Specific page references are intended for Lomax & Hahs-Vaughn, 3^rd ed.

As with the random-effects model, it is possible to calculate the mixed-effects summary table from the fixed-effects model summary table.  This is important to know because SPSS does not generate a correct mixed-effects ANOVA summary table in its output.  The authors correctly suggest that the SPSS output cannot be interpreted directly for the mixed-effects model.  However, they present a rather unclear rationale for the different F-ratios on pp. 490–491.  Note, they are operating with the first factor $\displaystyle {A}$ as fixed and the second factor $\displaystyle {B}$ as random.

A more specific presentation of the formulas at the bottom of p. 491 would be: $\displaystyle {E}{\left({M}{S}_{{A}}\right)}={\sigma_{\epsilon}^{{2}}}+{\left({1}-\frac{{K}}{{T}_{{B}}}\right)}\cdot{n}{\sigma_{{\alpha{b}}}^{{2}}}+{K}{n}{\left[{\sum_{{{j}={1}}}^{{J}}}\frac{{\alpha_{{j}}^{{2}}}}{{{J}-{1}}}\right]}$
$\displaystyle {E}{\left({M}{S}_{{B}}\right)}={\sigma_{\epsilon}^{{2}}}+{\left({1}-\frac{{J}}{{T}_{{A}}}\right)}\cdot{n}{\sigma_{{\alpha{b}}}^{{2}}}+{J}{n}{\sigma_{{b}}^{{2}}}$ These formulas include the scaling factor for population level sampling.  $\displaystyle {T}_{{A}}$ is the total number of possible levels for the first factor, and $\displaystyle {T}_{{B}}$ is the total number of possible levels for the second factor.  (Notice how the levels are “switched” with the scaling factor for $\displaystyle {A}$ depending on the number of levels for $\displaystyle {B}$ and vice versa.)  Under an assumption that $\displaystyle {A}$ is fixed, then we are using all of the possible levels in our analysis.  Thus, $\displaystyle {T}_{{A}}={J}$, which implies $\displaystyle \frac{{J}}{{T}_{{A}}}=\frac{{J}}{{J}}={1}$, which implies that the scaling factor for the second (random) effect is zero.  The long-and-short-of-the-matter is that the 2^nd term disappears from the second equation.  However, as $\displaystyle {B}$ is a random factor, it is assumed that we have very few of the (nearly infinite) possible levels.  In other words, we could let $\displaystyle {T}_{{B}}\rightarrow\infty$, which implies $\displaystyle \frac{{K}}{{T}_{{B}}}\rightarrow{0}$, which implies the scaling factor for the first (fixed) effect is one.  Thus, we obtain the 2 equations in the book.  To summarize, for our mixed-effects model, we should have:

$\displaystyle {E}{\left({M}{S}_{{A}}\right)}={\sigma_{\epsilon}^{{2}}}+{n}{\sigma_{{\alpha{b}}}^{{2}}}+{K}{n}{\left[{\sum_{{{j}={1}}}^{{J}}}\frac{{\alpha_{{j}}^{{2}}}}{{{J}-{1}}}\right]}$	Fixed Factor
$\displaystyle {E}{\left({M}{S}_{{B}}\right)}={\sigma_{\epsilon}^{{2}}}+{J}{n}{\sigma_{{b}}^{{2}}}$	Random Factor

This means that the F-ratios appear to be reversed:  the fixed effect equation looks more like the random effects model and the random effect equation looks more like the fixed effects model.  The denominator for the fixed effect should be $\displaystyle {M}{S}_{{{A}{B}}}$ (as with the random effects model) and the denominator for the random effect should be $\displaystyle {M}{S}_{{\text{with}}}$ (as with the fixed effects model).

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Here is the ANOVA summary table for a two-factor fixed-effects ANOVA where there are two levels of factor A (curriculum intervention) and six levels of factor B (teacher).  Each cell includes 12 students.

Source	$\displaystyle {S}{S}$	$\displaystyle {d}{f}$	$\displaystyle {M}{S}$	$\displaystyle {F}$	$\displaystyle {p}$
$\displaystyle {A}$	350.4	1	350.4	5.484	0.0207
$\displaystyle {B}$	1012.8	5	202.56	3.17	0.0098
$\displaystyle {A}\times{B}$	822.3	5	164.46	2.574	0.0295
Error	8434.8	132	63.9
TOTAL	10620.3	143

Calculate the mixed-effects test statistics and significances (with the first factor fixed and the second factor random).  For these problems, you are encouraged to use Excel's =FDIST() function to calculate significance values.

Report the mixed-effects results for the first (fixed) effect:
$\displaystyle {d}{f}_{{\text{denom}}}=$
$\displaystyle {F}_{{A}}=$
$\displaystyle {p}=$
(Please report the F-ratio accurate to 3 decimal places and the P-value accurate to 4 decimal places.)

Report the mixed-effects results for the second (random) effect:
$\displaystyle {d}{f}_{{\text{denom}}}=$
$\displaystyle {F}_{{B}}=$
$\displaystyle {p}=$
(Please report the F-ratio accurate to 3 decimal places and the P-value accurate to 4 decimal places.)