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Experimental Statistics - week 7. Chapter 15: Factorial Models (15.5). Chapter 17: Random Effects Models. Model for 2-factor Design. where. Hypothetical Cell Means. Auditory. Visual. 5 10 15. Auditory. Visual.

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experimental statistics week 7
Experimental Statistics - week 7

Chapter 15:

Factorial Models (15.5)

Chapter 17:

Random Effects Models

slide3

Hypothetical Cell Means

Auditory

Visual

5 10 15

Auditory

Visual

5 10 15

slide4

Sum-of-Squares Breakdown

(2-factor ANOVA)

SSA

SSB

SSAB

SSE

slide5

2-Factor ANOVA Table(2-Factor Completely Randomized Design)

Source SS df MS F

Main Effects

A SSA a -1

B SSB b- 1

Interaction

AB SSAB (a -1)(b- 1)

Error SSE ab(n -1)

Total TSS abn -1

(page 900)

slide6

Hypotheses:

Main Effects:

Interactions:

slide7

STIMULUS EXAMPLE:

Personal computer presents stimulus, and person responds.

Study of how RESPONSE TIME is effected by a WARNING given prior to the stimulus:

2-factors of interest:

Warning Type --- auditory or visual

Time between warning and stimulus -- 5 sec, 10 sec, or 15 sec. 

slide8

Auditory Visual

.204 .257

.170 .279

.181 .269

.167 .283

.182 .235

.187 .260

.202 .256

.198 .281

.236 .258

5 sec

WarningTime

10 sec

15 sec

slide9

GLM Output

Stimulus Data

The GLM Procedure

Dependent Variable: response

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 5 0.02554894 0.00510979 17.66 <.0001

Error 12 0.00347200 0.00028933

Corrected Total 17 0.02902094

R-Square Coeff Var Root MSE response Mean

0.880362 7.458622 0.017010 0.228056

Source DF Type I SS Mean Square F Value Pr > F

type 1 0.02354450 0.02354450 81.38 <.0001

time 2 0.00115811 0.00057906 2.00 0.1778

type*time 2 0.00084633 0.00042317 1.46 0.2701

slide10

Testing Procedure

2 factor CRD Design

Step 1. Test for interaction.

Step 2.

(a) IFthere IS NOT a significant interaction

- test the main effects

(b) IF there IS a significant interaction

- compare cell means

slide11

Stimulus Example

Test for Interaction:

Therefore we DO NOT reject the null hypothesis of no interaction.

slide13

Stimulus Example

Test for Interaction:

Therefore we DO NOT reject the null hypothesis of no interaction.

Thus - based on the testing procedure, we next test for main effects.

slide14

Testing Main Effects:

For each main effect (i.e. A and B)

Note:I’ll use LSD from this point on unless otherwise noted.

In General:

where N denotes the # of observations involved

in the computation of a marginal mean.

slide15

Auditory Visual

.204 .257

.170 .279

.181 .269

.167 .283

.182 .235

.187 .260

.202 .256

.198 .281

.236 .258

5 sec

WarningTime

10 sec

15 sec

slide16

Stimulus Example

Test for Main Effects:

A (type):

B (time):

Thus, there is a significant effect due to type but not time

- i.e. we can use LSD to compare marginal means for type

- we will do this here for illustration although MC not needed when there are only 2 groups

slide17

GLM Output -- Comparing “Types”

  The GLM Procedure

t Tests (LSD) for response

NOTE: This test controls the Type I comparisonwise error rate,

not the experimentwise error rate.

Alpha 0.05

Error Degrees of Freedom 12

Error Mean Square 0.000289

Critical Value of t 2.17881

Least Significant Difference 0.0175

Means with the same letter are not significantly different.

t Grouping Mean N type

A 0.264222 9 V

B 0.191889 9 A

slide18

GLM Output -- Comparing “Times”

The GLM Procedure

t Tests (LSD) for response

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

  Alpha 0.05

Error Degrees of Freedom 12

Error Mean Square 0.000289

Critical Value of t 2.17881

Least Significant Difference 0.0214

Means with the same letter are not significantly different.

t Grouping Mean N time

A 0.238500 6 15

A

A 0.226667 6 5

A

A 0.219000 6 10

slide19

Pilot Plant Data

Variable = Chemical Yield

Factors: A – Temperature (160, 180)

B – Catalyst (C1 , C2)

Temperature

59 74

61 70

50 69

58 67

50 81

54 85

46 79

44 81

Catalyst

slide23

DATA one;

INPUT temp catalyst$ yield;

datalines;

160 C1 59

160 C1 61

.

.

.

180 C2 79

180 C2 81

;

PROCGLM;

class temp catalyst;

MODEL yield=temp catalyst temp*catalyst;

Title 'Pilot Plant Example -- 2-way ANOVA';

MEANS temp catalyst/LSD;

RUN;

PROCSORT;BY temp catalyst;

PROCMEANS; BY temp catalyst; OUTPUTOUT=cells MEAN=yield;

RUN;

slide24

Pilot Plant -- GLM Output

Pilot Plant Example -- 2-way ANOVA

General Linear Models Procedure

Dependent Variable: YIELD

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 3 2525.0000000 841.6666667 58.05 0.0001

Error 12 174.0000000 14.5000000

Corrected Total 15 2699.0000000

R-Square C.V. Root MSE YIELD Mean

0.935532 5.926672 3.8078866 64.250000

Source DF Type I SS Mean Square F Value Pr > F

TEMP 1 2116.0000000 2116.0000000 145.93 0.0001

CATALYST 1 9.0000000 9.0000000 0.62 0.4461

TEMP*CATALYST 1 400.0000000 400.0000000 27.59 0.0002

slide25

RECALL: Testing Procedure

2 factor CRD Design

Step 1. Test for interaction.

Step 2.

(a) IFthere IS NOT a significant interaction

- test the main effects

(b) IF there IS a significant interaction

- compare cell means

slide26

Pilot Plant Example

Test for Interaction:

Therefore we reject the null hypothesis of no interaction

- and conclude that there is an interaction between temperature and catalyst.

Thus, we DO NOT test main effects

slide29

Since there is a significant interaction, we do not test for main effects!

- instead compare “Cell Means”

- NOTE: interaction plot is a plot of the cell means

slide30

Pilot Plant Data

Variable = Chemical Yield

Factors: A – Temperature (160, 180)

B – Catalyst (C1 , C2)

Temperature

59 74

61 70

50 69

58 67

50 81

54 85

46 79

44 81

Catalyst

slide31

Pilot Plant Data -- cell means

Temperature

57.0 70.0

48.5 81.5

Catalyst

slide32

Comparing Cell Means:

If there is significant interaction, then we compare the a x b cell means using the criteria below.

Procedure similar to that for comparing marginal means:

where N denotes the # of observations involved

in the computation of a cell mean.

slide33

GLM Output -- Comparing “Temps”

The GLM Procedure

t Tests (LSD) for yield

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

  Alpha 0.05

Error Degrees of Freedom 12

Error Mean Square 14.5

Critical Value of t 2.17881

Least Significant Difference 4.1483

Means with the same letter are not significantly different.

t Grouping Mean N temp

A 75.750 8 180

B 52.750 8 160

- disregard

slide34

GLM Output -- Comparing “Catalysts”

The GLM Procedure

  t Tests (LSD) for yield

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

Alpha 0.05

Error Degrees of Freedom 12

Error Mean Square 14.5

Critical Value of t 2.17881

Least Significant Difference 4.1483

Means with the same letter are not significantly different.

t Grouping Mean N catalyst

A 65.000 8 C2

A

A 63.500 8 C1

- disregard

slide35

Note:

- SAS does not provide a comparison of cell means

slide36

NOTE:

I will be out of the office tomorrow.

slide37

Testing Procedure Revisted

2 factor CRD Design

Step 1. Test for interaction.

Step 2.

(a) IFthere IS NOT a significant interaction

- test the main effects

(b) IF there IS a significant interaction

- compare a x b cell means (by hand)

Main Idea:We are trying to determine whether the factors effect the response either individually or collectively.

slide38

Statistics 5372: Experimental Statistics

Assignment Report Form

Name:

Lecture Assigned:

Data Set or Problem Description

Key Results of the Analysis

Conclusions in the Language of the Problem

Appendices:

A. Tables and Figures Cited in the Report

B. SAS Log from the Final SAS Run

Notes:

1. All assignments should be typed using a word processor according to the format above.

2. SAS output should consist only of tables and figures cited in the report. The report should refer to these tables and figures using numbers you assign, i.e. Table 1, etc.

3. The data should be listed somewhere in the report. (within SAS code is ok)

slide39

Homework Assignment

Due March 1, 2005

15.41, page 935

In this problem the authors consider two measures of the stability of a drug: MG/ML and pH. They ran a 2-factor ANOVA for each of these response variables using storage time and laboratory used in the analysis as the classification variables. There are 4 storage times considered and 2 labs. The data are in the table on page 935 and the resulting 2-factor ANOVA tables are shown on 935-936.

Using SAS, reproduce the ANOVA tables given in the book, and complete an assignment report form for the two analyses.

slide40

Auditory Visual

.204 .257

.170 .279

.181 .269

.167 .283

.182 .235

.187 .260

.202 .256

.198 .281

.236 .258

5 sec

WarningTime

10 sec

15 sec

slide41

Note:For balanceddesigns,

i.e. for STIMULUS data

.228 = (.227+.219+.239)/3

= (.192+.264)/2

slide42

Now Consider:

Auditory Visual

.204 .257

.170 .279

.181 .269

.167 .283

.182 .235

.187 .260

.202 .256

.198 .281

.236 .258

5 sec

WarningTime

10 sec

15 sec

slide43

Balanced Experimental Designs

  • Every Combination of the Factor Levels has an Equal Number of Repeats
  • Sums of Squares
    • Uniquely Calculated
      • Usual Textbook Formulas

Unbalanced Experimental Designs

  • Not Every Combination of the Factor Levels has an Equal Number of Repeats
  • Sums of Squares
    • Not Uniquely Calculated
      • Usual Textbook Formulas Are Not Valid
slide44

Unbalanced Experimental Designs

Many Software Programs Cannot Properly Calculate Sums of Squares for Unbalanced Designs

- they typically use “Textbook Formulas”

SAS:

- must Use Proc GLM, not Proc ANOVA

- Type I and Type III sums-of-squares results

will not generally agree

- use Type III sums of squares

-- analysis is closest to that for “Balanced Experiments”

slide45

Unbalanced Data -- GLM Output

The GLM Procedure

Dependent Variable: response

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 5 0.02547774 0.00509555 19.13 <.0001

Error 11 0.00293050 0.00026641

Corrected Total 16 0.02840824

R-Square Coeff Var Root MSE response Mean

0.896843 7.112913 0.016322 0.229471

Source DF Type I SS Mean Square F Value Pr > F

type 1 0.02309680 0.02309680 86.70 <.0001

time 2 0.00122742 0.00061371 2.30 0.1460

type*time 2 0.00115351 0.00057676 2.16 0.1611

Source DF Type III SS Mean Square F Value Pr > F

type 1 0.02367796 0.02367796 88.88 <.0001

time 2 0.00130085 0.00065042 2.44 0.1326

type*time 2 0.00115351 0.00057676 2.16 0.1611

slide46

Model for 3-factor Factorial Design

where

and also, the sum over any subscript of a 2 or 3 factor interaction is zero

slide47

Sum-of-Squares Breakdown

(3-factor ANOVA)

slide48

3-Factor ANOVA Table(3-Factor Completely Randomized Design)

Source SS df MS F

Main Effects

A SSA a -1

B SSB b - 1

C SSC c - 1

Interactions

AB SSAB (a -1)(b- 1)

AC SSAC (a -1)(c- 1)

BC SSBC (b -1)(c- 1)

ABC SSABC (a -1)(b- 1)(c- 1)

Error SSE abc(n -1)

Total TSS abcn -1

See page 908

slide49

Popcorn Data

Response variable --% of kernels that popped

  • Factors
  • (A) Brand (3 brands)
  • (B) Power of Microwave (500, 600 watts)
  • (C) 4, 4.5 minutes
  • n =2replications per cell
slide50

Popcorn Data

1 500 4.5 70.3

1 500 4.5 91.0

1 500 4 72.7

1 500 4 81.9

1 600 4.5 78.7

1 600 4.5 88.7

1 600 4 74.1

1 600 4 72.1

2 500 4.5 93.4

2 500 4.5 76.3

2 500 4 45.3

2 500 4 47.6

2 600 4.5 92.2

2 600 4.5 84.7

2 600 4 66.3

2 600 4 45.7

3 500 4.5 50.1

3 500 4.5 81.5

3 500 4 51.4

3 500 4 67.7

3 600 4.5 71.5

3 600 4.5 80.0

3 600 4 64.0

3 600 4 77.0

slide51

SAS GLM Code – 3 Factor Model

PROCGLM;

class brand power time;

MODEL percent=brand power time brand*power brand*time power*time brand*power*time;

Title 'Popcorn Example -- 3-Factor ANOVA';

MEANS brand power time/LSD;

RUN;

The Statement

MODEL percent=brand power time brand*power brand*time power*time brand*power*time

can be written as

MODEL percent=brand | power | time;

slide52

The GLM Procedure

Dependent Variable: percent

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 11 3589.988333 326.362576 2.71 0.0503

Error 12 1444.170000 120.347500

Corrected Total 23 5034.158333

R-Square Coeff Var Root MSE percent Mean

0.713126 15.27011 10.97030 71.84167

Source DF Type I SS Mean Square F Value Pr > F

brand 2 566.690833 283.345417 2.35 0.1372

power 1 180.401667 180.401667 1.50 0.2443

time 1 1545.615000 1545.615000 12.84 0.0038

brand*power 2 125.125833 62.562917 0.52 0.6074

brand*time 2 1127.672500 563.836250 4.69 0.0314

power*time 1 0.015000 0.015000 0.00 0.9913

brand*power*time 2 44.467500 22.233750 0.18 0.8336

slide53

Testing Procedure

3 factor CRD Design

Step 1. Test for 3rd order interaction.

IF there IS a significant 3rd order interaction

- compare cell means

IF there IS NOT a significant 3rd order interaction

- test 2nd order interactions

IF there IS a significant 2rd order interaction

- compare associated cell means

IF there IS NOT a sig. 2nd order interaction

- test the main effects

In general -- test main effects only for variables not involved in a significant 2nd or 3rd order interaction

slide54

The GLM Procedure

Dependent Variable: percent

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 11 3589.988333 326.362576 2.71 0.0503

Error 12 1444.170000 120.347500

Corrected Total 23 5034.158333

R-Square Coeff Var Root MSE percent Mean

0.713126 15.27011 10.97030 71.84167

Source DF Type I SS Mean Square F Value Pr > F

brand 2 566.690833 283.345417 2.35 0.1372

power 1 180.401667 180.401667 1.50 0.2443

time 1 1545.615000 1545.615000 12.84 0.0038

brand*power 2 125.125833 62.562917 0.52 0.6074

brand*time 2 1127.672500 563.836250 4.69 0.0314

power*time 1 0.015000 0.015000 0.00 0.9913

brand*power*time 2 44.467500 22.233750 0.18 0.8336

Examine brand x time cell means

Examine Power main effect

slide56

To complete the analysis:

1. The F-test for Power was not significant (.2443)

2. Compare the 6 cell means plotted in interaction plot using procedure analogous to the one used for pilot plant data.

PROCSORT data=one;BY brand time;

PROCMEANS mean std data=one;BY brand time; OUTPUT OUT=cells MEAN=percent;

Title 'Brand x Time Cell Means for Popcorn Data';

RUN;

Obs brand time _TYPE_ _FREQ_ percent

1 1 4 0 4 75.200

2 1 4.5 0 4 82.175

3 2 4 0 4 51.225

4 2 4.5 0 4 86.650

5 3 4 0 4 65.025

6 3 4.5 0 4 70.775

Conclusions:

slide57

Models with Random Effects

Fixed-Effects Models

-- the models we’ve studied to this point

-- factor levels have been specifically selected

- investigator is interested in testing effects of these specific levels on the response variable

Examples:

-- CAR data

- interested in performance of these 5 gasolines

-- Pilot Plant data

- interested in the specific temperatures (160o and 180o) and catalysts (C1 and C2)

slide58

Random-Effect Factor

-- the factor has a large number of possible levels

-- the levels used in the analysis are a random sample from the population of all possible levels

- investigator wants to draw conclusions about thepopulation from which these levels were chosen

(not the specific levels themselves)

slide59

Fixed Effects vs Random Effects

This determination affects

- the model

- the hypothesis tested

- the conclusions drawn

- the F-tests involved (sometimes)

slide61

Hypotheses:

Ho: sa2 = 0

Ha: sa2 0

Ho says (considering the variability of the yij’s) :

- the component of the variance due to “Factor” has zero variance

-- i.e. no factor-to-factor variation

- all of the variability observed is just unexplained subject-to-subject variation

slide62

DATA one;

INPUT operator output;

DATALINES;

1 175.4

1 171.7

1 173.0

1 170.5

2 168.5

2 162.7

2 165.0

2 164.1

3 170.1

3 173.4

3 175.7

3 170.7

4 175.2

4 175.7

4 180.1

4 183.7

;

PROC GLM;

CLASS operator;

MODEL output=operator;

RANDOM operator;

TITLE ‘Operator Data: One Factor Random Effects Model';

RUN;

These are data from an experiment studying the effect of four operators (chosen randomly) on the output of a particular machine.

slide63

One Factor Random effects Model

The GLM Procedure

Dependent Variable: output

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 3 371.8718750 123.9572917 14.91 0.0002

Error 12 99.7925000 8.3160417

Corrected Total 15 471.6643750

R-Square Coeff Var Root MSE output Mean

0.788425 1.674472 2.883755 172.2188

Source DF Type I SS Mean Square F Value Pr > F

operator 3 371.8718750 123.9572917 14.91 0.0002

The GLM Procedure

Source Type III Expected Mean Square

operator Var(Error) + 4 Var(operator)

slide64

Conclusion:

We rejectHo : sa2 = 0 (p = .0002)

and we conclude that there is

variability due to operator

Note:

Multiple comparisons are not used in random effects analyses

-- we are interested in whether there is variability due to operator

- not interested in which operators performed better, etc. (they were randomly chosen)

slide65

Note:

- The same F test is used to test Ho for the 1-factor Fixed Effects and 1-factor Random effects models are the same

- Interpretations differ