2nd Half Review
ANOVA (Ch. 11)
Non-Parametric (7.11, 9.5)
Regression (Ch. 12)
ANCOVA
Categorical (Ch. 10)
Correlation (Ch. 12)
The Exam
• Thursday, April 27, 9:00am
– TC 348  Abdi to Middleton
– TC 348a  Minto to Shetty
– TC 348b  Siddiqui to Zdravic
•
•
•
•
•
50 Questions
Not Cumulative
3hr
Bring a calculator
No formula Sheets
ANOVA
• Variance Partitions
– Total = Among + Within
• Grand Mean, Group Mean and associated
Deviations
• When do we reject based on variance
ratio???
ANOVA Table
Source
df
Among Treatments
k-1
SS
 n X
k
j
j 1
Within Treatments
n-k
k
j
X
MS

2
n
2
(
X

X
)
 ij j
j 1 i 1
Total
n-1
k
n
2
(
X

X
)
 ij
j 1 i 1
F
SSamong MSamong
dfamong MSwithin
SSwithin
dfwithin
ANOVA
•
•
•
•
When do we use??
Model I vs Model II vs Model III??
Multi-Factors??
Main Effects vs Interactions??
Example From Text
Question #11.40, p. 518
10 women in an aerobic exercise class, 10 women in a modern dance
class, and a control group of 9 women were studied. One
measurement made on each women was change in fat-free mass over
the course of the 16-week training period. Summary statistics are
given in the table. The ANOVA SS(between) is 2.465 and the
SS(within) is 50.133.
Aerobics Dance
Control
Mean
0
0.44
0.71
SD
1.31
1.17
1.68
n
10
10
9
a) State the null hypothesis
b) Construct the ANOVA
table and test the null
hypothesis (α = 0.05)
Example From Text
Question #11.57, p. 522
A new investigational drug was given to 4 male and 4 female dogs, at
doses 8 mg/kg and 25 mg/kg. The variable recorded was alkaline
phosphatase level (U/Li).
Dose (mg/kg)
Male
Female
8
171
150
154
127
104
152
143
105
143
133.5
80
101
149
113
138
161
131
197
124.5
143
Avg
25
Avg
(SS(sex) = 81, SS(dose) = 81,
SS(interaction) = 784, and
SS(within) = 12604
a) Construct the ANOVA Table
b) Carry out an F test for
interactions: use (α = 0.05).
c) Test the null hypothesis that
does has no effect on alkaline
phosphatase level. (α = 0.05)
Extra Questions from the Text
•
•
•
•
11.4-11.6
11.9-11.11
11.17, 11.19
11.42, 11.43, 11.50, 11.54
Non-Parametric
• When to use??
– Normality
– Homogenous of Variance
– Independent Observations
• What about Power??
• What do they use to compare data??
Mann-Whitney Test
• Compares two samples
• Replaces two-sample t-test
n1 ( n1  1)
U  n1n2 
 R1
2
n2 ( n2  1)
U   n2 n1 
 R2
2
If either U or U’ is greater than the critical value of U,
then you should reject the Ho
Critical U
UCritical  U0.05,( 2 ),n1 ,n2
UCritical  U0.05,( 2 ),n2 ,n1
if n1 < n2
if n1 > n2
If either U or U’ is greater than the critical value of U,
then you should reject the Ho
One-tailed Mann-Whitney U test
Use U or U’ depending on whether you expect
sample 1 or sample 2 to be bigger
Ho: G1  G2
HA: G1 < G2
Ho: G1  G2
HA: G1 > G2
Ranking is
low to high
U
U’
Ranking is
high to low
U’
U
Wilcoxon paired sample test
• Compares two paired samples
• Replaces Paired t-test
Deer
1
2
3
4
5
6
7
8
9
10
Front
Leg
142
140
144
144
142
146
149
150
142
148
Back
Leg
138
136
147
139
143
141
143
145
136
146
Diff
4
4
-3
5
-1
5
6
5
6
2
Rank
|d|
Signed
Rank |d|
Critical T
Sum the positive ranks - T+
Sum the negative ranks - TIf either T+ or T- is less than or equal to T0.05, (2),n then reject Ho
Can also do these one tailed:
Ho: Measurement 1  Measurement 2
HA: Measurement 1 > Measurement 2
--> reject Ho if T-  T0.05, (1),n
Ho: Measurement 1  Measurement 2
HA: Measurement 1 < Measurement 2
--> reject Ho if T+  T0.05, (1),n
Kruskal Wallis Test
• Test for three or more groups
• Replaces ANOVA
k
2
i
12
R
H
 3( N  1)

N ( N  1) i1 ni
• Rank each individual sample across all
groups
• Sum ranks within each group = R
• Critical value -  02.05,k 1
Correction factor for tied ranks
C  1
t
N N
18
 1 3
24  24
 1  0.0013
 0.9987
3
m
3
t

t
  i  ti
i 1
 23  2  23  2  23  2
 18
H 44123
.
Hc 

 44180
.
C 0.9987
Critical Value
Practice Questions
• Mann-Whitney
– 7.79, 7.80, 7.82-7.84
• Wilcoxon
– 9.30 – 9.33
• Kruskal-Wallis
– 11.54, 11.57  use Kruskal-Wallis instead of
ANOVA
Regression
• Two or more continuous variables
• Linear relationship between
Y   0  1 X
Intercept
Slope
Least Squares
• Line with smallest residual sum of squares
Residual
 is to ˆ
 Yi  Yˆ
as
 is to X
ANOVA Table
Source of
Variation
Regression
Residual
Total
SS
DF
2

(
Y

Y
)
 i
2

 (Yi  Y )
 (Y  Y )
2
i
1
n-2
n-1
MS
F
SSRe gression
MSRe gression
DFRe gression
MSRe sidual
SS Re sidual
DFRe sidual
Coefficient of Determination
r 
2
SS Re gression
SSTotal
- proportion of variation explained
i
a
c
d
e
d
f
i
t
c
B
e
i
M
t
E
g
1
(
7
3
4
1
D
2
0
8
3
0
a
D
Intercept
Slope

Y  1047
.
 0.058 X
Confidence Interval for Slope
ˆ
ˆ
ˆ
Y  0  1 X
95%CI for 1
 ˆ1  t ( 2 ),n 2 * sˆ
Practice Questions
• 12.45
• 12.49-12.54
ANCOVA
• Continuous Dependent
• Continuous and Discrete Independents
• Compares relationship of two variables
across two groups
Categorical Data
• Discrete Response variable
• Interested in Frequencies
Chi-Squared Test
• Observed vs Expected Frequencies
k
 
2
( f observed  f exp ected )
i 1
2
f exp ected
f expected is hypothesized ratio e.g. 50:50 males to females

2
critical

2
0.05, k 1
Contingency Table
• Test for independence among variables
• 2x2, 2x3, 3x3 etc.
• 2 variables with 2 levels, 2 variables with 3
levels, 3 variables with 3 levels etc.
k
 
2
( f observed  f exp ected )
i 1

2
critical
f exp ected

2
0.05, k 1
2
Column 1
Variable A
Column 2
Column 3
Total
Row 1
O11
O21
O31
R1
Row 2
O12
O22
O31
R2
C1
C2
C3
Total
Variable B
R1 = sum of observed in Row 1
R2 = sum of observed in Row 2
C1 = sum of observed in Column 1
C2 = sum of observed in Column 2
C3 = sum of observed in Column 3
Total = sum of all observed
Column 1
Variable A
Column 2
Column 3
Total
Row 1
E11
E21
E31
R1
Row 2
E12
E22
E32
R2
C1
C2
C3
Total
Variable B
Expected calculation
Eij  Ri * C j */ Total
Practice Questions
• 10.73-10.78
• 10.84-10.88
Logistic Regression
• Discrete dependent – usually dichotomous
• Continuous independent