Sum of Squares - How many types of Sum of Squares do we have

Hi
I have no idea where to put this thread. But its to do with software output so I put it in here.
I was using sas and saw that there are 4 types of Sum Of Squares,
So my question is does anyone know how many types of Sum of Squares do we have and what does each sum of square represent?

Thanks in advance

Re: Sum of Squares

fed-up,

This link might be of some help, "Explained Sum of Squares"

FYI, I am not familiar with SAS. But, thought the above link might be of some use.

Re: Sum of Squares

fed-up said:

Hi
I have no idea where to put this thread. But its to do with software output so I put it in here.
I was using sas and saw that there are 4 types of Sum Of Squares,
So my question is does anyone know how many types of Sum of Squares do we have and what does each sum of square represent?

Thanks in advance

Click to expand...

Hi fed-up and welcome to the Cove ,

I can't answer your question - but I did move your thread to the correct board.

Re: Sum of Squares

Frank T. said:

fed-up,

This link might be of some help, "Explained Sum of Squares"

FYI, I am not familiar with SAS. But, thought the above link might be of some use.

Click to expand...

Hi Frank,

Thanks for the link. It was very useful as it explained in very simple terms. Though I couldnt understand the TypeII SS. It said the following. Maybe some one can help me understand it as I haven't a clue as to what it means.

"Type two sum of squares are calculated for a variable after adjusting for every other factor of equal or lesser order. For example in the model Y = A + B + C + AB + BC + AC + ABC, the type II sum of squares for A is obtained by adjusting for B and C"

Thanks

Re: Sum of Squares

fed-up said:

"Type two sum of squares are calculated for a variable after adjusting for every other factor of equal or lesser order. For example in the model Y = A + B + C + AB + BC + AC + ABC, the type II sum of squares for A is obtained by adjusting for B and C"

Click to expand...

Try this link, "Types of Sums of Squares". It might give a better explanation.

FWIW, it talks about the pros and cons pertaining to the different types of Sums of Squares.

Hi Frank

Thanks for the informative link. It was very useful.
Though I have another question to ask. When I was doing some research on Type IV SS. It said the following:

"
Type IV sums of squares are not recommended for testing hypotheses for lower-order effects in ANOVA designs with missing cells, even though this is the purpose for which they were developed. This is because Type IV sum-of-squares are invariant to some but not all g2 inverses of X'X that could be used to solve the normal equations. Specifically, Type IV sums of squares are invariant to the choice of a g2 inverse of X'X given a particular ordering of the levels of the categorical predictor variables, but are not invariant to different orderings of levels
"

My question is that what is meant by the following line - "Type IV sums of squares are invariant to the choice of a g2 inverse of X'X given a particular ordering of the levels of the categorical predictor variables, but are not invariant to different orderings of levels"??


Also

Type V sums of squares can be illustrated by using a simple example. Suppose that the effects considered are A, B, and A by B, in that order, and that A and B are both categorical predictors with, say, 3 and 2 levels, respectively. The intercept is first entered into the model. Then A is entered into the model, and its degrees of freedom are determined (i.e., the number of non-redundant columns for A in X'X, given the intercept). If A's degrees of freedom are less than 2 (i.e., its number of levels minus 1), it is eligible to be dropped. Then B is entered into the model, and its degrees of freedom are determined (i.e., the number of non-redundant columns for B in X'X, given the intercept and A). If B's degrees of freedom are less than 1 (i.e., its number of levels minus 1), it is eligible to be dropped. Finally, A by B is entered into the model, and its degrees of freedom are determined (i.e., the number of non-redundant columns for A by B in X'X, given the intercept, A, and B). If B's degrees of freedom are less than 2 (i.e., the product of the degrees of freedom for its factors if there were no missing cells), it is eligible to be dropped.

Q: why did they say in effect ‘A’ that if the df is less than 2. How did they come up with this statement? I mean why did they choose 2? Similarly for Effect ‘B’ and ‘A by B’?
Q: In Effect ‘A by B’ will that be dropped or not. Since it has a df of 2, and the rule say that its eligible to drop if less than 2?

Note that Type V sums of squares involve determining a reduced model for which all effects remaining in the model have at least as many degrees of freedom as they would have if there were no missing cells

Q: so the total df of the complete model is dfA + dfB + dfA by B = 2+1+2 = 5
And the df of the model after the effects are dropped is dfA by B = 2
So how does that equal. Or am I being really stupid and interpreting this completely wrong.

PS: Does anyone know where I can find more information on Type V and Type VI SS?
Thanks

The strategy we used is to take a look at the different Types of SSs output. If all of the SSs are similar, it doesn't matter, so we went on to something else. If they are different, then we would have to take a closer look.

Most of the time we had better control of the experiment and could better decide the best calculation to use because we knew quite a bit about the process. Usually Type III was the best and people were more familiar with it.

I suggest taking stock of the data. Which variables are random and which are fixed? How much autocorrelation is occurring? Why are we getting these results? This takes a great deal of tedious plotting.

BTW, if your model has both fixed and random effects, check out PROC MIXED. SAS-L is also a place to get help with SAS questions.

Best wishes,

Tom

I wish this was an easy straightforward answer. Most statistical textbook cover Type I and Type II Sums of squares. Type I SS are the full model SS, while Type II are the partial SS (effect of having the factor in the model). Type III and IV SS were the PhD dissertation for Goodkinght (founder of SAS) when he was at NC State. If you have a strong mathematical background, read the section on Estimable Functions in the SAS Documentation (http://support.sas.com/onlinedoc/913/docMainpage.jsp).

Thank You Statistical Steven and Tom Slack, for helping me with my query.

But I still cant seem to find much information on Type V SS and type VI SS.

Can someone suggest some good book or website?

Thanks

fed-up said:

But I still cant seem to find much information on Type V SS and type VI SS.

Can someone suggest some good book or website?

Thanks

Click to expand...

May I ask why you need to understand this? is there a particular application or simple curiosity? I ask because the "Sum of the Squares" is a simple interim calculation in performing any analysis of variance to determine largest variation contributors. With software this is usually hidden.

These two "types" are fairly rare and so it begs the question....