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Experimental Statistics - week 7

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Chapter 15:

Factorial Models (15.5)

Chapter 17:

Random Effects Models

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.

Statistics 5372: Experimental Statistics

Assignment Report Form

Name:

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)

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

Note:For balanceddesigns,

i.e. for STIMULUS data

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

= (.192+.264)/2

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

Balanced Experimental Designs

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

- Uniquely Calculated

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

- Not Uniquely Calculated

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”

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

Model for 3-factor Factorial Design

where

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

Sum-of-Squares Breakdown

(3-factor ANOVA)

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

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

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

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;

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

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

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

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

LSD =

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

70.3+91.0+78.7+88.74

= 82.175

= cell mean for Brand 1 and Time 4.5

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)

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)

Fixed Effects vs Random Effects

This determination affects

- the model

- the hypothesis tested

- the conclusions drawn

- the F-tests involved (sometimes)

1-Factor Random Effects Model

Assumptions:

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 “level-to-level” variation

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

-- at least none is explained by variation due to the factor

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.

t =

n =

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)

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)

RECALL: 1-Factor (Fixed-Effects) ANOVA Table

(page 389)

Rationale for F-test and critical region:

estimates

estimates

+ constant ×

- if no factor effects, we expect F≈ 1;

- if factor effects, we expect F > 1

Expected Mean Squares for 1-Factor ANOVA’s (p.979)

EMS

Source SS df MS Fixed Effects Random Effects

Treatments SST t -1 MST

Error SSE t(n - 1) MSE

Total TSS tn -1

Rationale for Test Statistic and Critical Region is the Same: Fixed or Random

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.

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)

Estimating Variance Components

Solving for sa2we get:

so, we estimate sa2 by

Also,

For OPERATOR Data,

RECALL: 2-Factor Fixed-Effects Model

where

Expected Mean Squares for

2-Factor ANOVA with Fixed Effects:

Expected MS

F-test

A

MSA/MSE

B

MSB/MSE

AB

MSAB/MSE

Error

2-Factor Random Effects Model

Assumptions:

Sum-of-Squares obtained as in Fixed-Effects case

Expected Mean Squares for

2-Factor ANOVA with Random Effects:

Expected MS

A

B

AB

Error

To Test:

we use F =

we use F =

we use F =

Note: Test each of these 3 hypotheses (no matter whether Ho:sab2= 0 is rejected)

2-Factor Random Effects ANOVA Table

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

Estimating Variance Components

2-Factor Random Effects Model

(note error on page 986)

DATA one;

INPUT operator filter loss;

DATALINES;

1 1 16.2

1 1 16.8

1 1 17.1

1 2 16.6

1 2 16.9

1 2 16.8

.

.

.

4 1 14.9

4 2 15.4

4 2 14.6

4 2 15.9

4 3 16.1

4 3 15.4

4 3 15.6

;

PROC GLM;

CLASS operator filter;

MODEL loss=operator filter operator*filter;

TITLE ‘2-Factor Random Effects Model';

RANDOM operator filter operator*filter/test;

RUN;

Filtration Process:

Response - % material lost through filtration

A – Operator (randomly selected) (a = )

B – Filter (randomly selected) (b = )

n =

Operator

1 2 3 4

16.2 15.9 15.6 14.9

1 16.8 15.1 15.9 15.2

17.1 14.5 16.1 14.9

16.6 16.0 16.1 15.4

2 16.9 16.3 16.0 14.6

16.8 16.5 17.2 15.9

16.7 16.5 16.4 16.1

3 16.9 16.9 17.4 15.4

17.1 16.8 16.9 15.6

Filter

SAS Random-Effects Output

(Filtration Data)

2-Factor Random Effects Model

General Linear Models Procedure

Dependent Variable: LOSS

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 11 16.60888889 1.50989899 8.16 0.0001

Error 24 4.44000000 0.18500000

Corrected Total 35 21.04888889

R-Square C.V. Root MSE LOSS Mean

0.789062 2.664175 0.4301163 16.144444

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

OPERATOR 3 10.31777778 3.43925926 18.59 0.0001

FILTER 2 4.63388889 2.31694444 12.52 0.0002

OPERATOR*FILTER 6 1.65722222 0.27620370 1.49 0.2229

Source Type III Expected Mean Square

OPERATOR Var(Error) + 3 Var(OPERATOR*FILTER) + 9 Var(OPERATOR)

FILTER Var(Error) + 3 Var(OPERATOR*FILTER) + 12 Var(FILTER)

OPERATOR*FILTER Var(Error) + 3 Var(OPERATOR*FILTER)

SAS Random-Effects Output – continued

“../test” option

Tests of Hypotheses for Random Model Analysis of Variance

Dependent Variable: LOSS

Source: OPERATOR

Error: MS(OPERATOR*FILTER)

Denominator Denominator

DF Type III MS DF MS F Value Pr > F

3 3.4392592593 6 0.2762037037 12.4519 0.0055

Source: FILTER

Error: MS(OPERATOR*FILTER)

Denominator Denominator

DF Type III MS DF MS F Value Pr > F

2 2.3169444444 6 0.2762037037 8.3885 0.0183

Source: OPERATOR*FILTER

Error: MS(Error)

Denominator Denominator

DF Type III MS DF MS F Value Pr > F

6 0.2762037037 24 0.185 1.4930 0.2229