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I mean any reason aside from the convenience of being able to complete the analysis within a single procedure.

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Here's the thing: if there is an interaction it does make no sense to "not be interested in it", because you cannot meaningfully interpret main effects alone if there's an interaction. So in addition to the answer below, I would urge you to reconsider what you are doing. – Erik Apr 14 '14 at 6:48
up vote 19 down vote accepted

Yes, for several reasons!

1) Simpsons paradox. Unless the design is balanced, if one of the variables affects the outcome, you can't properly assess even the direction of the effect of the other one without adjusting for the first (see the first diagram at the link, in particular - reproduced below**). This illustrates the problem - the within-group effect is increasing (the two colored lines), but if you ignore the red-blue grouping you get a decreasing effect (the dashed, gray line) -- completely the wrong sign!

enter image description here

While that's showing a situation with one continuous and one grouping variable, similar things can happen when unbalanced two-way main effects ANOVA is treated as two one-way models.

2) Let's assume there's a completely balanced design. Then you still want to do it, because if you ignore the second variable while looking at the first (assuming both have some impact) then the effect of the second goes into the noise term, inflating it... and so biasing all your standard errors upward. In which case, significant - and important - effects might look like noise.

Consider the following data, a continuous response and two nominal categorical factors:

      y x1 x2
1  2.33  A  1
2  1.90  B  1
3  4.77  C  1
4  3.48  A  2
5  1.34  B  2
6  4.16  C  2
7  5.88  A  3
8  2.56  B  3
9  5.97  C  3
10 5.10  A  4
11 2.62  B  4
12 6.21  C  4
13 6.54  A  5
14 6.01  B  5
15 9.62  C  5

The two way main effects anova is highly significant (because it's balanced, order doesn't matter):

Analysis of Variance Table
Response: y
          Df Sum Sq Mean Sq F value    Pr(>F)    
x1         2 26.644 13.3220  24.284 0.0004000 
x2         4 38.889  9.7222  17.722 0.0004859 
Residuals  8  4.389  0.5486                      

But the individual one way anovas are not significant at the 5% level:

(1) Analysis of Variance Table
Response: y
          Df Sum Sq Mean Sq F value  Pr(>F)  
x1         2 26.687 13.3436  3.6967 0.05613 
Residuals 12 43.315  3.6096                  

(2) Analysis of Variance Table
Response: y
          Df Sum Sq Mean Sq F value  Pr(>F)  
x2         4 38.889  9.7222  3.1329 0.06511 
Residuals 10 31.033  3.1033                  

Notice in each case that the mean square for the factor was unchanged ... but the residual mean squares increased dramatically (from 0.55 to over 3 in each case). That's the effect of leaving out an important variable.

**(the above diagram was made by Wikipedia user Schutz, but placed in the public domain; while attribution isn't required for items in the public domain, I feel it's worthy of recognition)

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Yes. If the two independent variables are related and/or the ANOVA is not balanced, then a two way ANOVA shows you the effect of each variable controlling for the other one.

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