Estimating and Testing 2

1

Estimating and Testing 2

0

(n-1)s2/2 has a 2 distribution with n-1 degrees of freedom

Like other parameters, can create CIs and

hypothesis tests since we know the

distribution

2

Estimating and Testing 1

2 / 22

0

12/2

2 has an F distribution with n1-1 and n2-1 d.f.

Like other parameters, can create CIs and

hypothesis tests since we know the

distribution

3

One-Way ANOVA

• Example• As a training specialist, you want to determine whether three

different training methods are equally effective so you want to compare the mean time to complete a task after individuals are trained with the three different methods. (Random variable: completion time)

• One-Way ANOVA hypothesis test• H0: 1 = 2 = 3 = . . . = k • HA: The means are not all equal (at least one pair differs from

each other) • Three Requirements

• k independent random samples• Random variables are normally distributed• Equal population variances for the random variables

4

One-Way ANOVA Data Set

SampleOne

SampleTwo

. . . Samplek

Data

Values

. . .

Mean x1 x2 xk

Variance s12 s2

2 . . . sk2

Sample Size n1 n2 nk

x1,1

x1,2

x1,3

…

x2,1

x2,2

x2,3

…

xk,1

xk,2

xk,3

…

_ _ _

5

One-Way ANOVA

• One-Way ANOVA hypothesis test

• H0: 1 = 2 = 3 = . . . = k

• HA: The means are not all equal (at least one pair differs from each other)

• Test Statistic

• F = BSV / WSV with k-1, nT-k d.f.

• Decision Rule

• Reject H0 if F > F

• Intuition

• If BSV is “large” then H0 is unlikely to be true

Note: Always one-tailed and >

6

F Distribution

0?

P(F > ) = 0.05

P(F < ) = 0.95

9 and 10 d.f

7

Select F Distribution

5% Critical ValuesNumerator Degrees of Freedom

1 2 4 5 6 7 8 9 …1 161 199 225 230 234 237 239 2412 18.5 19.0 19.2 19.3 19.3 19.4 19.4 19.43 10.1 9.55 9.12 9.01 8.94 8.89 8.85 8.818 5.32 4.46 3.84 3.69 3.58 3.50 3.44 3.39

10 4.96 4.10 3.48 3.33 3.22 3.14 3.07 3.0211 4.84 3.98 3.36 3.20 3.09 3.01 2.95 2.9012 4.75 3.89 3.26 3.11 3.00 2.91 2.85 2.8018 4.41 3.55 2.93 2.77 2.66 2.58 2.51 2.46100 3.94 3.09 2.46 2.31 2.19 2.10 2.03 1.97

1000 3.85 3.00 2.38 2.22 2.11 2.02 1.95 1.89…D

enom

inat

or D

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es o

f F

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8

One-Way ANOVA: Example

• H0: 1 = 2 = 3

• HA: The means are not all equal

• Test Statistic• F = BSV / WSV with k-

1,nT-k d.f.

• Decision Rule

• Reject H0 if F > F

• Intuition

• If BSV is “large” then H0 is unlikely to be true

Sample One

Sample Two

Sample Three

10 16 20

10 14 20

11 16 20

10 14 20

9 20x

s2

n

_

nT = , x = =

9

One-Way ANOVA: Another Example

• H0: 1 = 2 = 3

• HA: The means are not all equal

• Test Statistic• F = BSV / WSV with k-

1,nT-k d.f.

• Decision Rule

• Reject H0 if F > F

• Intuition

• If BSV is “large” then H0 is unlikely to be true

Sample One

Sample Two

Sample Three

1 19 5

12 31 33

20 4 20

10 6 12

7 30x

s2

n

_

nT = , x = =

10

Data Analysis: ANOVA-Single

FactorGroups Count Sum Average Variance

Column 1 5 50 10 0.5

Column 2 4 60 15 1.333

Column 3 5 100 20 0

ANOVA

Source of Variation

SS df MS F P-value

F crit

Between Groups

250 2 125 229.167 0.000 3.98

Within Groups

6 11 0.545

Total 256 13

11

Two-Way ANOVA

• Example• Suppose 10 individuals are asked to judge the taste quality of

three beers: Budweiser, Bass Ale, and London Pride. Based on some complicated rating system, each individual assigns a numerical score to each beer (all 10 individuals taste each of the three beers).

• Note that we now have two factors: raters and beers• Questions

• Is the average taste rating equal for each beer?• Is the average taste rating equal for each rater?

• Can do an ANOVA hypothesis test for each question:• H0: 1 = 2 = 3 = . . . = k • HA: The means are not all equal (at least one pair differs from

each other)

12

Two-Way ANOVA: Example

• Suppose 10 individuals are asked to judge the taste quality of three beers: Budweiser, Bass Ale, and London Pride. Based on some complicated rating system, each individual assigns a numerical score to each beer (all 10 individuals taste each of the three beers). Once the data are collected, you estimate the following ANOVA table:

Source df

Sum of

Squares

Mean

Square

F

Beer 2 5.98 2.99 2.39

Rater 9 112.49 13.61 10.89

Error 18 22.50 1.25

13

Another Example

Treatments

A B C

Blocks

1 10 9 82 12 6 53 18 15 144 20 18 185 8 7 8

• These data values were obtained from a randomized block design experiment.

• First, Pretend that these data were collected using a completely randomized design. Can we conclude at the 5% significance level that there are differences in the treatment means?

• How does your answer change if you account for the randomized block design?

14

Pretend One-Way

Groups Count Sum Average Variance

Treatment A 5 68 13.60 26.80

Treatment B 5 55 11.00 27.50

Treatment C 5 53 10.60 27.80

ANOVA

Source of Variation

SS df MS F P-value

F crit

Between Groups

26.533 2 0.627 3.86

Within Groups

328.400 12

Total 354.933 14

15

But is Actually Two-Way

ANOVA: Two-Factor Without Replication

Source of Variation

SS df MS F P-value

F crit

Rows 312.267 4 78.067 38.711 0.000 3.84

Columns 26.533 2 13.267 6.579 0.020 4.46

Error 16.133 8 2.017

Total 354.933 14

16

One-Way ANOVA via Regression

Sample One

Sample Two

Sample Three

10 16 20

10 14 20

11 16 20

10 14 20

9 20

S1 S2 S3 Treatment1 0 0 101 0 0 101 0 0 111 0 0 101 0 0 90 1 0 160 1 0 140 1 0 160 1 0 140 0 1 200 0 1 20

… …

“Stack” data using dummy

variables

17

One-Way ANOVA via Regression

ANOVA (from ANOVA output)

Source of Variation

SS df MS F P-value

F crit

Between 250 2 125 229.167 0.000 3.98

Within 6 11 0.545

Total 256 13

ANOVA (from Regression output)

df SS MS F Significance

Regression 2 250 125 229.167 0.0000

Residual 11 6 0.545

Total 13 256

18

The RegressionANOVA Table

Regression Statistics

Multiple R 0.988

R Squared 0.977

Adj. R Squared 0.972

Standard Error 0.739

Obs. 14

ANOVA

df SS MS F Significance

Regression 2 250 125 229.167 0.0000

Residual 11 6 0.545

Total 13 256

Coeff. Std. Error t stat p value Lower 95% Upper 95%

Intercept 10 0.330 30.277 0.0000 9.273 10.727

S2 5 0.495 10.092 0.0000 3.910 6.090

S3 10 0.467 21.409 0.0000 8.972 11.028

F test for:Ho: 1= 2 = … = k = 0(excluding the intercept)HA: at least one i0

19

Regression and Two-Way ANOVA

Treatments

A B C1 10 9 82 12 6 53 18 15 144 20 18 185 8 7 8

Blo

cks

“Stack” data using dummy

variables

A B C B2 B3 B4 B5 Value1 0 0 0 0 0 0 101 0 0 1 0 0 0 121 0 0 0 1 0 0 181 0 0 0 0 1 0 201 0 0 0 0 0 1 80 1 0 0 0 0 0 90 1 0 1 0 0 0 60 1 0 0 1 0 0 150 1 0 0 0 1 0 180 1 1 0 0 0 1 70 0 1 0 0 0 0 8

… …

20

Regression and Two-Way ANOVA

Source | SS df MS Number of obs = 15----------+---------------------- F( 6, 8) = 28.00 Model | 338.800 6 56.467 Prob > F = 0.0001 Residual | 16.133 8 2.017 R-squared = 0.9545-------------+------------------- Adj R-squared = 0.9205 Total | 354.933 14 25.352 Root MSE = 1.4201

-------------------------------------------------------------treatment | Coef. Std. Err. t P>|t| [95% Conf. Int]----------+-------------------------------------------------- b | -2.600 .898 -2.89 0.020 -4.671 -.529 c | -3.000 .898 -3.34 0.010 -5.071 -.929 b2 | -1.333 1.160 -1.15 0.283 -4.007 1.340 b3 | 6.667 1.160 5.75 0.000 3.993 9.340 b4 | 9.667 1.160 8.34 0.000 6.993 12.340 b5 | -1.333 1.160 -1.15 0.283 -4.007 1.340 _cons | 10.867 .970 11.20 0.000 8.630 13.104-------------------------------------------------------------

Need Partial F test(later in course)