ANOVA II
3/30
Review
•
•
•
•
Which is more reliable (bigger F, smaller p)?
Greater between-groups variability
Smaller within-groups variability
Larger sample sizes
Mi:
X1
X2
X3
X4
X5
82
97
88
86
79
95
79
93
81
74
77
84
79
94
78
81
83
75
99
92
90
92
80
85
87
85
87
83
89
82
Mi:
Mi:
X1
X2
X3
X4
X5
82
X1
95
68
83
97
X2
79
97
88
88
X3
93
83
81
86
X4
81
86
88
79
X5
74
59
80
77
81
85
81
63
84
84
79
87
83
84
86
79
88
83
75
74
85
94
81
87
99
94
91
78
54
83
92
58
84
90
67
86
82
76
87
92
83
85
97
92
89
80
70
84
88
75
82
85
99
89
86
85
90
87
72
82
79
67
81
95
71
85
77
79
87
84
93
78
83
79
81
89
94
74
62
82
78
81
83
75
99
92
90
92
80
85
87
85
87
83
89
82
Partitioning Sum of Squares
•
General approach to ANOVA uses sums of squares
– Works with unequal sample sizes and other complications
– Breaks total variability into explained and unexplained parts
•
SStotal
– Sum of squares for all data, treated as a single sample
– Based on differences from grand mean, M
SStotal =
å
(X - M)
2
all groups
•
SSresidual (SSwithin)
– Variability within groups, just like with independent-samples t-test
– Not explainable by group differences
æ
ö
2
2
2
SSresidual = å çç å ( X i - M i ) ÷÷ =
å ( X 1 - M 1) +… + å ( X k - M k )
i=1 to k è group i
group 1
group k
ø
•
SStreatment (SSbetween)
– Variability explainable by difference among groups
– Treat each datum as its group mean and take difference from grand mean
æ
2ö
SStreatment = å ç ni × ( M i - M ) ÷
è
ø
i=1 to k
Magic of Squares
SStotal = SStreatment + SSresidual
M1
×5 M
×6
M2
SStotal = S(X – M)2
×5
×4
M3
SSresidual = Si(S(Xi – Mi)2)
M4
SStreatment = Si(ni(Mi – M)2)
Test Statistic for ANOVA
• Goal: Determine whether group differences account
for more variability than expected by chance
• Same approach as with regression
• Calculate Mean Square: MStreatment
• H0: MStreatment ≈ s2
• Estimate s2 using MSresidual =
MStreatment
• Test statistic: F =
MSresidual
SStreatment
=
df treatment
SSresidual
df residual
Lab Sections Revisited
• Two approaches
– Variance of section means
– Mean Square for treatment (lab section)
• Variance of group means
– Var(76.6 82.5 83.7 79.5 82.9) = 8.69
– Expected by chance: s2/22
– Var(M) vs. s2/22, or 22*var(M) vs. s2
• Mean Square for treatment
22
MStreatment =
mean(M)
åi n`i ( M i - M )
2
= 22× var( M )
5 -1
22× ( 76.6 - 81.0) + 22× ( 82.5 - 81.0) + 22× ( 83.7 - 81.0) + 22× (79.5 - 81.0) + 22× (82.9 - 81.0)
2
=
F=
191.1
191.1
=
= .96
MSresidual 199.1
2
2
4
2
2
= 191.1
Two Views of ANOVA
s2
0.8
• Simple: Equal sample sizes
var(M)
s
2
=
n×var(M)
0.4
F=
s2
0.0
p
n
n
-4
-2
n×var(M)
=
MSresidual
m
0
2
4
z
M1
M3 M2
Mk
• General: Using sums of squares
F=
MStreatment
MSresidual
MStreatment
If sample sizes equal:
SStreatment åi ni ( M i - M )
=
=
df treatment
k -1
2
åi ( M i - M )
2
MStreatment = n×
k -1
= n×var(M)
Degrees of Freedom
SStreatment = var(M)
dftreatment = k – 1
SStotal
= å( X - M )
2
dftotal = Sni – 1
SSresidual = å var(X i )
i
dfresidual = S(ni – 1)
= Sni – k .