The main purpose of this article is to develop statistical theory for analysing continuous and polytomous variables in several populations. A general multivariate model is defined with a set of identification conditions. Interpretations of these identification conditions are studied. To achieve the desirable asymptotic properties for statistical inferences, the maximum likelihood approach will be employed to estimate the unknown parameters in the model. Computationally, a program based on the Fletcher‐Powell algorithm is constructed to get the maximum likelihood estimates, and the information matrix is implemented to produce the standard error estimates. Statistical inference for various null hypotheses on comparisons of the means, variances, polychoric and polyserial correlations among the variables across or within different populations is discussed. A computationally more efficient partition maximum likelihood approach is also proposed. Finally, applications of the theory to some examples and a simulation study on the comparison of the maximum likelihood approach and partition maximum likelihood approach are presente