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Some arguments why the inequality $$\sum_{i=1}^N\log P(x_i|z_i,\mu,\Sigma) \ge \sum_{i=1}^N\log P(x_i|\hat{z}_i,\mu,\Sigma) \ge \sum_{i=1}^N\log P(x_i|\hat a,\hat{\mu},\hat{\Sigma})$$ does not always hold: the first term is one realisation of the random variable $$\sum_{i=1}^N\log P(x_i|Z_i,\mu,\Sigma)$$and the expectation of $$P(x_i|Z_i,\mu,\Sigma)$$ in \$...

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responses = np.array([ [0,1,2,2,2], [0,1,2,1,1], [0,1,2,0,0], [0,1,2,0,1], [0,1,2,1,0] ]).flatten() students = 5 questions = 5 categories = 3 with pm.Model() as model: z_student = pm.Normal("z_student", mu=0, sigma=1, shape=(students,categories)) z_question = pm.Normal("...

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No, PERMANOVA does not require homogeneity of variances, so long as your study has a balanced design (the test is very robust to heterogeneity of variance for experiments with balanced design). You can use ordinal data in PERMANOVA, such as responses to questionnaires, and it is not based on comparison of means, so the criticism of finding means and standard ...

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