10/8/2015ENGM 720: Statistical Process Control1 ENGM 720 - Lecture 12 Introduction to Designed...

04/21/23 ENGM 720: Statistical Process Control 1

ENGM 720 - Lecture 12

Introduction to Designed Experiments

04/21/23 TM 720: Statistical Process Control 2

Assignment:

Reading:• Chapter 13

• Section 13.1 through 13.4

Assignment:• None. Study for Final.

• Purposes for Experimental Design

• Experimental Design Terms


What is an Experiment?

Montgomery (2001): • A test or series of tests in which purposeful changes are

made to the input variables of a process or system so that we may observe or identify the reasons for changes that may be observed in the output response.

Strategic manipulation of a system in order to observe and understand its’ response.

Usually, sequential experiments are better:• One variable at a time – misses interactions

• All variable combinations – too expensive


Some Examples of Elements in Experimentation

Purpose:• Characterizing

• Screening

• Optimizing Strategy:

• One-factor-at-a-time

• Comprehensive

• Sequential Design:

• Simple Comparison

• Response Surface methods

• Factorial

• Fractional Factorial


Experimental Factors (Terms)

Design Factors

• Design (varied) Factors

• Constant (held-constant) Factors

• Allowed to Vary Factors

Nuisance Factors• Controllable

• Uncontrollable

• Noise


Identifying Factors & Ranges

Experience Team Approach Fishbone Diagrams

• Four M’s and an E

• Man

• Material

• Machine

• Method

• Environment

Trial/Pilot Runs


Some More Terms Primary Factors = Design Factors - manipulated levels

Treatments = Levels

Blocking - making comparisons under homogeneous conditions

Replications - all actions required to set the experimental conditions are taken for each observation.

Repeated Measures - observations that cannot be randomized in order.


Controlled Experimental Observation Control

• Blocking

• Randomization

• Replication

• Replication vs. Repeated Measures

Observation• Main Effects

• Interaction Effects

• Estimation

• Location

• Variation


7 Steps of Designed Experiments

1. Statement of problem

2. Selection of response variable

3. Choice of factors, levels, and ranges

4. Choice of experimental design

5. Perform the experiment protocol

6. Statistical analysis of the data

7. Conclusions & recommendations

Can do in any order


Example: Eye Drop Effectiveness

Purpose: Determine better of two eye drops Blocking Variable: Patients Why Block

• Variation (due to patients) is great, perhaps greater than effect of medication

To Block• Assign one medicine to an eye, and the other

medicine to the other eye

• Randomized variables? (left vs. right eye,…)


Example: Gas Mileage (Octane)

Purpose: Mileage w/ Fuel Quality Desired Blocking Variable: Time of Day

(Traffic Load) Why Repeated Measures, Not Replications

• Can’t empty tank and replace octane immediately

• Variation (due to sequence of trips) is just giving information on measurement accuracy for traffic

To Replicate• Repeat trip conditions (time, etc.) with both octanes

• Lurking variables? (summer vs. academic year,…)


Effects: Main

A Main Effect is the difference between responses at different levels of a Design Factor

Example: Intelligence Drug• Design Factors

• Drug

• Student

Yes

No

Avg Good

75%

84%

90%

99%


Effects: Interactions

An Interaction is the failure of a factor to produce the same effect on the response at different levels

Example: Intelligence Drug• Design Factors

• Drug

• School

Yes

No

ESU SDSMT

45%

75%

99%

95%


Introduction to Comparisons Comparisons usually look for an effect that is comparably

large with respect to the variation present

Visual Inference Testing• Dot Diagrams / Barcode Plots

• Applications• Small Data Sets

• Subjective?

Statistical Inference Testing• Applications

• Large Data Sets

• More powerful to find smaller effects

• Objective?


Visual Tests of Comparison Stragglers defined:

• Left stragglers are observations less than the larger of the two minima

• Right stragglers are observations greater than the smaller of the two maxima

• Total number of stragglers = Left stragglers + Right stragglers

Tukey’s Quick Test Tukey, J. W. (1959)

• If the total number of stragglers is 8 or more, then the locations can be judged statistically significant at the .05 level• Significance level is about .035 for larger sample sizes


Visual Tests of Comparison Three-Straggler Rule Lenth, R. V. (1994)

• If there are at least 3 left stragglers and at least 3 right stragglers, then the locations can be judged statistically significant at the .05 level

• Should have at least 5 observations in each set

• Significance is about .035 for larger sample sizes

Modified Quick Test Lenth, R. V. (1994)

• Conclude a statistical difference in location if the total number of stragglers is 8 or more, or if there are at least 3 stragglers at each end.

• Significance level is almost exactly .05


Ex 1: Popcorn Brand - Method Purpose:

• Determine the best process for popping corn seeds Response Variable:

• Number of un-popped seeds (50% unbroken by flower) Factors:

• Brand (design, two discrete levels)• Orville Redenbacher - Regular

• Jolly Time - Yellow

• Method (design, three discrete levels)• Microwave Bowl

• Hot Air

• Oil Skillet

• Time (constant, continuous 2:00 minutes, except as noted)


Experiment Data

Brand

Pilot Runs Orville Redenbacher - Regular

Jolly Time - Yellow

1OR 2JT 1 2 3 4 1 2 3 4

Meth

od

M.W 4* 5* 26 18 7 37 5 9 14 34

H.A. 64* 18* 22 26 32 38 20 22 20 17

Oil 446* 10* 10 110 86 14 9 224 4 81

* Time was 2:30 (min:sec); otherwise, time was 2:00 min


Run Sequence

Run Order (Brand)

Mthd PR1 PR2 E1 E2 E3 E4 E5 E6 E7 E8

MWOR

4*

JT

5*

JT

5

JT

9

JT

14

OR

26

OR

18

OR

7

JT

34

OR

37

HAOR

64*

JT

18*

OR

22

OR

26

JT

20

JT

22

JT

20

OR

32

OR

38

JT

17

OilOR

446*

JT

10*

JT

9

OR

10

OR

110

OR

86

JT

224

OR

14

JT

4

JT

81

* Time was 2:30 (min:sec); otherwise, time was 2:00 min


Basic Statistical Concept Noise results in variation = Experimental Error

• Should be unavoidable, certainly uncontrolled, and indicates that the measured value is a Random Variable (abbreviated r.v.).


Definitions (review from Lect 6)

Analysis-of-variance (ANOVA) is a statistical method used to test hypotheses regarding more than two sample means.



The strategy in an analysis of variance is to compare the variability between sample means to the variability within sample means. If they are the same, the null hypothesis is accepted. If the variability between is bigger than within, the null hypothesis is rejected.

Null Hypothesis

AlternativeHypothesis



An experimental unit is the item measured during an experiment. The errors in these measurements are described by random variables.

It is important that the error in measurement be the same for all treatments (random variables must be independent and have the same distribution).

The easiest way to assure the error is the same for all treatments is to randomly assign experimental units to treatment conditions.



The variable measured in an experiment is called the dependent variable.

The variable manipulated or changed in an experiment is called the independent variable.

Independent variables are also called factors, and the sample means within a factor are called levels or treatments.



Random samples of size n are selected from each of k different populations. The k different populations are classified on the basis of a single criterion or factor. (one-factor and k treatments)

It is assumed that the k populations are independent and normally distributed with means µ1, µ2, ... , µk, and a common variance 2.

Hypothesis to be tested is

H0 : 1 2 k

H1 : At least two of the means are not equal



A fixed effects model assumes that the treatments have been specifically chosen by the experimenter, and our conclusions apply only to the levels chosen

Fixed Effect Statistical Model:

where ij is a iid N(0,2). A random effects model assumes the treatments are random

samples from a larger population, and our conclusions apply to the larger population in general.

Because the fixed effects model assumes that the experiment is performed in a random manner, a one-way ANOVA with fixed effects is often called a completely randomized design.

.ijiijiijy Overall Mean

ith Treatment Effect

Error in MeasurementObserved Value



For a fixed effects model, if we restrict:

Then

is equivalent to:

01

k

ii

j)(i,pair oneleast at for :

: 210

jiA

k

H

H

i oneleast at for 0:

0: 210

iA

k

H

H


Analysis of the Fixed Effects Model (review from Lect 6)

Treatment

1 2 … i … k

y11 y21 … yi1 … yk1

y12 y22 … yi2 … yk2

y1n y2n … yin … ykn

Total T1 T2 … Ti … Tk T

Mean y1 y2 … yi … yk y

... ... ... ...



Sum of Squares Treatments:The sum of squares treatments is a measure of the variability between the factor levels.

Error Sum of Squares:The error sum of squares is a measure of the variability within the factor levels.

Factor level 1

Factor level 2

Factor level 3

X3 X1 X2

Sum of Squares Treatments (SSTr)

Sum of Squares Errors (SSE)


Analysis of the Fixed Effects Model (review from Lect 6) Sum of Squares Partition for One Factor Layout:

In a one factor layout the total variability in the data observations is measured by the total sum of squares SST which is defined to be

k

i

n

jij

k

i

n

jij

k

i

n

jij kn

yyyknyyySST

1 1

22

1 1

22

1 1

2

Total Sum of SquaresSST

Treatment Sum of SquaresSSTr

Error Sum of SquaresSSE


Analysis of the Fixed Effects Model (review from Lect 6) Sum of Squares Partition for One Factor Layout:

This can be partitioned into two componentsSST = SSTr + SSE,

where the sum of squares for treatments (SSTr)

measures the variability between the factor levels, and the sum of squares for error (SSE)

measures the variability within the factor levels.

kn

y

n

yyknynyynSSTr

k

i

ik

ii

k

ii

2

1

22

1

2

1

2

k

i

k

iin

jij

k

i

k

ii

n

jij

k

i

n

jiij k

y

yynyyySSE1

1

2

1

2

1 1

2

1

2

1 1

2



Source Degrees of Freedom

Sum of Squares

Mean Squares F-statistic p-value

Treatments k-1 SSTr 1

k

SSTrMSTr

MSE

MSTrF )( ,1 FFP kNk

Error N-k SSE kN

SSEMSE

Total N-1 SST


ANOVA Example (review from Lect 6)

The tensile strength of a synthetic fiber used to make cloth for men’s shirts is of interest to a manufacturer. It is suspected that strength is affected by the percentage of cotton in the fiber.

Five levels of cotton percentage are of interest: 15%, 20%, 25%, 30%, and 35%.

Five observations are to be taken at each level of cotton percentage and the 25 total observations are to be run in random order.



Percentage of Cotton

Observation 15 20 25 30 35

1 7 12 14 19 7

2 7 17 18 25 10

3 15 12 18 22 11

4 11 18 19 19 15

5 9 18 19 23 11

Total 49 77 88 108 54

Average 9.8 15.4 17.6 21.6 10.6

Tensile Strength of Synthetic Fiber (lb/in2)



Source ofVariation

Degrees ofFreedom

Sum ofSquares Mean Square F

% Cotton(Treatments)

475.76

Error 161.20

Total 636.96

5 -1= 4

24 -4= 20

5*5 -1= 24

475.76/4 = 118.94

161.20/20 = 8.06

118.94/8.06 = 14.75


Critical Points for the F-Distribution

1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 14 60 120 INF1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 243.90 245.95 248.02 249.05 250.10 245.36 252.20 253.25 254.302 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.42 19.48 19.49 19.503 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.71 8.57 8.55 8.534 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.87 5.69 5.66 5.635 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.64 4.43 4.40 4.376 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.96 3.74 3.70 3.677 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.53 3.30 3.27 3.238 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.24 3.01 2.97 2.939 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 3.03 2.79 2.75 2.71

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.86 2.62 2.58 2.5411 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.74 2.49 2.45 2.4112 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.64 2.38 2.34 2.3013 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.55 2.30 2.25 2.2114 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.48 2.22 2.18 2.1315 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.42 2.16 2.11 2.0716 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.37 2.11 2.06 2.0117 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.33 2.06 2.01 1.9618 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.29 2.02 1.97 1.9219 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.26 1.98 1.93 1.8820 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 2.22 1.95 1.90 1.8421 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 2.20 1.92 1.87 1.8122 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 2.17 1.89 1.84 1.7823 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 2.15 1.86 1.81 1.7624 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 2.13 1.84 1.79 1.7325 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 2.11 1.82 1.77 1.7126 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 2.09 1.80 1.75 1.6927 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 2.08 1.79 1.73 1.6728 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 2.06 1.77 1.71 1.6529 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 2.05 1.75 1.70 1.6430 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 2.04 1.74 1.68 1.6240 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.95 1.64 1.58 1.5160 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.86 1.53 1.47 1.39120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.78 1.43 1.35 1.26INF 3.84 3.00 2.61 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.69 1.32 1.22 1.03

Degrees of Freedom #1 (v1)DOF #2 (v2)

Alpha = 0.05



Anova: Single Factor

SUMMARYGroups Count Sum Average Variance

15 5 49 9.8 11.220 5 77 15.4 9.825 5 88 17.6 4.330 5 108 21.6 6.835 5 54 10.8 8.2

ANOVASource of Variation SS df MS F P-value F critBetween Groups 475.76 4 118.94 14.757 9E-06 2.866081Within Groups 161.2 20 8.06

Total 636.96 24



Mean Fiber Strength

0

5

10

15

20

25

30

15 20 25 30 35

Percentage Cotton

Ten

sile

Str

eng

th (l

b/in

^2)


Assumptions of ANOVA Models

Analysis of Variance models make the following assumptions with regard to the underlying structure of the data:• The error variance is a Normal random variable with

mean equal to zero and variance equal to 2.• The error variance is the same (homogeneous) for all

conditions.• The error variance is independent from trial to trial.

Violation of these assumptions can have only minor effects or could have very large effects - depending on the data set and the assumptions.


Residuals

Violations in the assumptions of ANOVA models are most often uncovered through examining the residuals:

iij

ijijij

yy

yye ˆ

-1 0 1 2 3 4 5 6 7 8 9

-6

-4

-2

0

2

4

6

8

10

12

14

Fitted Values of ACCURACY-NUMBER OF ERRORS

Res

idua

ls o

f AC

CU

RA

CY

-NU

MB

ER

OF

ER

RO

RS


Normality Assumption The normality assumption can be evaluated by comparing

residuals with values that would be expected from a Normal distribution.

If fewer residuals are available (more typical), then normal probability plots can be used. A good approximation to the expected value of the kth smallest observation in a random sample of size n is:

Not much can be done to correct for violations of this assumption. However, ANOVA’s are very robust with respect to this assumption.

25.0375.0

nk

zMSE


Equal Variance Assumption

The equal variance assumption is usually checked by plotting the residuals versus the predicted or fitted value. Characteristic patterns that indicate unequal variance are cone-shaped:

Residual

Fitted Value

Residual

Fitted Value


Equal Variance Example

-40

-30

-20

-10

0

10

20

30

40

50

430 440 450 460 470 480 490 500 510 520

Predicted Value

Res

idu

al


Factorial Experiments

Experiments are often performed to investigate the effects of two or more independent variables on a single dependent variable.

The simplest experimental design to accomplish this is called the factorial or full factorial experiment. When employing this design, each complete trial or replication is done at every possible combination of the independent variables.

Factors arranged a full factorial design are often said to be crossed.


Main Effects and InteractionsFactorial Experiment -- No Interaction

0

10

20

30

40

50

60

A1 A2Factor A

Res

pon

se

B1

B2

Factorial Experiment -- Interaction

0

5

1015

20

25

3035

40

45

A1 A2Factor A

Res

pon

seB1

B2

Factor B

Factor B

20 30

40.6 51

B1 B2

A1

A2

Factor B

Factor A

20 30

40.6 14

B1 B2

A1

A2

Factor B

Factor A


The Two-Factor Factorial Design

1 2 … b

1y111, y112, …, y11n

y121, y122, …, y12n

y1b1, y1b2, …, y1bn

2y211, y212, …, y21n

y221, y222, …, y22n

y2b1, y2b2, …, y2bn

a

ya11, ya12, …, ya1n

ya21, ya22, …, ya2n

yab1, yab2, …, yabn

Factor B

Factor A



Fixed Effect Statistical Model:

where ijk is an iid N(0,2) random variable. Hypotheses:

.ijkijjiijky

0 oneleast at :

0: 210

A

a

H

H

0 oneleast at :

0: 210

A

b

H

H

0 oneleast at :

allfor 0:0

ijA

ij

H

i,jH

Overall Mean

ith Factor A Effect

Error in Measurement

Observed Value

jth Factor B Effect i jth Interaction Effect



Total Sum of SquaresSSTO

SS Treatment ASSA

SS ErrorSSE

SS Treatment BSSB

SS Interaction ABSSAB



Sum of Squares:

abn

yySS

a

i

b

j

n

kijkTO

2

1 1 1

2

abn

y

bn

ySS

a

i

iA

2

1

2

abn

y

an

ySS

b

j

jB

2

1

2

abn

y

an

y

bn

y

n

ySS

b

j

ja

i

ia

i

b

j

ijAB

2

1

2

1

2

1 1

2

ABBATOE SSSSSSSSSS



Source of Variation

Degrees of Freedom

Sum of Squares

Mean Square F0 FCrit

A Main Effect

a-1 SSA

SSA

a-1

MSA

MSE

F ,a-1,ab(n-1)

B Main Effect

b-1 SSB

SSB

b-1

MSB

MSE

F ,b-1,ab(n-1)

AB Interaction

(a-1)(b-1) SSAB

SSAB

(b-1)(a-1)

MSAB

MSE

F(a-1)(b-1),ab(n-1)

Error ab(n-1) SSE

SSE

ab(n-1)

Total abn-1 SSTO

ANOVA Table


Two-Factor ANOVA Example 1

There are two different driving routes from the factory to the port, Route 1 and Route 2, and the time of the day when the truck leaves the factory is classified as being either in the morning, the afternoon, or the evening.

Driving route will be considered as Factor A with a=2 levels and period of day will be considered as Factor B with b=3 levels.



Morning Afternoon Evening

Route 1

X111=490X112=553X113=489X114=504X115=519

(X11=511.0)

X121=511X122=490X123=489X124=492X125=451

(X12=486.6)

X131=435X132=468X133=463X134=450X135=444

(X13=452.0)

(X1=483.2)

Route 2

X211=485X212=489X213=475X214=470X215=459

(X21=475.6)

X221=456X222=460X223=464X224=485X225=473

(X22=467.6)

X231=406X232=422X233=459X234=442X235=464

(X23=438.6)

(X2=460.6)

(X1=493.3) (X2=477.1) (X3=445.3) (X=471.9)

Period of Day


Two-Factor ANOVA Example 1Route 1 Route 2

Morning 490 485553 489489 475504 470519 459

Afternoon 511 456490 460489 464492 485451 473

Evening 435 406468 422463 459450 442444 464

ANOVASource of Variation SS df MS F P-value F critPeriod of Day 11925.6 2 5962.8 15.9561 3.9E-05 3.4028Route 3830.7 1 3830.7 10.2507 0.00383 4.2597Interaction 653.6 2 326.8 0.8745 0.42994 3.4028Error 8968.8 24 373.7

Total 25378.7 29



Period of Day Ev Af MoMean 445.3 477.1 493.3

Ev 445.3 0 -31.8 -48 p=3

Af 477.1 0 -16.2 p=2

Mo 493.3 0

3.4931.4773.445MorningAfternoonEvening yyy



400

420

440

460

480

500

520


Period of Day

Dri

ve T

ime

Route 1Route 2

420

430

440

450

460

470

480

490

500


Period of day

Dri

ve T

ime

445

450

455

460

465

470

475

480

485

Route 1 Route 2

Route

Dri

ve T

ime


Two-Factor ANOVA Example 2 An experimenter is interested in evaluating the relative

effectiveness of three drugs (Factor B) in bringing about behavioral changes in two categories schizophrenics and depressives, of patients (Factor A).

What is considered to be a random sample of 9 patients belonging to Category a1 (schizophrenics) is randomly divided into three subgroups with three patients in each subgroup. Each subgroup is then assigned to one of the drug conditions. An analogous procedure is followed for 9 patients belonging to Category a2 (depressives).

Criterion ratings are made of the behavior of each subject before and after the administration of the drugs.


Two-Factor ANOVA Example 2Drug b1 Drug b2 Drug b3

Category a1 8 4 0 10 8 6 8 6 4

Category a2 14 10 6 4 2 0 15 12 9

ANOVA

Source of Variation SS df MS F P-valueCategory 18 1 18 2.0377358 0.1789399Drugs 48 2 24 2.7169811 0.1063435Interaction 144 2 72 8.1509434 0.0058103Error 106 12 8.833

Total 316 17



0

2

4

6

8

10

12

14

Drug B1 Drug B2 Drug B3

Rat

ings

Dif

fere

nce

Category A1Category A2



0

2

4

6

8

10

12

14

Category A1 Category A2

Rat

ings

Dif

fere

nce

Drug B1Drug B2Drug B3


The Two Factor Factorial Effects Model The Fixed Effects Model is:

Estimated parameters for the Effects Model are:

ijkijjiijky

yyyy

yy

yy

y

yy

jiijij

jj

ii

ijijk

)(

ˆ

ˆ

ˆ

ˆ


Three-Factor Models and Beyond Model: Sum of Squares:

ijklijkjkikijkjiijkly

abcn

y

abn

y

acn

y

bcn

y

an

y

bn

y

cn

y

n

ySS

abcn

y

abn

y

acn

y

an

ySS

abcn

y

abn

y

bcn

y

bn

ySS

abcn

y

acn

y

bcn

y

cn

ySS

abcn

y

abn

ySS

abcn

y

acn

ySS

abcn

y

bcn

ySS

abcn

yySS

c

k

kb

j

ja

i

i

b

j

c

k

jka

i

c

k

kia

i

b

j

ija

i

b

j

c

k

ijkABC

b

j

c

k

kjb

j

c

k

jkBC

c

k

ka

i

ia

i

c

k

kiAC

b

j

ja

i

ia

i

b

j

ijAB

c

k

kC

b

j

jB

a

i

iA

a

i

b

j

c

k

n

lijklTO

2

1

2

1

2

1

2

1 1

2

1 1

2

1 1

2

1 1 1

2

2

1 1

22

1 1

2

2

1

2

1

2

1 1

2

2

1

2

1

2

1 1

2

2

1

22

1

22

1

2

2

1 1 1 1

2

,,,


Three-Factor Models and Beyond

TO

EE

E

ABCijkABCABC

E

BCjkBCBC

E

ACikACAC

SE

ABijABAB

E

CkCC

E

BkBB

E

AkAA

SSabcnTotal

MSSSnabcErrorMS

MSF

cba

nMSSScbaABC

MS

MSF

cb

anMSSScbBC

MS

MSF

ca

bnMSSScaAC

M

MSF

ba

cnMSSSbaAB

MS

MSF

c

abnMSSScC

MS

MSF

b

acnMSSSbB

MS

MSF

a

bcnMSSSaA

1

1111

111

1111

1111

1111

11

11

11

2

0

2

2

0

2

2

0

2

2

0

2

2

0

22

0

22

0

22

Source df SS MS E(MS) F


Questions & Issues

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10/8/2015ENGM 720: Statistical Process Control1 ENGM 720 - Lecture 12 Introduction to Designed...

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