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

• Variance Partitions– Total = Among + Within

• Grand Mean, Group Mean and associated Deviations

• When do we reject based on variance ratio???

ANOVA

ANOVA Table

Source df SS MS F

Among Treatments k-1 SSamong

dfamong

MSamong

MSwithin

Within Treatments n-k SSwithin

dfwithin

Total n-1

k

j

n

ijij XX

1 1

2)(

k

jjj XXn

1

2

k

j

n

iij XX

1 1

2)(

ANOVA

• When do we use??

• Model I vs Model II vs Model III??

• Multi-Factors??

• Main Effects vs Interactions??

Example From TextQuestion #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 TextQuestion #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

Avg 143 133.5

25 80 101

149 113

138 161

131 197

Avg 124.5 143

(SS(sex) = 81, SS(dose) = 81, SS(interaction) = 784, and SS(within) = 12604

a) Construct the ANOVA Tableb) 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

U n nn n

R

1 21 1

1

1

2

( )

U n n

n nR2 1

2 22

1

2

( )

If either U or U’ is greater than the critical value of U, then you should reject the Ho

Critical U

If either U or U’ is greater than the critical value of U, then you should reject the Ho

U UCritical n n 0 05 2 1 2. ,( ), , if n1 < n2

U UCritical n n 0 05 2 2 1. ,( ), , if n1 > n2

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

Deer12345678910

FrontLeg142140144144142146149150142148

BackLeg138136147139143141143145136146

Diff44-35-156562

Rank|d|

SignedRank |d|

Critical TSum the positive ranks - T+

Sum the negative ranks - T-

If 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 2HA: Measurement 1 > Measurement 2

--> reject Ho if T- T0.05, (1),n

Ho: Measurement 1 Measurement 2HA: Measurement 1 < Measurement 2

--> reject Ho if T+ T0.05, (1),n

Kruskal Wallis Test

• Test for three or more groups

• Replaces ANOVA

HN N

R

nNi

ii

k

12

13 1

2

1( )( )

• Rank each individual sample across all groups

• Sum ranks within each group = R

• Critical value - 21,05.0 k

Correction factor for tied ranks

Ct

N N

1

118

24 241 0 0013

0 9987

3

3

.

.

t t ti ii

m

3

1

3 3 32 2 2 2 2 2

18

HH

Cc 44123

0 998744180

.

..

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

XY 10

Intercept Slope

Least Squares

• Line with smallest residual sum of squares

YYiˆResidual

ˆ tois as X tois

ANOVA Table

Source of Variation SS DF MS F

Regression

Residual

Total

( )Y Yi 2

( )Y Yi 2

( )Y Yi 2

n-2

1

n-1

SS

DFgression

gression

Re

Re

SS

DFsidual

sidual

Re

Re

MS

MSgression

sidual

Re

Re

Coefficient of Determination

Total

gression

SS

SSr Re2

- proportion of variation explained

Coefficientsa

-1.047 .243 -4.314 .0015.80E-02 .010 .858 6.013 .000

(Constant)DIAM

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: STRENGTHa.

. .Y X 1047 0 058

SlopeIntercept

Confidence Interval for Slope

1for %95 CI

ˆ2),2(1 *ˆ st n

XY 10ˆˆˆ

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

22

1

( )exp

exp

f f

fobserved ected

ectedi

k

f expected is hypothesized ratio e.g. 50:50 males to females

21,05.0

2 kcritical

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.

22

1

( )exp

exp

f f

fobserved ected

ectedi

k

21,05.0

2 kcritical

Variable AColumn 1 Column 2 Column 3 Total

Row 1 O11 O21 O31 R1

Variable B

Row 2 O12 O22 O31 R2

C1 C2 C3 Total

R1 = sum of observed in Row 1R2 = sum of observed in Row 2C1 = sum of observed in Column 1C2 = sum of observed in Column 2C3 = sum of observed in Column 3Total = sum of all observed

E R C Totalij i j * *

Expected calculation

Variable A Column 1 Column 2 Column 3 Total

Row 1

E11 E21 E31 R1

Variable B Row 2

E12 E22 E32 R2

C1 C2 C3 Total

/

Practice Questions

• 10.73-10.78

• 10.84-10.88

Logistic Regression

• Discrete dependent – usually dichotomous

• Continuous independent