Consider an experiment where subjects reaction times are measured in three conditions. Because every subject participates in each condition, this is a typical within-subjects design. Now I'm wondering if a one-way within subjects anova is completely equivalent to a two-way anova with subject as the second factor. I already read some chapters on this topic but the difference between both procedures was never discussed explicitly.So,   Is there an advantage of using one compared to the other?  Is a multilevel approach with subject as the random effect and condition as a fixed effect even better?  What if an effect of time is assumed, (e.g. fatigue). Could this be modelled additionally as a "third" factor?   I know these are pretty vague/broad questions but I hope you could give me some hints anyway.  Thanks in advance
Consider repeated measures ANOVA (RM-ANOVA) with one within-subject factor A and several measurements per subject for each level of A.  It is closely related to two-way ANOVA with two factors: A and subject. They use identical decomposition of the sum of squares into four parts: A, subject, A⋅subject, and residual. However, two-way ANOVA tests the effect of A by comparing SS of A with the residual SS, while RM-ANOVA tests the effect of A by comparing SS of A with the A$\cdot$subject interaction SS.   Why the difference?    Does this difference automatically follow from the repeated-measures structure of the data, or is it some convention?  Does this difference between two-way ANOVA and RM-ANOVA correspond to testing two different nulls? If so, what exactly are they and why would we use different nulls in these two cases? Two-way ANOVA's test can be understood as an F-test between two nested models: the full model, and the model without A. Can RM-ANOVA be understood in a similar way?   (If there is only one measurement per subject for each level of A, then the distinction sort of disappears because A$\cdot$subject and residual variation cannot be disentangled: )    Demonstration  I will use toy data d2 generated in . The same webpage shows correct syntax for RM-ANOVA.  # Discarding between-subject factors and leaving only one within-subject factor d = d2[d2$Xb1=='CG' &amp; d2$Xb2 == 'f', c(1,4,6)]   (See reproducible version .) Data look like that:       id Xw1     Y 1    s1   A  28.6 2    s1   A  96.6 3    s1   A  64.8 4    s1   B 107.5 5    s1   B  77.3 6    s1   B 120.9 7    s1   C 141.2 8    s1   C 124.1 9    s1   C  88.0 10   s2   A  86.7 ...   Here is two-way ANOVA: summary(aov(Y ~ Xw1*id, d))               Df Sum Sq Mean Sq F value   Pr(&gt;F)     Xw1           2  95274   47637  16.789 3.73e-07 *** id           19  31359    1650   0.582    0.913     Xw1:id       38  71151    1872   0.660    0.929     Residuals   120 340490    2837                    Here is RM-ANOVA: summary(aov(Y ~ Xw1 + Error(id/Xw1), d))  Error: id           Df Sum Sq Mean Sq F value Pr(&gt;F) Residuals 19  31359    1650                 Error: id:Xw1           Df Sum Sq Mean Sq F value   Pr(&gt;F)     Xw1        2  95274   47637   25.44 9.73e-08 *** Residuals 38  71151    1872                       Error: Within            Df Sum Sq Mean Sq F value Pr(&gt;F) Residuals 120 340490    2837               Note the identical SS decomposition, but two-way ANOVA tests Xw1 against the residual, while RM-ANOVA tests Xw1 against the Xw1:id interaction.  Why?  This question is related to . If we try using Error(id) instead of Error(id/Xw1) in the example above, then Xw1 will get tested against Xw1:id interaction lumped together with the residual variation.  (The same issue arises in factorial RM-ANOVA with multiple within-subject factors, where each factor or interaction is tested against its own "error term" aka "error stratum". These error strata are always given by the corresponding interaction with the block/plot/subject variable id.)
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