# Analysis of Split-Plot Experiments

Document Sample

Design & Analysis of Split-Plot
Experiments (Univariate Analysis)
Elements of Split-Plot Designs
• Split-Plot Experiment: Factorial design with at least 2
factors, where experimental units wrt factors differ in
“size” or “observational points”.
• Whole plot: Largest experimental unit
• Whole Plot Factor: Factor that has levels assigned to
whole plots. Can be extended to 2 or more factors
• Subplot: Experimental units that the whole plot is split
• Subplot Factor: Factor that has levels assigned to
subplots
• Blocks: Aggregates of whole plots that receive all levels
of whole plot factor
Examples
• Agriculture: Varieties of a crop or gas may need to be
grown in large areas, while varieties of fertilizer or
varying growth periods may be observed in subsets of
the area.
• Engineering: May need long heating periods for a
process and may be able to compare several
formulations of a by-product within each level of the
heating factor.
• Behavioral Sciences: Many studies involve repeated
measurements on the same subjects and are analyzed as
a split-plot (See Repeated Measures lecture)
Design Structure
• Blocks: b groups of experimental units to be exposed
to all combinations of whole plot and subplot factors
• Whole plots: a experimental units to which the whole
plot factor levels will be assigned to at random within
blocks
• Subplots: c subunits within whole plots to which the
subplot factor levels will be assigned to at random.
• Fully balanced experiment will have n=abc
observations
Data Elements (Fixed Factors, Random Blocks)
• Yijk: Observation from wpt i, block j, and spt k
• m : Overall mean level
• a i : Effect of ith level of whole plot factor (Fixed)
• bj: Effect of jth block (Random)
• (ab )ij : Random error corresponding to whole plot
elements in block j where wpt i is applied
• g k: Effect of kth level of subplot factor (Fixed)
• (ag )ik: Interaction btwn wpt i and spt k
• (bc )jk: Interaction btwn block j and spt k (often set to 0)
• e ijk: Random Error= (bc )jk+ (abc )ijk
• Note that if block/spt interaction is assumed to be 0, e
represents the block/spt within wpt interaction
Model and Common Assumptions
• Yijk = m + a i + b j + (ab )ij + g k + (ag )ik + e ijk
a

a
i 1
i   0

b j ~ NID(0,  b )
2

( ab) ij ~ NID(0,  ab )
2

c

g
k 1
k   0
a                  c

 (a g )
i 1
ik     (a g ) ik  0
k 1

e ijk ~ NID(0. e2 )
COV (b j , ( ab) ij )  COV (b j , e ijk )  COV ((ab) ij , e ijk )  0
Mean and Covariance Structure (Fixed
Whole Plot and Subplot Factors)

E (Yijk )  m  a i  g k  (ag ik

2
b
     e i  i' j  j ' k  k '
2
ab
2


  b   ab i  i' j  j ' k  k '
2     2
COV (Yijk , Yi ' j 'k ' )  
      b i  i' j  j ' k , k '
2


       0 j  j ' i, i' , k , k '
Obtaining Variances of Sums & Means
 a b c           a b c
V   Yijk    V (Yijk     COV (Yijk , Yijk ' )
a   b    c

                
 i 1 j 1 k 1  i 1 j 1 k 1 i 1 j 1 k 1 k ' k
a          b     c    c                    a    a     b           c    c
  COV (Yijk , Yi ' jk ' )    COV (Yijk , Yi ' j 'k ' )
i 1 i ' i j 1 k 1 k '1                i 1 i '1 j 1 j ' j k 1 k '1

(      2
        (
 abc  b2   ab   e2  abc(c  1)  b2   ab 
2

a (a  1)bc 2 b2  a 2b(b  1)c 2 (0)
 a 2bc 2 b2  abc 2 ab  abc e2
2

(      a 2bc 2 b2  abc 2 ab  abc e2 1 2 1 2
2
1 2
 V Y ...                              2
  b   ab      e
(abc)               b     ab      abc
Variances of Other Means
( 
V Y i..
1 2 1 2
  b   ab   e
b       b
1 2
bc
( 
V Y . j.
1 2
  b   ab   e
2

a
1 2
ac
( 
V Y ..k
1 2 1 2
  b   ab   e
b        ab
1 2
ab
( 
V Y ij .
1 2
  b   ab   e
2     2

c
(  (
V Y i .k
1 2
  b   ab   e
b
2
2

( 
V Y . jk
1 2 1 2
  b   ab   e
2

a        a
(                                    
a                                                                a
Whole Plot Trt : bc Y i..  Y ...                                                                           bc Y i..  abcY ...
2                          2                        2

i 1                                                             i 1

(                                
b                                                                        b
Blocks : ac Y . j .  Y ...                                                  ac Y . j .  abcY ...
2                                      2                      2

j 1                                                                      j 1

(                                                            
a            b
Whole Plot Error : c  Y ij .  Y i..  Y . j .  Y ...
2

i 1 j 1
Analysis of            a       b                               a                                          a
 c  Y                          bc Y                                  ac Y . j .  abcY ...
2                                      2                                           2                     2
ij .                                   i ..
Variance             i 1 j 1                               i 1                                     j 1

(                                    
c                                                                        c
Subplot Trt : ab Y ..k  Y ...                                                                  ab Y ..k  abcY ...
2                                    2                      2

k 1                                                                  k 1

(                                                       
a          c
WP * SP Interaction : b Y i .k  Y i..  Y ..k  Y ...
2

i 1 k 1
a       c                                 a                                            c
 b Y                            bc Y                                   ab Y ..k  abcY ...
2                                        2                                         2                      2
i .k                                     i ..
i 1 k 1                                i 1                                        k 1

(                                                                   
b         c
Subplot Error : a  Y . jk  Y . j .  Y ..k  Y ...
2

j 1 k 1

 (Y                                                                                                                                                      
a       b       c                                                                                                                                              2

ijk      Y ij .  Y i.k  Y . jk  Y i..  Y . j .  Y ..k  Y ...
i 1 j 1 k 1
a       b       c                             a          b                                      a          c                           a
  Y  c  Y                                                                     b Y                                       bc Y i..
2                                          2                                          2                           2
ijk                                          ij .                                       i .k
i 1 j 1 k 1                             i 1 j 1                                           i 1 k 1                              i 1
Expected Values in Analysis of Variance
(          
2

 a bc  b  abc  ab  abc e  m
2        1 
EY   ...             2  2  2      2  2         2      2

 abc 

E(             bc  bc   bc   bc  (m  a )
2
2              1
    
2       2            2       2                 2                               2
Y   i ..                                     b                    ab                e                           i
    

E(              ac  a c   ac   ac  m
2
2              1
     
2       2       2            2        2                    2           2
Y   . j.                                         b                     ab                   e
     

E(              ab  a b  ab  ab  (m  g )
2
2              1
     
2       2                    2                     2                           2
Y   ..k                                      b                    ab                e                           k
     

E(             1  c   c   c  (m  a )
2
2
 
2       2           2    2                     2                           2
Y   ij .                             b                ab                    e                       i
c

E(              b  b  b  b  (m  a  g  (a g )
2
2              1
 
2               2                  2
Y   i .k                             b               ab                 e                       i           k           ik )2
 

E(               1  a   a  a  (m  g )
2
2
 
2       2            2                         2                       2
Y   . jk                                 b            ab                    e                       k
a
E (Yijk    b   ab   e2  ( m  a i  g k  (a g ) ik ) 2
2
2      2
Expected Mean Squares (Fixed WP&SP Trts)

E (MSBLOCKS   ac b2  c ab   e2
2
df BLOCKS  b  1

i1 a i2
a

E (MSWP   c ab   e2  bc
2
dfWP  a  1
a 1
E (MSBLK WP   c ab   e2
2
df BLK WP  (a  1)(b  1)

k 1 g k2
c

E (MSSP    e2  ab                     df SP  c  1
c 1
i 1 k 1 (a g ) ik
a      c      2

E (MSWPSP    e2  b                              dfWPSP  (a  1)(c  1)
(a  1)(c  1)
E (MSERROR    e2     df ERROR  a (b  1)(c  1)
Tests for Fixed Effects
Whole Plot Trt Effec ts: H 0 : a1    a a  0
MSWP
Test Statistic : FWP 
MSBLOCK *WP
P  P ( F  FWP | F ~ Fa 1.( a 1)(b 1) )
WP

Subplot Trt Effec ts: H 0 : g 1    g c  0
MSSP
Test Statistic : FSP 
MSERROR
P  P ( F  FSP | F ~ Fc 1.a ( b 1)(c 1) )
SP

WP  SP Interac tio : H 0 : (a g ) ik  0 i, k
n
MSWP SP
Test Statistic : FWP SP     
MSERROR
P  SP  P ( F  FWP SP | F ~ F( a 1)(c 1).a ( b 1)(c 1) )
WP
Comparing Whole Plot Trt Means
(
E Y i..  Y i '..                      1 c     1 c                       1 c     1 c          
  m  a i   g k   (ag )ik    m  a i '   g k   (ag )i 'k   a i  a i '
          bc k 1 bc k 1                   bc k 1 bc k 1      
V (Y    i ..    Y i '..     V (Y  V (Y  2COV (Y
i ..       i ..       i ..   , Y i '..   
 b c         b c        b c c
COV (Yi.. , Yi '..   COV    Yijk ,  Yi ' jk    COV (Yijk , Yi ' jk ' )
                       
 j 1 k 1  j 1 k 1   j 1 k 1 k '1
b               c      c
  COV (Yijk , Yi ' j 'k ' )  bc 2 b2  b(b  1)c 2 (0)
j 1 j ' j k 1 k '1

(    1
 COV Y i.. , Y i '..   b2
b

(            1
1 2 1              1  2 2 1 
 V Y i..  Y i '..  2  b2   ab   e2   b2    ab   e2 
b       b           bc     b  b     c 

(                    
2MSBLOCK WP
^
 V Y i..  Y i '.. 
bc

(
95% CI for (a i  a i '  : Y i..  Y i '..  t          
2MSBLOCK WP
bc
Comparing Subplot Trt Means
(
E Y ..k  Y ..k '              1 a           1 a                1 a             1 a           
  m   a i  g k   (ag ) ik    m   a i  g k '   (ag ) ik '   g k  g k '
    ab i 1       ab i 1            ab i 1         ab i 1       
V (Y   ..k    Y ..k '     V (Y  V (Y  2COV (Y
..k              ..k '                ..k   , Y ..k '   
 a b                   a b
COV (Y..k , Y..k ' )  COV    Yijk ,  Yijk '    COV (Yijk , Yijk '  
a b

                      
 i 1 j 1  i 1 j 1  i 1 j 1

 COV (Y                                                                                             
, Yi ' jk '     COV (Yijk , Yi ' j 'k '   ab  b2   ab  a (a  1)b b2  a 2b(b  1)(0)
a            b                                 a            a   b
2
ijk
i 1 i ' i j 1                               i 1 i '1 j 1 j ' j

(     1

1 2
 COV Y ..k , Y ..k '   b2   ab
b       ab

(
 V Y ..k  Y ..k ' 
2

(ab) 2
a b b  ab ab  ab e  a b b  ab ab 
2   2

2        2    2   2       2    2 e2
ab

(                  
2 MSERROR
^
 V Y ..k  Y ..k ' 
ab

95% CI for (g k  g k ' ) :               (Y   ..k      Y ..k '  t    2 MSERROR
ab
Comparing Subplot Effects in Same WPT
(                
E Y i.k  Y i.k '  m  a i  g k  (a g ) ik   m  a i  g 'k  (a g ) ik '  
(g k  g k '   ((a g )ik  (a g )ik ' 
(                 (  (                                (
V Y i.k  Y i.k '  V Y i.k  V Y i.k '  2COV Y i.k , Y i.k '                  
b                        b
COV (Yi.k , Yi.k ' )   COV (Yijk , Yijk ' )    COV (Yijk , Yij 'k ' ) 
j 1                     j 1 j ' j

                                             (
b  b   ab  b(b  1)(0)  COV Y i.k , Y i.k ' 
2     2
1 2
b
 b   ab
2
            
(
 V Y i .k  Y i .k '       2 2
b

 b   ab
2
  e2   b   ab 
2     2    2 e2

b
(                  2 MSERROR
^
 V Y i .k  Y i .k '       
b

95% CI for (g k  g k '   ((a g ) ik  (a g ) ik '  :        (Y   i .k           
 Y i .k '  t
2 MSERROR
b
Comparing WPT Effects in Same Subplot TRT
(                     
E Y i .k  Y i '.k  m  a i  g k  (a g ) ik   m  a i '  g 'k  (a g ) i 'k  
(a i  a i '   ((a g ) ik       (a g ) i 'k 
(                            (          (                  (
V Y i .k  Y i '.k  V Y i .k  V Y i '.k  2COV Y i .k , Y i '.k         
b                           b
COV (Yi .k , Yi '.k )   COV (Yijk , Yi ' jk )    COV (Yijk , Yi ' j 'k ) 
j 1                        j 1 j ' j

     2
b
1 2
b   b(b  1)(0)  COV Y i .k , Y i .k '   b
b
(                
(                                 2  ab   e2
 (            
2
2 2
 V Y i .k  Y i '.k   b   ab   e   b 
2      2     2

b                              b
2MSBLOCK WP  (c  1) MSERROR 
(               
^
 V Y i .k  Y i '.k 
bc

95% CI for (a i  a i '   ((a g ) ik  (a g ) ik '  :
2MSBLOCK WP  (c  1) MSERROR 
(Y   i .k           
 Y i '.k  t
bc
(df given below )
Approximate Degrees of Freedom
(Satterthwaite)

^

(c  1) MS ERROR  MS BLOCK WP 
2

 (c  1) MS ERROR  MS BLOCK WP  
2                 2

                                    
 a (b  1)( c  1)    (a  1)(b  1) 
Random WP & SP Effects Model
Yijk  m  ai  b j  (ab) ij  ck  (ac) ik  e ijk
( 
ai ~ NID 0,     2
a

b ~ NID(0,  
j
2
b

(ab) ~ NID(0,  
ij
2
ab

c ~ NID(0,  
k
2
c

(ac) ~ NID(0,  
ik
2
ac

e ~ NID(0,  
ijk
2
e

All random effects are assumed to be mutually pairwise independent
Random Effects Model: Mean/Variance
Structure
E (Yijk   m                i, j , k
 a   b2   ab   c2   ac   e2 i  i ' , j  j ' , k  k '
2             2             2


           a   b2   ab i  i ' , j  j ' , k  k '
2            2

           a   c2   ac i  i ' , j  j ' , k  k '
2            2


           b2   c2 i  i ' , j  j ' , k  k '
COV (Yijk , Yi ' j 'k '   

 a i  i' , j  j ' , k  k '
2

                   b2 i  i' , j  j ' , k  k '

                   c2 i  i' , j  j ' , k  k '
                   0 i  i' , j  j ' , k  k '



Expected Mean Squares (Random WP&SP Trts)

E (MSBLOCKS   ac  c   e
2
b
2
ab
2
df BLOCKS  b  1
E (MSWP   bc  c  b   e
2
a
2
ab
2
ac
2
dfWP  a  1
E (MSBLK WP   c ab   e2
2
df BLK WP  (a  1)(b  1)
E (MSSP   ab  b   e
2
c
2
ac
2
df SP  c  1
E (MSWPSP   b   e
2
ac
2
dfWPSP  (a  1)(c  1)
E (MSERROR    e2
df ERROR  a (b  1)(c  1)
Testing for WP Random Effects

H0 : a  0 H A : a  0
2            2

Note : E (MSWP   E (MSERR   E (MSBLK WP   E (MSWPSP    a
2

MSWP  MSERR
Test Statistic : FWP   
MSBLK WP  MSWPSP
Approximate Degrees of Freedom :

1 
(MSWP  MSERR       2
2 
(MSBLK WP  MSWPSP     2

(MSWP 2  (MSERR 2               (MSBLK WP 2  (MSWPSP 2
a 1      a(b  1)(c  1)         (a  1)(b  1)   (a  1)(c  1)
Testing for SP Effects and WP/SP Interaction

Subplot Effects: H 0 :  c2  0 H A :  c2  0
MSSP
Test Statistic : FSP 
MSWPSP
Degrees of Freedom : 1  c  1  2  (a  1)(c  1)

WP  SP Interaction : H 0 :  ac  0 H A :  ac  0
2              2

MSWPSP
Test Statistic : FWPSP   
MSERR
Degrees of Freedom : 1  (a  1)(c  1)  2  a (b  1)(c  1)
Mixed Effects Model (Fixed WP, Random SP)
Yijk  m  a i  b j  (ab) ij  ck  (ac) ik  e ijk
a

a
i 1
i   0

b j ~ NID 0,  b
2
(           
( ab) ij ~ NID 0,  ab
2
(           
ck ~ NID 0,  c2(           
( ac) ik ~ N 0,  ac
2
(           
 ac
2
COV ((ac) ik , (ac) i 'k                    i  i '
a 1
COV ((ac) ik , (ac) i 'k '   0 i, i ' , k  k '
e ijk ~ NID(0,  e2 
Mixed Effects (Fixed WP) -
Mean/Variance Structure
E (Yijk )  m  a i          i, j , k

 b2   ab   c2   ac   e2 i  i ' , j  j ' k  k '
2              2


            b2   ab i  i ' , j  j ' k  k '
2

            c2   ac i  i ' , j  j ' k  k '
2

                    1
  b2   c2            ac i  i ' , j  j ' k  k '
2

COV (Yijk , Yi ' j 'k '   
                  a 1

               b2 i  i ' , j  j ' , k  k '
                1
       c2          ac i  i ' , j  j ' , k  k '
2

               a 1

                 0 i, j  j ' , k  k '
Expected Mean Squares (Fixed WP/Random SP)

E (MSBLOCKS   ac  c   e
2
b
2
ab
2
df BLOCKS  b  1
a

 a i2
E (MSWP   c  b
a
2
 ac   e  bc i 1
2      2
dfWP  a  1
a 1                   a 1
ab

E (MSBLK WP   c ab   e2 df BLK WP  (a  1)(b  1)
2

E (MSSP   ab c2   e2   df SP  c  1

E (MSWPSP   b
a
 ac   e2 dfWPSP  (a  1)(c  1)
2

a 1
E (MSERROR    e2 df ERROR  a(b  1)(c  1)
Testing for WP Fixed Effects
H 0 : a1    a a  0 H A : Not all a i  0

Note : E (MSWP   E (MSERR   E (MSBLK WP   E (MSWPSP   bc
 a i2
a 1
MSWP  MSERR
Test Statistic : FWP   
MSBLK WP  MSWPSP
Approximate Degrees of Freedom :

1 
(MSWP  MSERR 2            2 
(MSBLK WP  MSWPSP 2
(MSWP 2  (MSERR 2               (MSBLK WP 2  (MSWPSP 2
a 1      a(b  1)(c  1)        (a  1)(b  1)   (a  1)(c  1)
Testing for SP Effects and WP/SP Interaction

Subplot Effects: H 0 :  c2  0 H A :  c2  0
MSSP
Test Statistic : FSP 
MSERR
Degrees of Freedom : 1  c  1  2  a (b  1)(c  1)

WP  SP Interaction : H 0 :  ac  0 H A :  ac  0
2              2

MSWPSP
Test Statistic : FWPSP   
MSERR
Degrees of Freedom : 1  (a  1)(c  1)  2  a (b  1)(c  1)
Mixed Effects Model (Random WP, Fixed SP)
Yijk  m  ai  b j  (ab) ij  g k  (ac) ik  e ijk
( 
ai ~ NID 0,  a
2

b ~ NID(0,  
j
2
b

(ab) ~ NID(0,  
ij
2
ab
c

g
k 1
k    0
2
(
(ac) ik ~ N 0,  ac       
 ag
2

COV ((ac) ik , (ac) ik '              k  k '
c 1
COV ((ac) ik , (ac) i 'k '   0 k , k ' , i  i '
e ijk ~ NID(0,  e2 
Mixed Effects (Fixed SP) -
Mean/Variance Structure
E (Yijk )  m  g k         i, j , k

 a   b2   ab   ac   e2 i  i ' , j  j ' , k  k '
2             2      2

 2                    1
 a   b   ab 
2      2
 ac i  i ' , j  j ' , k  k '
2

COV (Yijk , Yi ' j 'k '   
                     a 1
 a   ac i  i ' , j  j ' , k  k '
2     2

               b2 i  i ' , j  j ' , k , k '

               0 i  i ' , j  j ' , k , k '
Expected Mean Squares (Random WP/Fixed SP)
E (MSBLOCKS   ac b2  c ab   e2
2
df BLOCKS  b  1
E (MSWP   bc a  c ab   e2
2      2
dfWP  a  1
E (MSBLK WP   c ab   e2
2
df BLK WP  (a  1)(b  1)
c

 g k2
E (MSSP   b
a
 ac   e  ab k 1
2      2
df SP  c  1
a 1                  c 1
E (MSWPSP   b
c
 ac   e2 dfWPSP  (a  1)(c  1)
2

c 1
E (MSERROR    e df ERROR  a(b  1)(c  1)
2
Testing Main Effects and WP/SP Interaction

Whole plot Effects: H 0 :  a  0 H A :  a  0
2             2

MSWP
Test Statistic : FWP                 df : 1  a  1  2  (a  1)(b  1)
MSBLK WP
Subplot Effects: H 0 : g 1    g c  0 H A : Not all g k  0
MSSP
Test Statistic : FSP                 df : 1  c  1  2  (a  1)(c  1)
MSWPSP
WP  SP Interaction : H 0 :  ac  0 H A :  ag  0
2              2

MSWPSP
Test Statistic : FWPSP                df : 1  (a  1)(c  1)  2  a (b  1)(c  1)
MSERR

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