Hi all:
I have a quesion about ANOVA: Is SS(Sum of Square) of a specific factor constant with the number of factors changing?
dat1 includes one factor g1,and g1's SS is called SS_g1_dat1.
dat2 includes two factors g1,g2,and g1's SS is called SS_g1_dat2.
My quesion is: Is SS_g1_dat1 equals to SS_g1_dat2?
I have both "yes" and "no" reasons for the quesion,but don't know which one is correct,which need your precious help.
The reasion for SS_g1_dat1 equals to SS_g1_dat2:
The formula for computing SS is:sum(sample size of level(i)*(mean of level(i)-TotalMean)^2),with i refers to each level in SS_g1_dat1 and SS_g1_dat2.
Every element of the formula is constant,so SS is constant.
Using the dataset "warpbreaks" from R:
anova(lm(breaks~wool))
Analysis of Variance Table
Response: breaks
Df Sum Sq Mean Sq F value Pr(>F)
wool 1 450.7 450.67 2.6684 0.1084
Residuals 52 8782.1 168.89
anova(lm(breaks~wool+tension))
Analysis of Variance Table
Response: breaks
Df Sum Sq Mean Sq F value Pr(>F)
wool 1 450.7 450.67 3.3393 0.073614 .
tension 2 2034.3 1017.13 7.5367 0.001378 **
Residuals 50 6747.9 134.96
anova(lm(breaks~tension+wool))
Analysis of Variance Table
Response: breaks
Df Sum Sq Mean Sq F value Pr(>F)
tension 2 2034.3 1017.13 7.5367 0.001378 **
wool 1 450.7 450.67 3.3393 0.073614 .
Residuals 50 6747.9 134.96
>From above,wool's SS is always 450.7 not matter the number and order of factors.
The reasion for SS_g1_dat1 NOT equals to SS_g1_dat2:
The total SS is constant,so SS for each factor is decreasing with the number of factors increasing.
But when I use dataset "warpbreaks" to comfirm, it failed to confirm.The result shows that wool's SS is always 450.7 not matter the number and order of factors.
So which reason of the above two is correct then?
Many thanks for your help.
My best
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