Re: Type I-IV Sum of squares with empty cells

On Tue, 17 May 2005 isaac.neuhaus@BMS.COM wrote:

> I am trying to understand how the different types of sum of squares are
> calculated and I haven't been able to understand how SAS calculates the
> type III sum of squares when there are empty cells in the design. I
> looked at Donald Macnaughton SS.sas code which has been an excellent
> teaching resource but when it comes to a design with empty cells the
> type III sum of squares (HTI) do not agree with those produced by proc
> glm in SAS. Can anybody point to where I can learn how to manually
> calculate them or even better explain how?

Isaac,

First, you should read Chapters 13-15 of Milliken and Johnson, "Analysis
of Messy Data", Vol 1. They provide more detail and information than I
could hope to in this short note.

Here is a brief GLM analysis (though one could perhaps do even better with
MIXED) that is based on the approach from the above reference:

Proc tabulate shows the nature of the empty cells:

----------------------------
| | b |
| |-------------------|
| | 1 | 2 |
| |---------+---------|
| | N |Mean | N |Mean |
|------+---+-----+---+-----|
|a | | | | |
|1 | 1| 7.0| 2| 9.5|
|2 | 1| 1.0| .| .|
|3 | 1| 4.0| 2| 5.0|
|4 | .| .| 1| 5.0|
----------------------------

I've recoded A and B into one variable called lvl which is

lvl = 10*a + b;

so the cell means layout is now:

------------------
| | N |Mean |
|------+---+-----|
|lvl i | | |
|11 1 | 1| 7.0|
|12 2 | 2| 9.5|
|21 3 | 1| 1.0|
|31 4 | 1| 4.0|
|32 5 | 2| 5.0|
|42 6 | 1| 5.0|
------------------

The variable i is the index found in the ESTIMATE and CONTRAST statements
in the second GLM step:

With empty cells compare the TYPE III and TYPE IV sums of Squares:

proc glm data=one;
class a b;
model y = a b a*b / ss3 ss4;
run; quit;

Source DF Squares Mean Square F Value Pr > F

Model 5 57.00000000 11.40000000 45.60 0.0216
Error 2 0.50000000 0.25000000
Corrected Total 7 57.50000000


Source DF Type III SS Mean Square F Value Pr > F

a 3 36.30000000 12.10000000 48.40 0.0203
b 1 4.08333333 4.08333333 16.33 0.0561
a*b 1 0.75000000 0.75000000 3.00 0.2254

Source DF Type IV SS Mean Square F Value Pr > F

a 3* 28.80000000 9.60000000 38.40 0.0255
b 1* 4.08333333 4.08333333 16.33 0.0561
a*b 1 0.75000000 0.75000000 3.00 0.2254

* NOTE: Other Type IV Testable Hypotheses exist which may yield different SS.

Now how does one reproduce these Type IV SS?

Use the cell means model with GLM:

proc glm data=one;
class lvl;

model y = lvl / ss4;

/* Sums of Squares for A: assume the following contrasts
A 1 vs 2 in b=1
A 1 vs 3 in b=1,2
A 3 vs 4 in b=2
*/
contrast 'A' lvl 1 0 0 -1 0 0,
lvl 1 1 0 -1 -1 0,
lvl 0 0 0 0 1 -1;
ESTIMATE 'A 1 vs 2 in B=1' lvl 1 0 0 -1 0 0;
ESTIMATE 'A 1 vs 3 in B=1,2' lvl 1 1 0 -1 -1 0 / DIVISOR=2;
ESTIMATE 'A 3 vs 4 in B=2' lvl 0 0 0 0 1 -1;

/* one contrast for B= 1 vs 2 in A=1,3 */

CONTRAST 'B' lvl 1 -1 0 1 -1 0 ;
ESTIMATE 'B' lvl 1 -1 0 1 -1 0 /DIVISOR=2;

/* Interaction comes from A=1,3 in B=1,2 */

contrast 'AB' lvl 1 -1 0 -1 1 0;
ESTIMATE 'AB' lvl 1 -1 0 -1 1 0 /DIVISOR=2;

run; quit;

< edited output >

The GLM Procedure

Class Level Information

Class Levels Values
lvl 6 11 12 21 31 32 42


Dependent Variable: y

Sum of
Source DF Squares Mean Square F Value Pr > F

Model 5 57.00000000 11.40000000 45.60 0.0216
Error 2 0.50000000 0.25000000
Corrected Total 7 57.50000000

Source DF Type IV SS Mean Square F Value Pr > F
lvl 5 57.00000000 11.40000000 45.60 0.0216

Contrast DF Contrast SS Mean Square F Value Pr > F

A 3 28.80000000 9.60000000 38.40 0.0255
B 1 4.08333333 4.08333333 16.33 0.0561
AB 1 0.75000000 0.75000000 3.00 0.2254

NOTE: the contrast results are the same as the TYPE IV when the model
y = a + b + ab was entered in the first GLM step

You can also compute estimates from the above contrasts:

Standard
Parameter Estimate Error t Value Pr > |t|

A 1 vs 2 in B=1 3.00000000 0.70710678 4.24 0.0513
A 1 vs 3 in B=1,2 3.75000000 0.43301270 8.66 0.0131
A 3 vs 4 in B=2 -0.00000000 0.61237244 -0.00 1.0000
B -1.75000000 0.43301270 -4.04 0.0561
AB -0.75000000 0.43301270 -1.73 0.225


Robin High ("sometimes known to also have missing cells")
Univ. of Oregon

Robin High wrote:
> On Tue, 17 May 2005 isaac.neuhaus@BMS.COM wrote:
>
> > I am trying to understand how the different types of sum of squares
are
> > calculated and I haven't been able to understand how SAS calculates
the
> > type III sum of squares when there are empty cells in the design. I
> > looked at Donald Macnaughton SS.sas code which has been an
excellent
> > teaching resource but when it comes to a design with empty cells
the
> > type III sum of squares (HTI) do not agree with those produced by
proc
> > glm in SAS. Can anybody point to where I can learn how to manually
> > calculate them or even better explain how?
>
> Isaac,
>
> First, you should read Chapters 13-15 of Milliken and Johnson,
"Analysis
> of Messy Data", Vol 1. They provide more detail and information than
I
> could hope to in this short note.
>
> Here is a brief GLM analysis (though one could perhaps do even better
with
> MIXED) that is based on the approach from the above reference:
>
> Proc tabulate shows the nature of the empty cells:
>
> ----------------------------
> | | b |
> | |-------------------|
> | | 1 | 2 |
> | |---------+---------|
> | | N |Mean | N |Mean |
> |------+---+-----+---+-----|
> |a | | | | |
> |1 | 1| 7.0| 2| 9.5|
> |2 | 1| 1.0| .| .|
> |3 | 1| 4.0| 2| 5.0|
> |4 | .| .| 1| 5.0|
> ----------------------------
>
> I've recoded A and B into one variable called lvl which is
>
> lvl = 10*a + b;
>
> so the cell means layout is now:
>
> ------------------
> | | N |Mean |
> |------+---+-----|
> |lvl i | | |
> |11 1 | 1| 7.0|
> |12 2 | 2| 9.5|
> |21 3 | 1| 1.0|
> |31 4 | 1| 4.0|
> |32 5 | 2| 5.0|
> |42 6 | 1| 5.0|
> ------------------
>
> The variable i is the index found in the ESTIMATE and CONTRAST
statements
> in the second GLM step:
>
> With empty cells compare the TYPE III and TYPE IV sums of Squares:
>
> proc glm data=one;
> class a b;
> model y = a b a*b / ss3 ss4;
> run; quit;
>
> Source DF Squares Mean Square F Value Pr
> F
>
> Model 5 57.00000000 11.40000000 45.60
0.0216
> Error 2 0.50000000 0.25000000
> Corrected Total 7 57.50000000
>
>
> Source DF Type III SS Mean Square F Value Pr > F
>
> a 3 36.30000000 12.10000000 48.40 0.0203
> b 1 4.08333333 4.08333333 16.33 0.0561
> a*b 1 0.75000000 0.75000000 3.00 0.2254
>
> Source DF Type IV SS Mean Square F Value Pr > F
>
> a 3* 28.80000000 9.60000000 38.40 0.0255
> b 1* 4.08333333 4.08333333 16.33 0.0561
> a*b 1 0.75000000 0.75000000 3.00 0.2254
>
> * NOTE: Other Type IV Testable Hypotheses exist which may yield
different SS.
>
> Now how does one reproduce these Type IV SS?
>
> Use the cell means model with GLM:
>
> proc glm data=one;
> class lvl;
>
> model y = lvl / ss4;
>
> /* Sums of Squares for A: assume the following contrasts
> A 1 vs 2 in b=1
> A 1 vs 3 in b=1,2
> A 3 vs 4 in b=2
> */
> contrast 'A' lvl 1 0 0 -1 0 0,
> lvl 1 1 0 -1 -1 0,
> lvl 0 0 0 0 1 -1;
> ESTIMATE 'A 1 vs 2 in B=1' lvl 1 0 0 -1 0 0;
> ESTIMATE 'A 1 vs 3 in B=1,2' lvl 1 1 0 -1 -1 0 / DIVISOR=2;
> ESTIMATE 'A 3 vs 4 in B=2' lvl 0 0 0 0 1 -1;
>
> /* one contrast for B= 1 vs 2 in A=1,3 */
>
> CONTRAST 'B' lvl 1 -1 0 1 -1 0 ;
> ESTIMATE 'B' lvl 1 -1 0 1 -1 0 /DIVISOR=2;
>
> /* Interaction comes from A=1,3 in B=1,2 */
>
> contrast 'AB' lvl 1 -1 0 -1 1 0;
> ESTIMATE 'AB' lvl 1 -1 0 -1 1 0 /DIVISOR=2;
>
> run; quit;
>
> < edited output >
>
> The GLM Procedure
>
> Class Level Information
>
> Class Levels Values
> lvl 6 11 12 21 31 32 42
>
>
> Dependent Variable: y
>
> Sum of
> Source DF Squares Mean Square F Value Pr >
F
>
> Model 5 57.00000000 11.40000000 45.60
0.0216
> Error 2 0.50000000 0.25000000
> Corrected Total 7 57.50000000
>
> Source DF Type IV SS Mean Square F Value Pr
> F
> lvl 5 57.00000000 11.40000000 45.60
0.0216
>
> Contrast DF Contrast SS Mean Square F Value Pr
> F
>
> A 3 28.80000000 9.60000000 38.40
0.0255
> B 1 4.08333333 4.08333333 16.33
0.0561
> AB 1 0.75000000 0.75000000 3.00
0.2254
>
> NOTE: the contrast results are the same as the TYPE IV when the model
> y = a + b + ab was entered in the first GLM step
>
> You can also compute estimates from the above contrasts:
>
> Standard
> Parameter Estimate Error t Value Pr >
|t|
>
> A 1 vs 2 in B=1 3.00000000 0.70710678 4.24
0.0513
> A 1 vs 3 in B=1,2 3.75000000 0.43301270 8.66
0.0131
> A 3 vs 4 in B=2 -0.00000000 0.61237244 -0.00
1.0000
> B -1.75000000 0.43301270 -4.04
0.0561
> AB -0.75000000 0.43301270 -1.73 0.225
>
>
> Robin High ("sometimes known to also have missing cells")
> Univ. of Oregon

Thank you for the explanation. Is there a typo in the following
contrast?

/* Sums of Squares for A: assume the following contrasts
A 1 vs 2 in b=1
A 1 vs 3 in b=1,2
A 3 vs 4 in b=2
/
contrast 'A' lvl 1 0 0 -1 0 0,
lvl 1 1 0 -1 -1 0,
lvl 0 0 0 0 1 -1;

Shouldn't it be:

contrast 'A' lvl 1 0 -1 0 0 0,
lvl 1 1 0 -1 -1 0,
lvl 0 0 0 0 1 -1;

Do you also have an executive summary like this one for the type III
sum of squares?

Thanks again,

Isaac