Analysis of variance of global experiments with repeated measurements with practical application

International Journal of Multidisciplinary and Current
Educational Research (IJMCER)
ISSN: 2581-7027 ||Volume|| 2 ||Issue|| 1 ||Pages|| 09-21 ||2020||
9Pagewww.ijmcer.com|1| Issue2Volume|
Analysis of variance of global experiments with repeated
measurements with practical application
Zaineb Salih Hameed , Salam Abdulhussein Sehen, Sara Mahdi Abooud ,
Marwi Adil Mutlag, Yusra Rasim Jabbar
University of AL-Qadisiyah, Iraq
ABSTRACT: The global experiments in the presence of repeated measurements are represented in the case
of three overlapping factors and the third factor is represented by experimental units (Subjects). Repeated
measurements or treatments are taken for experimental units and these treatments are treated as a fourth factor.
This type of experiment has been analyzed by the parameterized methods represented by the F test. If the
conditions for variance analysis are available for repeated measurement experiments, and if the conditions are
not met, we use the non-parametric methods of converting to the ranks.
The purpose of this research is an analytical study of this type of experiment by non-parametric and non-
parametric methods and the application of this experiment to thalassemia in DhiQar
I. INTRODUCTION AND GOAL
Designs of repeated measurements is a method used in designs of experiments that are taken measurements
repeatedly for each experimental unit and there will usually be a link between observations within the
experimental unit. Repetitive measurements designs are used to increase the accuracy of experiments by
omitting or neglecting the differences between experimental units to estimate the effects of treatment and
experimental error. This type of designs is very useful when the experimental units are limited and this type of
analysis in the design of experiments is widespread, especially in psychology and in analytical experiences and
educational research.
In this paper, we assume that we have a nested experience of two factors, that is, the first factor (A) has (b)
levels and the second factor (B) and has (q) levels since the levels of the second factor (B) are nested within the
levels of the first factor (A) And then this relationship is denoted by the symbol (A) B and the second factor (B)
is called the Nested Factor and the first factor (A) by the factor Factor Nest then the experimental units are
taken for each level of the interfering factor (B) where these experimental units are considered as a third factor
(C) ) It has (n) levels, and the levels of the third factor are intertwined within the levels of both the first (A) and
the second (B) levels. (AB) It is called the three-stage cross-design, and then takes the responses for each
experimental unit with different time periods and this process is known as the repeated measurements where the
responses of the same experimental unit are combined and these repeated measurements will be considered as a
fourth factor (D) and has (r) from the levels where the levels of the fourth factor Intersecting with the levels of
the first factor (A) as well as intersecting with the levels of the second factor (B) within the levels of the first
factor (A) as well as intersecting with the levels of the third factor (C) within the levels of both the first factors
(A) and the second (B) therefore we will have an experiment Interfering operation with repeated measurements.
As shown in the following figure, where X means an intersection relationship and o means an interference
relationship

Analysis of variance of global experiments with repeated measurements with practical application
(‫.1الشكل‬ ‫متكررة‬ ‫قياسات‬ ‫بوجود‬ ‫متداخلة‬ ‫عامليه‬ ‫لتجربة‬ ‫مخطط‬ )
A
B
C
D
Factor D
Factor CFactor BFactor A
𝐝 𝟏 𝐝 𝟐 . . . 𝐝 𝐫
Y1111Y1112 . . . Y111r
Y1121Y1122 . . . Y112r
.. .
. . .
. . .
Y11n11Y11n12 . . . Y11n1r
1
2
.
.
.
𝐧 𝟏
𝐛 𝟏
𝐚 𝟏
Y1211Y1212. . . Y121r
Y1221Y1222. . . Y122r
. . .
. . .
. . .
Y12n21Y12n22. . . Y12n2r
1
2
.
.
.
𝐧 𝟐
𝐛 𝟐
Y1q′11Y1q′12 . . . Y1q′1r
Y1q′21Y1q′22 . . . Y1q′2r
. . .
. . .
.. .
Y1q′n
bq′1Y1q′n
bq′2 . . . Y1q′n
bq′r
1
2
.
.
.
𝐧 𝐛𝐪′
𝐛 𝐪′
Yp1′11Yp1′12 . . . Yp1′1r
Yp1′21Yp1′22 . . . Yp1′2r
. ..
. . .
. . .
Yp1′n
b1′1Yp1′n
b1′ 2 . . . Yp1′n
b1′r
1
2
.
.
.
𝐧 𝐛𝟏′
𝐛 𝟏′
𝐚 𝐩
Yp2′11Yp2′12. . . Yp2′1r
Yp2′21Yp2′22. . . Yp2′2r
. . .
. . .
. . .
Yp2′ 𝐧
𝐛𝟐′ 1Yp2′ 𝐧
𝐛𝟐′ 2. . . Yp2′ 𝐧
𝐛𝟐′ r
1
2
.
.
.
𝐧 𝐛𝟐′
𝐛 𝟐′
x

Table (1) is a chart of two experiments with repeated measurements
The aim of this research is an analytical study of interferential globalization experiments with the presence of
repeated measurements with the parameter methods represented by the F test as well as the nonparametric
methods by converting data to the ranks.
the theoretical side
This topic deals with the mathematical model and the variance analysis test (F) for this experiment
Mathematical model
A mixed linear model for this experiment can be written as follows:
Y_ijkL = μ + A_i + B_j (i) + C_ (k (ij)) + D_L + AD_ (iL) + DB_Lj (i) + 〖E〗 _kL (ij) ... (1)
i = 1,2,. . . , p; j = 1,2,. . ., q;
k = 1,2,. . ., n; L = 1,2,. . ., r;
whereas:
Y_ijkL: the value of the observation below the level (i) of the first factor (A), the level (j) of the second factor
(B) interfering within the level (i) of the first factor (A) and the level (k) of the third factor (C) interfering
within The levels (j, i) of the first factors (A) and the second (B), respectively, and the level (L) of the fourth
factor (D.)
:‫تمثل‬ represents the average mean effect and is constant and unknown.
A_i: Effect of Level (i) from Factor One (A.)
B_ (j (i)): effect of the level (j) of the second factor (B) interfering within the level (i) of the first factor (A.)
C_k (ij): effect of the level (k) of the third factor (C) intertwined within both levels (j, i) of the first factors (A)
and the second B)), respectively, which is a random variable
That is to say :-
〖C〗 _ (k (ij)) ~ N (0, σ_c ^ 2)
:D_L effect of level (L) of the fourth factor (D) representing repeated measurements.
AD_iL: effect of the interaction between level (i) of first factor (A) and level (L) of fourth factor (D.)
DB_ (Lj (i)): effect of the interaction between level (L) of the fourth factor D)) and the level (j) of the second
factor (B) interfering within the level (i) of the first factor (A.)
〖E〗 _ (kL (ij)): the effect of a random error resulting from the effect of the interaction between the level (L) of
the fourth factor (D) and the level (k) of the third factor C)) interfering under both levels (j, i) of the factor First
(A) and second factor (B)
ANOVA Table
The methods or mathematical formulas appropriate for calculating the sums of squares are obtained as follows:
SST
= ∑ ∑ ∑ ∑(YijkL – Y̅….)
2
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑ ∑(YijkL – Y̅ijk. + Y̅ijk. – Y̅….)
2
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ijk. − Y̅….)
2
n
k=1
q
j=1
p
i=1
+ 2 ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)( Y̅ijk. − Y̅….)
r
L=1
n
k=1
q
j=1
p
i=1
. . .
. . .
. . .
.
.
.
.
.
.
Ypq11Ypq12 . . . Ypq1r
Ypq21Ypq22 . . . Ypq2r
. . .
. . .
. . .
Ypqnbq1Ypqnbq2 . . . Ypqnbqr
1
2
.
.
.
𝐧 𝐛𝐪
𝐛 𝐪

= ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ijk. − Y̅….)
2
n
k=1
q
j=1
p
i=1
+ 2( 0 )
= SSWithin + SSBetween
‫أن‬ ‫بسبب‬ ‫وذلك‬
∑ ∑ ∑ ∑(YijkL − Y̅ijk.)( Y̅ijk. − Y̅….)
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑( Y̅ijk. − Y̅….)[∑(YijkL − Y̅ijk.)
r
L=1
]
n
k=1
q
j=1
p
i=1
= 0
SSW = ∑ ∑ ∑ ∑(YijkL − Y̅ijk.)
2
r
L=1
n
k=1
q
j=1
p
i=1
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk. − Y̅…. + Y̅…. − Y̅i..L + Y̅i..L − Y̅ij.L + Y̅ij.L − Y̅ij.. + Y̅ij.. − Y̅…L
r
L=1
n
k=1
q
j=1
p
i=1
+ Y̅…L − Y̅i… + Y̅i…)
2
= ∑ ∑ ∑ ∑(YijkL − Y̅ijk. − Y̅ij.L + Y̅ij..)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ ∑ ∑ ∑ ∑(Y̅ij.L − Y̅ij.. − Y̅i..L + Y̅i…)
2
r
L=1
n
k=1
q
j=1
p
i=1
+ ∑ ∑ ∑ ∑(Y̅i..L − Y̅i… − Y̅…L + Y̅….)2
r
L=1
n
k=1
q
j=1
p
i=1
+ ∑ ∑ ∑ ∑( Y̅…L − Y̅….)2
r
L=1
n
k=1
q
j=1
p
i=1
= SSD + SSAD + SSDB(A) + SSDC(AB)
SSB = r ∑ ∑ ∑( Y̅ijk. − Y̅….)
2
n
k=1
q
j=1
p
i=1
= r ∑ ∑ ∑( Y̅ijk. − Y̅…. − Y̅i… + Y̅i… − Y̅ij.. + Y̅ij..)
2
n
k=1
q
j=1
p
i=1
= r ∑ ∑ ∑( Y̅i… − Y̅….)2
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ij.. − Y̅i…)
2
n
k=1
q
j=1
p
i=1
+ r ∑ ∑ ∑( Y̅ijk. − Y̅ij..)
2
n
k=1
q
j=1
p
i=1
+ 2 ( 0 ) = SSA + SSB(A) + SSC(AB)
[1] =
Y….
2
pqnr
= C. F [2] = ∑ ∑ ∑ ∑ YijkL
2
r
L=1
n
k=1
q
j=1
p
i=1
[3] =
∑ Yi…
2p
i=1
nqr
[4] =
∑ Y.j..
2q
j=1
npr
[5] =
∑ Y..k.
2n
k=1
pqr
[6] =
∑ Y...L
2r
L=1
npq
[7] =
∑ ∑ Yij..
2q
j=1
p
i=1
nr
[8] =
∑ ∑ Yi.k.
2n
k=1
p
i=1
qr
[9] =
∑ ∑ Yi..L
2r
L=1
p
i=1
nq
[10] =
∑ ∑ Y.jk.
2n
k=1
q
j=1
pr
[11] =
∑ ∑ Y.j.L
2r
L=1
q
j=1
np
[12] =
∑ ∑ Y..kL
2r
L=1
n
k=1
pq
[13] =
∑ ∑ ∑ Yijk.
2n
k=1
q
j=1
p
i=1
r
[14] =
∑ ∑ ∑ Yij.L
2r
L=1
q
j=1
p
i=1
n
[15] =
∑ ∑ ∑ Y.jkL
2r
L=1
n
k=1
q
j=1
p
[16] =
∑ ∑ ∑ Yi.kL
2r
L=1
n
k=1
p
i=1
q
The table of variance analysis for the following design is shown in the table below:
The sum of the total squares will be:
SST = 2− 1 = SSB + SSW… (2)

Sum of squares between experimental units:
SSB = 13− 1 = SSA + SSB (A) + SSCAB… (3)
Sum of squares inside experimental units:
SSW = 2− 13 = SSD + SSAD + SSDB (A) + SSDC (AB) ... (4)
The sum of the squares of the first factor A will be:
SSA = 3− 1… (5)
The sum of squares of the third factor (experimental units) that are nested within the first factor (A) and the
second factor (B:)
SSC (AB) = 13− 7… (6)
The sum of the squares of the second factor (B) that overlap within the first factor (A) will be:
SSB (A) = 7− 3… (7)
The sum of the squares of the fourth factor (repeated measurements) (D) will be:
SSD = 6− 1… 8
The sum of the squares of the interaction between the first factor (A) and the fourth factor (D) is:
SSAD = 9− 3− 6+ 1… (9)
The sum of the squares of the interaction between the fourth factor (D) and the second factor (B) in the first
factor (A)
SSDB (A) = 14− 7−9 + 3… (10)
The sum of the squares of the error is:
SSError = SSDC (AB) = 2−13−14 + 7… (11)
As these interconnected global experiences are considered to be three balanced stages, whereas balanced
interrelated trials mean the number of levels of interfering factor equal within each level of interfering factor,
meaning that the levels of interfering factor (B) are equal within each level of interfering factor (A) as well as
factor levels The overlap (C) is equal within each level of the interference factor (B.)
Likewise:
𝟏 − SSB(A) = SSB + SSAB … (12)
∗ SSB =
∑ Y.j..
2q
j
pnr
−
Y….
2
pqnr
= [4] − [1]
∗ SSAB =
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
+
Y….
2
pqnr
= [7] − [3] − [4] + [1]
∗ SSB(A) =
∑ Y.j..
2q
j
pnr
–
Y….
2
pqnr
+
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
+
Y….
2
pqnr
= [4] − [1] + [7] − [3] − [4] + [1]
=
∑ ∑ Yij..
2q
j
p
i
nr
–
∑ Yi…
2p
i
qnr
= [7] − [3]
𝟐 − SSC(AB) = SSC + SSAC + SSBC + SSABC … (13)
∗ SSC =
∑ Y..k.
2n
k
pqr
−
Y….
2
pqnr
= [5] − [1]
∗ SSAC =
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ Yi…
2p
i
qnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
= [8] − [3] − [5] + [1]
∗ SSBC =
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
= [10] − [4] − [5] + [1]
∗ SSABC =
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
−
∑ ∑ Yij..
2q
j
p
i
nr
–
∑ ∑ Yi.k.
2n
k
p
i
qr
–
∑ ∑ Y.jk.
2n
k
q
j
pr
–
Y….
2
pqnr
= [13] + [3] + [4] + [5] − [7] − [8] − [10] − [1]

∴ SSC(AB) =
∑ Y..k.
2n
k
pqr
−
Y….
2
pqnr
+
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ Yi…
2p
i
qnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
+
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
+
Y….
2
pqnr
+
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
−
∑ ∑ Yij..
2q
j
p
i
nr
–
∑ ∑ Yi.k.
2n
k
p
i
qr
–
∑ ∑ Y.jk.
2n
k
q
j
pr
–
Y….
2
pqnr
= [5] − [1] + [8] − [3] − [5] + [1] + [10] − [4] − [5] + [1] + [13] + [3] + [4]
+ [5] − [7] − [8] − [10] − [1]
=
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ Yij..
2q
j
p
i
nr
= [13] − [7]
𝟑 − SSDB(A) = SSDB + SSDBA … (14)
∗ SSDB =
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ Y.j..
2q
j
pnr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
= [11] − [4] – [6] + [1]
∗ SSDBA =
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y…L
2r
L
npq
−
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y.j.L
2r
L
q
j
np
–
Y….
2
pqnr
= [14] + [3] + [4] + [6] − [7] − [9] − [11] − [1]
∴ SSDB(A) =
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ Y.j..
2q
j
pnr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
+
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
+
∑ Yi…
2p
i
qnr
+
∑ Y.j..
2q
j
pnr
+
∑ Y...L
2r
L
npq
−
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y.j.L
2r
L
q
j
np
−
Y….
2
pqnr
= [11] − [4] − [6] + [1] + [14] + [3] + [4] + [6] − [7] − [9] − [11] − [1]
=
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
+
∑ Yi…
2p
i
qnr
−
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ Yi..L
2r
L
p
i
nq
= [14] + [3] − [7] − [9]
𝟒 − SSDC(AB) = SSDC + SSACD + SSBCD + SSABCD … (15)
∗ SSDC =
∑ ∑ Y..kL
2r
L
n
k
pq
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
= [12] − [5] − [6] + [1]
∗ SSACD =
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
+
∑ Yi…
2p
i
qnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
−
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
= [16] + [3] + [5] + [6] − [8] − [9] − [12] − [1]
∗ SSBCD =
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
–
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
= [15] + [4] + [5] + [6] − [10] − [11] − [12] − [1]
∗ SSABCD = ∑ ∑ ∑ ∑ YijkL
2
−
r
L
n
k
q
j
p
i
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
−
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
−
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
∑ ∑ Yij..
2q
j
p
i
nr
+
∑ ∑ Yi.k.
2n
k
p
i
qr
+
∑ ∑ Yi..L
2r
L
p
i
nq
+
∑ ∑ Y.jk.
2n
k
q
j
pr
+
∑ ∑ Y.j.L
2r
L
q
j
np
+
∑ ∑ Y..kL
2r
L
n
k
pq
+
Y….
2
pqnr
= [2] − [13] − [14] − [16] − [15] − [3] − [4] – [5] − [6] + [7] + [8] + [9] + [10]
+ [11] + [12] + [1]

∴ SSDC(AB) =
∑ ∑ Y..kL
2r
L
n
k
pq
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
Y….
2
pqnr
+
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
+
∑ Yi…
2p
i
qnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
−
∑ ∑ Yi.k.
2n
k
p
i
qr
−
∑ ∑ Yi..L
2r
L
p
i
nq
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
+
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
+
∑ Y.j..
2q
j
pnr
+
∑ Y..k.
2n
k
pqr
+
∑ Y...L
2r
L
npq
–
∑ ∑ Y.jk.
2n
k
q
j
pr
−
∑ ∑ Y.j.L
2r
L
q
j
np
−
∑ ∑ Y..kL
2r
L
n
k
pq
−
Y….
2
pqnr
+ ∑ ∑ ∑ ∑ YijkL
2
−
r
L
n
k
q
j
p
i
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
−
∑ ∑ ∑ Yi.kL
2r
L
n
k
p
i
q
−
∑ ∑ ∑ Y.jkL
2r
L
n
k
q
j
p
−
∑ Yi…
2p
i
qnr
−
∑ Y.j..
2q
j
pnr
−
∑ Y..k.
2n
k
pqr
−
∑ Y...L
2r
L
npq
+
∑ ∑ Yij..
2q
j
p
i
nr
+
∑ ∑ Yi.k.
2n
k
p
i
qr
+
∑ ∑ Yi..L
2r
L
p
i
nq
+
∑ ∑ Y.jk.
2n
k
q
j
pr
+
∑ ∑ Y.j.L
2r
L
q
j
np
+
∑ ∑ Y..kL
2r
L
n
k
pq
+
Y….
2
pqnr
= [12] − [5] − [6] + [1] + [16] + [3] + [5] + [6] − [8] − [9] − [12] − [1] + [15]
+ [4] + [5] + [6] − [10] − [11] − [12] – [1] + [2] − [13] − [14] − [16] − [15]
− [3] − [4] – [5] − [6] + [7] + [8] + [9] + [10] + [11] + [12] + [1]
= ∑ ∑ ∑ ∑ YijkL
2
r
L
n
k
q
j
p
i
+
∑ ∑ Yij..
2q
j
p
i
nr
−
∑ ∑ ∑ Yijk.
2n
k
q
j
p
i
r
−
∑ ∑ ∑ Yij.L
2r
L
q
j
p
i
n
= [2] + [7] − [13] − [14]
S .O . V dF S .S M . S E M S F
Between(sub.
)
A
B(A)
Error(Bet.)
= C(AB)
Within(sub.)
D
AD
DB(A)
Error(Within
)
= C(AB
pq(i)n(ij) − 1
p − 1
P(q(i)– 1)
q(i)(n(ij) − 1)
pq(i)n(ij)(r−1)
r−1
(p−1)(r − 1)
P(r−1)(q(i) −
1)
pq(i)(r −
1)(nij − 1)
SSB
SSA
SSB(A)
SSE(B)
SSW
SSD
SSAD
SSDB(A)
SSError
SSA
P − 1
SSB(A)
P(q(i)– 1)
SSE(B)
pq(i)(n(ij) − 1)
SSD
r − 1
SSAD
(p − 1)(r − 1)
SSDB(A)
P(r − 1)(q(i) − 1)
SSE(W)
pq(i)(r − 1)(n(ij) − 1)
rσc
2
+
qnr∅A
rσc
2
+ nr∅B
rσc
2
σDC
2
+ pqn∅D
σDC
2
+ qn∅AD
σDC
2
+ n∅DB
MSA
MSE(B)
MSB(A)
MSE(B)
MSD
MSE(W)
MSAD
MSE(W)
MSDB(A)
MSE(W)
Total pq(i)n(ij)r−1 SST

Table (2) shows ANOVA for its experiments with the presence of repeated measurementsThis analysis is used
to test the hypothesis:
H_0 = T_1 = T_2 = ... = T_r ... (16)
Analysis of variance for repeated measurement experiments requires the following conditions to be met:
The main influences are aggregate: It means that the factors influence add to each other to determine the values
of observations
The random and independent random distribution of the experimental error: This condition assumes that the
errors are randomly and independently distributed with an average of zero and its value varies (σ ^ 2), meaning
that:
e_ij ~ NID (0, σ ^ 2) ... (17)
Homogeneity of variance: This condition means that random differences are equal within groups homogeneous
and therefore random differences are equal for different samples, which helps to obtain a reduced variance of all
groups.
There is no correlation between the mean and the contrast
Sphericality: This condition means the order of the experimental units does not change in the results of the
experiment as well as the association of any two treatments is the same for each pair of treatments and the
spherical condition indicates that there is no interaction between the factor of the experimental units and the
treatments (repeated measurements).
Analytical methods for this experiment
There are several statistical methods for analyzing the designs of repeated measurements, including parameter
and non-parametric methods and multivariate methods and each of the mentioned methods is implemented
according to certain conditions. In this research we will use the parameter methods represented by testing F and
nonparametric methods by means of Rank Transformation.
The practical side:
This study was applied to data collected from Al-Hussein Hospital _ Thalassemia Center in DhiQar from
patients with beta anemia of the Mediterranean type of beta or the so-called Thalassemia major, and the number
of observations (160) was represented by two groups (two medications) and each group included (20) Patient
(10 males, 10 females) and given treatment at four equal time periods each 30-day time period
Figure (2) Box-and-Whisker Plot plot of Thalassemia data, where no abnormal values are shown
The hypothesis that errors are distributed naturally is tested with an average (0) and a variance (σ2) that are:
H_0 = ε_ij ~ N (0, σ_ε ^ 2). . . (18)
H_1 = ε_ij≁ N (0, σ_ε ^ 2). . . (19)
First, we find the error values (ε_ij) according to the linear model of the design:
Y_ijkL = μ + A_i + B_j (i) + C_ (k (ij)) + D_L + AD_iL + DB_Lj (i) + 〖E〗 _kL (ij) ... (20)
ε_ (Lk (ij)) = Y_ijkL- μ_ (ijk.) - μ_ (ij.L) + μ_ (ij ..). . . (21)
σDC
2
P-Value
Test ValueTest
Box-and-Whisker Plot
0 2 4 6 8
(X 1000)
Col_5

Table (5) Natural Distribution Test.
Since the p-value of the Chi-Squared test is greater than α = 0.01, the null hypothesis is accepted, meaning that
errors are naturally distributed with mean (0) and variance (σ2.)
2- Test of Homogeneity of Variances
The homogeneity test for treatments is the second condition of the analysis of variance. This condition can be
formulated as follows:
H_0 ∶σ_1 ^ 2 = σ_2 ^ 2 = σ_3 ^ 2 = σ_4 ^ 2 V.S H_1 leastat least two (σ ^ 2) not equal. . . (22)
Using the Bartlett and Cochrane tests to check this condition as:
P-Value
Test valuetest
0.6888
0.50551
1.00567
0.30676
Bartlett's test
Cochran's test
Table (6): Bartlett and Cochran tests for homogeneity of variations
From the above table it appears that Bartlett and Cochran values where the value of p-value is greater than α =
0.01 This leads to acceptance of the null hypothesis as the variations are homogeneous.
3- Test of Correlation
The third condition of the variance analysis is the absence of an association between the mean and the variance
of the treatments
H_0: ρ = 0 VS H_1: ≠ ≠ 0… (23)
It was found that the p-value of the Pearson correlation coefficient is equal to (0.532) which is greater than α =
0.01, thus accepting the null hypothesis that there is no correlation.
Test of Sphericity - Spherical Test 4
To test the fourth condition of the variance analysis conditions for repeated measurements experiments is
H_0 ∶Σ is sphericity. . . (24)
H_1 ∶Σ is not sphericity. . . (25)
To test this hypothesis, we use:
Mauchlys test
This test is based on the Eigen value of the contrast and contrast matrix common to repeated measurements and
the test statistics are:
W = (∏▒λ_i) / [1 / (A-1) ∑▒λ_i] ^ (A-1) = 0.7987236078
χ _ ((w)) ^ 2 = - (1-f) (s-1) ln⁡ 〖( w)〗 = 18.465218557
f = (2 〖(A-1)〗 ^ 2 + A + 2) / (6 (A-1) (S-1)) = 0.03418803419
Tabular value χ _ ((α, v)) ^ 2 where:
V = A (A-1) = 6; α = 0.01
〖Χ〗 _table ^ 2 = χ _ ((α, v)) ^ 2 = χ _ ((0.01,6)) ^ 2 = 16.81
Since the calculated value of χ2 is greater than the tabular value of ‫لذلك‬2, it therefore rejects the null hypothesis
that the contrast and contrast matrix is spherical. Since the spherical condition is not fulfilled, the F test
becomes inaccurate and therefore the degrees of freedom must be adjusted for the F test as follows:
Green house - Geisser Correction - A, which is a value calculated using double Central, and this value is
denoted by (ε) ̂ and is calculated as follows:
ε ̂ = (∑_a▒S_ (a, a)) ^ 2 / ((A-1) ∑_ (a, a ') _S_ (a, a') ^ 2) = 0.888081355
Huynh - Feldt Correction -B
It is the most used formula for modifying degrees of freedom Test F because it is more powerful and it is based
or dependent on the value calculated from Green house - Geisser and symbolized by the symbol (ε ̃:)
ε ̃ = (S (A-1) ε ̂-2) / (A-1) [S-1- (A-1) ε ̂] = 0.9592916574
0.0229
0.1297
0.8111
0.0181
42.2750
0.9743
0.2389
2.3624
Chi-Squared
Shapiro –Wilk W
Skewness Z-Score
Kurtosis Z-Score

We now find a table of variance analysis for the data as follows:
The number of levels of factor I A (pharmacokinetics) p = 2
The number of levels of the second factor B (gender) q = 2
The number of levels of the third factor D (times) r = 4
The number of levels of the fourth factor C (experimental units) n = 10
S .O . V dF S .S M . S FC Ftable
Between(su
b.)
A
B(A)
Error(Bet.)
= C(AB)
Within(sub.
)
D
AD
DB(A)
Error(With
in)
= C(AB)
39
1
2
36
120
3
3
6
108
681492955
161885522.9
113997336.1
405610096
98061199
19081539
3122776
7601402
68255482
161885522.9
56998668.05
11266947.11
6360513
1040925.333
1266900.333
631995.203
14.368
5.058
10.064
1.647
2.005
F(0.01,1,36) = 7.31
F(0.01,2,36) = 5.18
F(0.01,3,104) = 3.95
F(0.01,3,104) = 3.95
F(0.01,6,104) = 2.96
Total 159 779554154
Table (7) Table of analysis of variance of the global experiment in the presence of repeated
measurementsThrough the results of the ANOVA table, we note that:
There are significant differences in levels between factor A (pharmacists.)
There were no significant differences between levels of factor II B (gender) within factor I (pharmacists.)
There are significant differences between factor D levels (repeated measurements.)
There were no significant differences for interaction between factor A (pharmacokinetics) and factor III D
(repeated measurements.)
-There were no significant differences for interaction between factor II B (gender) and factor IIID (repeated
measurements) within factor IA (pharmacokinetics.)
We are now taking the data rank, finding the table of variance analysis and studying the conditions of variance
analysis for repeated measurements experiments.
5- Test of Normality for errors
To test the hypothesis that errors are naturally distributed (18) using the Chi-Squared test after taking error
estimates(21)
P-ValueTest valuetest
0.20109
0.37615
0.96351
0.00037
31.7625
0.9797
0.0457
3.5566
Chi-Squared
Shapiro –Wilk W
Skewness Z-Score
Kurtosis Z-Score
Table (8) testing the normal distribution of data types.
Since the p-value of the Chi-Squared test is greater than α = 0.01, the null hypothesis is accepted, meaning that
errors are naturally distributed with mean (0) and variance (σ2.)
6- Test of Homogeneity of Variances
To test this condition of the conditions of analysis of variance, which is homogeneity of variations (22), by
using the Bartlet test and Cochran test we conclude:

P-Value
Test value
Test
0.93525
1.0
1.00275
0.26725
Bartlett's test
Cochran's test
Table (9): Bartlett and Cochran tests for homogenization of variations of data types.From the above table it
appears that Bartlett and Cochran values where the value of p-value is greater than α = 0.01 This leads to
acceptance of the null hypothesis as the variations are homogeneous.
7- Test of Correlation
To test the third condition of the conditions of the analysis of variance, which is the absence of a correlation
between the arithmetic mean and the variance as in the hypotheses (23) through the value of p-value of the
Pearson correlation coefficient where it is equal to (0.588) which is greater than 0.01 = α, therefore the null
hypothesis is not rejected, that is, there is no Correlation between mean and variance.
Test of Sphericity Spherical Test - 8
To test the spherical hypothesis (24), which is one of the conditions for analyzing variance for experiments,
repeat measurements:
Mauchlys test
This test is based on the roots of the Eigen value of the coefficient of variance and co-contrast of the treatments
and the test statistic:
W = (∏▒λ_i) / [1 / (A-1) ∑▒λ_i] ^ (A-1) = 0.7399671452
χ _ ((w)) ^ 2 = - (1-f) (s-1) ln⁡ 〖( w)〗 = 11.34329754
f = (2 〖(A-1)〗 ^ 2 + A + 2) / (6 (A-1) (S-1)) = 0.03418803419
The test statistic is compared to the tabular value of χ2 with the degree of freedom v where:
V = A (A-1) = 6 and α = 0.01
〖Χ〗 _table ^ 2 = χ _ ((α, v)) ^ 2 = χ _ ((0.01,6)) ^ 2 = 16.81
Since the computed value of χ2 is less than the tabular value of ‫لذلك‬2, the null hypothesis that the contrast and
contrast matrix is common is spherical. Therefore, the spherical condition has been met, and the F test becomes
accurate.
After checking the conditions of variance analysis, we find a table of variance analysis for the data types for
comparison with the table of variance analysis for the original data as:
The number of levels of factor I A (pharmacokinetics) p = 2
The number of levels of the second factor B (gender) q = 2
The number of levels of the third factor D (times) r = 4
The number of levels of the fourth factor C (experimental units) n = 10
S .O . V dF S .S M . S FC Ftable
Between(sub.
)
A
B(A)
Error(Bet.)
= C(AB)
Within(sub.)
D
AD
DB(A)
Error(Withi
n)
= C(AB)
39
1
2
36
120
3
3
6
108
297801.375
62805.625
53354.925
181640.825
43517.125
10087.7125
1018.587
2901
29509.825
62805.625
26677.4625
5045.57847
3362.5708
339.529
483.5
273.23912
12.447
5.287
12.306
1.242
1.769
F(0.01,1,36) = 7.31
F(0.01,2,36) = 5.18
F(0.01,3,108) = 3.95
F(0.01,3,108) = 3.95
F(0.01,6,108) = 2.96
Total 159 341318.5
Table (10): Table of variance analysis for the data typesThrough the results of the ANOVA table, we note that:
There are significant differences in levels between factor A (pharmacists).

There are significant differences between levels of factor II B (gender) within factor I (pharmacists).
There are significant differences between factor D levels (repeated measurements).
There were no significant differences for interaction between factor A (pharmacokinetics) and factor III D
(repeated measurements).
There were no significant differences for interaction between factor B (sex) and factor D (repeated
measurements) within factor I (pharmacokinetics).
• Conclusions
1- It was found that converting data from parameterized to non-parametric methods by grade resulted in
providing conditions for variance analysis for repeated measurements experiments such as normal distribution
of errors, homogeneity of variations, correlation between mean and variance, as well as the spherical condition.
2- Where it was observed that the conditions for normal distribution and homogeneity of variations, as well as
the condition for the absence of correlation between the mean and variance, were achieved and after a transfer
to the ranks, it improved the value of value - P for the condition of normal distribution from (0.0229) to
(0.2210) as well as for homogeneity of variance for Bartlett's test ( 0.688) to (0.935) and likewise for the
Cochran test, it changed from (0.505) to (1.0), and the correlation condition was a value equal to (0.532) and
changed to (0.588). As for the spherical condition, unfulfilled by the value of the test (18.465), which led to
adjusting the degrees of freedom for the F test From (3,108) to (3,104), after a conversion to the data levels, the
spherical condition was met and the test value was equal to (11.343). As for the results of the F test, it may
change. It ranges for the first factor (A) from (14.368) to (12.447), for the second factor (B) from (5.058) to
(5.287) and the fourth factor (D) from (10.069) to (12.306) and for the interaction between the first factor and
the fourth factor (AD From (1.647) to (1.212), and also with regard to the interaction between the second factor
and the fourth factor (DB), it has changed from (2.05) to (1.769).
3- In the practical application of overlapping factor experiments with the presence of repeated measurements
and there were no alternative methods for unparalleled parameters such as Friedman or others so the method we
used was the alternative method for these cases.
4- Among the medical conclusions, the majority of patients with blood types (A +) and (B +), as well as the
majority of patients from his parents who carry the disease and are relatives and it was also observed with poor
or uneducated academic achievement for one of the spouses (housewives, earners) and it was found that the
majority of patients have His residence is rural.
5- Through the four applications, we see that giving the dose of the first drug to patients led to an increase in the
amounts of iron for patients over time, and this type of treatment is used when the amounts of iron are high.
Likewise, when administering the dose of the second medication to patients, it led to increased amounts of iron
for patients over time, but with lower proportions than the first treatment, and this type of treatment is used
when the iron is weak in the patient’s blood.
• Recommendation
Through this research, we reached the following recommendations that they should be taken into consideration
and serve the scientific and human aspect and preserve the life of the individual, we recommend the following:
1. Applying the conditions of variance analysis for repeated measurements in the case of three or more factors,
and in their different situations and experiments.
2. Application of unbalanced overlapping globalization experiments with repeated measurements. Or missing
values.
3. Transferring data to non-parametric methods by converting ranks as one of the solutions in the absence of
conditions for variance analysis for repeated measurements experiments.
4. The use of multivariate methods for analyzing experiments, repeated measurements, and clarifying the
conditions for this type of analysis.
5. Applying repeated measurements in the case of medical data of all kinds in the service of the scientific
process, including psychological data and time-dependent psychology.
6. Giving the dose to patients resulted in increasing the amounts of iron for patients over time, so it is necessary
to carry out a study on different drugs or doses in order to lead to reducing the amounts of iron as well as
reducing complications for patients.
REFERENCES
[1] Amy Minke (1997): “Conducting Repeated Measures Analyses: Experimental Design Considerations“.
Texas A&M University.
[2] Brunner.E ,Puri.M.L and Munzel.U (1999) ‘‘Rank – Score Test in Factorial Designs with Repeated
Measures ‘‘ . Journal of Multivariate Analysis 70, p.p 286 – 317.
[3] Bryan. J.J (2009) ‘‘ Rank Transforms and Test of Interaction for Repeated Measures Experiments with
Various Covariance Structures ‘‘Oklahoma State University.

[4] Carriere .K.C and Reinsel.G.C. (1993):”Optimal Tow-Period Repeated Measurement Designs with two
or more Treatments”. Biometrika,Vol.80 , No.4,P.P 924-929.
[5] Carricre.K.C (1994):”In Complete Repeated Measures Data Analysis in the Pressence of Treatment
Effects ”. JASA ,Vol. 89, No.426, P.P 680-686.
[6] Hager .W (2007) “ Some Common features and Some differences between the Parametric ANOVA for
Repeated Measures and the Friedman ANOVA for Ranked data ”. Psychology Science , Vol. 49 P.P
209-222.
[7] Lin, P.Y. and Ying, Z., (2001):‘‘Semi parametric and Non parametric Regression Analysis of
Longitudinal Data ‘‘. JASA,Vol. 96 , No.453,
[8] Potvin, C.; Lechowicz, M. J. and Tardif, S., (1990): ‘‘The Statistical Analysis of Ecophysidagical
Response Curves Obtained from Experiments Involving Repeated Measures ‘‘.Ecological Society of
America. Vol. 71, No. 4, PP. 1389-1400.
[9] Shaffer, J. P., (1981): ‘‘The Analysis of Variance Mixed Model with Allocated Observations:
Application to Repeated Measurement Design‘‘. JASA, Vol. 76, No. 375, PP. 607-611.
[10] Shukla, G, and Kumar, V. (2012): ‘‘Different Methods of Analyzing Multiple Samples Repeated
Measures Data’’. Journal of Reliability and Statistical Studies, Vol. 5, PP. 83-93.
[11] Weinberg, J. M. and Lagakos, S. W., (2001): “Efficiency Comparison of rank and Permutation test
based on Summary Statistics Computed from Repeated Measures data”. Statist. 20, PP. 705-731.

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