Indeed, because the z j used with PCGSE are proportional to the PC loadings, they have an asymptotic multivariate normal distribution [ 38], assuming multivariate normality for the underlying genomic data, with significant correlation present between the loadings associated with the genes that have high pair-wise correlations [ 3].
Assessment of multivariate normality of distribution of data in AMOS, using recommendations for item skewness (± 3) and kurtosis (± 7) [ 55] indicated that the data did not significantly deviate from normality.
Estimated marginal distributions are then used to transform variates to univariate normality; the transformed variates are assumed to have a multivariate normal distribution.
Asymptotic multivariate normality of the MLEs then follows from Lemma 1 and Theorem 2 of Self and Liang (1987), which indicates that this centered vector converges to a multivariate normal distribution as stated below.
Review of multivariate normal distribution theory.
We did this to account for the binomial distribution of our fitness variable (survival), which likely violates the assumption of normal errors and multivariate normality in least-square calculations [ 52, 53].
Examination of univariate and multivariate normality for each questionnaire item was highly suggestive of non-normal distributions in the population.
To adjust for non-Gaussian scale distributions and likely violations of multivariate normality, we estimated our SEM with ADF estimation and undertook a bootstrap of n = 1000 samples to generate robust standard errors.
Due to the lack of multivariate normality, it was difficult to use the chi-square distribution to interpret the (non-central) chi-square statistic, and although the RMSEA depends on the normality assumption as well, it is still possible to use the statistic to compare the fit of different models to the same data.
In our approach we assumed multivariate normality, that is normality of all marginal and conditional distributions of the measurements and, related to this, linearity between the conditional mean expression of a gene and the expression levels of its neighboring genes (equation (2)).
