A Large Sample Variances for ˆδ ITT, ˆδ IV, ˆδ PP, and ˆδ AT
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1 Web-based Supplementary Materials for A Comparison of Methods for Estimating the Causal Effect of a Treatment in Randomized Clinical Trials Subject to Noncompliance by Roderick Little, Qi Long and Xihong Lin. A Large Sample Variances for ˆδ ITT, ˆδ IV, ˆδ PP, and ˆδ AT In this section, we derive large sample variances for ˆδ ITT, ˆδIV, ˆδPP, and ˆδ AT under ER. ˆδ ITT, ˆδ IV, ˆδ PP, and ˆδ AT are of the form g(y (c+a), y n, y (c+n), y a, ˆπ n, ˆπ a ), and assuming constant variance within cells (y (c+a), y n, y (c+n), y a, ˆπ n, ˆπ a ) are asymptotically independent with variances: var(y (c+a) ) = σ + π cπ a mα(π c + π a ) var(y n ) = var(y (c+n) ) = mαπ n + π c π n m( α)(π c + π n ) var(y a ) = m( α)π a var(ˆπ n ) = π n( π n ) mα var(ˆπ a ) = π a( π a ) m( α) where = µ c µ n and = µ c µ a. Hence, their large sample variances can be computed using the Delta method. Since the resulting expressions are complex and do not provide much insight, we examine variances in more detail for the case when π a =, which is the case in our example since there are no always-takers. When π a =, ˆδ ITT, ˆδ IV, ˆδ PP, and
2 ˆδ AT are functions of (y c, y n, y +, ˆπ c ) with variances: var(y c ) = var(y n ) = mαπ c mα( π c ) var(y + ) = σ + π c ( π c ) m( α) var(ˆπ c ) = π c( π c ) mα Hence we can apply delta method to obtain variances of δ ITT, δ IV, δ PP, and δ AT, that is, ( ) ( ) ( ) ( ) var(g) = var(y c) + var(y n) + var(y +) + var(ˆπ c ) ˆπ c y c y n y + Simple algebra leads to formulae ()-(4) for the variances. For example, and ˆδ AT = y c ( α)y + + α( π c )y n αˆπ c = y c y + + α( π c)(y + y n ) αˆπ c y c =, = α( ˆπ c), y n αˆπ c = α, y + αˆπ c = α( α)π c ˆπ c ( αˆπ c ) So ( ) ( ) ( ) ( ) Var(ˆδ AT ) = var(y y c) + var(y c y n ) + var(y n y +) + var(ˆπ c ) + ˆπ c [ = σ + α( ˆπ ] [ c) mαπ c αˆπ c mα( π c ) + α ] σ + π c( π c ) αˆπ c m( α) [ ] α( α)πc + π c( π c ) ( αˆπ c ) mα = [ σ + α( π c) ( α)σ + m απ c ( απ c ) ( απ c ) + π c( π c )( α) { } ] ( απc ) + α( α)π ( απ c ) 4 c = [ ] σ + σ + π c ( π c )( α) ( απ m απ c απ c ( απ c ) 4 c + απc ) B Relationships Between Var(ˆδ IV ), Var(ˆδ PP ), and Var(ˆδ AT ) B. Var(ˆδ IV ) Var(ˆδ PP ) We first show Var(ˆδ IV ) Var(ˆδ PP )
3 We know ( ) Var(ˆδ IV ) = + π mα( α)πc c ( π c ) = A + A ( Var(ˆδ PP ) = (απc + α) + απc mα( α)π ( π ) c) = B + B c where A and B are the terms involving, and A and B are the terms involving. We first show that Let us consider A/B A = B = mα( α)πc mα( α)π c (απ c + α). mα( α)πc A B = mα( α)π c (απ c + α) = π c ( α( π c )) where the last inequality holds due to < α, π c <. Similarly we can show A B that is, π c( π c) A B = mα( α)πc απc ( πc) mα( α)π c = απc where the last inequality holds due to < α, π c <. Hence Var(ˆδ IV ) Var(ˆδ PP ) holds. B. Var(ˆδ IV ) Var(ˆδ PP Var(ˆδ AT ) when = We know Var(ˆδ AT ) = ( m απ c ( απ c ) σ + ( α)π ) c( π c )( απ c + απc) ( απ c ) 4 = C + C We can show that B C, that is, B C = mα( α)π c (απ c + α) m απ σ c( απ c)
4 = ( απ c)( α + απ c ) α = α + α π c ( π c ) α where the last inequality holds since < α, π c <. We know when =, A = B = C =. Hence, Var(ˆδ IV ) Var(ˆδ PP ) Var(ˆδ AT ) when =. B. Relationship Between Var(ˆδ AT ) and Var(ˆδ IV ) When We consider A C = π c( π c ) / ( α)π c( π c )( απ c + απc) mα( α)π c m( απ c ) 4 ( απ c ) 4 = α( α) π c ( απ c + απc ) = ( απ c ) 4 π c ( α) [α( απ c ) + α πc ] If απ c, equivalently, απ c, then A C = π c ( απ c ) 4 ( α) [α( απ c ) + α π c] π c ( απ c ) 4 ( α) [( απ c ) + α π c] = π c ( απ c ) 4 ( α) ( απ c ) = π c ( απ c ) ( α) where the first inequality is due to α < and απ c, and the last inequality is due to < α, π c <. Hence we have A C and A C when απ c ; in other words, when απ c, Var(ˆδ IV ) Var(ˆδ AT ). An implication of this result is that when the compliance rate is low, say less than 5%, or for a relatively balanced design, say α is close to.5, δ AT is more efficient than δ IV. 4
5 B.4 Numerical Evaluation of General Relationships Between Var(ˆδ AT ) and Var(ˆδ IV )(Var(ˆδ PP )) When, the general relationship between Var(ˆδ AT ) and Var(orˆδ IV )(Var(ˆδ PP )) is considerably more complicated and involved, since the relative size of C and B or C and A can change for different α and π c values. We conducted numerical evaluations to examine the ratios, Var(ˆδ AT ) and Var(ˆδ AT ). Figures and show the numerical results for the ratio of Var(ˆδ IV ) Var(ˆδ PP ) Var(ˆδ AT ), when = Var(ˆδ IV /σ = and 5, respectively. Figures and 4 show the numerical results ) for the ratio of Var(ˆδ AT ) Var(ˆδ PP ), when = /σ = and 5, respectively. Based on our numerical results, we conclude that:. If α and π c are close to, C will start dominating C as will A and B. As a result, Var(ˆδ AT ) may exceed Var(ˆδ PP ) or even Var(ˆδ IV ). For example, when α =.9595, π =.9898, and =, Var(ˆδ AT )/Var(ˆδ IV ) =.6.. We also observe when α is away from, then Var(ˆδ IV ) Var(ˆδ AT ) and Var(ˆδ PP ) Var(ˆδ AT ) hold for all π c. Combining this observation with the results in Section B., we conclude that Var(ˆδ IV ) Var(ˆδ PP ) Var(ˆδ AT ) holds in most realistic settings. 5
6 (δ AT ) Figure : Var Var(δ IV ) When = /σ = (δ AT ) Figure : Var Var(δ IV ) When = /σ = 5 6
7 Var(δ AT ) When = /σ = Figure : Var (δ P P ) Var(δ AT ) When = /σ = 5 Figure 4: Var (δ P P ) 7
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