances, equal covariances
I am not sure if you can specify this model using MCLUST (can use mclust::mclustModelNames to see all available models)
Also not sure about the Rmixmod model
\[ \left[ \begin{matrix} { \sigma }_{ 1p }^{ 2 } & { \sigma }_{ 21 } & { \sigma }_{ 31 } & { \sigma }_{ 41 } \\ { \sigma }_{ 12 } & { \sigma }_{ 2p }^{ 2 } & { \sigma }_{ 23 } & { \sigma }_{ 24 } \\ { \sigma }_{ 13 } & { \sigma }_{ 12 } & { \sigma }_{ 3p }^{ 2 } & { \sigma }_{ 33 } \\ { \sigma }_{ 14 } & { \sigma }_{ 12 } & { \sigma }_{ 12 } & { \sigma }_{ 4p }^{ 2 } \end{matrix} \right] \]
E. Varying means, variances, and covariances
corresponds to the MCLUST model “VVV”, “ellipsoidal, varying volume, shape, and orientation”
corresponds to the Rmixmod model “Gaussian_pk_Lk_Ck”
\[ \left[ \begin{matrix} { \sigma }_{ 1p }^{ 2 } & { \sigma }_{ 21p } & { \sigma }_{ 31p } & { \sigma }_{ 41p } \\ { \sigma }_{ 12p } & { \sigma }_{ 2p }^{ 2 } & { \sigma }_{ 23p } & { \sigma }_{ 24p } \\ { \sigma }_{ 13p } & { \sigma }_{ 12p } & { \sigma }_{ 3p }^{ 2 } & { \sigma }_{ 33p } \\ { \sigma }_{ 14p } & { \sigma }_{ 12p } & { \sigma }_{ 12p } & { \sigma }_{ 4p }^{ 2 } \end{matrix} \right] \]