Matrix Approach to 1-Way ANOVA Comparison of Sexual Side Effects in 4 Antidepressants JG Modell, et al (1997). "Comparative Sexual Side Effects of Bupropion, Fluoxetine, Paroxetine, and Sertraline ", Clinical Pharmacology and Therapeutics, Vol.61(4),pp476-487.
Matrix Approach to 1-Way ANOVA
Jan 07, 2016
Matrix Approach to 1-Way ANOVA. Comparison of Sexual Side Effects in 4 Antidepressants JG Modell, et al (1997). "Comparative Sexual Side Effects of Bupropion, Fluoxetine, Paroxetine, and Sertraline ", Clinical Pharmacology and Therapeutics, Vol.61(4),pp476-487. Model. - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Matrix Approach to 1-Way ANOVA
Comparison of Sexual Side Effects in 4 Antidepressants
JG Modell, et al (1997). "Comparative Sexual Side Effects of Bupropion, Fluoxetine, Paroxetine, and Sertraline ", Clinical Pharmacology and Therapeutics, Vol.61(4),pp476-487.
Model
2
1 1
1,..., 1,...,
~ 0, 1,...,
Problem: 1 parameters and only group means
Possible Constraints (Last one isn't helpful in extended models):
) 0
ij i ij i ij i
ij i i
t t
i t ii i
Y i t j r
NID i t
t t
i
1
1 1
1 1 1
) 0
) 0
) 0
t t ti
i i t t i i t ii i i t
t t
i i
rii r r r
r
iii
iv
Matrix Approach with no constraints (t=3)
2
3
2
3
1
2
1
2
3
1
2
3
1
2
3
( )
i
i
i
ir
r
ij i i r r
r
r r r r
r r r r
r r r r
r
r
r
Y
Y
Y
E Y E E
E
1
i
1 1 1 1
2 2 2 2
3 3 3 3
1
1
2 i
3
i
Y
Y Y Y
Y
1
Y 1 Y 1
1
1 1 0 0
β X 1 0 1 0
1 0 0 1
1
Y Xβ 1
1
Problem: Last 3 columns of sum to first column so ' is not full rank!
X X X
Applying Constraint 1 – Part 1
3
3 1 21
1
2
1
2
1 2
^
1 2 3 1 3 2 3
1 3 1 3 3
2 3 3 2 3
0
( )
ii
r r r
r r r
r r r
r
r
r
E
r r r r r r r
r r r r r
r r r r r
1 1 1
2 2 2
3 3 3
1
2
3
1 1
1 1
1T T1 1 11
T T1 1 1
1 1 0
β X 1 0 1
1 1 1
1
Y X β 1
1
β X X X Y
X X X Y1 2 3
1 3
2 3
Y Y Y
Y Y
Y Y
Applying Constraint 1 – Part 2
1 2 3 1 3 2 3 1 3 2 3 3
2 2 22 3 1 3 1 3 2 3 1 2 3 3
2 21 2 1 3
2 21 2 1 2 3 1 3
2 22 3 2 3
det
2
(1 1) (1 1)
(1 1) ((1 1 1 2) ( 2 2)) ((1 1 2) (1 2 1))
(1 1) ((1 1 2) ( 2 1
r r r r r r r r r r r r
r r r r r r r r r r r r
r r r r
r r r r r r r
r r r r
T1 1X X
33
1 2 3
1 3 3 1 3 3 1 3 1 3
3 2 3 2 3 2 3 2 3 3
1 3 2 3 1 2 3 2 3 1
3 2 3 2 3 2 3
1)) ((1 2) (1 1 1))
9
1
det
det det det
det det det
r
r r r
r r r r r r r r r r
r r r r r r r r r r
r r r r r r r r r r
r r r r r r r
1T1 1 T
1 1
X XX X
2 3 1 3
2 3 3
1 3 2 3 1 2 3 2 3 1 2 3 1 3
1 3 3 1 3 3 1 3 1 3
det det det
r r r r
r r r
r r r r r r r r r r r r r r
r r r r r r r r r r
Applying Constraint 1 – Part 3
1 2 3
1 2 1 3 2 3 2 3 1 2 1 3 1 3 1 2 2 3
2 3 1 2 1 3 1 2 1 3 2 3 1 3 2 3
1 3 1 2 2 3 1 3 2 3 1 2 1 3 2 3
^
^
1
^
2
1 2
1
9
2 2
2 4 2 2
2 2 2 4
1
9
r r r
r r r r r r r r r r r r r r r r r r
r r r r r r r r r r r r r r r r
r r r r r r r r r r r r r r r r
r r
1T1 1
^ 1T T1 1 11
X X
β X X X Y
3
1 2 1 3 2 3 2 3 1 2 1 3 1 3 1 2 2 3 1 2 3
2 3 1 2 1 3 1 2 1 3 2 3 1 3 2 3 1 3
1 3 1 2 2 3 1 3 2 3 1 2 1 3 2 3 2 3
2 2
2 4 2 2
2 2 2 4
r
r r r r r r r r r r r r r r r r r r Y Y Y
r r r r r r r r r r r r r r r r Y Y
r r r r r r r r r r r r r r r r Y Y
Applying Constraint 1 – Part 4
1 1 2 1 3 2 3 2 3 1 2 1 3^
2 1 2 1 3 2 3 1 3 1 2 2 31 2 3
3 1 2 1 3 2 3 2 3 1 2 1 3 1 3 1 2 2 3
1. 2. 3.1 2 3 2 1 3 3 1 21 2 3
^
21
29
2 2
1 13 3 3
9 3
Y r r r r r r r r r r r r
Y r r r r r r r r r r r rr r r
Y r r r r r r r r r r r r r r r r r r
Y r r Y r r Y r r Y Y Yr r r
1 2 3 1 2 1 3 1 2 1 3 2 3
1 2 2 3 1 2 1 3 1 2 1 3 2 31 2 3
3 2 3 1 2 1 3 1 2 1 3 2 3 1 2 1 3 2 3
1 21 2 3 2 1 3 3 1 21 2 3
2 41
2 2 29
2 4 2 2
1 2 16 3 3
9 3 3
Y r r r r r r r r r r r r
Y r r r r r r r r r r r rr r r
Y r r r r r r r r r r r r r r r r r r
Y r r Y r r Y r r Y Yr r r
3
^ ^
2 1 3 3 1 22 3
1
3
2 1 1 2 1 1
3 3 3 3 3 3
Y
Y Y Y Y Y Y
Applying Constraint 2 – Part 1
31 2
3 3 1 1 2 2 3 1 21 3 3
1
1 2
2 1 2 1 21 2
3 3 3 3
^
0
( )
i ii
r r r r
r r r r
r r r r
r rr r r r
r r
E
r r r rr r r r
1 1 1 1
2 2 2 2
3 3 3 3
2 2 2 2
1T T2 2 22
1 1 0 1
β X 1 0 1 Y X β 1
1 1 1 1
β X X X Y
1 2 3 1 2 3
1 1 2 11 1 3
3 3 3
21 2 22 32
33 3
0 0
0 1
0 1
r r r Y Y Y
r r r rr Y Y
r r r
rr r r Y Yr rr r
T T2 2 2X X X Y
Applying Constraint 2 – Part 2
3
1 21 2 3 1 2
3 3
2
1 21 2 3
3
1 2 1 21 2 3 1 2 2
3 3
1 2
det
1 1 0 0
0 0
1 1
r rr r r r r
r r
r rr r r
r
r r r rr r r r r
r r r
N r r
T2 2X X
1 2 3 1 22
3 3
r r r r rN
r r
Applying Constraint 2 – Part 3
1 21 1 2 11 1
33 3 3
2 1 21 2 222
3 33 3
1 2 22 2
3 3
1
det
01 0 1
det det det
0 1 01
0 0 0
det det1 0 1
r rr r r rr rrr r r
r r rr r r rrr rr r
N
r r rr r
r r
1T2 2 T
2 2
X XX X
1 22
33
1 21 1 2 11 1
33 3 3
1 2
3
2 232
1 2
0
det0
0 0 00
det det det01 0 1
0 0
0
N
rrrrr
NN
r rr r r rr r
rr r r
Nr r
r
Nr rr
N r r
3 2 31 2
3 3 1
1 31 1 31 2
23 3
10 0
10
100
Nr r rNr r
r r Nr N
r rNr r rNr rN Nrr r
Applying Constraint 2 – Part 4
^
1 2 3
^2 3 1
1 1 31 3^
221 3
2 332
^ 1 2 31 2 3 1 2 3
^2 3 1
1 1 31 3
10 0
10
10
Y Y YNr r r
Y YNr N r
rr rY Y
rN Nr
Y Y Y rY r Y r YY
N N
r r rY Y
Nr r
^ 1T T2 2 22β X X X Y
2 3 32 21 2 3 12 3
3
^1 3 1 3 31 2 1
2 1 3 22 1 3 2 33 2 3
^1 2 1 2
3 1 2 33
1
1
r r rr rY Y Y Y Y Y Y
N r N N N
r r r r rr r rY Y Y Y Y Y Y Y Y
N r Nr r N N N
r r r rY Y Y Y Y
N N N
Applying Constraint 3 – Part 1
3 3
1
2
1
2
1 2 3 1 2 1 2 3
1 1 1
2 2 2
0
( )
0
0
r r r
r r r
r r r
r
r
r
E
r r r r r Y Y Y
r r Y
r r Y
1 1 1
2 2 2
3 3 3
1
2
3
3 3
3 3
^ 1T T3 3 33
T T3 3 3
1 1 0
β X 1 0 1
1 0 0
1
Y X β 1
1
β X X X Y
X X X Y
Applying Constraint 3 – Part 2
2 23 3 1 2 3 1 2 1 2 2 1 1 2 3
3 3
3 3
1 1 1 1
1 2 2 2
1 2 2 1
2 2 2 2
1 2 2
1 1
det 0 0 0
1
det
0 0det det det
0 0
det det det0 0
det det de0 0
r r r r r r r r r r r r
r r r r
r r r r
r r N r N r
r r r r
r r N r
r r
T
1T
T
X X
X XX X
1
1 1
1 2 1 2 1 2 3 3 32
1 2 2 2 1 2 3 1 3 1 3 31 2 3 2
1 2 1 2 1 1 3 3 2 3 2 3
t
1 1 11
1 1
1 1
N r
r r
r r r r r r r r r
r r Nr r r r r r r r r rr r r
r r r r Nr r r r r r r r
Applying Constraint 3 – Part 3
^
^11
^
2
3 3 3 1 2 3
3 1 3 1 3 3 1
3 3 2 3 2 3 2
^
33 1 2 3 3 1 3 2
^
1 31 3 1 2 3 3 1 1 3 2
1 1 1
1 1
1 1
1 1 1
1 1 1 1
r r r Y Y Y
r r r r r r Y
r r r r r r Y
r Y Y Y r Y r Y Y
r Y Y Y r r Y r Y Y Y
^T T3 3 33β X X X Y
^
2 32 3 1 2 3 3 1 3 2 21 1 1 1r Y Y Y r Y r r Y Y Y
Applying Constraint 4
11
2 2
3 3
1 1
2 2
3 3
1
2
0
( )
0 0
0 0
0 0
1 0 0
0 1 0
0 0 1
i i
r r r
r r r
r r r
E
r Y
r Y
r Y
r
r
1 1 1 1
2 2 2 2
3 3 3 3
n
4 1 4 4 n
n
^ 1T T T T4 4 4 4 4 44
^ 1T T4 4 44
1 0 0 1
β X 0 1 0 Y X β 1
0 0 1 1
β X X X Y X X X Y
β X X X Y
11
22
33 3
YY
Y Y
r Y Y
1
1
2
1
1
11
11
11 1
2
1
1...
1 1
Constraint 1: 0 :
1 1
1...
Constraint 2: 0 :
t
t
iti
ii
t
t i
ii t
tt
ti
iti
i ii
Y Ytt
Y Yt t
tY Y
t t
r Y r Y YN
rN rY Y Y
N Nn
^
1
^
2
β
β
11 1
11
1
1 2
1
Constraint 3: 0 : Constraint 4: 0 :
t
tt i
t i t t
ii t
t
t
t
t t t
Y
N r rY Y Y Y
N N
Y Y
Y Y Y
Y Y Y
^ ^
3 4β β
SUMMARY
Estimable Functions
• A linear function of the parameter vector , h’ is said to be estimable if h’=t’X. That is, if h’ can be written as a linear function of the rows of X
1
2
3
'1 1
Some commonly estimated estimable functions:
) ) ) 0 . . 0
Some non-estimable functions:
) )
r r r r
r r r r
r r r r
t t
i i i i i ii i
i
i ii iii c s t c
iv v
1 1 1 1
2 2 2 2
3 3 3 3
1 1 0 0
β X 1 0 1 0
1 0 0 1
Uniqueness of Estimates of Estimable Functions
1
1
1 11
2
1 1 11
1
11 111
2
Estimable Function: :
Constraint 1: 0 :
1 1 1...
1 1
Constraint 2: 0 :
1...
t
ii
t
t i
i
t
i ii
ti
t iti
tY Y Y Y
t t t
tY Y Yt t
n
rN rr Y rY Y Y
N N N
1 11 1 11
1 11 1
1 11 1
Constraint 3: 0 :
Constraint 4: 0 : 0
t tt
r N rY Y Y
N N
Y Y Y Y
Y Y
Application: Comparison of t=4 Antidepressants
• Response: Y = Change in libido scores for patients on antidepressants
• Treatments: t=4 Brands of antidepressants Prozac (r1 = 37)
Paxil (r2 = 21)
Zoloft (r3 = 27)
Wellbutrin (r4 = 22) N=107, t=4, Prozac, Paxil, Zoloft are SSRI’s, Wellbutrin not
• Will make generalizations of case t=3
Data and Analysis of VarianceBrand (i) n(i) Ybar(i) SD(i) Ybar SSTRT SSEProzac (1) 37 -0.49 0.97 -0.3751 0.4881 33.8724Paxil (2) 21 -0.90 0.73 -0.3751 5.7850 10.6580Zoloft (3) 27 -0.49 1.25 -0.3751 0.3562 40.6250W e llbutrin (4) 22 0.46 0.80 -0.3751 15.3441 13.4400Total 107 #N /A #N /A #N /A 21.9735 98.5954
AN OVASource df SS M S F F(.05) P-valueBrands 3 21.97 7.32 7.65 2.69 0.0001Error 103 98.60 0.96Total 106 120.57
H0 : Brand Means are all equalConclude brand means are not all equal Note: s2 = MSE = 0.96
Parameter Estimates – Constraint 1 X1'X1 X1'Y
107 15 -1 5 -40.1415 59 22 22 -28.25-1 22 43 22 -29.025 22 22 49 -23.35
(X1'X1)^-1 be ta10.009821 -0.00306 0.002084 -0.00056 -0.355-0.00306 0.023335 -0.00884 -0.00619 -0.1350.002084 -0.00884 0.033631 -0.01134 -0.545-0.00056 -0.00619 -0.01134 0.02834 -0.135
Parameter Estimates – Constraint 2 X2'X2 X2'Y
107.0000 0.0000 0.0000 0.0000 -40.140.0000 99.2273 35.3182 45.4091 -35.150.0000 35.3182 41.0455 25.7727 -28.560.0000 45.4091 25.7727 60.1364 -25.65
(X2'X2)^-1 be ta20.009346 0.000000 0.000000 0.000000 -0.375140.000000 0.017681 -0.009346 -0.009346 -0.114860.000000 -0.009346 0.038273 -0.009346 -0.524860.000000 -0.009346 -0.009346 0.027691 -0.11486
Parameter Estimates – Constraint 3 X3'X3 X3'Y
107.0000 37.0000 21.0000 27.0000 -40.1437.0000 37.0000 0.0000 0.0000 -18.1321.0000 0.0000 21.0000 0.0000 -18.927.0000 0.0000 0.0000 27.0000 -13.23
(X3'X3)^-1 be ta30.045455 -0.045455 -0.045455 -0.045455 0.46-0.045455 0.072482 0.045455 0.045455 -0.95-0.045455 0.045455 0.093074 0.045455 -1.36-0.045455 0.045455 0.045455 0.082492 -0.95
Parameter Estimates – Constraint 4 X4'X4 X4'Y
37.0000 0.0000 0.0000 0.0000 -18.130.0000 21.0000 0.0000 0.0000 -18.90.0000 0.0000 27.0000 0.0000 -13.230.0000 0.0000 0.0000 22.0000 10.12
(X4'X4)^-1 be ta40.027027 0.000000 0.000000 0.000000 -0.490.000000 0.047619 0.000000 0.000000 -0.90.000000 0.000000 0.037037 0.000000 -0.490.000000 0.000000 0.000000 0.045455 0.46
Related Documents
