The traditional cell-mean (or structural) model of one-way within-subject (repeated-measures) ANOVA  is

Suppose we want to model a new one-way within-subject (repeated-measures) ANOVA:

Y_ij = μ_i+ β_j + ε_ij                                                                                     (II)

where all terms bear the same meaning as in model (I) except for factor effects (constants) {μ_i, i = 1, 2, ..., a}. And the new assumptions are the same as before except E(Y_ij) = μ_i.

The absence of a constant in model (II) is because we don't want a common mean removed from each effect and would like to test the following null hypothesis

H₀: μ₁ = 0, μ₂ = 0, ..., and μ_a = 0                                                          (III)

instead of the main effect test of factor A in model (I),

H₀: α₁ = 0, α₂ = 0, ..., and α_a = 0   (no difference among factor levels)
        
(III) is an extension of one-sample t test to a whole set of factor effects instead of one specific simple effect μ_i.

I failed to find any discussion about model (II) and (III) in any text books I could access to, so I tried the following myself. Solving (II) as a general linear model is very straightforwad by coding factor A with a dummy variable, and we get an F test with F = {[(residuals SS of the reduced model) - (residuals SS of full)]/DF1}/[(residuals SS of full)/DF2].  The nice thing about this test is that there is only one F and the degrees of freedom are accurate.

However I would like to analyze it in a more computationally economical way by calculating sums of squares as in the traditional ANOVA approach.

First I started with the reduced model under null hypothesis (III)

Y_ij = β_j + ε_ij                                                                                            (IV)

The errors under model (II) and (IV) are respectively,

ε_ij = Y_ij - μ_i - β_j

ε_ij = Y_ij - β_j

But I could not go any further from here, thus failed to work out a formula for hypothesis (III). Any suggestions about this approach?

Then I tried the following method.

Intuitively we would pick SSA* = bΣ_iY_i.² (thereafter a dot in the place of an index indicates the term as the mean over that index) as the sums of squares for hypothesis (III). It seems SSA* is not really a perfect candidate for hypothesis (III) since

Y_i. = (1/b)Σ_jY_ij= μ_i+ β_. + ε_i.,

which still contains subject effect β_., but I failed to come up with a better solution with some combination of Y_i., Y_.j, Y_ij, and Y_... The expected value of MSA* can be derived as the following

E(SSA*)
= bE(Σ_iY_i.²)

= bE[Σ_i(μ_i+ β_. + ε_i.)²]

= bE[Σ_i(μ_i²+ β_.² + ε_i.²+2μ_iβ_. + 2μ_iε_i.+ 2β_.ε_i.)]

= bΣ_i[μ_i²+ Eβ_.² + Eε_i.²+2E(μ_iβ_.) + 2E(μ_iε_i.) + 2E(β_.ε_i.)]

= bΣ_i(μ_i²+ σ_p²/b + σ²/b)

= bΣ_iμ_i²+ a(σ_p² + σ²)

As the degrees of freedom for SSA* is a, we have

E(MSA*) = (bΣ_iμ_i²)/a+ (σ_p² + σ²)

Similar derivation can be applied to MSS (mean squares for Subject) and MSAS (mean squares for interaction term between factor A and Subject), which are the same as in traditional one-way within-subject ANOVA,

E(MSS) = aσ_p² + σ²,

Based on the above 3 terms, we can construct the following F tests for (III),

F₁ = aMSA*/[MSS+(a-1)MSAS]                                             
F₂ = (MSA* - MSS)/[MSAS/(1-a)]

F₂ is not a good candidate since the numerator might be negative. With a quasi-F statistic F₁  with a composite mean square term in the denominator, MSS+(a-1)MSAS, the degrees of freedom for the denominator can be approximated as

df_denom = [MSS+(a-1)MSAS]²/{MSS²/(b-1)+(a-1)²MSAS²/[(a-1)(b-1)]}  (IV)

And the degrees of freedom for F₁ is F(a, df_denom).

Questions:

(1) Is everything above correct?
(2) I am not so sure about the multiplier (a-1)² in (IV): Is it correct?
(3) One thing I don't feel comfortable with this approach is that there might have multiple choices of F, and I have to approximate the degrees of freedom for those composite sums of squares. And I am not so sure whether this approach would be equivalent to the one with general linear model.
(4) Any better choices of F statistic for hypoethesis (III)?
(5) Hypothesis (III) would be denied (rejected) if either
      (i) the main effect of factor A in model (I) is significant (because this denies  that the α_i are equal at all), or
      (ii) the hypothesis that the grand mean μ in model (I) is zero is rejected?

This seems very nice because it is much simpler than the above approach with composite sums of squares, but my sub-questions are:

A. Yes, hypothesis (III) would be denied (rejected) if either (i) or (ii) is rejected, but is (III) equivalent to the following 2 hypotheses in the traditional ANOVA combined? Or in other words, is it true that rejecting (III) would lead to at least one of the following 2 hypotheses for model (I) being rejected?

H₀ : the main effect of factor A is 0, or α_i = 0, i = 1, 2, ..., a

and

H₀ : μ = 0

B.  One annoying part of this new approach is, I will have to have 2 separate F tests, lack of the nice feature of only one statistic for the hypothesis.