RBF-pq
Schematic with Example Data
Level s a1 a2 a3
b1 s1
s2
s3
s4
s5
37
42
33
29
24
39
30
34
26
21
31
21
20
18
10
b2 s1
s2
s3
s4
s5
43
44
36
27
25
35
40
31
22
27
41
50
39
36
34
b3 s1
s2
s3
s4
s5
48
47
29
38
28
46
36
45
27
26
64
52
53
42
49
Or in the abbreviated form,
| Level | a1 | a2 | a3 |
| b1 | S1 n = 5 | S1 n = 5 | S1 n = 5 |
| b2 | S1 n = 5 | S1 n = 5 | S1 n = 5 |
| b3 | S1 n = 5 | S1 n = 5 | S1 n = 5 |
Linear Model
Yijk = μ + αj + βk + αβjk + πi + εijk
μ = overall poulation mean
αj = the effect of treatment level j
βk = the effect of treatment level k
αβjk = the joint effects of treatment level j and k
πi = the effect of block i
εijk = experimental error
Hypotheses
Assumptions
| 1. | Independence |
| 2. | Normality |
| 3. | Homogeneity of variance |
| 4. | Non-additivity |
| 5. | Variance-covariance matrix a. compound symmetry b. circularity c. sphericity |
Expected Mean Squares
E(MS a) = σ2ε + nσ2α E(MS b) = σ2ε + nσ2β E(MS a*b) = σ2ε + nσs2αβ E(MS blks) = σ2ε + pσ2π E(MS res) = σ2ε
ANOVA Summary Table
| Source | SS | df | MS | F | p-value | Error | |
| 1 | A | 190.000 | 2 | 95.000 | 4.79 | .0152 | [5] |
| 2 | B | 1543.333 | 2 | 771.666 | 38.99 | .0000 | [5] |
| 3 | A*B | 1236.667 | 4 | 309.167 | 15.58 | .0000 | [5] |
| 4 | Blocks (Subjects) | 1615.111 | 4 | 403.778 | 20.35 | .0000 | [5] |
| 5 | Residual | 634.889 | 32 | 19.840 | |||
| Total | 5220.000 | 44 |
Omega-Squared
ω2A*B = (1236.6667 - 4*19.84)/(634.889 + 19.84) = 0.2209
ω2B = (1543.333 - 2*19.84)/(634.889 + 19.84) = 0.2870
ω2A = (1190.0 - 2*19.84)/(634.889 + 19.84) = 0.0287
Using Stata
input s a b y x1 x2 x3 x4 s1 s2 s3 s4
1 1 1 37 1 1 1 1 1 1 1 1
2 1 1 42 1 1 1 1 -1 1 1 1
3 1 1 33 1 1 1 1 0 -2 1 1
4 1 1 29 1 1 1 1 0 0 -3 1
5 1 1 24 1 1 1 1 0 0 0 -4
1 1 2 43 1 1 -1 1 1 1 1 1
2 1 2 44 1 1 -1 1 -1 1 1 1
3 1 2 36 1 1 -1 1 0 -2 1 1
4 1 2 27 1 1 -1 1 0 0 -3 1
5 1 2 25 1 1 -1 1 0 0 0 -4
1 1 3 48 1 1 0 -2 1 1 1 1
2 1 3 47 1 1 0 -2 -1 1 1 1
3 1 3 29 1 1 0 -2 0 -2 1 1
4 1 3 38 1 1 0 -2 0 0 -3 1
5 1 3 28 1 1 0 -2 0 0 0 -4
1 2 1 39 -1 1 1 1 1 1 1 1
2 2 1 30 -1 1 1 1 -1 1 1 1
3 2 1 34 -1 1 1 1 0 -2 1 1
4 2 1 26 -1 1 1 1 0 0 -3 1
5 2 1 21 -1 1 1 1 0 0 0 -4
1 2 2 35 -1 1 -1 1 1 1 1 1
2 2 2 40 -1 1 -1 1 -1 1 1 1
3 2 2 31 -1 1 -1 1 0 -2 1 1
4 2 2 22 -1 1 -1 1 0 0 -3 1
5 2 2 27 -1 1 -1 1 0 0 0 -4
1 2 3 46 -1 1 0 -2 1 1 1 1
2 2 3 36 -1 1 0 -2 -1 1 1 1
3 2 3 45 -1 1 0 -2 0 -2 1 1
4 2 3 27 -1 1 0 -2 0 0 -3 1
5 2 3 26 -1 1 0 -2 0 0 0 -4
1 3 1 31 0 -2 1 1 1 1 1 1
2 3 1 21 0 -2 1 1 -1 1 1 1
3 3 1 20 0 -2 1 1 0 -2 1 1
4 3 1 18 0 -2 1 1 0 0 -3 1
5 3 1 10 0 -2 1 1 0 0 0 -4
1 3 2 41 0 -2 -1 1 1 1 1 1
2 3 2 50 0 -2 -1 1 -1 1 1 1
3 3 2 39 0 -2 -1 1 0 -2 1 1
4 3 2 36 0 -2 -1 1 0 0 -3 1
5 3 2 34 0 -2 -1 1 0 0 0 -4
1 3 3 64 0 -2 0 -2 1 1 1 1
2 3 3 52 0 -2 0 -2 -1 1 1 1
3 3 3 53 0 -2 0 -2 0 -2 1 1
4 3 3 42 0 -2 0 -2 0 0 -3 1
5 3 3 49 0 -2 0 -2 0 0 0 -4
end
table a, cont(freq mean y sd y) by(b)
----------+-----------------------------------
b and a | Freq. mean(y) sd(y)
----------+-----------------------------------
1 |
1 | 5 33 6.964194
2 | 5 30 6.964194
3 | 5 20 7.516648
----------+-----------------------------------
2 |
1 | 5 35 8.803409
2 | 5 31 6.964194
3 | 5 40 6.204837
----------+-----------------------------------
3 |
1 | 5 38 9.513149
2 | 5 36 9.513149
3 | 5 52 7.968688
----------+-----------------------------------
histogram y, by(a b) normal
anova y a b a*b s, repeated(a b)
Number of obs = 45 R-squared = 0.8784
Root MSE = 4.45424 Adj R-squared = 0.8328
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 4585.11111 12 382.092593 19.26 0.0000
|
a | 190.00 2 95.00 4.79 0.0152
b | 1543.33333 2 771.666667 38.89 0.0000
a*b | 1236.66667 4 309.166667 15.58 0.0000
s | 1615.11111 4 403.777778 20.35 0.0000
|
Residual | 634.888889 32 19.8402778
-----------+----------------------------------------------------
Total | 5220.00 44 118.636364
Between-subjects error term: s
Levels: 5 (4 df)
Lowest b.s.e. variable: s
Repeated variable: a
Huynh-Feldt epsilon = 0.7892
Greenhouse-Geisser epsilon = 0.6319
Box's conservative epsilon = 0.5000
------------ Prob > F ------------
Source | df F Regular H-F G-G Box
-----------+----------------------------------------------------
a | 2 4.79 0.0152 0.0237 0.0331 0.0438
Residual | 32
-----------+----------------------------------------------------
Repeated variable: b
Huynh-Feldt epsilon = 0.9493
Greenhouse-Geisser epsilon = 0.6954
Box's conservative epsilon = 0.5000
------------ Prob > F ------------
Source | df F Regular H-F G-G Box
-----------+----------------------------------------------------
b | 2 38.89 0.0000 0.0000 0.0000 0.0000
Residual | 32
-----------+----------------------------------------------------
Repeated variables: a*b
Huynh-Feldt epsilon = 1.4947
*Huynh-Feldt epsilon reset to 1.0000
Greenhouse-Geisser epsilon = 0.5901
Box's conservative epsilon = 0.2500
------------ Prob > F ------------
Source | df F Regular H-F G-G Box
-----------+----------------------------------------------------
a*b | 4 15.58 0.0000 0.0000 0.0001 0.0043
Residual | 32
-----------+----------------------------------------------------
omega2 a*b
omega squared for a*b = 0.2209
effect size = 0.5324
omega2 b
omega squared for b = 0.2870
effect size = 0.6344
omega2 a
omega squared for a = 0.0287
effect size = 0.1719
quietly anova y a b a*b /* run without s to get a plot of cell means */
anovaplot b a, scatter(msym(none))
quietly anova y a b a*b s /* rerun the original anova to get correct mse */
sme a b
Test of a at b(1): F(2/32) = 11.676584
Test of a at b(2): F(2/32) = 5.1242562
Test of a at b(3): F(2/32) = 19.152958
Critical value of F for alpha = .05 using ...
--------------------------------------------------
Dunn's procedure = 4.1487813
Marascuilo & Levin = 4.6658516
per family error rate = 4.6658516
simultaneous test procedure = 9.4358325
fhcomp a if b==1
Fisher-Hayter pairwise comparisons for variable a
studentized range critical value(.05, 2, 32) = 2.8806588
mean critical
grp vs grp group means dif dif
-------------------------------------------------------
1 vs 2 33.0000 30.0000 3.0000 5.7383
1 vs 3 33.0000 20.0000 13.0000* 5.7383
2 vs 3 30.0000 20.0000 10.0000* 5.7383
fhcomp a if b==3
Fisher-Hayter pairwise comparisons for variable a
studentized range critical value(.05, 2, 32) = 2.8806588
mean critical
grp vs grp group means dif dif
-------------------------------------------------------
1 vs 2 38.0000 36.0000 2.0000 5.7383
1 vs 3 38.0000 52.0000 14.0000* 5.7383
2 vs 3 36.0000 52.0000 16.0000* 5.7383
generate x5 = x1*x3
generate x6 = x1*x4
generate x7 = x2*x3
generate x8 = x2*x4
regress y x1 x2 x3 x4 x5 x6 x7 x8 s1 s2 s3 s4
Source | SS df MS Number of obs = 45
-------------+------------------------------ F( 12, 32) = 19.26
Model | 4585.11111 12 382.092593 Prob > F = 0.0000
Residual | 634.888889 32 19.8402778 R-squared = 0.8784
-------------+------------------------------ Adj R-squared = 0.8328
Total | 5220 44 118.636364 Root MSE = 4.4542
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | 1.5 .8132297 1.84 0.074 -.1564948 3.156495
x2 | -1.166667 .4695184 -2.48 0.018 -2.123044 -.210289
x3 | -3.833333 .8132297 -4.71 0.000 -5.489828 -2.176839
x4 | -3.5 .4695184 -7.45 0.000 -4.456378 -2.543622
x5 | -.25 .9959989 -0.25 0.803 -2.278783 1.778783
x6 | .25 .5750403 0.43 0.667 -.9213187 1.421319
x7 | 3.083333 .5750403 5.36 0.000 1.912015 4.254652
x8 | 1.916667 .3319996 5.77 0.000 1.240406 2.592928
s1 | 1.222222 1.049875 1.16 0.253 -.9163033 3.360748
s2 | 1.962963 .6061457 3.24 0.003 .7282847 3.197641
s3 | 2.509259 .4286097 5.85 0.000 1.63621 3.382309
s4 | 1.972222 .3319996 5.94 0.000 1.295961 2.648483
_cons | 35 .6639993 52.71 0.000 33.64748 36.35252
------------------------------------------------------------------------------
test x1 x2
( 1) x1 = 0
( 2) x2 = 0
F( 2, 32) = 4.79
Prob > F = 0.0152
test x3 x4
( 1) x3 = 0
( 2) x4 = 0
F( 2, 32) = 38.89
Prob > F = 0.0000
test x5 x6 x7 x8
( 1) x5 = 0
( 2) x6 = 0
( 3) x7 = 0
( 4) x8 = 0
F( 4, 32) = 15.58
Prob > F = 0.0000
test s1 s2 s3 s4
( 1) s1 = 0
( 2) s2 = 0
( 3) s3 = 0
( 4) s4 = 0
F( 4, 32) = 20.35
Prob > F = 0.0000
xi3: regress y r.a*r.b r.s
r.a _Ia_1-3 (naturally coded; _Ia_1 omitted)
r.b _Ib_1-3 (naturally coded; _Ib_1 omitted)
r.s _Is_1-5 (naturally coded; _Is_1 omitted)
Source | SS df MS Number of obs = 45
-------------+------------------------------ F( 12, 32) = 19.26
Model | 4585.11111 12 382.092593 Prob > F = 0.0000
Residual | 634.888889 32 19.8402778 R-squared = 0.8784
-------------+------------------------------ Adj R-squared = 0.8328
Total | 5220 44 118.636364 Root MSE = 4.4542
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Ia_2 | -3 1.626459 -1.84 0.074 -6.31299 .3129895
_Ia_3 | 3.5 1.408555 2.48 0.018 .6308669 6.369133
_Ib_2 | 7.666667 1.626459 4.71 0.000 4.353677 10.97966
_Ib_3 | 10.5 1.408555 7.45 0.000 7.630867 13.36913
_Ia2Xb2 | -1 3.983996 -0.25 0.803 -9.115134 7.115134
_Ia2Xb3 | 1.5 3.450241 0.43 0.667 -5.527912 8.527912
_Ia3Xb2 | 18.5 3.450241 5.36 0.000 11.47209 25.52791
_Ia3Xb3 | 17.25 2.987997 5.77 0.000 11.16365 23.33635
_Is_2 | -2.444444 2.09975 -1.16 0.253 -6.721496 1.832607
_Is_3 | -5.888889 1.818437 -3.24 0.003 -9.592924 -2.184854
_Is_4 | -10.03704 1.714439 -5.85 0.000 -13.52923 -6.544839
_Is_5 | -9.861111 1.659998 -5.94 0.000 -13.24242 -6.479805
_cons | 35 .6639993 52.71 0.000 33.64748 36.35252
------------------------------------------------------------------------------
describe _Ia_2 - _Is_5
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-)
_Ia2Xb2 float %9.0g a(2 vs. 1)*b(2 vs. 1)
_Ia2Xb3 float %9.0g a(2 vs. 1)*b(3 vs. 2-)
_Ia3Xb2 float %9.0g a(3 vs. 2-)*b(2 vs. 1)
_Ia3Xb3 float %9.0g a(3 vs. 2-)*b(3 vs. 2-)
_Is_2 double %10.0g s(2 vs. 1)
_Is_3 double %10.0g s(3 vs. 2-)
_Is_4 double %10.0g s(4 vs. 3-)
_Is_5 double %10.0g s(5 vs. 4-)
test _Ia_2 _Ia_3
( 1) _Ia_2 = 0
( 2) _Ia_3 = 0
F( 2, 32) = 4.79
Prob > F = 0.0152
test _Ib_2 _Ib_3
( 1) _Ib_2 = 0
( 2) _Ib_3 = 0
F( 2, 32) = 38.89
Prob > F = 0.0000
test _Ia2Xb2 _Ia2Xb3 _Ia3Xb2 _Ia3Xb3
( 1) _Ia2Xb2 = 0
( 2) _Ia2Xb3 = 0
( 3) _Ia3Xb2 = 0
( 4) _Ia3Xb3 = 0
F( 4, 32) = 15.58
Prob > F = 0.0000
test _Is_2 _Is_3 _Is_4 _Is_5
( 1) _Is_2 = 0
( 2) _Is_3 = 0
( 3) _Is_4 = 0
( 4) _Is_5 = 0
F( 4, 32) = 20.35
Prob > F = 0.0000
Linear Statistical Models Course
Phil Ender, 25apr06, 12Feb98