Gui Citovsky Julie Heymann Jessica Sopp Jin Lee Qi Fan Hyunhwan Lee Jinzhu Yu Lenny Horowitz Shuvro Biswas Outline TwoFactor Experiments with Fixed Crossed Factors 2 ID: 759606
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Slide1
Multifactor Experiments
November 26, 2013
Gui
Citovsky, Julie Heymann, Jessica
Sopp
, Jin Lee, Qi Fan,
Hyunhwan
Lee,
Jinzhu
Yu,
Lenny Horowitz,
Shuvro
Biswas
Slide2Outline
Two-Factor Experiments with Fixed Crossed Factors
2
k
Factorial Experiments
Other Selected Types of Two-Factor Experiments
Slide3Two-Factor Experiments with Fixed Crossed Factors
Slide4First, single factor
Comparison of two or more treatments (groups)Single treatment factorExample: A study to compare the average flight distances for three types of golf balls differing in the shape of dimples on them: circular, fat elliptical, thin ellipticalTreatments circular, fat elliptical, thin ellipticalTreatment factor type of ball
Slide5Single factor continued
Slide6Two-Factor Experiments With Fixed Crossed Factors
Two fixed factors, A with
a ≥ 2
levels and B with
b ≥ 2
levels
ab
treatment combinations
If there are
n
observations obtained under each treatment combination (
n
replicates)
,
then there is a total of
abn
experimental units
Slide7Two-Factor Experiments With Fixed Crossed Factors
Example: Heat treatment experiment to evaluate the effects of a
quenching medium
(two levels: oil and water) and
quenching temperature
(three levels: low, medium, high) on the surface hardness of steel
2 x 3 = 6 treatment combinations
If 3 steel samples are treated for each combination, we have
N =
18 observations
Slide8Model and Estimates of its Parameters
Let
y
ijk=kth observation on the (i,j)th treatment combination, i=1,2,…,a , j=1,2,…,b, and k=1,2,…,n.Let random variable Yijk correspond to observed outcome yijk.Basic Model: and independentwhere
Slide9Table format
Slide10Parameters
Grand Mean:
i
th
Row Average:
j
th
Column Average:
(
i,j
)
th
Row Column Interaction
i
th Row Main Effect:
j
th
Column Main Effect:
Slide11Least Squares Estimates
Slide12Variance
Sample variance for (i, j)th cell is:Pooled estimate for σ2:
Slide13Example
Experiment to study how mechanical bonding strength of capacitors depends on the type of substrate (factor A) and bonding material (factor B).3 substrates: Al2O3 with bracket, Al2O3 no bracket, BeO no bracket4 types of bonding material: Epoxy I, Epoxy II, Solder I and Solder IIFour capacitors were tested at each factor level combination
Slide14Example continued
Pooled sample variance:
Slide15Example continued: Sample Means
Slide16Example continued: Other Model Parameters
Slide17Two- Way Analysis of Variance
We define the following sum of squares:
Slide18Analysis of Variance
Degrees of Freedom:SST: N – 1SSA: a – 1 SSB: b – 1 SSAB: (a – 1)(b – 1)SSE: N – abSST = SSA + SSB + SSAB + SSE.Similarly, the degrees of freedom also follow this identity, i.e.
Slide19Analysis of Variance
Mean squares =
Slide20Hypothesis Test
We test three hypotheses:
Not all
Not all
Not all
If all interaction terms are equal to zero, then the effect of one factor on the mean response does not depend on the level of the other factors.
Slide21When do we reject H0?
Use F-statistics to test our hypotheses by taking the ratio of the mean squares to the MSE. Reject Reject RejectWe test the interaction hypothesis H0AB first.
Slide22Summary (Table 13.5)
Source of Variation (Source)Sum of Squares (SS)Degrees of Freedom (d.f.)Mean Square (MS)FMain Effects Aa – 1Main Effects Bb – 1Interaction AB(a – 1)(b – 1)ErrorN – abTotalN – 1
Slide23Example: Bonding Strength of Capacitors
Data
Capacitors;
input
Bonding $ Substrate $ Strength @@;
Datalines
;
Epoxy1 Al203 1.51 Epoxy1 Al203 1.96 Epoxy1 Al203 1.83 Epoxy1 Al203 1.98
…
;
proc
GLM
plots
=diagnostics
data
=Capacitors;
TITLE
"Analysis of Bonding Strength of Capacitors"
;
CLASS
Bonding Substrate;
Model
Strength = Bonding | Substrate
;
run
;
Slide24Bonding Strength of Capacitors ANOVA Table
At α=0.05, we can reject
H
0B
and
H
0AB
but fail to reject
H
0A
.
The main effect of bonding material and the interaction between the bonding material and the substrate are both significant.
The main effect of substrate is not significant at our α.
Slide25Main Effects Plot
Definition: A main effects plot is a line plot of the row means of factor and A and the column means of factor B.
Slide26Interaction Plot
Slide27Model Diagnostics with Residual Plots
Why do we look at residual plots? Is our constant variance assumption true?Is our normality assumption true?
Slide282
k
Factorial Experiments
Slide292k Factorial Experiments
2
k
factorial experiments is a class of multifactor experiments consists of design in which each factor is studied at 2 levels.
If there are k factors, then we have 2
k
treatment combinations
2-factor
and
3-factor
experiments can be generalized
to >3-factor experiments
Slide3022 experiment
22 Experiment: experiment with factors A and B, each at two levels.
ab = (A high, B high) a = (A high, B low)b = (A low, B high) (1) = (A low, B low)
Slide3122 experiment cont’d
Yij ~ N(µi, σ2) i = (1), a, b, ab j = 1, 2, … , n
Assume a balanced design with n observations for each treatment combinations, denote these observations by y
ij
Slide3222 experiment cont’d
Main effect of factor A (): difference in the mean response between the high level of A and the low level of A, averaged over the levels of BMain effect of factor B (: difference in the mean response between the high level of B and the low level of B, averaged over the levels of AInteraction effect of AB (): difference between the mean effect of A at the high level of B and at the low level of B
= = ( =
Slide3322 experiment cont’d
Est. Main Effect A = = Est. Main Effect B = = Est. Interaction AB = =
The least square estimates of the main effects and the interaction effects are obtained by replacing the treatment means by the corresponding cell sample means.
Slide34Contrast Coefficients for Effects in a 22 ExperimentTreatmentcombinationEffectIABAB(1)+--+a++--b+-+-ab++++
*Notice that the term-by-term products of any two contrast vectors equal the third one
2
2
experiment cont’d
Slide3523 experiment
23 Experiment: experiment with factors A, B, and C with n observations. Yij ~ N(µi, σ2), i = (1), a, b, ab, c, ac, bc, abc j = 1, 2, … , n.
Est. Main Effect A =
Est. Main Effect B =
Est. Main Effect C =
Slide3623 experiment cont’d
Est. Interaction Effect AB =
Est. Interaction Effect BC =
Est. Interaction Effect AC =
Est. Interaction Effect ABC =
Slide37Contrast coefficients for Effects in a 23 ExperimentTreatment CombinationEffectIABABCACBCABC(1)+--+-++-a++----++b+-+--+-+ab++++----c+--++--+ac++--++--bc+-+-+-+-abc++++++++
2
3
experiment cont’d
Slide3823 experiment example
Factors affecting bicycle performance:Seat height (Factor A): 26" (-), 30" (+)Generator (Factor B): Off (-), On(+) Tire Pressure (Factor C): 40 psi (-), 55 psi (+)
Slide3923 experiment example cont’d
Travel times
from Bicycle Experiment
Factor
Time
(
Secs
.)
A
B
C
Run 1
Run 2
Mean
-
-
-
51
54
52.5
+
-
-
41
43
42.0
-
+
-
54
60
57.0
+
+
-
44
43
43.5
-
-
+
50
48
49.0
+
-
+
39
39
39.0
-
+
+
53
51
52.0
+
+
+
41
44
42.5
Slide4023 experiment example cont’d
A =
= -10.875B = = 3.125C = = -3.125AB = = -0.625AC = = 1.125BC = = 0.125ABC = = 0.875
significant
Slide412k experiment
2k experiments, where k>3.n iid observations yij (j = 1,2,…n) at the ith treatment combination and its sample mean yi (i = 1,2,…, 2k) has the following estimated effect.
Est. Effect =
Slide42Statistical Inference for 2k Experiments Basic Notations and Derivations
d.f.
Slide43CI and Hypotheses Test with t Test
Therefore a CI for any population effect is given byThe t-statistic for testing the significance of any estimated effect is
Slide44Hypotheses Test with F Test
Equivalently, we can use F test to do itThe estimated effect is significant at level if
Slide45Sums of Squares for Effects
The effects are mutually orthogonal contrasts.
Slide46Regression Approach to 2k Experiments
a 22 experimentMultiple regression model
Slide47Regression Approach to 2k Experiments
23 experimentIf all interactions are dropped from the model, the new fitted model is
Slide48Regression Approach to 2k Experiments
The interpolation formula
Slide49Bicycle Example: Main Effects Model
main effects model minimum travel time
A(seat height)= -10.875 B(generator) = 3.125 C(tire pressure) = -3.125 = 47.1875
Slide50Bicycle Example: Main Effects Model
= 1.56+5.0625+0.0625+4.1875 = 10.875 d.f. = 4Pure SSE = 33.5 d.f = 8pooled SSE = 33.5 + 10.875 d.f. = 12 (total)MSE = = 3.698
Sums of squares for omitted interactions effects
Slide51Bicycle Example:
Residual Diagnostics
To check model assumptions
proc
glm
plots=diagnostics data = biker;class A B C;model travel= A|B|C;run;
Residuals
Normality
Equal error variance
Slide52Single Replicated Case
Unreplicated case: n =1Problems in statistical testing0 degrees of freedom for error, cannot use formal tests and C.I. to estimate of error and assess effectsPotential solutionsPooling high-order interactions to estimate errorGraphical approach: normal plot against effectsEstimated effectsIndependent, orthogonal, normally distributed, common variance (
Unusual response? Noise? Spoiling
the
results?
Slide53Single Replicated Case
Effect
Sparsity
principle
If number of effects is large (e.g. k= 4, 15 effects), a majority of them are small ~N (0,
σ
2
), few a large and more
influential ~ (u≠0,
σ
2
)
Reduced model
retaining only significant effects, omitting non-significant ones
Obtain sums of squares for omitted effects => pooled error sum of squares (SSE) (Error due to ignoring negligible effects)
Error
d.f.
= # pooled omitted effects
MSE = SSE/error
d.f.
Perform formal statistical inferences
Slide54Other Types of Two-Factor Experiments
Section 13.3
Slide55Two-Factor Experiments with (Crossed and) Mixed Factors
A is fixed factor with a levels
B is random factor with b levels
Assume a balanced design with n ≥ 2
obs’s
at each of (a x b) treatment combinations
Slide56Example:
Compare three testing laboratories
Material tested comes in batches
Several samples from each batch tested in each laboratory
Laboratories represent a fixed factor
Batches represent
a
random factor
Two factors are crossed, since samples are tested from each batch in each laboratory
Model?
Slide57Mixed Effects Model
Y
ijk
= µ +
τ
i
+ ß
j
+ (
τ
ß)
ij
+
Є
ijk
µ
,
τ
i
are fixed parameters
ß
j
, (
τ
ß)
ij
are random parameters
Є
ijk
i.i.d
. N(0,
σ
2
) random errors
Slide58The (Probability) Distribution of the Random Effects
The random βj are the main effects of B, which are assumed to be i.i.d. N(0, σβ2) where σβ2 is called the variance component of the B (random factor) main effect. The distribution of βj would therefore be f βj (x) = exp(-x2/2σβ2 )
Slide59+Variance Components ModelSST = SSA + SSB +SSAB +SSE (same as fixed-effects model)
Variance Components Model
Slide60Expected Mean Squares
E(MSA
) =
σ
2
+ n
σ
2
AB
+ n
Σ
i
a
τ
i
2
/(a-1
)
E(MSB) =
σ
2
+ n
σ
2
AB
+ an
σ
2
B
E(MSAB) =
σ
2
+ n
σ
2
AB
E(MSE) =
σ
2
Slide61Unbiased estimators of variance components
2 = MSE2AB = (MSAB - 2 )/n2B = (MSB - 2 - n 2AB) /an
Slide62Common tests
H
0A
:
τ
1
=
… =
τ
a
= 0
vs. H
1A
: At least one
τ
i
≠ 0
H
0B
:
σ
2
B
= 0
vs. H
1B
:
σ
2
B
>
0
H
0AB
:
σ
2
AB
= 0
vs. H
1AB
:
σ
2
AB
> 0
Slide63Common tests: results
Reject H
0A
if F
A
= MSA/MSAB > f
a-1,(a-1)(b-1),
α
Reject H
0B
if F
B
= MSB/MSAB > f
b-1,(a-1)(b-1),
α
Reject H
0AB
if F
AB
= MSAB/MSE > f
(a-1)(b-1),v,
α
Slide64Two-Factor Experiments w. Nested and Mixed Factors
Model:
Where,
Slide65Two-Factor Experiments w. Nested and Mixed Factors
Orthogonal Decomposition of Sum of Squares
Slide66Two-Factor Experiments w. Nested and Mixed Factors
ANOVA Table
Slide67Illustrative Example
Consider the Following Experiment:~ A Concentration of Reactant~ B Concentration of Catalyst
Slide68Analysis with SAS
Code
Slide69Analysis with SAS
Selected Output
Slide70Summary
Two factor experiments with multiple levelsModel:We can decompose the Sum of Squares as:And compute test statistics under Ho, as:
Slide71Summary
2^k Factorial Experimentsk factors, 2 levels eachCalculate the Sum of Squares due to an effect as
Slide72Acknowledgements
Tamhane
,
Ajit
C., and Dorothy D. Dunlop. "Analysis of Multifactor Experiments."
Statistics and Data Analysis: From Elementary to Intermediate
. Upper Saddle River, NJ: Prentice Hall, 2000
.
Cody, Ronald P., and Jeffrey K. Smith. "Analysis of Variances: Two Independent Variables."
Applied Statistics and the SAS Programming Language
. 5th ed. Upper Saddle River, NJ: Prentice Hall, 2006
.
Prof. Wei Zhu
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