1 Experimental Statistics - week 5 Chapters 8, 9: Miscellaneous topics Chapter 14: Experimental...

1

Experimental StatisticsExperimental Statistics - week 5 - week 5Experimental StatisticsExperimental Statistics - week 5 - week 5

Chapters 8, 9: Miscellaneous topics

Chapter 14: Experimental design concepts Chapter 15: Randomized Complete Block Design (15.3)

2

1-Factor ANOVA Model

yij = iij

yij = iij

or

unexplained partmean for ith treatment

observed data

3

0 1 2:

:

The hypotheses:

at least 2 means a unequalt

a

H

H

0 1 2: 0

: 0

at least one t

a i

H

H

were rewritten as:

4

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

TSS SSB SSW Notation:

In words:

TSS(total SS) = total sample variability among yij values

SSB(SS “between”) = variability explained by differences in group means

SSW(SS “within”) = unexplained variability (within groups)

5

Analysis of Variance TableAnalysis of Variance TableAnalysis of Variance TableAnalysis of Variance Table

Note: unequal sample sizes allowed

2

0 2( 1, )B

TW

sH F F t n t

s We reject at significance level if

6

Extracted from From Ex. 8.2, page 390-391

3 Methods for Reducing Hostility

12 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded.

Method 1 96 79 91 85

Method 2 77 76 74 73

Method 3 66 73 69 66

Test: 0 1 2 3:H

7

ANOVA Table Output – extracted hostility data - calculations done in class

Source SS df MS F p-value

Between 767.17 2 383.58 16.7 <.001 samples

Within 205.74 9 22.86 samples

Totals 972.91

Protected LSD: Preceded by an F-test for overall significance.

1 2

1 2

22

1 2

1 1( )α/ W

y y

y y

t sn n

and are significantly different if

| | LSD

where

LSD = +

and within (error) df

Unprotected: Not preceded by an F-test (like individual t-tests).

Only use the LSD if F is significant.

Fisher’s Least Significant Fisher’s Least Significant Difference (LSD)Difference (LSD)

X

9

Hostility Data - Completely Randomized Design The GLM Procedure t Tests (LSD) for score

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 9 Error Mean Square 22.86111 Critical Value of t 2.26216 Least Significant Difference 7.6482 Means with the same letter are not significantly different.

t Grouping Mean N method

A 87.750 4 1 B 75.000 4 2 B B 68.500 4 3

10

Ex. 8.2, page 390-391

3 Methods for Reducing Hostility

24 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded.

Method 1 96 79 91 85 83 91 82 87

Method 2 77 76 74 73 78 71 80

Method 3 66 73 69 66 77 73 71 70 74

Test: 0 1 2 3:H

Notice unequal sample sizes

11

ANOVA Table Output – full hostility data


Between 1090.6 2 545.3 29.57 <.0001 samples

Within 387.2 21 18.4 samples

Totals 1477.8 23

12

The GLM Procedure t Tests (LSD) for score

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 21 Error Mean Square 18.43878 Critical Value of t 2.07961 Comparisons significant at the 0.05 level are indicated by ***.

Difference method Between 95% Confidence Comparison Means Limits 1 - 2 11.179 6.557 15.800 *** 1 - 3 15.750 11.411 20.089 *** 2 - 1 -11.179 -15.800 -6.557 *** 2 - 3 4.571 0.071 9.072 *** 3 - 1 -15.750 -20.089 -11.411 *** 3 - 2 -4.571 -9.072 -0.071 ***

Notice the different format since there is not one LSD value with which to make all pairwise comparisons.

13

Duncan's Multiple Range Test for score

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.

Alpha 0.05 Error Degrees of Freedom 21 Error Mean Square 18.43878 Harmonic Mean of Cell Sizes 7.91623

NOTE: Cell sizes are not equal.

Number of Means 2 3 Critical Range 4.489 4.712

Means with the same letter are not significantly different.

Duncan Grouping Mean N method

A 86.750 8 1

B 75.571 7 2

C 71.000 9 3

Note: Duncan’s test (another multiple comparison test) avoids the issue of different sample sizes by using the harmonic mean of the ni’s.

14

Some Multiple Comparison Techniquesin SAS

FISHER’S LSD (LSD)

BONFERONNI (BON) DUNCAN

STUDENT-NEWMAN-KEULS (SNK)

DUNNETT RYAN-EINOT-GABRIEL-WELCH (REGWQ) SCHEFFE TUKEY

15

1122.4 2324.6 3120.3 4419.8 5324.3 6222.2 7228.5 8225.7 9320.210119.611228.812424.013417.114419.315324.216115.817218.318117.519418.720322.921116.322414.023416.624218.125218.926416.027220.128322.529316.030119.331115.932320.3

Balloon Data Col. 1-2 - observation number Col. 3 - color (1=pink, 2=yellow, 3=orange, 4=blue) Col. 4-7 - inflation time in seconds

16

1122.4 2324.6 3120.3 4419.8 5324.3 6222.2 7228.5 8225.7 9320.210119.611228.812424.013417.114419.315324.216115.817218.318117.519418.720322.921116.322414.023416.624218.125218.926416.027220.128322.529316.030119.331115.932320.3

Balloon Data Col. 1-2 - observation number Col. 3 - color (1=pink, 2=yellow, 3=orange, 4=blue) Col. 4-7 - inflation time in seconds

17

ANOVA --- Balloon Data General Linear Models Procedure Dependent Variable: TIME Sum of MeanSource DF Squares Square F Value Pr > F Model 3 126.15125000 42.05041667 3.85 0.0200 Error 28 305.64750000 10.91598214 Corrected Total 31 431.79875000 R-Square C.V. Root MSE TIME Mean 0.292153 16.31069 3.3039343 20.256250 MeanSource DF Type I SS Square F Value Pr > F Color 3 126.15125000 42.05041667 3.85 0.0200

18

ANOVA --- Balloon Data

The GLM Procedure

t Tests (LSD) for time NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 28 Error Mean Square 10.91598 Critical Value of t 2.04841 Least Significant Difference 3.3839

Means with the same letter are not significantly different.

t Grouping Mean N color

A 22.575 8 2 A A 21.875 8 3 B 18.388 8 1 B B 18.188 8 4

19

Experimental Design: Concepts and Terminology

Designed Experiment- an investigation in which a specified framework is used to compare groups or treatments

Factors

- up to this point we’ve only looked at experiments with a single factor

- any feature of the experiment that can be varied from trial to trial

20

Experimental Units- subjects, material, etc. to which treatment factors are randomly assigned

- there is inherent variability among these units irrespective of the treatment imposed

Replication- we usually assign each treatment to several experimental units

- these are called replicates

- conditions constructed from the factors (levels of the factor considered, etc.)

Treatments

21

Examples:

Car Data

Hostility Data

Balloon Data

1. factor

2. treatments

3. experimental units

4. replicates

22

1122.4 2324.6 3120.3 4419.8 5324.3 6222.2 7228.5 8225.7 9320.210119.611228.812424.013417.114419.315324.216115.817218.318117.519418.720322.921116.322414.023416.624218.125218.926416.027220.128322.529316.030119.331115.932320.3

Balloon Data Col. 1-2 - observation number (run order) Col. 3 - color (1=pink, 2=yellow, 3=orange, 4=blue) Col. 4-7 - inflation time in seconds

Why randomize run order? i.e. why not blow up all the pink balloons first, blue balloons next, etc?

Question:

23

t i me

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

i d

0 10 20 30 40

Scatterplot Using GPLOT

What do we learn from this plot?

Run Order

Time

24

RECALL: 1-Factor ANOVA Model

2 ' are (0, )ij s NID

- random errors follow a Normal (N) distribution, are independently distributed (ID), and have zero mean and constant variance

( )i i ij i ijy

-- i.e. variability does not change from group to group

25

Model Assumptions:

Checking Validity of Assumptions

1. F-test similar to 2-sample case - Hartley’s test (p.366 text) - not recommended

2. Graphical - side-by-side box plots

- equal variances- normality

Equal Variances

26

Graphical Assessment of Equal Variance Assumption

27

Note:Optional approaches if equal variance assumption is violated:

1. Use Kruskal Wallis nonparametric procedure – Section 8.6 2. Transform the data to induce more nearly equal variances – Section 8.5 -- log -- square root

Note: These transformations may also help induce normality

28

yij = iij

Assessing Normality of Errors

ij = yij (i)so

ij is estimated by

.ij ij ie y y

= yij i

The e ij’s are called residuals.

29

proc glm; class color; model time=color; title 'ANOVA --- Balloon Data';

output out=new r=resball;means color/lsd;

run;proc sort; by color;run;proc boxplot; plot time*color; title 'Side-by-Side Box Plots for Balloon Data';run;proc univariate; var resball; histogram resball/normal; title 'Histogram of Residuals -- Balloon Data'; run;proc univariate normal plot; var resball; title 'Normal Probability Plot for Residuals - Balloon Data'; run;proc gplot; plot time*id; title 'Scatterplot of Time vs ID (Run Order)'; run;

SAS Code for Balloon Data

30

Normal Probability Plot 6.5+ +*+ | * *+++ | *+++ | +*+ | *** | **** 0.5+ ***+ | ++** | ++*** | ***** | +*+ | *+*+* -5.5+ * ++++ +----+----+----+----+----+----+----+----+----+----+

-2 -1 0 +1 +2

.ij ij ie y y Graphical Assessment of Normality of Residuals

- 6 - 3 0 3 6

0

5

10

15

20

25

30

35

40

Percent

r esbal l

31

Caution: Chapter 15 introduces some new notation - i.e. changes notation already defined

32

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y

Recall: Sum-of-Squares Identity 1-Factor ANOVA


In words:

Total SS = SS between samples + within sample SS

33

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y



- new notation for Chapter 15

34

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y


TSS S SSWST Notation:


35

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y


TSS SST SSE Notation:


In words:

Total SS = SS for “treatments” + SS for “error”

36

Revised ANOVA Table for 1-Factor ANOVA(Ch. 15 terminology - p.857)

Source SS df MS F

Treatments SST t 1

Error SSE N t Total TSS N

/( 1)MST SST t

/( )MSE SSE N t

/MST MSE

1 2 t

N

nt

n n n

total # of observations

(if equal # obs.)

37

Recall 1-factor ANOVA (CRD) Model for Gasoline Octane Data

yij = iij

yij = iij

or

unexplained partmean for ith gasoline

observed octane

-- car-to-car differences-- temperature-- etc.

38

Gasoline Octane Data

Question:

What if car differences are obscuring gasoline differences?

Similar to diet t-test example: Recall: person-to-person differences obscured effect of diet

39

Possible Alternative Design for Octane Study:

Test all 5 gasolines on the same car

- in essence we test the gasoline effect directly and remove effect of car-to-car variation

Question:How would you randomize an experiment with 4 cars?

40

Blocking an Experiment

- dividing the observations into groups (called blocks) where the observations in each block are collected under relatively similar conditions

- comparisons can many times be made more precisely this way

41

Terminology is based on Agricultural Experiments

Consider the problem of testingfertilizers on a crop - t fertilizers - n observations on each

42

Completely Randomized DesignCompletely Randomized DesignCompletely Randomized DesignCompletely Randomized Design

A

A

BB

C

C

B

A

C

C

B

A

A

B

C

t = 3 fertilizersn = 5 replications

- randomly select 15 plots- randomly assign fertilizers to the 15 plots

43

Randomized Complete Block Randomized Complete Block StrategyStrategy

Randomized Complete Block Randomized Complete Block StrategyStrategy

B | A | C

A | C | B

C | A | B

A | B | C C | B | A

t = 3 fertilizers

- select 5 “blocks”- randomly assign the 3 treatments to each block

Note: The 3 “plots” within each block are similar - similar soil type, sun, water, etc

44

Randomized Complete Block Design Randomly assign each treatment once to every block

Car Example Car 1: randomly assign each gas to this car

Car 2: ....

etc.

Agricultural Example Randomly assign each fertilizer to one of the 3 plots within each block

45

yij = ijij

Model For Randomized Complete Block (RCB) Design

effect of ith treatment

effect of jth block

unexplained error

(car)(gasoline)

1 1

0t b

i ji j

As before:

-- temperature-- etc.

46

Previous Data Table from Chapter 8 for 1-factor ANOVA

column averages don’t make any sense

47

Back to Octane data:

Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car.

“Restructured” Data

A 91.7 91.2 90.9 90.6B 91.7 91.9 90.9 90.9C 92.4 91.2 91.6 91.0D 91.8 92.2 92.0 91.4E 93.1 92.9 92.4 92.4

Old Data Format1 2 3 4

Car

Gas

A 91.7 91.2 90.9 90.6B 91.7 91.9 90.9 90.9C 92.4 91.2 91.6 91.0D 91.8 92.2 92.0 91.4E 93.1 92.9 92.4 92.4

Gas

48

2 2 2.. . .. .

1 1 1 1 1

( ) ( ) ( )t n t t n

ij i ij ii j i i j

y y n y y y y


TSS SST SSE Notation:

- using new notation for Chapter 15

In words:

Total SS = SS for “treatments” + SS for “error”

49

2 2 2 2.. . .. . .. . . ..

1 1 1 1 1 1

( ) ( ) ( ) ( )t b t b t b

ij i j ij i ji j i j i j

y y b y y t y y y y y y

A New Sum-of-Squares Identity

TSS SST SSB SSE Not atio n:

In words:

Total SS = SS for treatments + SS for blocks + SS for error

50

Hypotheses:

To test for treatment effects - i.e. gas differenceswe test

0 1 2: tH

To test for block effects - i.e. car differences (not usually the research hypothesis)we test

0 1 2: bH

51

Randomized Complete Block Design ANOVA Table

Source SS df MS F

Treatments SST t 1

Blocks SSB

Error SSE Total TSS bt

/( 1)MST SST t

/( 1)( 1)MSE SSE b t

/MST MSE

See page 866

( 1)( 1)b t

1b /( 1)MSB SSB b /MSB MSE

52

0

( 1,( 1)( 1))

H

MSTF F t b t

MSE

We reject at significance level if

0 1 2:

: 0t

a i

H

H

at least one

Test for Treatment Effects

Note:2MSE estimates

2 2

1

1

1

t

ii

MSTt

estimates

1F - if no treatment effects, we expect ; 1F - if treatment effects, we expect

53

0

( 1,( 1)( 1))

H

MSBF F b b t

MSE

We reject at significance level if

Test for Block Effects

0 1 2:

: 0b

a j

H

H

at least one

54

The first variable (A - E) indicates gas as it did with the CompletelyRandomized Design. The second variable (B1 - B4) indicates car.

A B1 91.7A B2 91.2A B3 90.9A B4 90.6B B1 91.7B B2 91.9B B3 90.9B B4 90.9C B1 92.4C B2 91.2C B3 91.6C B4 91.0D B1 91.8D B2 92.2D B3 92.0D B4 91.4E B1 93.1E B2 92.9E B3 92.4E B4 92.4

“Restructured” CAR Data - SAS Format

55

SAS file - Randomized Complete Block Design for CAR Data

INPUT gas$ block$ octane;PROC GLM; CLASS gas block; MODEL octane=gas block; TITLE 'Gasoline Example -Randomized Complete Block Design'; MEANS gas/LSD;RUN;

56

1-Factor ANOVA Table Output - octane data


Gas 6.108 4 1.527 6.80 0.0025 (treatments)

Error 3.370 15 0.225

Totals 9.478 19

57

1-Factor ANOVA Table Output - car data


Gas 6.108 4 1.527 15.58 0.0001 (treatments)

Cars 2.194 3 0.731 7.46 0.0044 (blocks)

Error 1.176 12 0.098

Totals 9.478 19

58

Dependent Variable: OCTANE Sum of MeanSource DF Squares Square F Value Pr > F Model 7 8.30200000 1.18600000 12.10 0.0001 Error 12 1.17600000 0.09800000 Corrected Total 19 9.47800000 R-Square C.V. Root MSE OCTANE Mean 0.875923 0.341347 0.3130495 91.710000 Source DF Anova SS Mean Square F Value Pr > F GAS 4 6.10800000 1.52700000 15.58 0.0001BLOCK 3 2.19400000 0.73133333 7.46 0.0044

SAS Output -- RCB CAR Data

1 2

1 2

y y

y y

and are significantly different if

| |

Multiple Comparisons in RCB AnalysisMultiple Comparisons in RCB Analysis

22

( )α/MSE

tb

(LSD)

(2 )2

( )α/ mMSE

tb

(Bonferroni)

60

t Grouping Mean N gas A 92.7000 4 E B 91.8500 4 D B C B 91.5500 4 C C B C B 91.3500 4 B C C 91.1000 4 A

t Grouping Mean N gas A 92.7000 4 E B 91.8500 4 D B C B 91.5500 4 C C C 91.3500 4 B C C 91.1000 4 A

CAR Data -- LSD Results

CRD Analysis

RCB Analysis

61

Bon Grouping Mean N gas A 92.7000 4 E A B A 91.8500 4 D B B 91.5500 4 C B B 91.3500 4 B B B 91.1000 4 A

CAR Data -- Bonferroni Results

CRD Analysis

RCB Analysis

Bon Grouping Mean N gas

A 92.7000 4 E

B 91.8500 4 D B B 91.5500 4 C B B 91.3500 4 B B B 91.1000 4 A

1 Experimental Statistics - week 5 Chapters 8, 9: Miscellaneous topics Chapter 14: Experimental...

Documents

Transcript of 1 Experimental Statistics - week 5 Chapters 8, 9: Miscellaneous topics Chapter 14: Experimental...