ED230B/C

Completely Randomized Factorial Design

CRF-pq -- Fixed Effects Model

AKA - Two-way ANOVA or Factorial ANOVA.

Schematic with Example Data

IV B

b1 b2 b3

A a1 24
33
37
29
42 44
36
25
27
43 38
29
28
47
48
a2 30
21
39
26
34 35
40
27
31
22 26
27
36
46
45
a3 21
18
10
31
20 41
39
50
36
34 42
52
53
49
64

Or in abbreviated form

IV B

b1 b2 b3

A a1 S₁
n=5 S₂
n=5 S₃
n=5
a2 S₄
n=5 S₅
n=5 S₆
n=5
a3 S₇
n=5 S₈
n=5 S₉
n=5

Where each S_j is an independent randomly assigned group of subjects.

Linear Model

Y_ijkl = μ + α_j + β_k + γ_l + αβ_jk + αγ_jl + βγ_kl + αβγ_jkl + ε_i(jkl)

where,

Y_ijk is the score for the ith observation in the jkth treatment combination
μ is the overall population mean (grand mean)
α_j is the effect of A treatment level j which is equal to μ_j. - μ
β_k is the effect of B treatment level k which is equal to μ_.k - μ
αβ_jk is the joint effect of treatment levels j and k which is equal to μ_jk - μ_j. - μ_.k + μ
ε_i(jk) is the error effect associated with Y_ijk and is equal to Y_ijk - μ - α_j - β_k - αβ_jk. The error effect is a random variable that is distributed NID(0,s²_ε)

Further:

Σα_j = 0 over j
Σβ_k = 0 over k
Σαβ_jk = 0 over j
Σαβ_jk = 0 over k

Hypotheses

A Main Effect
H₀: α_j = 0 for all j
H₁: α_j ~= 0 for at least one j

B Main Effect
H₀: β_k = 0 for all k
H₁: β_k ~= 0 for at least one k

A*B Interaction
H₀: αβ_jk = 0 for all jk combinations
H₁: αβ_jk ~= 0 for at least one jk combination

Assumptions

1. The linear model reflects all sources of variation.
2. The experiment contains all the treatment levels of interest.
3. The ε_i(jk) are independent of each other.
4. The ε_i(jk) are normally distributed in the population.
5. The ε_i(jk) have equal variance in the population.

ANOVA Summary Table

Source SS df MS F p-value

A Main effect 190.000 2 95.00 1.52 .2324

B Main effect 1543.333 2 771.67 12.35 .0001

A*B Interaction 1236.667 4 309.17 4.95 .0028

Within Cells 2250.000 36 62.50

Total 5220.000 44

Table of the F-distribution

Within Cell = Error or Residual

Fixed-Effects Expected Mean Squares

Cell Means & Standard Deviations

b1 b2 b3
a1 33
6.96 35
8.80 38
9.51

a2 30
6.96 31
6.96 36
9.51

a3 20
7.52 40
6.20 52
7.97

egen cell=group(a b)

tablist cell a b, clean

    cell   a   b   Freq  
       1   1   1      5  
       2   1   2      5  
       3   1   3      5  
       4   2   1      5  
       5   2   2      5  
       6   2   3      5  
       7   3   1      5  
       8   3   2      5  
       9   3   3      5 

tabstat y, by(cell) stat(n mean sd var)

Summary for variables: y
     by categories of: cell (group(a b))

    cell |         N      mean        sd  variance
---------+----------------------------------------
       1 |         5        33  6.964194      48.5
       2 |         5        35  8.803408      77.5
       3 |         5        38  9.513149      90.5
       4 |         5        30  6.964194      48.5
       5 |         5        31  6.964194      48.5
       6 |         5        36  9.513149      90.5
       7 |         5        20  7.516648      56.5
       8 |         5        40  6.204837      38.5
       9 |         5        52  7.968689      63.5
---------+----------------------------------------
   Total |        45        35  10.89203  118.6364
--------------------------------------------------

Graph of Cell Means

Strength of Association

In this example, variables A and B are fixed effects and the appropriate measure of association is the partial omega squared (see Kirk page 397).

For the CRF33 example:

_Y|B:A,A*B

_Y|A*B:A,B

If ω² is negative set ω² to equal zero.

Model for Orthogonal Coding

    A Main  B Main  A*B Interaction
A B X1 X2   X3 X4   X5 X6 X7 X8
1 1  1  1    1  1    1  1  1  1
1 2  1  1   -1  1   -1  1 -1  1
1 3  1  1    0 -2    0 -2  0 -2
2 1 -1  1    1  1   -1 -1  1  1
2 2 -1  1   -1  1    1 -1 -1  1
2 3 -1  1    0 -2    0  2  0 -2
3 1  0 -2    1  1    0  0 -2 -2
3 2  0 -2   -1  1    0  0  2 -2
3 3  0 -2    0 -2    0  0  0  4

Stata Computer Example

input a b y x1 x2 x3 x4
1 1 24  1  1  1  1
1 1 33  1  1  1  1
1 1 37  1  1  1  1
1 1 29  1  1  1  1
1 1 42  1  1  1  1
1 2 44  1  1 -1  1
1 2 36  1  1 -1  1
1 2 25  1  1 -1  1
1 2 27  1  1 -1  1
1 2 43  1  1 -1  1
1 3 38  1  1  0 -2
1 3 29  1  1  0 -2
1 3 28  1  1  0 -2
1 3 47  1  1  0 -2
1 3 48  1  1  0 -2
2 1 30 -1  1  1  1
2 1 21 -1  1  1  1
2 1 39 -1  1  1  1
2 1 26 -1  1  1  1
2 1 34 -1  1  1  1
2 2 35 -1  1 -1  1
2 2 40 -1  1 -1  1
2 2 27 -1  1 -1  1
2 2 31 -1  1 -1  1
2 2 22 -1  1 -1  1
2 3 26 -1  1  0 -2
2 3 27 -1  1  0 -2
2 3 36 -1  1  0 -2
2 3 46 -1  1  0 -2
2 3 45 -1  1  0 -2
3 1 21  0 -2  1  1
3 1 18  0 -2  1  1
3 1 10  0 -2  1  1
3 1 31  0 -2  1  1
3 1 20  0 -2  1  1
3 2 41  0 -2 -1  1
3 2 39  0 -2 -1  1
3 2 50  0 -2 -1  1
3 2 36  0 -2 -1  1
3 2 34  0 -2 -1  1
3 3 42  0 -2  0 -2
3 3 52  0 -2  0 -2
3 3 53  0 -2  0 -2
3 3 49  0 -2  0 -2
3 3 64  0 -2  0 -2
end

generate x5 = x1*x3
generate x6 = x1*x4
generate x7 = x2*x3
generate x8 = x2*x4

or

use http://www.gseis.ucla.edu/courses/data/crf33a, clear

table b,cont(freq mean y sd y) by(a)

----------+-----------------------------------
  a and b |      Freq.     mean(y)       sd(y)
----------+-----------------------------------
1         |
        1 |          5          33    6.964194
        2 |          5          35    8.803409
        3 |          5          38    9.513149
----------+-----------------------------------
2         |
        1 |          5          30    6.964194
        2 |          5          31    6.964194
        3 |          5          36    9.513149
----------+-----------------------------------
3         |
        1 |          5          20    7.516648
        2 |          5          40    6.204837
        3 |          5          52    7.968688
----------+-----------------------------------

parcoord a b y  /*  available from Stata STB 29  */



histogram y, by(a b) normal 



anova y a b a*b

                      Number of obs =      45     R-squared     =  0.5690
                      Root MSE      = 7.90569     Adj R-squared =  0.4732

             Source |  Partial SS    df       MS           F     Prob > F
         -----------+----------------------------------------------------
              Model |     2970.00     8      371.25       5.94     0.0001
                    |
                  a |      190.00     2       95.00       1.52     0.2324
                  b |  1543.33333     2  771.666667      12.35     0.0001
                a*b |  1236.66667     4  309.166667       4.95     0.0028
                    |
           Residual |     2250.00    36       62.50   
         -----------+----------------------------------------------------
              Total |     5220.00    44  118.636364 

omega2 b

 omega squared for b =  0.3352
 effect size   = 0.7101

omega2 a*b

 omega squared for a*b = 0.2597
 effect size   = 0.5923

Plotting Cell Means

anovaplot b a, scatter(msym(none))  /* findit anovaplot */

anovaplot a b, scatter(msym(none))  /* findit anovaplot */

Stata Regression Results

regress y x1 x2 x3 x4 x5 x6 x7 x8

  Source |       SS       df       MS                  Number of obs =      45
---------+------------------------------               F(  8,    36) =    5.94
   Model |     2970.00     8      371.25               Prob > F      =  0.0001
Residual |     2250.00    36       62.50               R-squared     =  0.5690
---------+------------------------------               Adj R-squared =  0.4732
   Total |     5220.00    44  118.636364               Root MSE      =  7.9057

------------------------------------------------------------------------------
       y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
      x1 |        1.5   1.443376      1.039   0.306      -1.427302    4.427302
      x2 |  -1.166667   .8333333     -1.400   0.170      -2.856745    .5234117
      x3 |  -3.833333   1.443376     -2.656   0.012      -6.760635   -.9060318
      x4 |       -3.5   .8333333     -4.200   0.000      -5.190078   -1.809922
      x5 |       -.25   1.767767     -0.141   0.888      -3.835198    3.335198
      x6 |        .25   1.020621      0.245   0.808      -1.819915    2.319915
      x7 |   3.083333   1.020621      3.021   0.005       1.013419    5.153248
      x8 |   1.916667   .5892557      3.253   0.002       .7216008    3.111733
   _cons |         35   1.178511     29.698   0.000       32.60987    37.39013
------------------------------------------------------------------------------

test x1 x2

 ( 1)  x1 = 0.0
 ( 2)  x2 = 0.0

       F(  2,    36) =    1.52
            Prob > F =    0.2324

test x3 x4

 ( 1)  x3 = 0.0
 ( 2)  x4 = 0.0

       F(  2,    36) =   12.35
            Prob > F =    0.0001

test x5 x6 x7 x8

 ( 1)  x5 = 0.0
 ( 2)  x6 = 0.0
 ( 3)  x7 = 0.0
 ( 4)  x8 = 0.0

       F(  4,    36) =    4.95
            Prob > F =    0.0028

xi3: regress y r.a*r.b
r.a               _Ia_1-3             (naturally coded; _Ia_1 omitted)
r.b               _Ib_1-3             (naturally coded; _Ib_1 omitted)
r.a*r.b           _IaXb_#_#           (coded as above)

      Source |       SS       df       MS              Number of obs =      45
-------------+------------------------------           F(  8,    36) =    5.94
       Model |     2970.00     8      371.25           Prob > F      =  0.0001
    Residual |     2250.00    36       62.50           R-squared     =  0.5690
-------------+------------------------------           Adj R-squared =  0.4732
       Total |     5220.00    44  118.636364           Root MSE      =  7.9057

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _Ia_2 |         -3   2.886751    -1.04   0.306    -8.854603    2.854603
       _Ia_3 |        3.5        2.5     1.40   0.170    -1.570235    8.570235
       _Ib_2 |   7.666667   2.886751     2.66   0.012     1.812064    13.52127
       _Ib_3 |       10.5        2.5     4.20   0.000     5.429765    15.57023
   _IaXb_2_2 |         -1   7.071068    -0.14   0.888    -15.34079    13.34079
   _IaXb_2_3 |        1.5   6.123724     0.24   0.808    -10.91949    13.91949
   _IaXb_3_2 |       18.5   6.123724     3.02   0.005     6.080511    30.91949
   _IaXb_3_3 |      17.25   5.303301     3.25   0.002     6.494407    28.00559
       _cons |         35   1.178511    29.70   0.000     32.60987    37.39013
------------------------------------------------------------------------------

describe _Ia_2 - _IaXb_3_3

              storage  display     value
variable name   type   format      label      variable label
-------------------------------------------------------------------------------
_Ia_2           double %10.0g                 a(2 vs. 1)
_Ia_3           double %10.0g                 a(3 vs. 2-)
_Ib_2           double %10.0g                 b(2 vs. 1)
_Ib_3           double %10.0g                 b(3 vs. 2-)
_IaXb_2_2       double %10.0g                 a(2 vs. 1) & b(2 vs. 1)
_IaXb_2_3       double %10.0g                 a(2 vs. 1) & b(3 vs. 2-)
_IaXb_3_2       double %10.0g                 a(3 vs. 2-) & b(2 vs. 1)
_IaXb_3_3       double %10.0g                 a(3 vs. 2-) & b(3 vs. 2-)

test _Ia_2 _Ia_3

 ( 1)  _Ia_2 = 0.0
 ( 2)  _Ia_3 = 0.0

       F(  2,    36) =    1.52
            Prob > F =    0.2324

test _Ib_2 _Ib_3

 ( 1)  _Ib_2 = 0.0
 ( 2)  _Ib_3 = 0.0

       F(  2,    36) =   12.35
            Prob > F =    0.0001

test _IaXb_2_2 _IaXb_2_3 _IaXb_3_2 _IaXb_3_3

 ( 1)  _IaXb_2_2 = 0.0
 ( 2)  _IaXb_2_3 = 0.0
 ( 3)  _IaXb_3_2 = 0.0
 ( 4)  _IaXb_3_3 = 0.0

       F(  4,    36) =    4.95
            Prob > F =    0.0028

Formulas

Linear model,

Prediction model,

where,

thus,

Linear Statistical Models Course

Phil Ender, 11apr06, 12Feb98

IV		B
		b1	b2	b3
A	a1	24 33 37 29 42	44 36 25 27 43	38 29 28 47 48
	a2	30 21 39 26 34	35 40 27 31 22	26 27 36 46 45
	a3	21 18 10 31 20	41 39 50 36 34	42 52 53 49 64

IV		B
		b1	b2	b3
A	a1	S₁ n=5	S₂ n=5	S₃ n=5
	a2	S₄ n=5	S₅ n=5	S₆ n=5
	a3	S₇ n=5	S₈ n=5	S₉ n=5

Source	SS	df	MS	F	p-value
A Main effect	190.000	2	95.00	1.52	.2324
B Main effect	1543.333	2	771.67	12.35	.0001
A*B Interaction	1236.667	4	309.17	4.95	.0028
Within Cells	2250.000	36	62.50
Total	5220.000	44

	b1	b2	b3
a1	33 6.96	35 8.80	38 9.51
a2	30 6.96	31 6.96	36 9.51
a3	20 7.52	40 6.20	52 7.97