I want to perform ANOVA, but now, I cannot come up with any idea.

**Motivation**

In Radiology, *modality* means imaging methods , MRI ,CT, PET,…etc,

and *reader* try to detect lesions from images taken by each modality, and resulting data contain True Positives: TP and False Positives: FP for each reader and modality. To compare modality, we use the AUC calculated for each reader and each modality, that is, AUC is indexed by two subscripts representing reader and modality.

E.g., the AUC for MRI is higher than that of CT, then we can say that MRI is better than CT. If number of modality is two, then it does not need ANOVA, I want to consider the more than 3 modality case.

**Notation**

Data has form y_{m,r}, where subscript indicates r-th reader and

m-th modality, respectively. And estimated characteristic \theta_{m,r}, indicating AUCs for each r-th reader and m-th modality.

**Detail**

I want to test the null hypothesis that

H_0: \theta_{1,r} = \theta_{2,r} = \cdots =\theta_{m,r}=\cdots=\theta_{M,r}

for all r-th reader.

Or I want to perform ANOVA with \theta_{m,r}.

**My poor Guess**

However, \theta_{m,r} is a two indexed family of posterior distributions,

I am not sure, but take a posterior mean of \theta_{m,r}, I get a family of deterministic real numbers as follows ;

y'_{m,r}:= \text{Posterior mean of } \theta_{m,r}

So, I want to perform the ANOVA for the data y'_{m,r}.

I am not sure how to model and how to test. I read Gelman’s book but I cannot understand.

I think it needs something like y'_{m,r} =\mu + \alpha_m + \beta_r +\epsilon_{m,r} but I am not sure. Please let me know How to modeling the two way ANOVA and how to test.