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1 Plaig Evaluatios of Itervetios with Required ad Optioal Compoets: Istrumetal Variable (IV) Estimatio ad Sample Size Requiremets E. C. Hedberg, Ph.D.* Seior Research Scietist NORC at the Uiversit of Chicago Daa Log, MD, FAAP Medical Director Ceter for Commuit Health ad Egagemet UCSF Beioff Childre's Hospital Oaklad *Correspodig Author

2 Cotext ad Purpose This paper origiates from a project where a cliet wished to evaluate a itervetio with two compoets. The itervetio s mai compoet that would appl to all treatmet participats was the opeig of a college savigs accout (CSA) for their child. This elemet of the itervetio was required, but more importatl, participatio was passive i that all participats radomized to treatmet would receive accouts. I additio, the cliet s itervetio also icluded a fiacial coachig program (Collis, Baker, & Gore, 007). B the ver ature of coachig, participats caot be forced ito participatio (Delgadillo, 04). This compoet of the itervetio was ot passive, ad required asset o the part of the participats to receive the treatmet. To estimate the effects of the mai (CSA) ad optioal (coachig) compoets, the authors of this paper decided to use a three-group radomizatio strateg: oe group would be radomized ito a cotrol coditio, a secod group would be radomized ito a treatmet coditio with the mai CSA compoet ol, ad a third group would be radomized ito a treatmet group that icluded both the mai CSA compoet ad, additioall, ecouragemet for the optioal compoet. The evaluatio team decided to use istrumetal variable (IV) methods to istrumet uptake of the optioal compoet with assigmet to the third group. This paper cosiders the estimatio techique ad the sample size requiremets of this desig. Research Desig The IV estimator of the effects through SLS ca be expressed i matrix form as (Camero & Trivedi, 005) () ( ) b = Z X Z'. IV where ZZ is the matrix of predictors ad istrumets, XX is the matrix of predictors ad variables to be istrumeted, ad is the vector of outcome measuremets for each uit. The asmptotic (large sample) samplig variace-covariace matrix for these estimated slopes is defied as (Camero & Trivedi, 005) () var { } = ( ) ( ) b ZX ZZ XZ, IV where is the variace of the residuals from the secod stage regressio. I this paper, we cosider to be the populatio variace of the depedet variable without itervetios ad assume it is costat across idividuals. Essetial values for computatio The cross-product matrixes described i this paper that are used to estimate the treatmet effects ad their samplig variaces make use of several importat quatities. To uderstad the mea of each group, suppose a regressio model was fit to data i () = α + αt + α A + α T. i 0 i i i

3 Here, αα 0 is the average of the outcome for the cotrol group. To reduce terms, we assume the data are cetered o the cotrol group average so αα 0 = 0. We also assume uit variace of. Next αα is the itet to treat effect of the mai treatmet compoet, or (4) α = ( µ T =, A = 0, T = 0) ( µ T = 0, A = 0, T = 0), ad αα is, (5) α = ( µ T =, A =, T = 0) ( µ T =, A = 0, T = 0). Fiall, αα is (6) α = ( µ T =, A =, T = ) ( µ T =, A =, T = 0). These expressios allow us to relate the OLS estimated mea differeces () to the effects estimated b the IV procedure (). The ext parameter is the compliace rate of those ecouraged to use the optioal compoet of usig the optioal compoet of the itervetio. This is defied as (7) C Pr( T A ) = = =. Note that this is ot the overall mea of TT, but istead the mea of TT ol for cases where AA =. Results ad Coclusio Estimates of the effects ad tests without a covariate It ca be show that β0 = 0 biv = ZX Z = β = α. α β = α + C NoCov (8) ( ) The samplig variaces ad covariaces of these slopes are estimated b 0 biv = ZX ZZ XZ = C. 0 C C NoCov (9) var { } ( ) ( ) The test of the mai treatmet effect ca be expressed usig scale free parameters (effect sizes) as follows

4 (0) z = α = δ, where δ is defied as a Cohe s d effect size (99), represetig the differece divided b the α populatio stadard deviatio, δ =. Expressio (0) is essetiall the test for a two-group balaced desig (Cohe, 977), except i the case of this test, the total sample size is ad ot. The local average treatmet effect estimate for the optioal compoet is a combiatio of the OLS effects of usig the optioal compoet (αα ) ad beig ecouraged to use (but ot usig) the optioal compoet α (αα ), β = α +. C The test of the optioal treatmet ca also be expressed i effect size uits () α α + z = C = δ C, C where δ is defied as α α + δ C =. Estimates of the effects ad tests with a covariate It ca be show that β0 = 0 β α = biv = ZX Z = α, β = α + C β = ρx UCorCov () ( ) which are the same slope estimates for treatmet effects (8) as i the first sceario, with the additio of the slope for the covariate ( β ). The samplig variaces ad covariaces of these slopes also take a similar form as i the first case (9)

5 () var UCorCov { biv } = ( ) ( UCorCov ZX ZZ XZ) UCorCov UCorCov 0 0 UCorCov UCorCov UCorCov 0 C = UCorCov UCorCov 0 0 C C UCorCov The samplig variaces of the slopes () are essetiall the same as i the case without a covariate (9), except ow the samplig variaces of the slopes use a differet residual variace ( UCorCov vs. ) from the model residuals. For the scale free test presetatios to maitai the same defied effect size (the effect over the populatio stadard deviatio, ), the withi-group outcome variace explaied b the predictor ca be emploed, R w, to adjust the effect sizes as a divisor (Lips, 998). This makes the z-test of the mai effect. (4) z α = = = ( R ) w Rw α δ Rw, where δ is defied as above. The test of the optioal treatmet is also improved b the covariate (5) α α + z = C = δ C ( Rw ) Rw C where δδ is defied as above, It is plausible that the covariate emploed i the test of treatmet effects is correlated with the participat s choice to take up the optioal treatmet compoet. However, this correlatio has o effect o the tests of the mai effect or optioal compoet, sice the covariate i icluded i all predictor matrixes, ad thus the correlatio with choice is removed based o the coditioal model. The paper will coclude b usig these formulas for sample size calculatios, e.g., δ = M = z z = = ( )

6 Refereces Camero, A. C., & Trivedi, P. K. (005). Microecoometrics: methods ad applicatios: Cambridge uiversit press. Cohe, J. (977). Statistical power aalsis for the behavioural scieces (Revised editio). New York, 7. Collis, M. J., Baker, C., & Gore, R. (007). Fiacial Coachig: A New Approach for Asset Buildig? Delgadillo, L. M. (04). Fiacial clarit: Educatio, literac, capabilit, couselig, plaig, ad coachig. Famil ad Cosumer Scieces Research Joural, 4(), 8-8. Lips, M. (998). Desig sesitivit: statistical power for applied experimetal research. Hadbook of applied social research methods. Thousad Oaks, CA: Sage Publicatios, 9-68.