Unbalanced 2-Factor Studies
KNNL – Chapter 23
Unequal Sample Sizes
• When sample sizes are unequal, calculations and
parameter interpretations (especially marginal ones)
become messier
• Observational studies often have unequal sample sizes
due to availability of sampling units for certain
combinations of factor levels (villagers of certain types in
a rural study for instance)
• Experimental studies, even when planned with equal
sample sizes can end up unbalanced through technical
problems or “drop outs”
• Some conditions may be cheaper to measure than
others, and will have larger sample sizes
• Some situations have particular contrasts of higher
importance
Regression Approach - I
Sample Sizes: # of Cases when Factor A is at level i, B @ j: nij
b
ni   nij
j 1
a
n j   nij
i 1
a
nij
b
nT   nij
Yij    Yijk
i 1 j 1
k 1
a
b
a
b
i 1
j 1
i 1
j 1
Restrictions on Effects:   i    j    ij    ij  0
  a  1   2  ...   a 1


nij
 ijk ~ N  0,  2  (independent)
Model: Yijk     i   j   ij   ijk
 b   1   2  ...   b 1
Y ij  
Yij 
 ib    i1   i 2  ...   i ,b 1
 aj    1 j   2 j  ...   a 1 j
Regression Approach - II
Regression Model:
Yijk    1 X ijk 1  ...   a 1 X ijk ,a 1  1 X ijka  ...  b 1 X ijk ,a b  2 
 11 X ijk1 X ijka  ...   a 1,b1 X ijk ,a 1 X ijk ,a b 2   ijk
 1 if case from level 1 of factor A

where: X ijk 1   1 if case from level a of factor A
0
otherwise

1 if case from level a-1 of factor A

X ijk ,a 1   1 if case from level a of factor A
0
otherwise

where: X ijka
X ijk ,a b  2
 1 if case from level 1 of factor B

  1 if case from level b of factor B
0
otherwise

1 if case from level b-1 of factor B

  1 if case from level b of factor B
0
otherwise

Regression Approach – Example I
Writer Type (B)
Style (Factor A)
Poets (B1)
Conceptualists (A1)
Eliot
(Finders)
Cummings
Plath
Pound
Wilbur
n & Mean
n11=5
Experimentalists (A2) Bishop
(Seekers)
Moore
Williams
Lowell
Stevens
Frost
n & Mean
n21=6
Year at Peak
23
26
30
30
34
28.60
29
32
40
41
42
48
38.67
Novelists (B2)
Fitzgerald
Hemingway
Melville
Lawrence
Joyce
n12=5
James
Faulkner
Dickens
Woolf
Conrad
Twain
Hardy
n22=7
Year at Peak
29
30
32
35
40
33.20
38
39
41
45
47
50
51
44.43
Yijk    1 X ijk 1  1 X ijk 2   11 X ijk 1 X ijk 2   ijk
 2  1
 2   1
 12   21    11
 22   11
X_ijk1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
X_ijk2
1
1
1
1
1
-1
-1
-1
-1
-1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
X1X2
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
Testing Strategies – Models Fit
1) Model 1: all Factor A, Factor B, and Interaction AB Effects
2) Model 2:all Factor A, Factor B Effects (Remove
Interaction)
3) Model 3: all Factor B,Interaction AB Effects (Remove A)
4) Model 4:all Factor A,Interaction AB Effects (Remove B)
5) To test for Interaction Effects, Model 1 is Full Model,
Model 2 is Reduced dfNumerator=(a-1)(b-1) dfden=nT-ab
6) Testing for Factor A Effects, Full=Model 1,
Reduced=Model 3 dfNumerator=(a-1) dfden=nT-ab
7) Testing for Factor B Effects, Full=Model 1,
Reduced=Model 4 dfNumerator=(b-1) dfden=nT-ab
Regression Approach – Example - Continued
Yijk    1 X ijk 1  1 X ijk 2   11 X ijk 1 X ijk 2   ijk
Model 1: E Yijk     1 X ijk 1  1 X ijk 2   11 X ijk 1 X ijk 2
^
Y  36.22  5.32 X 1  2.59 X 2  0.29 X 1 X 2
Model 2: E Yijk     1 X ijk 1  1 X ijk 2 Y  36.23  5.33 X 1  2.63 X 2
^
Model 3: E Yijk     1 X ijk 2   11 X ijk 1 X ijk 2 Y  36.90  2.77 X 2  0.47 X 1 X 2
^
Model 4: E Yijk     1 X ijk 1   11 X ijk 1 X ijk 2 Y  36.31  5.41X 1  0.62 X 1 X 2
^
ANOVA
Regression
Residual
Total
Model1
df
3
19
22
SS
827.91
557.05
1384.96
Model2
df
2
20
22
SS
826.01
558.95
1384.96
Model3
df
2
20
22
SS
188.77
1196.19
1384.96
Model4
df
2
20
22
SS
676.58
708.37
1384.96
Regression Approach – Example - Continued
H 0 :  11   12   21   22  0
SSE  R   558.95 df E  R   20
H A : Interaction Exists
SSE  F   557.05 df E  F   19
 SSE  R   SSE  F    df E  R   df E  F   
TS : F 

 SSE  F  df E  F  
 558.95  557.05  20  19  
*
 0.065 RR : FAB
 F .95,1,19   4.381
557.05 19
*
AB
ANOVA
Regression
Residual
Total
Model1
df
3
19
22
SS
827.91
557.05
1384.96
Model2
df
2
20
22
SS
826.01
558.95
1384.96
Model3
df
2
20
22
SS
188.77
1196.19
1384.96
Model4
df
2
20
22
SS
676.58
708.37
1384.96
Regression Approach – Example - Continued
H 0 : 1   2  0 H A : Factor A Effects Exist:
SSE  R   1196.19 df E  R   20
1196.19  557.05   20  19  
F 
 21.80 RR : FA*  F .95,1,19   4.381
557.05 19
*
A
H 0 : 1   2  0 H A : Factor B Effects Exist:
SSE  R   708.37 df E  R   20
 708.37  557.05   20  19  
F 
 5.16 RR : FB*  F .95,1,19   4.381
557.05 19
*
B
ANOVA
Regression
Residual
Total
Model1
df
3
19
22
SS
827.91
557.05
1384.96
Model2
df
2
20
22
SS
826.01
558.95
1384.96
Model3
df
2
20
22
SS
188.77
1196.19
1384.96
Model4
df
2
20
22
SS
676.58
708.37
1384.96
Estimating Treatment and Factor Level Means/Contrasts
Treatment Means:
nij
Parameter: ij
^
Estimator:  ij 
Y
 
ijk
k 1
nij
MSE
nij
^
Estimated Standard Error: s  ij 
= Y ij 
Factor A Means:
b
Parameter: i =

j 1
b
ij
^
Estimator:  i 
b
Y
 
ij 
^
j 1
Estimated Standard Error: s  i 
b
MSE b 1

b 2 j 1 nij
Factor B Means:
a
Parameter:  j =
 ij
i 1
a
a
^
Estimator:   j 
Y
 
ij 
^
Estimated Standard Error: s   j 
i 1
a
Contrast or Linear Function of Factor A Means:
a
Parameter: LA   ci i
i 1
a
^
 
^
^
Estimator: L A   ci  i Estimated Standard Error: s L A 
i 1
Contrast or Linear Function of Factor B Means:
b
Parameter: LB   c j  j
j 1
b
^
MSE a 1

a 2 i 1 nij
MSE a 2 b 1
 ci 
b 2 i 1 j 1 nij
 
^
MSE b 2 a 1
cj 
a 2 j 1 i 1 nij
^
Estimator: L B   c j   j Estimated Standard Error: s L B 
j 1
Contrast or Linear Function of Treatment Means:
a
b
Parameter: LAB   cij ij
i 1 j 1
^
a
b
 
^
a
b
Estimator: L AB   cij Y ij  Estimated Standard Error: s L AB  MSE 
i 1 j 1
i 1 j 1
cij2
nij
Standard Error Multipliers
Single Comparisons:
t 1   / 2  ; nT  ab 
General Multiple Comparisons of Treatment (Cell) Means :
Scheffe: S 
Bonferroni:
 ab  1 F 1   ; ab  1, nT  ab 
B  t 1    2 g   , nT  ab 
Tukey (all pairs of treatment means): T 
1
q 1   ; ab, nT  ab 
2
General Multiple Comparisons of Factor Level Means :
 a  1 F 1   ; a  1, nT  ab  Factor B:
Factor A or Factor B : B  t 1    2 g   , nT  ab 
Scheffe: Factor A: S A 
Bonferroni:
Tukey: Factor A: TA 
1
q 1   ; a, nT  ab 
2
Factor B: TB 
SB 
 b  1 F 1   ; b  1, nT  ab 
1
q 1   ; b, nT  ab 
2
Creative Life Cycles – Comparing Treatment Means
Comparing all 4 Treatment Means(athough no interaction was present):
 
MSE
29.32

 2.42
n11
5
Y 12  33.20 n12  5 s Y 12 
 
MSE
29.32

 2.21
n21
6
Y 22  44.43 n22  7 s Y 22 
Y 11  28.60 n11  5 s Y 11 
Y 21  38.67 n21  6 s Y 21 
T
 
 
MSE
29.32

 2.42
n12
5
MSE
29.32

 2.05
n22
7
 1
1
1
1 
q  0.95, 4, 23  4  19  
 3.977   2.812 s Y ij  Y i ' j '  MSE   
2
2
 nij ni ' j ' 




1 1
Y 11  Y 12  28.60  33.20  4.60 s Y 11  Y 12  29.32     3.43
5 5


HSD  2.812(3.28)  9.22


HSD  2.812(3.17)  8.92


1 1
Y 11  Y 21  28.60  38.67  10.07 s Y 11  Y 21  29.32     3.28
5 6
1 1
Y 11  Y 22  28.60  44.43  15.83 s Y 11  Y 22  29.32     3.17
5 7
1 1
Y 12  Y 21  33.20  38.67  5.47 s Y 12  Y 21  29.32     3.28
5 6




1 1
Y 12  Y 22  33.20  44.43  11.23 s Y 12  Y 22  29.32     3.17
5 7
1 1
Y 21  Y 22  38.67  44.43  5.76 s Y 21  Y 22  29.32     3.01
6 7
Conceptualists/Poets
HSD  2.812(3.43)  9.65
HSD  2.812(3.28)  9.22
HSD  2.812(3.17)  8.92
HSD  2.812(3.01)  8.47
Conceptualists/Novelists Experimentalists/Poets Experimentalists/Novelists
Creative Life Cycles – Comparing Factor Level Means
Factor A (Style):
b
^
 1 
Y
1 j
j 1
b

Y 11  Y 12 28.60  33.20

 30.90
2
2

Y 21  Y 22 38.67  44.43

 41.55
2
2
b
^
 2 
^
Y
2 j
j 1
b
^
 1   2  30.9  41.55  10.65

^

^
29.32 2
1 1 1 1
1  (1) 2        10.402  3.23

2
2
5 5 6 7
s  1   2 
t  0.975, 23  4  19   2.093
95% CI for 1  2 (Conceptualists - Experimentalists): -10.7  (2.093)(3.23)  10.65  6.75 
Factor B (Writer Type):
a
^
 1 
Y
i1
i 1
a

Y 11  Y 21 28.60  38.67

 33.635
2
2

Y 12  Y 22 33.20  44.43

 38.815
2
2
b
^
 2 
^
Y
i 2
j 1
b
^
 1   2  33.635  38.815  5.18

^
^

s  1   2 
29.32 2
1 1 1 1
1  (1) 2        10.402  3.23

2
2
5 5 6 7
t  0.975, 23  4  19   2.093
95% CI for 1  2 (Poets - Novelists): - 5.18  (2.093)(3.23)  5.18  6.75 
 11.93,1.57 
 17.40, 3.90 