We address the problem of parameter estimation in multivariate distributions under ignorable non-monotone missing data. The factoring likelihood method for monotone missing data, termed by Rubin (1974), is applied to a more general case of non-monotone missing data. The proposed method is asymptotically equivalent to the Fisher scoring method from the observed likelihood, but avoids the burden of computing the first and second partial derivatives of the observed likelihood. Instead, the maximum likelihood estimates and their information matrices for each partition of the data set are computed separately and combined naturally using the generalized least squares method. A numerical example is presented to illustrate the method.