How to interpret a multi-interaction ANOVA table using only the sum of squares and mean square columns?

Question

I read in a paper that for an ANOVA table, it is sufficient to only look at the sum of squares and mean square columns. The paper contained an ANOVA table that tested 4 factors up to a 4-way interaction term. The table is as follows for variable factors $X,Y,Z,W$:

\begin{array}{|lrrr|} \hline Value & Df & Sum Sq & Mean Sq \\ \hline X & 2 & 2.66 & 1.45 \\ Y & 2 & 2.45 & 1.43 \\ Z & 2 & 1.31 & 0.44 \\ W & 2 & 0.01 & 0.01 \\ X:Y & 4 & 0.99 & 0.32 \\ X:Z & 4 & 0.60 & 0.15 \\ Y:Z & 4 & 0.66 & 0.17 \\ X:W & 4 & 0.01 & 0.005 \\ Y:W & 4 & 0.05 & 0.01 \\ Z:W & 4 & 0.00 & 0.00 \\ X:Y:Z & 8 & 0.34 & 0.12 \\ X:Y:W & 8 & 0.02 & 0.00 \\ X:Z:W & 8 & 0.00 & 0.00 \\ Y:Z:W & 8 & 0.00 & 0.00 \\ X:Y:Z:W & 16 & 0.00 & 0.00 \\ \hline \end{array}

I am wondering how I can interpret the above values using the sum of squares column? What can we generally say if the sum of squares for a factor is many times larger than another factor? I am having difficulty interpreting this as it involves interactions.

Answer 1

I am wondering how I can interpret the above values using the sum of squares column?

Be careful interpreting the ANOVA table: The column for the regression sum-of-squares in the table shows the contribution to this quantity for each of the factors in the model. The crucial thing to bear in mind about this is that the order of inputting the factors matters ---i.e., each value shows the contribution of that factor, conditional on having put the previous factors into the model already. Taking each factor in the ANOVA table in order, we can say that:

• The base factor $X$ (two model parameters) adds $2.66$ to the regression sum-of-squares, compared to the null model;

• The base factor $Y$ (two model parameters) adds $2.45$ to the regression sum-of-squares, compared to a model that already has $X$ in it;

• The base factor $Z$ (two model parameters) adds $1.31$ to the regression sum-of-squares, compared to a model that already has $X$ and $Y$ in it;

• The base factor $W$ (two model parameters) adds $0.01$ to the regression sum-of-squares, compared to a model that already has $X$, $Y$ and $Z$ in it;

• The interaction factor $X:Y$ (four model parameters) adds $0.99$ to the regression sum-of-squares, compared to a model that already has $X$, $Y$, $Z$ and $W$ in it;

$\quad \quad \vdots$

• The interaction factor $X:Y:Z:W$ (sixteen model parameters) adds $0.00$ to the regression sum-of-squares, compared to a model that already has all the previous base terms and interaction factors in it.

What can we generally say if the sum of squares for a factor is many times larger than another factor?

The regression sum-of-squares for each factor is showing the contribution of that factor to the overall regression sum-of-squares after inclusion of the previous terms. It is hard to compare the size of the regression sum-of-squares for different factors, since they enter the model in a different order. If a later term has a lower regression sum-of-squares than an earlier term, than could be because it has less of a relationship with the response, or it could be because it is measured already including more earlier terms. In this case the comparison is ambiguous and we can't really say anything very useful. If a later term has a higher regression sum-of-squares than an earlier term then that shows a stronger relationship with the response.

Answer 2

This ANOVA table provides very limited information. The residual SS was not provided. Total SS = model SS + residual SS. If residual SS is large, say 100, then this model is not good, because 4 factors can explain maybe less than 10% variation of the response variable, and if simple size is not too large, maybe all of them are not statistically significant. If the residual SS is small, say 0.01, then the 4 factors explain nearly all of the variation of the response variable, and even the 4 way interaction term is significant (even the table displays the SS of XYZW is 0.00, maybe it is 0.001).

This study has 81 cells (combinations of X,Y,Z,W). We do not know if there are some numbers of subjects in each cell. If not, we need to know what type of SS is present, type 1 or type 3. Different type of SS has different explanation, apart from last one XYZW. This table does not give the type of SS.

To totally understand the results of analysis, the point estimates of the effects (or regression coefficients are needed also.

When high order interaction (4 in this case) is included in the model, it is difficult to interpret the results. In practice, the interactions with little contribution should be excluded and new model should be fit.

From this table, with low confidence, I can say that factors X, Y, Z and their 2-way interactions have contribution to the explanation of the variation of response variable, and factor Z seems having no relation with response variable.