รูปแบบต่างๆ ของ Balanced repeated measures (Kleinbaum, 1998)

Analysis of Repeated Measures Data

1. A balanced repeated measures design with one crossover factor.

2. A balanced repeated measures design with two crossover factors.

3. A balanced repeated measures design with one nest factor.

4. A balanced repeated measures design with one crossover factor and one nest factor.

4. A Balanced Repeated Measures Design With One Crossover Factor And One Nest Factor.

 Table 8 Data layout for balanced repeated measures design with one crossover factor and one nest factor.

 

 

Factor B

Factor A

Subject Number

1

2

…

B

1

(1,1)

(2,1)

:

(s,1)

Y111

Y211

:

Ys11

Y112

Y212

:

Ys12

 

Y11b

Y21b

:

YS1b

2

(1,2)

(2,2)

:

(s,2)

Y121

Y221

:

Ys21

Y122

Y222

:

Ys22

 

Y12b

Y22b

:

YS2b

 

:

:

:

…

:

a

(1,a)

(2, a)

:

(s, a)

Y1a1

Y2a1

:

Ysa1

Y1a2

Y2a2

:

Ysa2

 

Y1ab

Y2ab

:

Ysab

 Anova model treating both Factor A with and Factor B fixed factors :

 

            Yijk = m + Si(j) + aj + bk + djk + Ek(ij)                                         (4)

 

Where

            i = 1,…,s          (s = Number of subject observe at each level of Factor A)

            j = 1,…,a          (a = Number of levels of Factor A, with s different subject per level)

            k = 1,…,b       (b = Number of levels of Factor B, with each subject observed at all levels of Factor B )

            m = Overall mean

            aj = Fixed effect of level j of Factor A (and åaj =0)

            bk = Fixed effect of level k of Factor B (and åbk =0)

            djk = Fixed interaction effect of level j Factor A with level k of Factor B

            (and ådjk = 0 for each k, ådjk = 0 for each j)

            SI(j) = Random effect of subject i within level j of Factor A

            Ek(ij) = Random error for kth level of Factor B on subject i within level j of Factor A

And it is assumed that

            { Si(j)} and { Ek(ij} are mutually independent

            Si(j)  is distributed as N(0,sS2)

            Ek(ij is distributed as N(0,s2)  

Figure 4 Partitioning the total sums of squares for balanced repeated measures design with one crossover and one nest factor.

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คลิกขยายภาพ

 

Table 9 ANOVA table for balanced repeated measure design with One Crossover Factor and One nest factor.

Source

d.f.

MS

F (Both Factor Fixed)

F (One or Both Factors Random)

Between Subjects

Sa-1

MSS

 

 

Factor A

a-1

MSA

MSA /MSS(A)

MSA /MSS(A)adj

Subjects (within A)

a(s-1)

MSS(A)

 

 

 

 

 

 

 

Within Subjects

sa(b-1)

MSW

 

 

Factor B

(b-1)

MSB

MSB /MSError

MSB /MSAB

A x B

(a-1)(b-1)

MSAB

MSAB /MSError

MSAB /MSError

Subjects x B (within A)

(i.e.Error)

a(b-1)(s-1)

MSError

 

 

Total (corrected)

sab

 

 

 

        (ตัวอย่างการวิเคราะห์ สามารถศึกษาตัวอย่างการวิเคราะห์ด้วย SPSS ได้ที่ Using SPSS: Two-way Mixed-design ANOVA)

 

 

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