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

hi, Isaac:

You can bulid the IV type SS tests in SAS. For example. If you want to
bulid a type III test contrast for the 'a' effect; with the following
statement:

proc mixed data=test method=type3;
class a b;
model y=a b a*b/e3; /* get the type 3 coefficients for a */
run;

from the output, you will find this parts for the a effect:

Type 3 Coefficients for a

Effect a b Row1 Row2 Row3

Intercept
a 1 1
a 2 1
a 3 1
a 4 -1 -1 -1
b 1
b 2
a*b 1 1 0.25 -0.5 -0.25
a*b 1 2 0.75 0.5 0.25
a*b 2 1 1
a*b 3 1 -0.25 -0.5 0.25
a*b 3 2 0.25 0.5 0.75
a*b 4 2 -1 -1 -1


the row1 row2 and row3 are the contrast for test a effect, in this
sample, the contrast is:
intercept 0 a 1 0 0 -1 b 0 0 a*b .25 .75 0 -.25 .25 -1,
intercept 0 a 0 1 0 -1 b 0 0 a*b -.5 .5 1 -.5 .5 -1,
intercept 0 a 0 0 1 -1 b 0 0 a*b -.25 .25 0 .25 .75 -1;

the result as :


Type 3 Tests of Fixed Effects

Num Den
Effect DF DF F Value Pr > F

a 3 2 48.40 0.0203
b 1 2 16.33 0.0561
a*b 1 2 3.00 0.2254


Contrast

Num Den
Label DF DF F Value Pr > F

type III test 3 2 48.40 0.0203


you can bulid type I , II, III and IV SS test by using this method.


Baogong



On 5/18/05, Isaac Neuhaus <isaac.neuhaus@bms.com> wrote:
> Thank you. How do you build the contrasts?
> Isaac
>
>
> baogong jiang wrote:
> Hi, Isaac,

The SAS type I-IV SS test is very useful in a balanced design
> without
missing cells. If your data have missing cells, then depends on
> what
hypotheses you want to test, maybe none of the 4 type SS test
> is
right.

in proc mixed or proc glm, with the e or e1 e2 e3 e4 options in
> the
model statement, it's every easy to find out what hypothese the 4
> type
SS test are testing.

proc mixed method=type3;
class a b;
model y=a b
> a*b/e3;
run;


here is a program i wrote to compare the 4 type SS test. also
> I wrote
a macro PMMixed to calculate population marginal means when
> data
have missing cells. (SAS lsmeans will give out no-est. results.),
> if
you want the macro, just let me know!

data one;
do a=1 to 2;
do b=1 to
> 3;
do c=1 to 2;
do d=1 to 3;
do e=1 to
> 5;
score=10+30*abs(rannor(234323));
output;
end;end;end;end;end;
run;
data
> two;
set one;
if a=1 and b=2 then delete;
if b=2 and c=1 then
> delete;
run;

data three;
set two;
if a=1 and b=3 and d>2 then delete;
if
> a=2 and b=1 and d=5 then delete;
run;

proc tabulate data=three;
class a b c
> d;
var score;
table a,b;run;

proc mixed data=three;
class a b c d;
model
> score=a|b/solution ddfm=satterthwaite;
lsmeans a|b/e;
run;
proc glm
> data=three;
class a b c d;
model score=a|b/solution e1 e2 e3 e4;
contrast
> 'type 1 test for a effect' a 10 -10 b 2 -2 0
a*b 6 4 -4 -2 -4/e;
contrast
> 'type 2 test for a effect' a 1 -1 b 0 0 0
a*b .5556 .4444 -.5556 0
> -.4444/e;
contrast 'type 3 & 4 test for a effect' a 1 -1 b 0 0 0
a*b .5 .5
> -.5 0 -.5/e;
/* contrast LSmeans test is not estimable */;
contrast '
> LSmeans test ' a 6 -6 b 0 0 0 a*b 3 3 -2 -2 -2/e;
contrast 'type 1 & 2 test
> for b effect' b 1 0 -1 a*b .4444 -.4444
..5556 0 -.5556,
b 0 1 -1 a*b .2222
> -.2222
-.2222 1 -.7778/e ;

contrast 'type 3 test for b effect' b 1 0 -1
> a*b .5 -.5 .5 0 -.5,
b 0 1 -1 a*b .25 -.25 -.25 1 -.75/e ;
contrast 'type 4
> test for b effect' b 1 0 -1 a*b .5 -.5 .5 0 -.5,
b 0 1 -1 a*b 0 0 0 1 -1/e
> ;

lsmeans a|b/e;
run;
%pmmmixed( data=three, stmts=%str(
Class a b c
> d;
Model Score = a| b /solution e ddfm=Satterthwaite;
Lsmeans a| b / e
> );
run;
/* test macro result */;
proc mixed data=three;
class a b c
> d;
model score=a*b/e1 e2 e3;
contrast 'macro a effect' a*b 3 3 -2 -2
> -2/e;
contrast 'macro b effect' a*b .5 0 .5 -1 0,
a*b .5 -.5 .5 0
> -.5/e;
run;
/*

The GLM Procedure

Dependent Variable: score

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

Model 4 1649.69294
> 412.42324
1.34 0.2606

Error 120 37047.30146 308.72751

Corrected Total
> 124 38696.99440


R-Square Coeff Var Root MSE score Mean

0.042631
> 52.46313 17.57064 33.49141


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

a 1 576.6347140 576.6347140
1.87 0.1743
b 2 459.3012397 229.6506198
> 0.74 0.4775
a*b 1 613.7569905 613.7569905
1.99 0.1611


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

a 1 696.5114296 696.5114296
2.26 0.1357
b
> 2 459.3012397 229.6506198
0.74 0.4775
a*b 1 613.7569905 613.7569905
1.99
> 0.1611


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

a 1
> 551.0947151 551.0947151
1.79 0.1841
b 2 538.9807615 269.4903808
0.87
> 0.4204
a*b 1 613.7569905 613.7569905
1.99 0.1611


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

a 1* 551.0947151
551.0947151 1.79 0.1841
b 2*
> 335.2225480
167.6112740 0.54 0.5825
a*b 1 613.7569905
613.7569905 1.99
> 0.1611

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

Contrast DF Contrast SS Mean
Square F Value Pr > F
type 1
> test for a effect 1 576.6347140
576.6347140 1.87 0.1743
type 2 test for a
> effect 1 696.6283890
696.6283890 2.26 0.1357
type 3 & 4 test for a effect
> 1 551.0947151
551.0947151 1.79 0.1841
type 1 & 2 test for b effect 2
> 459.2409919
229.6204960 0.74 0.4775
type 3 test for b effect 2 538.9807615
>
269.4903808 0.87 0.4204
type 4 test for b effect 2 335.2225480
>
167.6112740 0.54 0.5825

Num Den
Label DF DF F Value Pr > F

macro a
> effect 1 120 0.79 0.3754
macro b effect 2 120 0.54 0.5815

*/;





On
> 5/17/05, isaac.neuhaus@bms.com <isaac.neuhaus@bms.com> wrote:

> 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


>

>


--
Baoogng Jiang
Department of Agronomy
Lousisana State University