Pertanika 7(2), 71-81 (1984) A Geometric Look At Repeated Measures Design with Missing Observations AHMAD BIN ALWI and CJ. MONLEZUN 1 Department of Mathematics, Universiti Pertanian Malaysia Serdang, Selangor, Malaysia, Key words: Repeated measures design; subspaces; noncentrality parameters; orthogonal: orthonormal RINGKASAN Di dalam kertas ini kami akan memberi gambaran geometri bagi Rekabentuk Sukatan Berulang untuk bilangan subjek yang tak sama serawatan yang mempunyai kehilangan cerapan. Untuk pembentukan geometri, kami menghadkan rekabentuk ini kepada tiga tahap bagi faktor A dan empat tahap bagi faktor B, Tujuan kertas ini ialah untuk membentuk ujian statistik bagi hipotesis yang dikehendaki iaitu tiada kesan utama A, tiada kesan utama B, dan tiada tindakan bersaling AB. SUMMARY In this paper, we will provide a geometric view of Repeated Measures Design for unequal number of subjects per treatment that has missing observations. For our geometric development we restrict our design to three levels of factor A and four levels of factor B. The purpose of this paper is to develop a test statistics for hypotheses of interest i.e. no main effect A, no main effect B, and no AB interaction. 1. INTRODUCTION The data for a two-factor Repeated Measures Design is collected and tabulated in a data table as shown in Figure 1. Let Y... be the measurement made on subject i (l<3^n!) at level j (l£j<a) of factor A and level k (l^k<b) of factor B. For illustrative purposes, we let a=3, b=4, n x =3, n 2 =2, n 3 =4. 111 211 Y Y 1 112 113 114 Y Y Y X 212 213 214 V V Y Y 311 312 313 314 A 2 Y 121 221 Y Y 231 331 Y 431 122 Y 222 Y 132 232 Y 332 432 Y 123 Y 223 Y 133 Y 233 Y 333 Y 433 Y 124 Y 224 Y 134 Y 234 Y 334 434 Figure 1: Data table for observations. We arbitrarily set the observations Y , Y Y 123' Y 232' Y 233 and Y 33 4 ^ missing. We model our experiment as: iik ik ii iik \ 1 * 1 ) 1 Assoc. Professor, Dept of Experimental Statistics, Louisiana State University, U.S. A Key to author's name: A. Ahmad. 71
Transcript
Pertanika 7(2), 71-81 (1984)
A Geometric Look At Repeated Measures Design with MissingObservations
AHMAD BIN ALWI and C J . MONLEZUN1
Department of Mathematics,Universiti Pertanian MalaysiaSerdang, Selangor, Malaysia,
Di dalam kertas ini kami akan memberi gambaran geometri bagi Rekabentuk Sukatan Berulanguntuk bilangan subjek yang tak sama serawatan yang mempunyai kehilangan cerapan. Untuk pembentukangeometri, kami menghadkan rekabentuk ini kepada tiga tahap bagi faktor A dan empat tahap bagi faktorB, Tujuan kertas ini ialah untuk membentuk ujian statistik bagi hipotesis yang dikehendaki iaitu tiadakesan utama A, tiada kesan utama B, dan tiada tindakan bersaling AB.
SUMMARY
In this paper, we will provide a geometric view of Repeated Measures Design for unequal numberof subjects per treatment that has missing observations. For our geometric development we restrict ourdesign to three levels of factor A and four levels of factor B. The purpose of this paper is to develop a teststatistics for hypotheses of interest i.e. no main effect A, no main effect B, and no AB interaction.
1. INTRODUCTION
The data for a two-factor Repeated MeasuresDesign is collected and tabulated in a datatable as shown in Figure 1. Let Y... be themeasurement made on subject i (l<3^n!) at levelj (l£j<a) of factor A and level k (l^k<b) offactor B. For illustrative purposes, we let a=3,b=4, nx=3, n2=2, n3=4.
111
211
Y Y1 112 113 114
Y Y YX212 213 214
V V Y Y311 312 313 314
A 2
Y121
221
Y
Y231
331
Y431
122
Y222
Y132
232
Y332
432
Y123
Y223
Y133
Y233
Y333
Y433
Y124
Y224
Y134
Y234
Y334
434
Figure 1: Data table for observations.
We arbitrarily set the observations Y , YY123'Y232' Y233 a n d Y 33 4 ^ missing. We modelour experiment as:
i ik ik ii iik \1*1)
1 Assoc. Professor, Dept of Experimental Statistics, Louisiana State University, U.S. A
Key to author's name: A. Ahmad.
71
AHMAD BIN ALWI AND C.J. MONLEZUN
TABLE 1
Set of vectors that span the cell means space, C
10
0
1
0
0
0
1
0
0
0
0
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
W 1 2
0
1
0
0
1
0
0
0
0
0
0
0
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
" 1 .
0
0
0
0
0
1
0
01
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
W 1 4
0
0
1
0
0
0
1
0
0
1
0
0
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
w21
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
W 2 2
0
0
0
0
0
0
0
0
0
0
0
1
0
01
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
W 2 3
0
0
0
0
0
0
0
0
0
0
0
0
0
00
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
W 2 4
0
0
0
0
0
0
0
0
0
0
0 '0
1
00
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
W 3 1
0
0
0
0
0
0
0
0
0
0
0
0
0
00
0
0
1
0
0
0
1
0
1
0
0
1
0
0
0
W 3 2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0100
W 3 3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00
10
0
0
0
0
1
0
0
1
0
W 3 4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c0
0
0
0
0
1
0
1
0
0
0
0
001
72
A GEOMETRIC LOOK AT REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS
where { S.., E..k } are 9+30 = 39 mutually We now define subspaces of R30 which facilitate
independent normal random variables each having h f imean zero, with Var(S.j) = »§, Var(E.jk) -*£< A = s p a n
t h e construction of test statistics. Let
Alternatively we can write the model as
(1.2)
and
B = span { bv b 2 , b 3 } ,
AB = span { (ab) l x , (ab)12 , (ab)13 (ab)21, (ab)^,
(a^)23 / '
'Within subject space' , Wg = span { »-!, s21 ,
S 3 1 ' S 1 2 * S 2 2 * S 1 3 * S 23> S 3 3 * S 4 3 > w h e r e S i j ' S
are defined in Table 4, and
T = W E B E AB.The observational vector is written as:
T is the smallest subspace containing both CYij = t Y i ir Yii2> Y i j3 ' Yij4 1 for ij=21.22,13,43 and Wg. We defined the Error space, E, as
the space orthogonal both to C and Ws i.e.E = span { e1, e2 ,..., e 1 2 } where e's are defined
2. GEOMETRIC DEVELOPMENT
Y =
Y23 =
Y31 =
[
(
[
[
Y
Y
Y
Y
111'
121'
231'
311*
Y112*
Y122*
Y
Y313
114
Y124
]'
• Y 3 1 4
in Table 5.
3.
IH
HYPOTHESES TESTING
The hypotheses of interest are:
7 Y Y 1331* 332* 333 J HB' k '
V = fv'1 I- X 1 1 '
33
2 1 ' x 3 1 '
4 3 J
22Y' Y'
1 3 ' 23* HA B
Y is a vector in the Euclidean space withdimension 30, R3 0 .
The cell means vector is written as:3 4
E(Y) = 2 2 U.. w where w is defined inj=lk=l ] k ] k JR
Table 1
The set of w.fe vectors form a basis for the cellmeans space, C, having dimension 12. If we
In general, when there are missing observationsfor subjects an exact test of HA is not available.
Why not have an exact test for HA?
The hypothesis for no main effect A in Ujk is
parameterized Ujk= U + a. + b ,
j K(ab)r
subjected to the conditions
$ a . = 2 b k = ? (ab) jk= 2(ab) j k = 0
then the cell means space, C, has a basis theset of vectors { l 3 0 , aT, a2, b 1 ( b 2 , b 3 , (ab) i x ,
(ab)12, (ab)13 , (ab)21 , (ab)22 , (aD)2 3 } as
defined in Table 2.
<==> a. = 0 <==> KA 'u = 0 < = = > A ( )
- 0 <==> E(Y) C WA = 13Q E B H AB (3.1)
( K . , U, and G. are defined as in Table 6).A A
To assure a central V distribution when H^ istrue, we need the numerator space for calculatingSum of squares for A, N'A, to be orthogonal to
If we want NTA, to be orthogonal to Wg also we
would define NA = T 0 [ B E AB B Wg ] .
But T = [ B H AB E W ] , therefore, XA -
Var(Y) = oil + olj where I is the n.. x n.. { } and we do not have test statistics. If
identity matrix and J is a matrix defined in w e w a n t N A l WS <as i n t h e c a s e w h e n a11 ,°b-Table 3. servations on a subject are present;, note that
73
AHMAD BIN ALWI AND CJ. MONLEZUN
TABLE 2
After reparamaterization, alternative basis for C
ho
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
h
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
- 1
- 1
- 1
- 1
- 1
- 1
— 1
- 1
- 1
- 1
- 1
- 1
- 1
a 2
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
- 1
- 1- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 1
~l
b l
1
0
- 1
1
0
0
- 1
1
0
- 1
1
0- 1
1
0
0
— 1
1
0
0
- 1
1
- 1
1
0
0
1
0
0
- 1
b 2
0
1
- 1
0
1
0
- 1
0
0
- 1
0
1- 1
0
1
0
— 1
0
1
0
- 1
0
- 1
0
1
0
0
1
0
- 1
b 3
0
0
- 1
0
0
1
- 1
0
1
- 1
0
0- 1
0
0
1
- 1
0
0
1
- 1
0
- 1
0
0
1
0
0
1
1
0
- 1
1
0
0
- 1
1
0
- 1
0
00
0
0
0
0
- 1
0
0
1
- 1
1
- 1
0
0
- 1
0
0
1
(ab),
0
1
- 1
0
1
0
- 1
0
0
- 1
0
00
0
0
0
0
0- 1
0
1
0
1
0
- 1
0
0
- 1
0
1
2 <*>>,
0
0
- 1
0
0
1
- 1
0
1
- 1
0
00
0
0
0
0
0
0
- 1
1
0
1
0
0
- 1
0
0
- 1
1
(ab)21
0
0
0
0
0
0
0
0
0
0
1
0
- 1
1
0
0
- 1
j,0
0
1
- 1
1
- 1
0
0
- 1
0
0
1
(ab)22
0
0
0
0
0
0
0
0
0
0
0
1- 1
0
1
0
- 1
0
- 1
0
1
0
1
0
- 1
0
0
- 1
0
1
(ab)23
0
0
0
0
0
0
0
0
0
0
0
0
- 1
0
0
1
- 1
0
0
- 1
1
0
1
0
0
- 1
0
0
- 1
1
74
A GEOMETRIC LOOK AT REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS
A GEOMETRIC LOOK AT REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS
e l
2- 1- 1- 1
11
- 1- 1- 1
20000000000
o0
ooooooo0
C2
20
_2- 1
001
- 101000000000000
ooooooo0
e 3
10
-1-2
00210
- 10000000000000ooo0
o00
e 4
00010
- 10
- 11000000000000000000000
A
e 5
00000000001
- 10
- 11000000000000000
TABLE 5basis for
e 6
0000000000
- 101100
- 10000000000000
the error
e 7
000000000000000001
00
- 11
- 1- 2
200
- 202
space, E
e 8
0000000000000000002
- 20000
- 110
- 100
e 9
00000000000000000001
- 10000000
- 11
e i o
00000000000000000010
- 1000000
- 101
e l l
000000000000000000000001
- 10
- 1100
e i 2
0000000000
o000000000
- 1- 1
1
u00 .0000
77
AHMAD BIN ALWI AND C.J. MONLEZUN
TABLE 6
The matrices used in formulating HA
TABLE 7The spanning vectors for S = W 0 ( l 3 0 0 A )
G A
•
in
VsxhVi
%Vz¥2
Vs
v
9k1/3
-ft- > / 2
-V*
_1/2
-<*
- 1
00
00
00
000
0000
%xhVs
%V2V2VA
V2
Vs
000
00
00
-44
~x—lA-Vs
-%-V3
-Vs
-lA
"! /_y
general, there may
K A =
u =
not be any*1T / _ 1 J .L
1111
- 1- 1- 1- 1
000
0
U l l
U 1 2
U 1 3
U14
u21
u22U23
T TU 3 1
U 3 2
U 3 3
U 3 4
vectors
I1110000
- 1- 1- 1
- 1
inW
v31/3
i / 3
-VA
-lA-VA
000
000
0000
0000
00
000
0000
S 2
111
0000
- 1- 1- 1
000
0000
0000
00
000
0000
S 3
000
0000
000
VsV3
v$-VA
-VA
0000
00
000
0000
S 4
000
0000
000
000
0000
VA
VA
VA
VA
-VI
-V2
000
0000
S 5
000
0000
000
000
0000
VA
VA
VA
VA
00
-a
- V 3
0000
S 6
000
0000
000
000
0000
1111
00
000
- 1- 1- 1- 1
vector, s6, in Table 7 is in Wg andorthogonal to WA in our case). In addition,J = 2PM + 3PM + 4PM behaves differently
2 3 4
on different vectors in Wg. Thus no exact testis available for testing H .
The hypothesis for no main effect B in \5 is
H B : U fc = U k .
<==> bk = 0 <==> K'B U = 0 <==>
* 0 O = > E(Y) C WB = 13Q S A H AB (3.2)
(KB,andGfi are defined in Table 8).
Now G C C C T and Go -L Wo but GD J -D B B B
Wg. Therefore, we take our numerator spaces asNB = T Q [ Wg E AB ] . Then Nfi -L WB and
78
A GEOMETRIC LOOK AT REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS
TABLE 8
The matrices used in formulating HB
GR=
K\k0
M-1/2
00
1/3
00
V*-1/2
0
1/2
-1/2
00
1/4
-Vs00
%0
1/4
-1/3
0
l /4
00
1/3
00
V40
-1/2
0
Vs-1/2
0
1/2
00
1/2
0- 1
0
1/4
0-1/3
0
l /4
0
1/4
0-1/3
1/4
0- y 3
0
1/3
0-Vs
1/3
00
-1/3
1/3
0-1/3
1/2
0-1/2
1/2
00
-1/2
1/4
00
-1/3
1/4
-X/4
1/4
00
%00
-1/3
1- 1
001
KB= - 1001
- 100
10
- 1010
- 1010
- 10
100
- 1100
- 1100
- 1
NR
as Y P
Ws. The sum of squares for B is defined
( the transpose of vector Y is to• B
multiply the orthogonal projection onto the NRspace and then multiply again to the vector YJ.Toseehow sum of squares of B is distributed we let{ b. } be the orthonormal basis for N_. Then
b-iYfP Y = (b/Y)a (3.3)
We note that b Y is distributed as a Normalrandom variable with mean b. E(Y), variance
b.' [ a | l + (TgJ ]b. = a^since Jb. * 0, and
Cov(b.'Y,b.,Y) = 0 for jfj'. If we divide b.'Y
by a£ , then the result is a Normal randomvariable with mean b̂ E(Y) and variance 1. There-fore, Y'PN Y/ol is X2(b-1,A) (3.4)
B
It may appear that we are testing the hypothesisN R ' E ( Y ) • 0. However, we show below thatE(Y) -LGB if and only if E(Y) _L Nfi.
Toshow: E(Y)_L ==> E(Y) 1 NB
Note that E(Y) C C. Let v e C . Then v ±_
if and only if v 1 NR .
Proof:(only if) Let v - I - Gfi
N —L WR by defination, then vthen v eWB. Since
(if) Let S = Ws 0 ( l 3 0 S A ). Then S - span{ sx, s2 ,..., sg } and S is linearly independent
- Nfi, Then v € AB
S. Therefore, v =
of C. Let v € T and v
E Wg = 13Q ffi A S AB
kl3Q + a + w + s where kl3Q € l 3 0 , a e A,w €AB, and s e S. If v e C, then s = 0 and v eWB ==> v e GB.
The hypothesis for no AB interaction in Ujk is
\ JtS IK
<==> (ab)V -
0 - • K A B UGAB'E(Y) = 0 <==> E(Y) e WAB = l 3 0 H
WAB and N A B ws.The sum of square for AB is defined as Y PN Y.
A B
Let { mfe } be an orthonormal basis for N
Then
Y'P.N Y _A B 2
k - 1
' ( < Y , 2 (3.6)
79
AHMAD BIN ALWI AND CJ. MONLEZUN
TABLE 9
The matrices used in formulating H A f i
-V*
0
- V i00
00
- V 2Vx0
G A B =
o000
00
000
0000
Vs00
0
o
Vs
0
V200
o10
0000
00
000
0000
0Vs
Vs00
0
o0
0000
00
000
0000
Vs- y 2
0
V200
Vs,00
000
0000
Vs00
- V A
0
00
Vs00
Vs0
000
0000
0y2
o
0
Vx00
o
0
000
0000
00
-VA
-VA
0
0
0
0
Vs
KAB
1- 1
00
- 1= 1
000000
1- 1
000000
- 1100
10
- 10
- 10100000
10
- 100000
- 1010
100
- 1- 1
0010000
100
- 10000
- 1001
With similar reasoning, m 'Y is N (m.'E(Y), a*)
and Cov (mk'Y,mk,'Y) = 0 for k i k'.
^ Y / a» is X2 ( (a-l)(b-l), X )
(3.7)
Therefore,
The noncentrality parameter is due to the non-zero mean in the Normal random variable and
we can define it asX = E(Y)'PN E(Y) I 2 o\ for X
xB , AB
Both hypotheses share the same sum of squarefor Error. Let { e. } be an orthonormal basis forE lE.
Then Y'P£ Y = t-aM (3.8)
where t is the dimension of the observational
80
A GEOMETRIC LOOK AT REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS
space, n. = n 1 + n 2 + n 3 , e.'Y is N(0, o\ ) .
CovCe.'Y^./Y) = 0 for ifi', Cov(e.'Y,b.'Y) =
0 , and Cov(e/Y,mk 'Y) = 0.
i m E ) (3.9Therefore, Y'PEY is
The test statistics for Hx is
v'p Y I df* ' x where X = NR, NA R .W = "B' " A B '
/ df.
W is distributed as a noncentral F with dfx anddf£ as its degree of freedom and X as itsnoncentrality parameter. When H x is true, X• 0 and thus W is distributed as central F. In thiscase we can find a critical value W such that
Pr( reject Hxl H x true ) = a and
Pr( reject Hxl H x true ) > a .
Computation for sum of square:
Let us define some matrices as follows:
S " [ *1V s21,..., s33 , s43 ]
AB - [ (ab> l l t (ab)13,... ( (ab)22, (ab)23 ] ,
B = [ b r b 2 , b3 ],
E = I ei» C2» •"' e i2 ^J
D - [ SllABllB ] His symbol for concatenation.
The ] I operator produces a new matrix byhorizontally joining two matrices say A and Bwhich must have the same number of rows.
C = [ S||AB ] , and G - [ S||B ] .
SSB = Y'PN YB
= Y P[T0(AB S WS
' Y'PTY _ Y'P ( A B
= Y'D(D'D) ^D'Y - Y'C(C'C)^
= Y;P
= v'p
SSAB = Y ;PM YNAB
p v[T 0 ( B B Wa)] *
Y'PTY - Y 'P ( B a W ^
Y'D(D'D)1D'Y -
or = Y' (I - D(D /D)-1Df) Y
Table for Analysis of Variance
Source df SS MS
A not available
Error A not available
B dim NB SSB MSB=SSB/dfB MSB/MSE
SSE - Y'PEY
AB dim N A B SSAB MSAB=SSAB/dfARMSAB/MSE
Error dim E SSE MSE=SSE/df£
( Note that the degree of freedom B, AB, Error is equalto the dimension of NR, NA R , E respectively. )
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GRAYBILL, FRANKLIN A., (1976): Theory and appli-cation of the Linear Models. New York: Holt,Rhinehart, and Winston, Inc.
GREENHOUSE, S.W., GEISSER, SM (1959): On methodsin the analysis of profile data. Psychometrika 24-2:95-112.
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