CRF-pq -- Fixed Effects Model
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 Sj is an independent randomly assigned group of subjects.
Linear Model
Yijkl = μ + αj + βk + γl + αβjk + αγjl + βγkl + αβγjkl + εi(jkl)
where,
Yijk is the score for the ith observation in the
jkth treatment combination
μ is the overall population mean (grand mean)
αj is the effect of A treatment level j which is equal to
μj. - μ
βk is the effect of B treatment level k which is equal to
μ.k - μ
αβjk is the joint effect of treatment levels j and k
which is equal to
μjk - μj. -
μ.k + μ
εi(jk) is the error effect associated with Yijk
and is equal to Yijk - μ - αj
- βk - αβjk.
The error effect is a random variable that is
distributed NID(0,s2ε)
Further:
Σαj = 0 over j
Σβk = 0 over k
Σαβjk = 0 over j
Σαβjk = 0 over k
Hypotheses
Assumptions
1. The linear model reflects all sources of variation.
2. The experiment contains all the treatment levels of interest.
3. The εi(jk) are independent of each other.
4. The εi(jk) are normally distributed in the population.
5. The εi(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 |
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:
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.philender.com/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
----------+-----------------------------------
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 8 371.25 5.94 0.0001
|
a | 190 2 95 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 36 62.5
-----------+----------------------------------------------------
Total | 5220 44 118.636364
effectsize b
anova effect size for b with dep var = y
total variance accounted for
omega2 = .26849661
eta2 = .29565773
Cohen's f = .60584458
partial variance accounted for
partial omega2 = .33523734
partial eta2 = .40685413
effectsize a#b
anova effect size for a#b with dep var = y
total variance accounted for
omega2 = .18678025
eta2 = .23690932
Cohen's f = .47924933
partial variance accounted for
partial omega2 = .25970608
partial eta2 = .35468451Plotting Cell Means
anovaplot b a, scatter(msym(none)) /* findit anovaplot */Stata Regression Resultsanovaplot a b, scatter(msym(none)) /* findit anovaplot */
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
Stata Regression with anovalator
regress y i.a##i.b
Source | SS df MS Number of obs = 45
-------------+------------------------------ F( 8, 36) = 5.94
Model | 2970 8 371.25 Prob > F = 0.0001
Residual | 2250 36 62.5 R-squared = 0.5690
-------------+------------------------------ Adj R-squared = 0.4732
Total | 5220 44 118.636364 Root MSE = 7.9057
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
a |
2 | -3 5 -0.60 0.552 -13.14047 7.14047
3 | -13 5 -2.60 0.013 -23.14047 -2.85953
|
b |
2 | 2 5 0.40 0.692 -8.14047 12.14047
3 | 5 5 1.00 0.324 -5.14047 15.14047
|
a#b |
2 2 | -1 7.071068 -0.14 0.888 -15.34079 13.34079
2 3 | 1 7.071068 0.14 0.888 -13.34079 15.34079
3 2 | 18 7.071068 2.55 0.015 3.65921 32.34079
3 3 | 27 7.071068 3.82 0.001 12.65921 41.34079
|
_cons | 33 3.535534 9.33 0.000 25.8296 40.1704
------------------------------------------------------------------------------
anovalator a b, main 2way fratio
anovalator main-effect for a
chi2(2) = 3.04 p-value = .21871189
scaled as F-ratio = 1.52
anovalator main-effect for b
chi2(2) = 24.693333 p-value = 4.344e-06
scaled as F-ratio = 12.346667
anovalator two-way interaction for a#b
chi2(4) = 19.786667 p-value = .00055023
scaled as F-ratio = 4.9466667
FormulasLinear model,
Prediction model,
where,
thus,
Linear Statistical Models Course
Phil Ender, 11apr06, 12Feb98