Round 523 to the nearest hundred

F-1

Stat 423, Stat 523 Formulas Chapter 7 Sections 7.1, 7.2, 7.3 We take a random sample X1, …, Xn from N(µ,s2) Two-Sided 100(1-a)% Confidence Intervals for µ

Requirements Confidence Interval Normal, s known

Normal, s unknown

Chapter 8 Sections 8.1, 8.2, 8.4 Steps in Testing Hypotheses 1. null hypothesis H0 and alternative hypothesis Ha H0: µ = µ0 Ha: µ > µ0, µ < µ0 or µ ¹ µ0 where µ0 is the known hypothesized value of µ. 2. test statistic

Requirements Test Statistic Reference Distribution Normal, s known

N(0,1)

Normal, s unknown

tn-1

3. rejection region or P-value (a = level of significance)

Ha

Rejection Region (RR) Z test T test

µ > µ0 z ³ za t ³ ta,n-1 µ < µ0 z £ -za t £ -ta,n-1 µ ¹ µ0 z ³ za/2 or z £ -za/2 t ³ ta/2,n-1 or t £ -ta/2,n-1

Ha

P-value

Z test R Command T test R Command

µ > µ0 1 - P(Z £ z) 1-pnorm(z) P(tn-1 ³ t) 1-pt(t,n-1)

µ < µ0 P(Z £ z) pnorm(z) P(tn-1 £ t) pt(t,n-1)

µ ¹ µ0 2[1 - P(Z £ |z|)] 2*(1-pnorm(abs(z))) 2P(tn-1 ³ |t|) 2*(1-pt(abs(t),n-1)) 4. Conclusion: Reject H0 at the a level of significance if:

• test statistic is inside the RR or P-value < a.

÷÷ ø

ö çç è

æ s ×+

s ×- aa

n zx ,

n zx

22

÷÷ ø

ö çç è

æ ×+×- -- aa n s

tx, n

s tx 1n,1n, 22

n

x z 0

s µ-

=

n s

x t 0

µ- =

g zg 0.100 1.282 0.050 1.645 0.025 1.960 0.010 2.326 0.005 2.576 0.001 3.09

F-2

Chapter 9 Section 9.1 z Tests and CIs Assumptions

• X1, X2, ..., Xm = data from population 1 with mean µ1 and variance s12 • Y1, Y2, ..., Yn = data from population 2 with mean µ2 and variance s22

• data:

Case I Normal Populations with Known Variances Hypothesis Test:

1. H0: µ1 - µ2 = D0 vs. Ha: where D0 is a known constant (zero usually).

2. Test statistic:

3. Rejection region and P-value Ha Rejection Region P-value P-value in R µ1 - µ2 > D0 z ³ za 1 - P(Z £ z) 1-pnorm(z) µ1 - µ2 < D0 z £ -za P(Z £ z) pnorm(z) µ1 - µ2 ¹ D0 z £ -za/2 or z ³ za/2 2[1 - P(Z £ |z|)] 2*(1-pnorm(abs(z)))

100(1-a)% Confidence Intervals for µ1 - µ2:

2-sided CI:

1-sided CIs: ,

where za is defined like in Chapters 7 and 8. Case II Large-Sample Procedures (s1 and s2 are unknown, m>30, n>30) Replace s1 and s2 in Case I with standard deviations s1 and s2.

ïî

ï í ì

===

===

nsize sample ,sdeviation standard ,y mean:2 Sample

msize sample ,sdeviation standard ,x mean:1 Sample

2

1

ï î

ï í

ì

D¹µ-µ D<µ-µ D>µ-µ

021

021

021

)1,0(N~

nm

)yx( z

2 2

2 1

0

s +

s

D-- =

nm z)yx(

2 2

2 1

2

s +

s ±- a

, nm

z)yx( 2 2

2 1

÷ ÷

ø

ö

ç ç

è

æ ¥+

s +

s -- a nm

z)yx( , 2 2

2 1

÷ ÷

ø

ö

ç ç

è

æ s +

s +-¥- a

9A

9B

F-3

Section 9.2 t Test and Confidence Interval • Normal populations, s1 and s2 are unknown, and sample sizes are small.

Case III t-based Procedures

, round down to the nearest integer.

t Test:

1. H0: µ1 - µ2 = D0 vs. Ha: 2. Test statistic:

3. Rejection region and P-value Ha Rejection Region P-value P-value in R µ1 - µ2 > D0 t ³ ta, n P(tn ³ t) 1-pt(t,n) µ1 - µ2 < D0 t £ -ta, n P(tn £ t) pt(t,n) µ1 - µ2 ¹ D0 t £ -ta/2, n or t ³ ta/2, n 2P(tn ³ |t|) 2*(1-pt(abs(t),n)) 100(1-a)% Confidence Intervals for µ1 - µ2:

2-sided CI:

1-sided CI: ,

1n

n s

1m

m s

n

s

m

s

22 2

22 1

22 2

2 1

-

÷÷ ø

ö çç è

æ

+ -

÷÷ ø

ö çç è

æ

÷ ÷ ø

ö ç ç è

æ +

=n

ï î

ï í

ì

D¹µ-µ D<µ-µ D>µ-µ

021

021

021

2 2

2 1

0 t~

n

s

m

s

yx t n

+

D-- = !

n

s

m

s t)yx(

2 2

2 1

,2 +±- na

, n

s

m

s t)yx(

2 2

2 1

, ÷ ÷

ø

ö

ç ç

è

æ ¥++-- na n

s

m

s t)yx( ,

2 2

2 1

, ÷ ÷

ø

ö

ç ç

è

æ ++-¥- na

9C

F-4

Chapter 10 Section 10.1 Single-Factor ANOVA (Equal Sample Sizes) • I = total number of treatments • J = common number of replications of each treatment • µi = mean of treatment i (for i = 1, 2, ..., I) • Xij = random variable that represents the measurement from the jth EU under

treatment i (for i = 1, ..., I and j = 1, ..., J) The One-Way Fixed Model: Xij = µi + Îij where Îi1, Îi2, ..., ÎiJ are iid N(0,s2). Definition Sums of Squares (SS)

Treatment i average: Grand Average:

• Total SS = Treatment i standard deviation = si

• Treatment (Among) SS =

• Error (Within) SS =

Þ ,

-------------------------------------------------------------------------------------- Alternative (Working) Formulas

Let , .

•

•

• SSE = SST – SSTr Remarks: • is called a residual and eij estimates Îij. • SST = SSTr + SSE Þ SSE = SST – SSTr. -------------------------------------------------------------------------------------- ANOVA table:

J

x

x

J

1j ij

.i

å ==

IJ

x

x

I

1i

J

1j ij

..

å å = ==

( )å å = =

-= I

1i

J

1j

2 ..ij xxSST

( )å =

-= I

1i

2 ...i xxJSSTr

( )å å = =

-= I

1i

J

1j

2 .iij xxSSE

( ) )1/( 1

. 2 --= å

= Jxxs

J

j iiji å

=

-= I

1i

2 is)1J(SSE I/sMSE Error Squared Mean

I

1i

2 i ÷÷ ø

ö çç è

æ == å

=

åå å == =

== J

1j ij.i

I

1i

J

1j ij.. xx ,xx IJ

x CF factor correction

2 ..==

CFxSST I

1i

J

1j

2 ij -= å å

= =

CF J

x

SSTr

I

1i

2 .i

-= å =

.iijij xxe -=

Source of Variation

degrees of freedom (df)

Sum of Squares (SS)

Mean Square (MS)

Test Statistic F

P-value

P-value in R

Treatments (Among)

I-1

SSTr

1-pf(F,I-1,I(J-1))

Error (Within)

I(J-1) SSE

Total IJ-1 SST

1I SSTr

MSTr -

= MSE MSTr

F = ( )FFP )1J(I,1I >--

)1J(I

SSE MSE

- =

10A

10B

10C

F-5

When H0: µ1 = µ2 = ... = µI is true, ~ FI-1,I(J-1).

Hypothesis Testing H0: µ1 = µ2 = ... = µI vs. Ha: H0 is false

• F-statistic:

• P-value: P-value = (In R: 1-pf(F,I-1,I*(J-1)))

• rejection region: RR = {F > Fa,I-1,I(J-1)} ------------------------------------------------------------------------------------- Section 10.2 Multiple Comparison in ANOVA (Equal Treatment Reps J) Tukey's Procedure for Simultaneous 100(1-a)% CIs for µi-µj:

T Method for Significant Differences

1. Compute .

2. List the sample means in increasing order. 3. Underline groups of means that do not differ by more than w. --------------------------------------------------------------------------------------

Contrast where .

Hypothesis Test (Equal Sample Sizes J)

1. H0: C = c0 vs. Ha:

2. Test Statistic

3. Rejection Region and P-value Ha Rejection Region P-value P-value in R C > c0 t ³ ta, I(J-1) P(tI(J-1) ³ t) 1-pt(t,I*(J-1)) C < c0 t £ -ta, I(J-1) P(tI(J-1) £ t) pt(t,I*(J-1)) C ¹ c0 t £ -ta/2, I(J-1) or t ³ ta/2, I(J-1) 2P(tI(J-1) ³ |t|) 2*(1-pt(abs(t),I*(J-1)))

---- F Test for H0: C = 0 vs. Ha: C ¹ 0

, Test Statistic

• Rejection Region ; • P-value = , (in R) 1-pf(F,1,I*(J-1)))

MSE MSTr

F =

MSE MSTr

F =

( )FFP )J(I,I >-- 11

( ) J MSE

Qxx )1J(I,I,ji -a±-

J MSE

Qw )1J(I,I, -a=

.ix

å =

µ= I

1i iicC 0c

I

1i i =å

=

ï î

ï í

ì

¹ < >

0

0

0

cC

cC

cC

)1J(I I

1i

2 i

0 t~

J

cMSE

cĈ t -

= å

- =

å =

´= I

1i

2 i

2

c

Ĉ J)C(SS )1J(I,1F~MSE

)C(SS F -=

}FF{RR )1J(I,1, -a>=

( )1, ( 1)I JP F F- >

10D

10E

10F

In R: qtukey(1-a,I,I*(J-1))

F-6

100(1-a)% CIs for Contrast C (Equal Sample Sizes)

2-sided:

1-sided:

------------------------------------------------------------------------------------- Section 10.3 ANOVA for Unequal Sample Sizes Ji = sample size for treatment i, n = SJi (total sample size).

Treatment i total: , Treatment i average:

Grand Average:

• Total Sum of Squares:

• Treatment Sum of Squares:

• Error Sum of Squares: SSE = SST – SSTr Treatment i standard deviation = si

ANOVA table: Source df SS MS F P-value P-value in R Treatments

I-1 SSTr

1-pf(F,I-1,n-I)

Error n-I SSE

Total n-1 SST • Reject Region = {F ³ Fa,I-1,n-I}

-------------------------------------------------------------------------------------- T Method for Significant Differences (Unequal Treatment Reps)

1. Compute for all pairs i,j where i¹j.

2. List the sample means in increasing order. 3. Underline and if they do not differ by more than wij.

J

cMSE

tĈ

I

1i

2 i

)1J(I,2

å =

-a±

J

cMSE

tĈ ,, , J

cMSE

tĈ

I

1i

2 i

)1J(I,

I

1i

2 i

)1J(I,

÷ ÷ ÷ ÷ ÷ ÷

ø

ö

ç ç ç ç ç ç

è

æ

+¥-

÷ ÷ ÷ ÷ ÷ ÷

ø

ö

ç ç ç ç ç ç

è

æ

¥+- åå =

-a =

-a

å =

= iJ

1j ij.i xx

i

.i .i J

x x =

n

x

x

I

1i

J

1j ij

..

i

å å = ==

( )å å = =

-= I

1i

J

1j

2 ..ij

i

xxSST n

x x

2 ..

I

1i

J

1j

2 ij

i

-= å å = =

( )å =

-= I

1i

2 ...ii xxJSSTr n

x

J

x 2.. I

1i i

2 .i -= å

=

å =

-= I

1i

2 ii s)1J(SSE

1I SSTr

MSTr -

= MSE MSTr

F = ( )FFP In,1I >--

In SSE

MSE -

=

J 1

J 1

2 MSE

Qw ji

In,I,ij ÷ ÷ ø

ö ç ç è

æ += -a

.ix

.ix .jx

10G

10H

10I

F-7

F-8

Hypothesis Test with Contrasts (Unequal Sample Sizes)

1. H0: C = c0 vs. Ha:

2. Test Statistic

3. Rejection Region and P-value Ha Rejection Region P-value P-value in R C > c0 t ³ ta, n-I P(tn-I ³ t) 1-pt(t,n-I)) C < c0 t £ -ta, n-I P(tn-I ³ |t|) pt(t,n-I) C ¹ c0 t £ -ta/2, n-I or t ³ ta/2, n-I 2P(tn-I ³ |t|) 2*(1-pt(abs(t),n-I)) 100(1-a)% CIs for Contrast C (Unequal Sample Sizes)

2-sided:

1-sided:

Special Case:

-------------------------------------------------------------------------------------- A Random Effects Model: Xij = µ + Ai + Îij where A1, A2, ..., AI are iid N(0,sA2) and Îi1, Îi2, ..., ÎiJ are iid N(0,s2). E(MSTr) = s2 + rsA2, E(MSE) = s2 where r=(n-SJi2/n)/(I-1). • F=MSTr/MSE tests H0: sA2 = 0 versus Ha: sA2 ¹ 0.

• Estimates: and where r=(n-SJi2/n)/(I-1).

• V(Xij) = s2 + sA2 = total variance observed in measurements • Estimate of V(Xij) =

• % of total variance explained by differences among treatments = %

ï î

ï í

ì

¹ < >

0

0

0

cC

cC

cC

In I

1i i

2 i

0 t~

J

c MSE

cĈ t -

= å

- =

}FF{RR

F~tF

0C:H vs. 0C:H

In,1,

In,1 2

a0

-a

-

>=

=

¹=

å =

-a± I

1i i

2 i

In,2 J

c MSEtĈ

, J

c MSEtĈ,

J

c MSEtĈ ,

I

1i i

2 i

In,

I

1i i

2 i

In, ÷÷ ÷

ø

ö

çç ç

è

æ ¥+-

÷÷ ÷

ø

ö

çç ç

è

æ +¥- åå

= -a

= -a

. J 1

with J

c replace,C If

j

I

1i i

2 i

j å =

µ=

MSEˆ2 =s r

MSEMSTrˆ2A -

=s

( ) 2A2ij ˆˆXV̂ s+s=

2 A

2

2 A

ˆˆ

ˆ 100

s+s

s ´

10J

10K

F-9

Chapter 11 Formulas Set Section 11.1 Two-Factor ANOVA with No Replications Notation • A = 1st factor, I = number of levels of A • B = 2nd factor, J = number of levels of B • Xij = the measurement from the combination of the ith level of A and jth level of B • xij = actual (observed) value of Xij Two-Way Additive Fixed Model Model equation and assumptions are

Xij = µ + ai + bj + Îij

where , and Îij's are iid N(0,s2). The average response at the

level i of A and level j of B is µij = E(Xij) = µ + ai + bj .

-------------------------------------------------------------------------------------- Parameter Estimates

Factor A, level i total and average:

Factor B, level j total and average: ,

Grand Average:

Parameter Estimate

µ

ai

bj

= + + is the predicted or fitted value.

eij = xij - is a residual which estimates Îij.

0 ,0 J

1j j

I

1i i åå

==

=b=a

J

x x ,xx .i.i

J

1j ij.i == å

=

I

x x ,xx j.j.

I

1i ijj. == å

=

IJ

x

x

I

1i

J

1j ij

..

å å = ==

..xˆ =µ

...ii xxˆ -=a

..j.j xx ˆ -=b

ijx̂ µ̂ iâ jb̂

ijx̂

11A

11B

F-10

Hypothesis Tests • Factor A: H0: a1 = a2 = ... = aI = 0 vs. Ha: at least one ai ¹ 0 • Factor B: H0: b1 = b2 = ... = bJ = 0 vs. Ha: at least one bj ¹ 0

Sums of Squares df

IJ-1

I-1

J-1

(I-1)(J-1)

ANOVA Table

Source df SS MS F P-value P-value in R Factor A

I-1

SSA

1-pf(F,I-1,(I-1)*(J-1))

Factor B

J-1 SSB

1-pf(F,J-1,(I-1)*(J-1))

Error (I-1)(J- 1)

SSE

Total IJ-1 SST

, SSE = SST - SSA – SSB

• Factor A: RR = {F=MSA/MSE > Fa,I-1,(I-1)(J-1)} • Factor B: RR = {F=MSB/MSE > Fa,J-1,(I-1)(J-1)}

------------------------------------------------------------------------------------- T Method for Significant Differences Compute

.

Apply • wA to

or • wB to

Block designs: ANOVA, T Method the same as above with Factor A = Blocks. Two-Way Additive Random Model: Xij = µ + Ai + Bj + Îij

( ) å åå å = == =

-=-= I

1i

2 ..

J

1j

2 ij

I

1i

J

1j

2 ..ij IJ

x xxxSST

( ) IJ

x x

J 1

xxJSSA 2 ..

I

1i

2 .i

I

1i

2 ...i -=-= åå

==

( ) IJ

x x

I 1

xxISSB 2 ..

J

1j

2 j.

J

1j

2 ..j. -=-= åå

==

( )å å = =

+--= I

1i

J

1j

2 ..j..iij xxxxSSE

1I SSA

MSA -

= MSE MSA

F = ( )FFP )1J)(1I(,1I >---

1J SSB

MSB -

= MSE MSB

F = ( )FFP )1J)(1I(,1J >---

)1J)(1I( SSE

MSE --

=

å å = =

= I

1i

J

1j

2 ijeSSE

åå ==

b -

+s=a -

+s=s= J

1j

2 j

2 I

1i

2 i

22

1J I

)MSB(E , 1I

J )MSA(E ,)MSE(E

scomparison B factor for I MSE

Qw

scomparison Afactor for J MSE

Qw

)1J)(1I(,J,B

)1J)(1I(,I,A

--a

--a

=

=

.I.2.1 x,...,x,x

J.2.1. x,...,x,x

11C

11D

F-11

Two-Way Additive Random Model: Xij = µ + Ai + Bj + Îij where the Ai's are iid N(0,sA2), the Bj's are iid N(0,sB2), and Îij's are iid N(0,s2).

tests H0: sA2 = 0 vs. Ha: sA2 ¹ 0.

tests H0: sB2 = 0 vs. Ha: sB2 ¹ 0.

Estimates:

total variance = .

--- Two-Way Additive Mixed Model: Xij = µ + Ai + bj + Îij where the Ai's are iid N(0,sA2), Sbj=0, and Îij's are iid N(0,s2).

tests H0: sA2 = 0 vs. Ha: sA2 ¹ 0.

tests H0: b1 = b2 = ... = bJ = 0 vs. Ha: at least one bj ¹ 0.

Estimates: , total variance =

-------------------------------------------------------------------------------------- Section 11.2 Two-Way ANOVA with Replications Two-Way Interaction Fixed Effects Model Xijk = kth observation for level i of A and level j of B.

Xijk = µ + ai + bj + gij + Îijk for i=1, ..., I, j=1, ...,J, k=1, ..., K and where

, for all i, for all j,

and Îij's are iid N(0,s2). The mean response at the level i of A and level j of B is µij = E(Xij) = µ + ai + bj + gij .

-------------------------------------------------------------------------------------- Estimates

, ,

, ,

Parameter Estimate

fitted value residual

µ

ai

bj

gij

2 B

22 A

22 I)MSB(E ,J)MSA(E ,)MSE(E s+s=s+s=s=

MSE MSA

F =

MSE MSB

F =

I MSEMSBˆ,

J MSEMSAˆ,MSEˆ 2B

2 A

2 -=s -

=s=s

( ) 2B2A2ij ˆˆˆXV̂ s+s+s=

å =

b -

+s=s+s=s= J

1j

2 j

22 A

22

1J I

)MSB(E ,J)MSA(E ,)MSE(E

MSE MSA

F =

MSE MSB

F =

J MSEMSAˆ,MSEˆ 2A

2 -=s=s ( ) 2A2ij ˆˆXV̂ s+s=

0 ,0 J

1j j

I

1i i åå

==

=b=a 0 J

1j ijå

=

=g 0 I

1i ijå

=

=g

å å å = = =

= I

1i

J

1j

K

1k ijk... xx IJK

x x ...... = K

x

x

K

1k ijk

.ij

å ==

J

x

x ,xx

J

1j .ij

..i

J

1j

K

1k ijk..i

å å å = = =

== I

x

x ,xx

I

1j .ij

.j.

I

1i

K

1k ijk.j.

å å å = = =

==

.ijijjiij xˆ ˆˆˆx̂ =g+b+a+µ=

ijijkijk x̂xe -=

...xˆ =µ

.....ii xxˆ -=a

....j.j xx ˆ -=b

....j...i.ijij xxxxˆ +--=g

11E

11F

11G

F-12

Hypothesis Tests • Factor A: H0: a1 = a2 = ... = aI = 0 vs. Ha: at least one ai ¹ 0 • Factor B: H0: b1 = b2 = ... = bJ = 0 vs. Ha: at least one bj ¹ 0 • Interaction: H0: gij = 0 for all i,j vs. Ha: at least one gij ¹ 0

ANOVA Table (Two-Way Interaction Fixed Model)

Source df SS MS F P-value P-value in R A

I-1 SSA

1-pf(F,I-1,I*J*(K-1))

B J-1 SSB

1-pf(F,J-1,I*J*(K-1))

Interaction (I- 1)(J-1)

SSAB

1-pf(F,(I-1)*(J-1),I*J*(K-1))

Error IJ(K-1) SSE

Total IJK-1 SST

• Factor A: RR = {F=MSA/MSE > Fa,I-1,IJ{K-1)} • Factor B: RR = {F=MSB/MSE > Fa,J-1,IJ(K-1)} • Interaction: RR = {F=MSAB/MSE > Fa,(I-1)(J-1),IJ(K-1)}

-------------------------------------------------------------------------------------- T Method for Factor Levels (Use only when interactions are not significant.) Note that I=# of A levels, J=# of B levels, K=# of replications.

Section 11.3 Three-Factor Fixed Effects ANOVA

Xijkl = µ + ai + bj + dk + gABij + gACik + gBCjk +gijk + Îijk for i=1, ..., I, j=1, ...,J, k=1, ..., K, l=1, ..., L, where Îijk's are iid N(0,s2) and the sum of parameters over any subscript is 0:

= = = =

= = = = .

The mean response at level i of A, j of B and k of C is µijk = µ + ai + bj + dk + gABij + gACik + gBCjk +gijk .

1I SSA

MSA -

= MSE MSA

F = ( )FFP )1K(IJ,1I >--

1J SSB

MSB -

= MSE MSB

F = ( )FFP )1K(IJ,1J >--

)1J)(1I( SSAB

MSAB --

= MSE MSAB

F = ( )FFP )1K(IJ),1J)(1I( >---

)1K(IJ SSE

MSE -

=

ï ï î

ïï í

ì

=

=

-a

-a

.J..2..1.)1K(IJ,J,B

I..2..1..)1K(IJ,I,A

x,...,x,x to apply IK MSE

Qw

x,...,x,x to apply JK MSE

Qw

=d=b=a ååå ===

K

1k k

J

1j j

I

1i i

I

1i

AB ijå

=

g J

1j

AB ijå

=

g I

1i

AC ikå

=

g å =

g K

1k

AC ik

J

1j

BC jkå

=

g

K

1k

BC jkå

=

g I

1i ijkå

=

g J

1j ijkå

=

g 0 K

1k ijkå

=

=g

11I

11J

11H

F-13

Test of Hypotheses • Factor A: H0: a1 = a2 = ... = aI = 0 vs. Ha: at least one ai ¹ 0 • Factor B: H0: b1 = b2 = ... = bJ = 0 vs. Ha: at least one bj ¹ 0 • Factor C: H0: d1 = d2 = ... = dK = 0 vs. Ha: at least one dk ¹ 0 • AB Interaction: H0: all gABij = 0 vs. Ha: at least one gABij ¹ 0 • AC Interaction: H0: all gACik = 0 vs. Ha: at least one gACik ¹ 0 • BC Interaction: H0: all gBCjk = 0 vs. Ha: at least one gBCjk ¹ 0 • ABC Interaction: H0: all gijk = 0 vs. Ha: at least one gijk ¹ 0

Assume that there are L observations from each ABC level combination (balanced data). Total sample size is IJKL. ANOVA Table (3 Factors Fixed Effects Model) Source df SS MS F P-value* A

I-1 SSA

B J-1 SSB

C K-1 SSC

AB Interaction

(I-1)(J-1) SSAB

AC Interaction

(I-1)(K-1) SSAC

BC Interaction

(J-1)(K-1) SSBC

ABC Interaction

(I-1) ´(J-1)(K-1)

SSABC

Error IJK(L-1) SSE

* In R, 1-pf(F,m,n) gives P(Fm,n > F).

Total IJKL-1 SST ! There should be at least L=2 observations per treatment to test for all interactions. If L=1, there is no MSE and, hence, no F-test of interactions. !

• Factor A: RR = {F=MSA/MSE > Fa,I-1,IJK{L-1)} • Factor B: RR = {F=MSB/MSE > Fa,J-1,IJK(L-1)} • Factor C: RR = {F=MSC/MSE > Fa,K-1,IJK(L-1)} • AB Interaction: RR = {F=MSAB/MSE > Fa,(I-1)(J-1),IJK(L-1)} • AC Interaction: RR = {F=MSAC/MSE > Fa,(I-1)(K-1),IJK(L-1)} • BC Interaction: RR = {F=MSBC/MSE > Fa,(J-1)(K-1),IJK(L-1)} • ABC Interaction: RR = {F=MSABC/MSE > Fa,(I-1)(J-1)(K-1),IJK(L-1)}

T Method for Factor Levels (use when no interaction is significant)

where {total reps per level} = JKL for factor A = IKL for factor B = IJL for factor C

Coefficient of Determination: , Adjusted R2:

1I SSA

MSA -

= MSE MSA

F = ( )FFP )1L(IJK,1I >--

1J SSB

MSB -

= MSE MSB

F = ( )FFP )1L(IJK,1J >--

1K SSC

MSC -

= MSE MSC

F = ( )FFP )1L(IJK,1K >--

)1J)(1I( SSAB

MSAB --

= MSE MSAB

F = ( )FFP )1L(IJK),1J)(1I( >---

)1K)(1I( SSAC

MSAC --

= MSE MSAC

F = ( )FFP )1L(IJK),1K)(1I( >---

)1K)(1J( SSBC

MSBC --

= MSE MSBC

F = ( )FFP )1L(IJK),1K)(1J( >---

)1K)(1J)(1I( SSABC

MSBC ---

= MSE

MSABC F = ( )FFP )1L(IJK),1K)(1J)(1I( >----

)1L(IJK SSE

MSE -

=

level per reps total

MSE Qw df} {MSE levels}, factor of {#, ´= a

SST SSE

R -= 12 SST SSE

Radj ´÷ ø ö

ç è æ-=

df error df total

12

11K

F-14

Latin Squares Design Model Assumptions

Xij(k) = µ + ai + bj + dk + eij(k)

where and eij(k)’s are iid N(0,s2).

N = # of factor levels (note that N=I=J=K)

, , ,

, , ,

Sums of Squares (Latin Squares Design)

Sums of Squares df

N2-1

N-1

N-1

N-1

(N-1)(N-2)

Note: SSE = SST – SSA – SSB – SSC

T Method for Factor Levels: For all factors, use .

-------------------------------------------------------------------------------------- Section 11.4 2p Factorial Experiments, Factor Effects, Yates Algorithm 23 Factorial Model: Xijkl = µ + ai + bj + dk + gABij + gACik + gBCjk +gijk + Îijkl for i=1,2, j=1,2, k=1,2, l=1, ..., L Estimates • • Fitted main effects of factors A, B and C

• Fitted 2-way interactions

• Fitted 3-way interactions

0kji =d=b=aå åå

å= j

)k(ij..i xx å= i

)k(ij.j. xx å= j,i

)k(ijk.. xx å= j,i

)k(ij... xx

N

x x ..i..i = N

x x .j..j. = N

x x k..k.. = 2

... ...

N

x x =

( ) 2

2 ...

N

1i

N

1j

2 )k(ij

N

1i

N

1j

2 ...)k(ij

N

x xxxSST -=-= å åå å

= == =

( ) 2

2 ...

N

1i

2 ..i

N

1i

2 .....i

N

x x

N 1

xxNSSA -=-= åå ==

( ) 2

2 ...

N

1j

2 .j.

N

1j

2 ....j.

N

x x

N 1

xxNSSB -=-= åå ==

( ) 2

2 ...

N

1k

2 k..

N

1k

2 ...k..

N

x x

N 1

xxNSSC -=-= åå ==

( ) 2N

1i

N

1j ...k...j...i)k(ij x2xxxxSSE å å

= =

+---=

N

MSE Qw df} {MSE N,, ´= a

....xˆ =µ

.....k..k......j.j.......ii xx ˆ xxˆ xxˆ -=d-=b-=a

.....k....j..jk. BC jk

.....k.....i.k.i AC ik

......j....i..ij AB ij

xxxxˆ

xxxxˆ

xxxxˆ

+--=g

+--=g

+--=g

.....k....j....i.jk..k.i..ij.ijkijk xxxxxxxxˆ -+++---=g

11L

11M

F-15

Yates Algorithm 1. List sample means (xbars) in Yates standard order.

• Start with (1) then a. • "Multiply by b" the previous treatments to get b and ab. • "Multiply by c" the previous treatments to get c, ac, bc, abc. etc. There should be 2p treatments in the list.

2. The next column is obtained by adding the numbers in the previous column in pairs and subtracting in pairs (2nd minus 1st). Repeat this process p times.

3. Divide the pth new column by 2p. The results are the overall mean and fitted effects (with all factors at the 2nd level).

Reverse the sign of the fitted effect if you change an odd number of subscripts. -------------------------------------------------------------------------------------- Section 11.4 Fractional Factorial Studies A. Choice of 1/2q Fraction of a 2p Factorial 1. Pick any p-q factors and list all their level combinations using -'s and +'s. 2. Pick q different groups of these "first" factors and multiply the signs of the

members of each group. Use the q products to determine the levels of the remaining q factors.

B. Determining the "Alias Structure" of the 1/2q Fraction Multiplication Rules:

• A*A = B*B = ... = I • I*A = A, I*B = B, etc.

1. Take the q generators and apply multiplication so that I is on the left-hand-side of the equation.

2. Multiply (LHS x LHS and RHS x RHS) the new equations in pairs, then in triples, then in sets of four, etc.

(2q - 1) factor products are equivalent to I. Factor effects are aliased in 2p-q groups of 2q members. C. Analyzing a 2p-q Fractional Factorial 1. Initially ignore the "last" q factors and treat the data as a full factorial in

the "first" p-q factors. Estimate the factor effects (e.g. using formulas in Section 11.4 or by Yates algorithm = p-q cycles and divide last cycle by 2p-q) and judge their statistical significance. a. (with replication)

• Compute

or get them from the ANOVA table.

• Compute .

A 100(1-a)% CI for an effect is . Note that an effect is judged not statistically significant at the a level

if .

b. If no replication, do a normal probability plot of fitted effects (exclude ). 2. Interpret the estimates in the light of the alias structure.

1)- size (sample of sumdf 1)- size (sample of sum

]s ingcorrespond1)- size [(sample of sum MSE

2

=

´ =

sizes sample all of sreciprocal of sum 2

1 MSEt)r(

q-pdf,2 ´´=a a

)(r effect ^

a±

)(r effect ^

a<

µ̂

11N

11O

F-16

Chapter 12 Formulas Linear Model: where e's are iid N(mean=0,variance=s2).

• b1 = average change in Y for every unit change in x • µy•x* = b0 + b1x* = average response at x* • s2y•x* = variance of Y at x=x*

--------------------------------------------------------------------------------------

Least-Squares Estimates:

Fitted Value:

Residual:

or .

Estimate of s2: mean square error = .

-------------------------------------------------------------------------------------- Numerical Diagnostics

Sample Correlation:

sx and sy are the sd's of x and y => .

Coefficient of Determination: , SSR = SST-SSE

Adjusted R2 = where MST = SST/(n-1)

ii10i xY e+b+b=

( )

( ) n

x x

n yx

yxˆ 2

i2 i

ii ii

1

åå

ååå -

- =b xˆyˆ 10 b-=b

i10i x ˆˆŷ b+b=

iii ŷye -=

( ) åå ==

=-= n

1i

2 i

n

1i

2 ii eŷySSE ii1i0

2 i yx

ˆyˆySSE ååå b-b-=

2n SSEˆMSE 2 -

=s=

( ) ( )

( ) ( )

( )( ) n

yx yxyyxxS

n

y yyyS or SST

n

x xxxS

ii iiiixy

2 i2

i 2

iyy

2 i2

i 2

ixx

åååå

ååå

ååå

-=--=

-=-=

-=-=

yyxx

xy

SS

S r =

y

x 1 s

sˆr b=

SST SSR

SST SSE

1R2 =-=

MST MSE

1 2n

1R)1n( R

2 2 adj -=-

-- =

12A

12B

12C

F-17

Section 12.3 Inference for b1

, .

100(1-a)% CI for b1 :

Hypothesis Test 1. H0: b1 = b10, Ha: b1 > b10, b1 < b10 or b1 ¹ b10

2. test statistic

3. rejection region and P-value (a = level of significance) Ha Rejection Region P-value P-value in R

b1 > b10 t ³ ta,n-2 1 - P(tn-2 £ t) 1-pt(t,n-2)) b1 < b10 t £ -ta,n-2 P(tn-2 £ t) pt(t,n-2) b1 ¹ b10 t £ -ta/2,n-2 or t ³ ta/2,n-2 2[1 - P(tn-2 £ |t|)] 2*(1-pt(abs(t),n-2)) ANOVA table with F-test to test H0: b1 = 0 versus b1 ¹ 0 (model utility test) See Formulas 12B and 12C for SSR, SSE and SST.

Source df SS MS=SS/df F P-value P-value in R Regression 1 SSR MSR F=MSR/MSE P(F1,n-2 > F) 1-pf(F,1,n-2) Error n-2 SSE MSE Total n-1 SST

The rejection region is {F=MSR/MSE ³ Fa,1,n-2} --------------------------------------------------------------------------------------

Section 12.4 CI for Mean Response µy.x and Prediction Interval at x=x* CI for Mean Response At x=x*, the mean response is µy.x* = b0 + b1x*. 100(1-a)% CI for µy.x*:

where

Prediction Interval 100(1-a)% PI for a response Y at x=x*:

or

MSEˆ2 =s xx

2 2 ˆ

S

ˆ s

1

s =

b

1 ˆ2n,2

1 st ˆ

b-a ×±b

1 ˆ

101

s

ˆ t

b

b-b =

ŷ2n,2 stŷ ×± -a

( ) xx

2*

ŷ S xx

n 1

ss -

+=

( ) xx

2*

2n,2 S xx

n 1

1stŷ -

++×± -a 2 ŷ

2 2n,2

sstŷ +×± -a

12D

12E

F-18

Section 13.1 More on Residuals ith residual (random version) is

•

where

• Standardized residual where

Diagnostic Plots 1. ei* (or ei) versus xi (no pattern) 2. ei* (or ei) versus yi (no pattern) 3. (linear) 4. normal probability plot of ei* (or ei) (linear) Section 13.2 Transformed Variables • intrinsically linear models - function of x and y that can be transformed

as y' = b0 + b1x'

where y' = {function of y only} and x' = {function of x only} Sections 13.4, 13.3 Multiple and Polynomial Regression Model: Y = b0 + b1x1 + b2x2 + ... + bkxk + e where the e's are independently distributed N(0,s2) Data: (x11 , x21, ..., xk1, y1), (x12 , x22, ..., xk2, y2), ..., (x1n , x2n, ..., xkn, yn)