Approach to One-Way ANOVA Comparison of Antidepressants' Sexual Side Effects

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Explore a matrix approach used in a study comparing sexual side effects of four antidepressants - Bupropion, Fluoxetine, Paroxetine, and Sertraline. The study delves into model constraints, application of constraints, and implications for statistical analysis in determining the impact of these medications on sexual side effects.

  • ANOVA
  • Antidepressants
  • Sexual Side Effects
  • Matrix Approach
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  1. 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.

  2. Model = = + = + + = 1,..., 1,..., Y i g j n ij i ij i ij i ( g ) = + = 2 ~ 0, 1,..., NID i g ij i i + Problem: Possible Constraints (Last one isn't helpful in extended models): = = = = = = 1 parameters and only group means g 1 g g = = ) 0 i i g i = = 1 1 i i 1 1 g g g n n i i = ) 0 i ii n n n i g g i g i = = = 1 1 1 i i i g ( ) ) 0 iii 1 1 ( ) ) 0 iv i i

  3. Matrix Approach with no constraints (g=3) Y Y 1 i Y Y Y 1 2 i = = Y Y 2 i 3 Y in i ( ( ( ) ) ) + + + 1 1 1 1 n 1 ( ) ( ) ( ) Y = + = + = Y 1 ( ) E Y E E i 2 ij i i n n i 2 3 n 3 1 1 0 0 n n n n 1 1 1 1 1 = = X 1 1 0 0 1 0 0 1 n n n n 2 2 2 2 2 n n n n 3 3 3 3 3 ( ( ( ) ) ) + + + 1 1 n 1 ( ) Y = = X E 1 1 2 n 2 3 n 3 Problem: Last 3 columns of sum to first column so ' X X X is not full rank!

  4. Applying Constraint 1 Part 1 3 = = 0 3 1 2 i = 1 i 1 1 0 n n n 1 1 1 = = X 1 1 0 1 1 1 1 n n n 2 2 2 1 1 2 n n n 3 3 3 ( ( ) ) + + 1 1 n 1 = = ( ) Y X E 1 1 1 2 n 2 ( ) 1 1 2 n 3 ( ) + ^ 1 = T 1 T 1 X X X Y 1 1 + + + + n n n n n n n n n Y Y Y 1 2 3 1 3 2 3 1 2 3 = = T 1 T 1 X X X Y n n n n n + Y Y Y Y 1 1 3 1 3 3 1 3 n n n 2 3 3 2 3 2 3

  5. Applying Constraint 1 Part 2 ( ) = T 1 X X det 1 ( ( ) ( ) ( ( )( )( ) ( )( ) + + + + + 2 n n n n n n n n n n n n 1 2 3 1 3 2 3 1 3 2 3 3 ) ) ( ) ( + ) ( ) ( + ) 2 2 + + + + 2 3 n n n n n n n n n n n n 2 3 1 3 1 3 2 3 1 2 3 (1 1) + + + (1 1) + 2 1 2 1 n n n n 2 2 3 = ((1 1 1 2) ( 2 2)) + + + + + + + ((1 1 2) (1 2 1)) + + + + 2 (1 1) n n 1 2 3 n n n n n 1 3 n n 3 3 ((1 2) (1 1 1)) n + 1 2 2 2 n n 9 n n n + 2 (1 1) ((1 1 2) ( 2 1 1)) 3 2 3 = 1 2 3 1 X X ( ) 1 = T 1 X X ( ) 1 T 1 det 1 = + + n n n + n n + n n + n + n n n n n n 1 3 3 1 3 3 1 3 1 3 det det det n n n n n + n 3 2 n n 3 n n 2 n + 3 n 2 n n 3 n n 2 + 3 + 3 n + n n n n n n n n n 1 3 2 3 1 2 3 2 3 1 2 3 1 3 det det det n + n n n n + n + 3 n n 2 3 2 3 + 2 n 3 n 2 n 3 n 3 n n + n n n n n n n n 1 3 2 3 1 2 3 2 3 1 2 3 1 3 det det det n n n n n n 1 3 3 1 3 3 1 3 1 3

  6. Applying Constraint 1 Part 3 1 ( ) 1 = T 1 X X 1 9 + 1 2 3 n n n n n n n n n + 2 2 1 2 n n n n n n 1 3 n n n n n n 2 3 n n n n 1 2 n n n n n n 1 3 n n n n n n n n 1 2 n n 2 3 n n n n n n + 2 3 1 3 2 + + 2 2 4 2 n n 2 3 1 2 1 3 1 2 1 3 2 3 1 3 4 + 2 3 2 2 1 2 n n 1 3 n n 1 3 1 2 2 3 1 3 = 2 3 2 3 ^ ( ) ^ ^ 1 = = T 1 T 1 X X X Y 1 1 1 ^ 2 1 9 1 2 n n n n n n n n n 3 + + + + 2 2 1 3 n n n n n n 2 3 n n n n n n 2 3 n n n n 1 2 n n n n n n 1 3 n n n n n n n n 1 2 n n 2 3 n n n n n n + Y Y Y 1 2 1 3 2 1 2 3 + + 2 2 4 2 n n Y Y Y Y 2 3 1 2 1 3 1 2 1 3 2 3 1 3 4 + 2 3 1 3 2 2 1 2 n n 1 3 n n 1 3 1 2 2 3 1 3 2 3 2 3 2 3

  7. Applying Constraint 1 Part 4 ( ( ( ) ) ( ( ( ) ( ) ( ) ( ) ) ) ( + + + + 2 Y 1 2 n n 1 3 n n 2 3 n n 2 3 n n 1 2 n n 1 3 n n 1 1 ^ = + + + + 2 Y 1 2 n n 1 3 n n 2 3 n n 1 3 n n 1 2 n n 2 3 n n 2 9 1 2 3 n n n ) ) + + 2 2 Y 1 2 n n 1 3 n n 2 3 n n 2 3 n n n n 1 3 n n 1 3 n n 1 2 n n 2 3 n n 3 1 2 1 3 1 ( ) ( ) ( ) = + + = + + 3 3 3 Y 2 3 n n Y 1 3 n n Y 1 2 n n Y Y Y 1. 2. 3. 1 2 3 9 1 2 3 n n n ( ( ( ) ( ( ( ) ( ) ( ) ( ) + + + + 2 4 Y 2 3 n n 1 2 n n 1 3 n n 1 2 n n 1 3 n n 2 3 n n 1 1 ^ ) ) = + + 2 2 2 Y 2 3 n n 1 2 n n 1 3 n n 1 2 n n 1 3 n n 2 3 n n 1 2 9 1 2 3 n n n ) ) ( 1 3 ) + + 2 4 2 2 Y 2 3 n n 1 2 n n 1 3 n n 1 2 n n 1 3 n n 2 3 n n 1 2 n n 1 3 n n 2 3 n n 3 1 2 3 1 3 ( ) ( ) ( ) = + + = 6 3 3 Y 2 3 n n Y 1 3 n n Y 1 2 n n Y Y Y 1 2 3 1 2 3 9 1 2 3 n n n 2 3 1 3 1 3 2 3 1 3 1 3 ^ ^ = = Y Y Y Y Y Y 2 1 3 3 1 2 2 3

  8. Applying Constraint 2 Part 1 3 n n n n i = = 1 = 0 n n n n 1 2 3 3 1 2 2 3 1 2 i = 1 i 3 3 ( ( ) ) + + 1 1 0 1 1 n n n n 1 1 1 1 = = = = ( ) Y E X 1 0 1 X 1 2 2 2 2 1 2 n n n n 2 2 2 2 n n n n n n n n 2 1 1 1 1 2 1 1 2 n n n 1 2 n 3 3 3 3 3 3 3 3 ( ) ^ 1 = T 2 T 2 X X X Y 2 2 + + 0 0 n n n + + Y Y Y 1 2 3 1 2 n n n n 3 n n 1 2 n n n = + = T 2 T 2 X X X Y 0 1 n Y Y 1 1 2 1 1 3 3 3 3 1 2 n n n n n Y Y 2 + 0 1 n 2 2 3 2 3 3 3

  9. Applying Constraint 2 Part 2 ( ) = T 2 X X det 2 n n n n ( ) + + + + + + 1 1 0 0 n n n n n 1 2 1 2 3 1 2 3 3 2 1 2 n n n ( ) + + + + 0 0 n n n 1 2 3 3 ) ( ) n n n n n n ( )( ) = + + + + + 1 1 n n n n n 1 2 1 2 2 n 1 2 3 1 2 3 3 3 ( + + + n n n n n n ( ) = + = 1 2 3 1 2 2 N n n N 1 2 n 3 3

  10. Applying Constraint 2 Part 3 1 X X ( ) 1 = T 2 X X ( ) 2 T 2 det 2 = 1 2 n n n n n 1 2 n n n n n + 0 1 n 1 + 0 1 n 1 1 1 3 3 3 3 det det det n n 1 2 n n n 1 2 n n n n n + 0 1 n 2 + 0 1 n 2 2 2 3 3 3 0 3 0 0 N 0 N det det det 1 2 n n n 1 2 n n n n n n n + + 0 1 0 1 n n 2 2 2 2 3 3 3 3 0 0 0 N 0 N det det det 1 2 n n n n n 1 2 n n n n n + + 0 1 0 1 n n 1 1 1 1 3 3 3 3 Nn n 1 N 0 0 1 2 n 0 0 3 ( ) ( ) + + Nn n n n n 1 N + n Nn n = = 2 2 3 2 Nn 3 0 0 3 1 2 n 2 N n n n 1 2 3 3 + 1 ( ) ( ) n n 1 N Nn n n Nn n 1 Nn 3 0 1 1 3 0 1 2 n n 2 3 3

  11. Applying Constraint 2 Part 4 1 N 0 0 ^ + + Y Y Y 1 2 n n n n 3 ( ) + n n 1 N + ( ) ^ ^ 1 = = = T 2 T 2 2 Nn 3 0 Y Y 1 X X X Y 1 2 2 1 3 1 3 ^ ( ) n n 1 N 2 Y Y 2 1 Nn 3 0 2 3 3 2 + + + + Y Y Y n Y n Y n Y ^ 1 2 3 = = = 1 2 3 1 2 3 Y N N 1 N ( ) ( ) + + n n n n n N n n n n n N ^ = = = 2 Nn 3 2 3 3 Y Y Y Y Y Y Y Y Y 1 2 2 1 2 3 1 1 1 3 2 3 N 1 3 3 ( ) ( ) + + n n n n 1 N n N n n n n n N ^ = + = = 1 Nn 3 1 3 3 Y Y Y Y Y Y Y Y Y 1 2 1 2 1 3 2 2 1 3 2 3 N 3 2 3 ( ) + n n n N n N ^ = = 1 2 Y Y Y Y Y 1 2 3 1 2 3 3 N

  12. Applying Constraint 3 Part 1 = = = + + 3 = 0 1 1 1 0 0 n n n 1 1 1 = X 1 1 0 3 3 2 n n n 2 2 2 1 0 1 3 n n n 3 3 3 1 n 1 ( ( ) ) = ( ) Y X E 1 3 3 2 n 2 1 3 n ( ) + ^ 1 = T 3 T 3 X X X Y 3 3 + + + n n n n n n n n Y Y Y Y Y 1 2 3 2 3 1 2 3 = = T 3 T 3 X X X Y 0 n 3 2 2 2 0 3 3 3

  13. Applying Constraint 3 Part 2 ( ) ) ( ) ( ( ) = + + 0 0 + + + + = T 2 2 X X det 0 n n n n n 2 3 n n 3 2 n n 1 2 3 n n n 3 3 1 2 3 2 3 1 X X ( ) 1 = T X X ( ) 3 3 T det 3 3 = 0 n 0 n n n n n n n 2 2 2 2 det det det 0 0 3 n n 3 3 3 N n n N n n n n 2 3 3 2 det det det 0 0 3 3 N n 3 n 3 n n n n n n n N n n n 2 3 3 2 det det de t 0 0 2 2 2 2 1 1 1 2 3 n n n n n n 2 3 n n n n n 2 3 1 n n 1 1 1 ( ) ( n ) = = + 2 3 1 1 1 n Nn n n n 1 2 n n n 2 3 3 2 3 1 1 2 1 1 2 3 n n n ( ) ( ) + 2 2 1 2 3 n n Nn n n 1 3 n n 2 3 2 1 1 1 3

  14. Applying Constraint 3 Part 3 ^ ( ) ^ ^ 1 = = = T 3 T 3 3 X X X Y 2 3 ^ 3 + + 1 1 1 n n n Y Y Y Y Y 1 n n 1 1 1 2 3 ( ) ( n ) + 1 1 1 n n n 1 2 n n n 1 1 2 1 2 ( ) ( ) + 1 n 1 3 n n 1 1 1 3 3 ^ ( )( ) ( )( ) ( )( ) ) = + + = 1 1 1 n Y Y Y n Y n Y Y 1 1 1 2 3 1 2 1 3 ^ ( ( )( ) ( ) ( + ) ( ) ) = + + + + = 1 1 1 1 n Y Y Y n n Y n Y Y Y 2 1 2 1 1 2 3 1 2 2 1 3 ^ ( ( )( ) ( + ) ( ) ( + ) = + + + = 1 1 1 1 n Y Y Y n Y n n Y Y Y 3 1 3 1 1 2 3 1 2 3 1 3

  15. Applying Constraint 4 = 3 3 = = 0 i i 1 0 0 1 n 1 n n n 1 1 1 1 1 = = = ( ) Y E X 0 0 1 0 0 1 X 1 1 4 1 4 4 n 2 2 n n n 2 2 2 2 3 n 3 n n n 3 3 0 n 0 0 n n Y Y Y 1 1 ( ) ^ 1 = = = T 4 T 4 T 4 T 4 0 0 X X X Y X X X Y 4 4 4 2 2 0 3 3 Y = 1 0 n 0 0 n n Y Y Y 1 1 1 ( ) ^ 1 = = T 4 T 4 0 0 1 Y X X X Y 2 4 4 2 2 0 1 Y 3 3 3

  16. 1 g + + ... Y Y 1 g S U M M A R Y g 1 1 g g Y Y 1 i g ^ g = = Constraint 1: 0: = 2 i 1 i = 1 i g 1 1 g g Y Y 1, g i g = 1 i i g 1 1 N + + = ... n Y n Y Y 1 g 1 g g n N N n = i Y Y Y Y 1 1 1 i g ^ N i = = Constraint 2: 0: = n 2 i 2 i = 1 i N n g n N 1 g = i Y Y Y Y 1, 1 g i g N = 1 i i g 1 Y Y 1 1 Y Y Y ^ ^ 2 1 2 = = = = Constraint 3: 0: Constraint 4: 0: 3 4 1 Y Y Y 1 g g

  17. 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 1 0 0 n n n n 1 1 1 1 1 = = X 1 1 0 0 1 0 0 1 n n n n 2 2 2 2 2 n n n n 3 3 3 3 3 Some commonly estimated estimable functions: g g + i = ) ) ) . . 0 i ii iii c st c ' i i i i i = = 1 1 i i Some non-estimable functions: ) ) i iv v vi + ) ' i i

  18. Uniqueness of Estimates of Estimable Functions + Estimable Function: : 1 g = Constraint 1: 0: i = 1 i g 1 g 1 1 g g = + + = ... Y Y Y Y 1 1 g i 1 g = 2 i 1 g 1 g + = + = Y Y Y 1 1 1 1 g g i = Constraint 2: 0: n i = 1 i g 1 N n N N n = + + = ... i n Y n Y Y Y 1 1 1 t i 1 1 t N = 2 i = n N N n + = + = Y Y Y 1 1 1 1 1 1 N = = + = Constraint 3: 0: Y Y Y Y 1 2 1 2 2 2 1 = = = + = Constraint 4: 0: 0 Y Y 1 1 1 1

  19. Application: Comparison of g=4 Antidepressants Response: Y = Change in libido scores for patients on antidepressants Treatments: g=4 Brands of antidepressants Prozac (n1 = 37) Paxil (n2 = 21) Zoloft (n3 = 27) Wellbutrin (n4 = 22) N=107, g=4, Prozac, Paxil, Zoloft are SSRI s, Wellbutrin not Will make generalizations of case g=3

  20. Data and Analysis of Variance Brand (i) Prozac (1) Paxil (2) Zoloft (3) Wellbutrin (4) Total n(i) 37 21 27 22 107 Ybar(i) -0.49 -0.90 -0.49 0.46 #N/A SD(i) 0.97 0.73 1.25 0.80 #N/A Ybar -0.3751 -0.3751 -0.3751 -0.3751 #N/A SSTRT 0.4881 5.7850 0.3562 15.3441 21.9735 SSE 33.8724 10.6580 40.6250 13.4400 98.5954 ANOVA Source Brands Error Total df 3 103 106 SS MS 7.32 0.96 F F(.05) 2.69 P-value 0.0001 21.97 98.60 120.57 7.65 H0 : Brand Means are all equal Conclude brand means are not all equal Note: s2 = MSE = 0.96

  21. Parameter Estimates Constraint 1 X1'X1 107 15 -1 5 X1'Y -40.14 -28.25 -29.02 -23.35 15 59 22 22 -1 22 43 22 5 22 22 49 (X1'X1)^-1 0.009821 -0.00306 0.002084 -0.00056 -0.00306 0.023335 -0.00884 -0.00619 0.002084 -0.00884 0.033631 -0.01134 -0.00056 -0.00619 -0.01134 beta1 -0.355 -0.135 -0.545 -0.135 0.02834

  22. Parameter Estimates Constraint 2 X2'X2 107.0000 0.0000 0.0000 0.0000 X2'Y -40.14 -35.15 -28.56 -25.65 0.0000 99.2273 35.3182 45.4091 0.0000 35.3182 41.0455 25.7727 0.0000 45.4091 25.7727 60.1364 (X2'X2)^-1 0.009346 0.000000 0.000000 0.000000 0.000000 0.017681 -0.009346 -0.009346 0.000000 -0.009346 0.038273 -0.009346 0.000000 -0.009346 -0.009346 0.027691 beta2 -0.37514 -0.11486 -0.52486 -0.11486

  23. Parameter Estimates Constraint 3 X3'X3 107.0000 21.0000 27.0000 22.0000 X3'Y -40.14 -18.9 -13.23 10.12 21.0000 21.0000 0.0000 0.0000 27.0000 0.0000 27.0000 0.0000 22.0000 0.0000 0.0000 22.0000 (X3'X3)^-1 0.027027 -0.027027 -0.027027 -0.027027 -0.027027 0.074646 0.027027 0.027027 -0.027027 0.027027 0.064064 0.027027 -0.027027 0.027027 0.027027 0.072482 beta3 -0.49 -0.41 0.00 0.95

  24. Parameter Estimates Constraint 4 X4'X4 37.0000 0.0000 0.0000 0.0000 X4'Y -18.13 -18.9 -13.23 10.12 0.0000 21.0000 0.0000 0.0000 0.0000 0.0000 27.0000 0.0000 0.0000 0.0000 0.0000 22.0000 (X4'X4)^-1 0.027027 0.000000 0.000000 0.000000 0.000000 0.047619 0.000000 0.000000 0.000000 0.000000 0.037037 0.000000 0.000000 0.000000 0.000000 0.045455 beta4 -0.49 -0.9 -0.49 0.46

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