document: [ secintro ] dyadic data are common in psychosocial and behavioral studies [ @xcite ] . 
   many social phenomena , such as dating and marital relationships , are interpersonal by definition , and , as a result , related observations do not refer to a single person but rather to both persons involved in the dyadic relationship . members of dyads often influence each other s cognitions , emotions and behaviors , which leads to interdependence in a relationship . 
   for example , a husband s ( or wife s ) drinking behavior may lead to lowered marital satisfaction for the wife ( or husband ) . 
   a consequence of interdependence is that observations of the two individuals are correlated . 
   for example , the marital satisfaction scores of husbands and wives tend to be positively correlated . 
   one of the primary objectives of relationship research is to understand the interdependence of individuals within dyads and how the attributes and behaviors of one dyad member impact the outcome of the other dyad member .    in many studies , 
   dyadic outcomes are measured over time , resulting in longitudinal dyadic data . 
   repeatedly measuring dyads brings in two complications . 
   first , in addition to the within - dyad correlation , repeated measures on each subject are also correlated , that is , within - subject correlation . when analyzing longitudinal dyadic data , it is important to account for these two types of correlations simultaneously ; otherwise , the analysis results may be invalid . 
   the second complication is that longitudinal dyadic data are prone to the missing data problem caused by dropout , whereby subjects are lost to follow - up and their responses are not observed thereafter . in psychosocial dyadic studies , 
   the dropouts are often nonignorable or informative in the sense that the dropout depends on missing values . in the presence of the nonignorable dropouts 
   , conventional statistical methods may be invalid and lead to severely biased estimates [ @xcite ] . 
   there is extensive literature on statistical modeling of nonignorable dropouts in longitudinal studies . based on different factorizations of the likelihood of the outcome process and 
   the dropout process , @xcite identified two broad classes of likelihood - based nonignorable models : selection models [ @xcite ; @xcite ; follman and wu ( @xcite ) ; @xcite ] and pattern mixture models [ @xcite ; little ( @xcite , @xcite ) ; hogan and laird ( @xcite ) ; @xcite ; @xcite ] . 
   other likelihood - based approaches that do not directly belong to this classification have also been proposed in the literature , for example , the mixed - effects hybrid model by @xcite and a class of nonignorable models by @xcite . 
   another general approach for dealing with nonignorable dropouts is based on estimation equations and includes @xcite , @xcite , @xcite and @xcite . 
   recent reviews of methods handling nonignorable dropouts in longitudinal data can be found in @xcite , @xcite , little ( @xcite ) , @xcite and @xcite . in spite of the rich body of literature noted above , to the best of our knowledge 
   , the nonignorable dropout problem has not been addressed in the context of longitudinal dyadic data . 
   the interdependence structure within dyads brings new challenges to this missing data problem . 
   for example , within dyads , one member s outcome often depends on his / her covariates , as well as the other member s outcome and covariates . 
   thus , the dropout of the other member in the dyad causes not only a missing ( outcome ) data problem for that member , but also a  missing ( covariate ) data problem for the member who remains in the study.=-1    we propose a fully bayesian approach to deal with longitudinal dyadic data with nonignorable dropouts based on a selection model . 
   specifically , we model each subject s longitudinal measurement process using a transition model , which includes both the patient s and spouse s characteristics as covariates in order to capture the interdependence between patients and their spouses . 
   we account for the within - dyad correlation by introducing dyad - specific random effects into the transition model . to accommodate the nonignorable dropouts , we take the selection model approach by directly modeling the relationship between the dropout process and missing outcomes using a discrete time survival model .    the remainder of the article is organized as follows . in section  [ sec2 ] 
   we describe our motivating data collected from a longitudinal dyadic breast cancer study . in section  [ sec3 ] 
   we propose a bayesian selection - model - based approach for longitudinal dyad data with informative nonresponse , and provide estimation procedures using a gibbs sampler in section  [ sec4 ] . in section  [ sec5 ] we present simulation studies to evaluate the performance of the proposed method . in section  [ sec6 ] we illustrate our method by analyzing a breast cancer data set and we provide conclusions in section  [ sec7 ] . 
   our research is motivated by a single - arm dyadic study focusing on physiological and psychosocial aspects of pain among patients with breast cancer and their spouses [ @xcite ] . for individuals with breast cancer , spouses are most commonly reported as being the primary sources of support [ @xcite ] , and spousal support is associated with lower emotional distress and depressive symptoms in these patients [ @xcite ] . 
   one specific aim of the study is to characterize the depression experience due to metastatic breast cancer from both patients and spouses perspectives , and examine the dyadic interaction and interdependence of patients and spouses over time regarding their depression . 
   the results will be used to guide the design of an efficient prevention program to decrease depression among patients . 
   for example , conventional prevention programs typically apply interventions to patients directly . 
   however , if we find that the patient s depression depends on both her own and spouse s previous depression history and chronic pain , when designing a prevention program to improve the depression management and pain relief , we may achieve better outcomes by targeting both patients and spouses simultaneously rather than targeting patients only . in this study 
   ,    female patients who had initiated metastatic breast cancer treatment were approached by the project staff . 
   patients meeting the eligibility criteria ( e.g. , speak english , experience pain due to the breast cancer , having a male spouse or significant other , be able to carry on pre - disease performance , be able to provide informed consent ) were asked to participate the study on a voluntary basis . 
   the participation of the study would not affect their treatment in any way . 
   depression in patients and spouse was measured at three time points ( baseline , 3 months and 6 months ) using the center for epidemiologic studies depression scale ( cesd ) questionnaires . 
   however , a substantial number of dropouts occurred . 
   baseline cesd measurements were collected from 191 couples ; however , at 3 months , 101 couples ( 105 patients and 107 spouses ) completed questionnaires , and at 6 months , 73 couples ( 76 patients and 79 spouses ) completed questionnaires . 
   the missingness of the cesd measurements is likely related to the current depression levels of the patients or spouses , thus an nonignorable missing data mechanism is assumed for this study . 
   consequently , it is important to account for the nonignorable dropouts in this data analysis ; otherwise , the results may be biased , as we will show in section  [ sec6 ] . 
   consider a longitudinal dyadic study designed to collect @xmath0 repeated measurements of a response @xmath1 and a vector of covariates @xmath2 for each of @xmath3 dyads . 
   let @xmath4 , @xmath5 and @xmath6 denote the outcome , @xmath7 covariate vector and outcome history , respectively , for the member @xmath8 of dyad @xmath9 at the @xmath10th measurement time with @xmath11 . 
   we assume that @xmath2 is fully observed ( e.g. , is external or fixed by study design ) , but @xmath1 is subject to missingness due to dropout . 
   the random variable  @xmath12 , taking values from @xmath13 to @xmath14 , indicates the time the member @xmath8 of the @xmath9th dyad drops out , where @xmath15 if the subject completes the study , and @xmath16 if the subject drops out between the @xmath17th and @xmath10th measurement time , that is , @xmath18 are observed and @xmath19 are missing . 
   we assume at least 1 observation for each subject , as subjects without any observations have no information and are often excluded from the analysis .    when modeling longitudinal dyadic data , we need to consider two types of correlations : the within - subject correlation due to repeated measures on a subject , and the within - dyad correlation due to the dyadic structure 
   . we account for the first type of correlation by a transition model , and the second type of correlation by dyad - specific random effects @xmath20 , as follows : @xmath21 
   regression parameters in this random - effects transition model have intuitive interpretations similar to those of the actor  partner interdependence model , a conceptual framework proposed by @xcite to study dyadic relationships in the social sciences and behavior research fields . 
   specifically ,  @xmath22 and @xmath23 represent the `` actor '' effects of the patient , which indicate how the covariates and the outcome history of the patient ( i.e. , @xmath24 and @xmath25 ) affect her own current outcome , whereas @xmath26 and @xmath27 represent the `` partner '' effects for the patient , which indicate how the covariates and the outcome history of the spouse ( i.e. , @xmath28 and @xmath29 ) affect the outcome of the patient . similarly , @xmath30 and @xmath31 characterize the actor effects and @xmath32 and @xmath33 characterize the partner effects for the spouse of the patient . 
   estimates of the actor and partner effects provide important information about the interdependence within dyads . 
   we assume that residuals @xmath34 and @xmath35 are independent and follow normal distributions @xmath36 and @xmath37 , respectively ; and @xmath34 and @xmath35 are independent of random effects @xmath20 s . the parameters @xmath38 and @xmath39 are intercepts for the patients and spouses , respectively .    in many situations , 
   the conditional distribution of @xmath4 given @xmath40 and  @xmath5 depends only on the @xmath41 prior outcomes @xmath42 and @xmath5 . 
   if this is the case , we obtain the so - called @xmath41th - order transition model , a type of transition model that is most useful in practice [ @xcite ] . 
   the choice of the model order @xmath41 depends on subject matters . in many applications , 
   it is often reasonable to set @xmath43 when the current outcome depends on only the last observed previous outcome , leading to commonly used markov models . the likelihood ratio test can be used to assess whether a specific value of  @xmath41 is appropriate [ @xcite ] . 
   auto - correlation analysis of the outcome history also can provide useful information to determine the value of @xmath41 [ @xcite ; @xcite ] . 
   define @xmath44 and @xmath45 for @xmath46 . given \{@xmath47 and the random effect @xmath20 , the joint log likelihood of @xmath48 for the @xmath9th dyad under the @xmath41th - order ( random - effects ) transition model 
   is given by @xmath49 where @xmath50 is the likelihood corresponding to model ( [ transition ] ) , and @xmath51 is assumed free of @xmath52 , for @xmath46 . 
   an important feature of model ( [ transition ] ) that distinguishes it from the standard transition model is that the current value of the outcome @xmath1 depends on not only the subject s outcome history , but also the spouse s outcome history . 
   such a `` partner '' effect is of particular interest in dyadic studies because it reflects the interdependence between the patients and spouses . 
   this interdependence within dyads also makes the missing data problem more challenging . 
   consider a dyad consisting of subjects @xmath53 and  @xmath54 and that  @xmath54 drops out prematurely . because the outcome history of @xmath54 is used as a  covariate in the transition model of @xmath53 , when @xmath54 drops out , we face not only the missing outcome ( for @xmath54 ) but also missing covariates ( for @xmath53 ) . 
   we address this dual missing data problem using the data augmentation approach , as described in section  [ sec4 ] .    to account for nonignorable dropouts , 
   we employ the discrete time survival model [ @xcite ] to jointly model the missing data mechanism . 
   specifically , we assume that the distribution of @xmath12 depends on both the past history of the longitudinal process and the current outcome @xmath4 , but not on future observations . 
   define the discrete hazard rate @xmath55 . 
   it follows that the probability of dropout for the member @xmath8 in the @xmath9th dyad is given by @xmath56    we specify the discrete hazard rate @xmath57 using the logistic regression model : @xmath58 where @xmath59 is the random effect accounting for the within - dyadic correlation , and @xmath60 and @xmath61 are unknown parameters . in this dropout model 
   , we assume that , conditioning on the random effects , a subject s covariates , past history and current ( unobserved ) outcome , the dropout probability of this subject is independent of the characteristics and outcomes of the other member in the dyad . 
   the spouse may indirectly affect the dropout rate of the patient through influencing the patient s depression status ; however , when conditional on the patient s depression score , the dropout of the patient does not depend on her spouse s depression score .    in practice 
   , we often expect that , given @xmath4 and @xmath62 , the conditional dependence of @xmath12 on @xmath63 will be negligible because , temporally , the patient s ( current ) decision of dropout is mostly driven by his ( or her ) current and the most recent outcome statuses . using the breast cancer study as an example , we do not expect that the early history of depression plays an important role for the patient s current decision of dropout ; instead , the patient drops out typically because she is currently experiencing or most recently experienced high depression . 
   the early history may influence the dropout but mainly through its effects on the current depression status . 
   once conditioning on the current and the most recent depression statuses , the influence from the early history is essentially negligible . 
   thus , we use a  simpler form of the discrete hazard model @xmath64 
   [ secestimation ] under the bayesian paradigm , we assign the following vague priors to the unknown parameters and fit the proposed model using a gibbs sampler : @xmath65 where @xmath66 denote an inverse gamma distribution with a shape parameter @xmath67 and a scale parameter @xmath68 . 
   we set @xmath67 and @xmath68 at smaller values , such as 0.1 , so that the data dominate the prior information . 
   let @xmath69 and @xmath70 denote the observed and missing part of the data , respectively . 
   considering the @xmath8th iteration of the gibbs sampler , the first step of the iteration is `` data augmentation '' [ @xcite ] , in which the missing data  @xmath70 are generated from their full conditional distributions . without loss of generality , 
   suppose for the @xmath9th dyad , member 2 drops out of the study no later than member 1 , that is , @xmath71 , and let @xmath72 . 
   assuming a  first - order ( @xmath73 ) transition model ( or markov model ) and letting @xmath74 denote a  generic symbol that represents the values of all other model parameters , the data augmentation consists of the following 3 steps :    for @xmath75 , draw @xmath76 from the conditional distribution @xmath77 where @xmath78    draw @xmath79 from the conditional distribution @xmath80    draw @xmath81 from the conditional distribution @xmath82    now , with the augmented complete data @xmath83 , the parameters are drawn alternatively as follows :    for @xmath84 , draw random effects @xmath20 from the conditional distribution @xmath85 where @xmath86    draw @xmath87 from the conditional distribution @xmath88 where @xmath89    draw @xmath90 from the conditional distribution @xmath91    draw @xmath92 from the normal distribution @xmath93 where @xmath94 and @xmath95    similarly , draw @xmath96 from the conditional distribution @xmath97 where @xmath98 and @xmath99 are defined in a similar way to @xmath100 and @xmath101    draw @xmath102 and @xmath103 from the conditional distributions @xmath104 
   draw random effects @xmath59 from the conditional distribution @xmath105    draw @xmath106 from the conditional distribution @xmath107 
   [ secsimu ]    we conducted two simulation studies ( a and b ) . 
   simulation a consists of 500 data sets , each with 200 dyads and three repeated measures . for the @xmath9th dyad 
   , we generated the first measurements ,  @xmath108 and @xmath109 , from normal distributions @xmath110 and @xmath111 , respectively , and generated the second and third measurements based on the following random - effects transition model : @xmath112 where @xmath113 , @xmath114 , @xmath115 , and covariates  @xmath116 and @xmath117 were generated independently from @xmath118 . 
   we assumed that the baseline ( first ) measurements @xmath108 and @xmath109 were observed for all subjects , and the hazard of dropout at the second and third measurement times depended on the current and last observed values of @xmath1 , that is , @xmath119 under this dropout model , on average , 24% ( 12% of member 1 and 13% of member 2 ) of the dyads dropped out at the second time point and 45% ( 26% of member  1 and 30% of member  2 ) dropped out at the third measurement time . 
   we applied the proposed method to the simulated data sets . 
   we used 1,000 iterations to burn in and made inference based on 10,000 posterior draws . for comparison purposes 
   , we also conducted complete - case and available - case analyses . 
   the complete - case analysis was based on the data from dyads who completed the follow - up , and the available - case analysis was based on all observed data ( without considering the missing data mechanism ) . 
   [ tab1 ]    @ld2.2ccd2.2ccd2.2cc@ & & & + & & & + * parameter * & & * se * & * coverage * & & * se * & * coverage * & & * se * & * coverage * + @xmath120 & -0.03 & 0.06 & 0.93 & -0.01 & 0.05 & 0.94 & -0.01 & 0.05 & 0.95 + @xmath121 & -0.06 & 0.05 & 0.81 & -0.03 & 0.04 & 0.88 & 0.07 & 0.04 & 0.96 + @xmath122 & -0.16 & 0.12 & 0.72 & -0.10 & 0.10 & 0.81 & 0.05 & 0.08 & 0.94 + @xmath123 & -0.17 & 0.12 & 0.75 & -0.10 & 0.10 & 0.78 & 0.02 & 0.09 & 0.97 + @xmath124 & -0.06 & 0.06 & 0.89 & -0.06 & 0.05 & 0.84 & 0.08 & 0.05 & 0.97 + @xmath125 & -0.04 & 0.05 & 0.87 & -0.00 & 0.04 & 0.95 & -0.04 & 0.06 & 0.96 + @xmath126 & -0.17 & 0.12 & 0.73 & -0.10 & 0.10 & 0.84&-0.01 & 0.12 & 0.95 + @xmath127 & -0.17 & 0.12 & 0.72 & -0.10 & 0.10 & 0.81 & 0.01 & 0.09 & 0.97 +    table [ tbtransmodel1 ] shows the bias , standard error ( se ) and coverage rate of the 95% credible interval ( ci ) under different approaches . 
   we can see that the proposed method substantially outperformed the complete - case and available - case analyses . 
   our method yielded estimates with smaller bias and coverage rates close to the 95% nominal level . 
   in contrast , the complete - case and available - case analyses often led to larger bias and poor coverage rates . 
   for example , the bias of the estimate of @xmath122 under the complete - case and available - case analyses were @xmath128 and @xmath129 , respectively , substantially larger than that under the proposed method ( i.e. , 0.05 ) ; the coverage rate using the proposed method was about 94% , whereas those using the complete - case and available - case analyses were under 82% . 
   the second simulation study ( simulation b ) was designed to evaluate the performance of the proposed method when the nonignorable missing data mechanism is misspecified , for example , data actually are missing at random ( mar ) . 
   we generated the first measurements , @xmath108 and @xmath109 , from normal distribution @xmath130 independently , and generated the second and third measurements based on the same transition model as in simulation a. we assumed the hazard of dropout at the second and third measurement times depended on the previous ( observed ) value of @xmath1 quadratically , but not on the current ( missing ) value of @xmath1 , that is , @xmath131 under this mar dropout model , on average , 37% ( 21% of member 1 and 21% of member 2 ) of the dyads dropped out at the second time point and 27% ( 24% of member 1 and 33% of member 2 ) dropped out at the third measurement time . to fit the simulated data , we considered two nonignorable models with different specifications of the dropout ( or selection ) model . 
   the first nonignorable model assumed a flexible dropout model @xmath132 which included the true dropout process ( [ simu2 ] ) as a specific case with @xmath133 ; and the second nonignorable model took a misspecified dropout model of the form @xmath134    table [ simulationc ] shows the bias , standard error and coverage rate of the 95% ci under different approaches . 
   when the missing data were mar , the complete - case analysis was invalid and led to biased estimates and poor coverage rates because the complete cases are not random samples from the original population . 
   in contrast , the available - case analysis yielded unbiased estimates and coverage rates close to the 95% nominal level . 
   for the nonignorable models , the one with the flexible dropout model yielded unbiased estimates and reasonable coverage rates , whereas the model with the misspecified dropout model led to biased estimates ( e.g. , @xmath135 and @xmath136 ) and poor coverage rates . 
   this result is not surprising because it is well known that selection models are sensitive to the misspecification of the dropout model [ @xcite ; @xcite ] . for nonignorable missing data , 
   the difficulty is that we can not judge whether a specific dropout model is misspecified or not based solely on observed data because the observed data contain no information about the ( nonignorable ) missing data mechanism . to address this difficulty , 
   one possible approach is to specify a flexible dropout model to decrease the chance of model misspecification . 
   alternatively , maybe a better approach is to conduct sensitivity analysis to evaluate how the results vary when the dropout model varies . 
   we will illustrate the latter approach in the next section . 
   =    [ tab2 ]    @ld2.2cccccd2.2ccd2.2cc@ & & & & & & & & + & & & & + & & & & + * parameter * & & * se * & * coverage * & * bias * & * se * & * coverage * & & * se * & * coverage * & & * se * & * coverage * + @xmath120 & -0.06 & 0.08 & 0.86 & 0.00 & 0.06 & 0.95 & -0.01 & 0.06 & 0.95 & 0.14 & 0.06 & 0.78 + @xmath121 & -0.09 & 0.08 & 0.82 & 0.00 & 0.05 & 0.96 & 0.07 & 0.05 & 0.97 & -0.01 & 0.05 & 0.95 + @xmath122 & -0.11 & 0.14 & 0.84 & 0.00 & 0.10 & 0.95 & 0.04 & 0.08 & 0.96 & 0.03 & 0.08 & 0.94 + @xmath123 & -0.13 & 0.14 & 0.84 & 0.00 & 0.10 & 0.96 & 0.02 & 0.09 & 0.97 & 0.02 & 0.09 & 0.98 + @xmath124 & -0.07 & 0.08 & 0.87 & 0.00 & 0.06 & 0.96 & 0.02 & 0.06 & 0.97 & 0.12 & 0.06 & 0.79 + @xmath125 & -0.10 & 0.08 & 0.78 & 0.00 & 0.07 & 0.96 & 0.00 & 0.06 & 0.96 & -0.08 & 0.06 & 0.93 + @xmath126 & -0.14 & 0.14 & 0.82 & 0.00 & 0.10 & 0.96&0.01 & 0.12 & 0.94 & 0.01 & 0.12 & 0.95 + @xmath127 & -0.14 & 0.13 & 0.83 & 0.01 & 0.10 & 0.96 & 0.01 & 0.09 & 0.97 & 0.01 & 0.09 & 0.98 + 
   [ secapplication ]    we applied our method to the longitudinal metastatic breast cancer data . 
   we used the first - order random - effects transition model for the longitudinal measurement process . in the model 
   , we included 5 covariates : chronic pain measured by the multidimensional pain inventory ( mpi ) and previous cesd scores from both the patients and spouses , and the patient s stage of cancer . in the discrete - time dropout model 
   , we included the subject s current and previous cesd scores , mpi measurements and the patient s stage of cancer as covariates . 
   age was excluded from the models because its estimate was very close to 0 and not significant . 
   we used 5,000 iterations to burn in and made inference based on 5,000 posterior draws . 
   we also conducted the complete - case and available - case analyses for the purpose of comparison . 
   [ tab3 ]    @lcd2.14d2.14d2.14@ & & & & + & intercept & 2.53  ( -1.71 , 6.77 ) & 0.99  ( -2.55 , 4.52 ) & 5.10  ( 3.31 , 6.59 ) + & patient cesd & 0.43  ( 0.29 , 0.58 ) & 0.56  ( 0.44 , 0.68)&0.87  ( 0.80 , 0.93 ) + & spouse cesd & 0.07  ( -0.06 , 0.20 ) & 0.06  ( -0.06 , 0.17 ) & 0.14  ( 0.09 , 0.19 ) + & patient mpi & 0.94  ( 0.22 , 1.67 ) & 0.82  ( 0.21 , 1.43 ) & 1.24  ( 0.83 , 1.64 ) + & spouse mpi & 1.06  ( 0.29 , 1.82 ) & 0.90  ( 0.31 , 1.48 ) & 0.62  ( 0.40 , 0.84 ) + & cancer stage & 0.39  ( -0.81 , 1.60 ) & 0.59  ( -0.43 , 1.60 ) & 0.10  ( -0.47 , 0.66 ) + [ 4pt ] spouses & intercept & 3.68  ( -0.55 , 7.92 ) & 2.00  ( -1.63 , 5.64 ) & 8.16  ( 4.26 , 11.9 ) + & patient cesd & -0.05  ( -0.19 , 0.09 ) & 0.01  ( -0.11 , 0.13 ) & 0.68  ( 0.63 , 0.74 ) + & spouse cesd & 0.77  ( 0.64 , 0.90 ) & 0.78  ( 0.66 , 0.89 ) & 0.76  ( 0.71 , 0.81 ) + & patient mpi & 0.43  ( -0.29 , 1.15 ) & 0.27  ( -0.27 , 0.81 ) & 0.53  ( 0.33 , 0.73 ) + & spouse mpi & 0.55  ( -0.22 , 1.31 ) & 0.58  ( -0.04 , 1.20)&0.36  ( -0.64 , 1.15 ) + & cancer stage & -0.42  ( -1.63 , 0.79)&-0.21  ( -1.23 , 0.80)&-0.50  ( -0.92 , 0.09 ) +    as shown in table  [ tab3 ] , the proposed method suggests significant `` partner '' effects for the patients . specifically , the patient s depression increases with her spouse s mpi [ @xmath137 and 95% @xmath138 and previous cesd [ @xmath139 and 95% @xmath140 . 
   in addition , there are also significant `` actor '' effects for the patients , that is , the patient s depression is positively correlated with her own mpi and previous cesd scores . for the spouses , we observed similar significant `` partner '' effects : the spouse s depression increases with the patient s mpi and previous cesd scores . 
   the `` actor '' effects for the spouses are different from those for the patients . 
   the spouse s depression correlates with his previous cesd scores but not the mpi level , whereas the patient s depression is related to both variables . 
   based on these results , we can see that the patients and spouses are highly interdependent and influence each other s depression status . 
   therefore , when designing a prevention program to reduce depression in patients , we may achieve better outcomes by targeting both patients and spouses simultaneously . 
   as for the dropout process , the results in table  [ tab4 ] suggest that the missing data for the patients are nonignorable because the probability of dropout is significantly associated with the patient s current ( missing ) cesd score . 
   in contrast , the missing data for the spouse appears to be ignorable , as the probability of dropout does not depend on the spouse s current ( missing ) cesd score . for the variance components , the estimates of residuals variances for patients and spouses are @xmath141 [ 95% @xmath142 and @xmath143 [ 95% @xmath144 , respectively 
   . the estimates of the variances for the random effects    @xmath20 and @xmath59 are @xmath145 [ 95% @xmath146 and @xmath147 [ 95% @xmath148 , respectively , suggesting substantial variations across dyads . 
   compared to the proposed approach , both the complete - case and available - case analyses fail to detect some `` partner '' effects . for example , for spouses , the complete - case and available - case analyses assert that the spouse s cesd is correlated with his own previous cesd scores only , whereas the proposed method suggested that the spouse s cesd is related not only to his own cesd but also to the patient s cesd and mpi level . 
   in addition , for patients , the `` partner '' effect of the spouse s cesd is not significant under the complete - case and available - case analyses , but is significant under the proposed approach . 
   these results suggest that ignoring the nonignorable dropouts could lead to a failure to detect important covariate effects . 
   nonidentifiability is a common problem when modeling nonignorable missing data . in our approach , 
   the observed data contain very limited information on the parameters that link the missing outcome with the dropout process , that is , @xmath149 and @xmath150 in the dropout model . 
   the identification of these parameters is heavily driven by the untestable model assumptions [ @xcite ; @xcite ] . in this case 
   , a sensible strategy is to perform a sensitivity analysis to examine how the inference changes with respect to the values of @xmath151 and @xmath152 [ daniels and hogan ( @xcite , @xcite ) ; @xcite ] . 
   we conducted a bayesian sensitivity analysis by assuming informative normal prior distributions for @xmath149 and @xmath153 with a small variance of 0.01 and the mean fixed , successively , at various values . 
   figures [ figsensitivitygibbs1 ] and [ figsensitivitygibbs2 ] show the parameter estimates of the measurement models when the prior means of @xmath149 and @xmath150 vary from @xmath154 to 3 . 
   in general , the estimates were quite stable , except that the estimate of cancer stage in the measurement model of patient ( figure [ figsensitivitygibbs1 ] ) and the estimate of spouse s mpi in the measurement model of spouse ( figure [ figsensitivitygibbs2 ] ) demonstrated some variations .     and @xmath150 with a mean varying from @xmath154 to 3 and a fixed variance of 0.01 . ] 
   [ fig1 ]     and @xmath150 with a mean varying from @xmath154 to 3 and a fixed variance of 0.01 . ]    [ fig2 ]    we conducted another sensitivity analysis on the prior distributions of  @xmath155 , @xmath156 , @xmath90 and @xmath106 . 
   we considered various inverse gamma priors , @xmath157 , by setting @xmath158 and 5 . 
   as shown in table [ tablepriorvar ] , the estimates of the measurement model parameters were stable under different prior distributions , suggesting the proposed method is not sensitive to the priors of these parameters . 
   [ seccon ] we have developed a selection - model - based approach to analyze longitudinal dyadic data with nonignorable dropouts . 
   we model the longitudinal outcome process using a transition model and account for the correlation within dyads using random effects . in the model , we allow a  subject s outcome to depend on not only his / her own characteristics but also the characteristics of the other member in the dyad . as a  result , 
   the parameters of the proposed model have appealing interpretations as `` actor '' and `` partner '' effects , which greatly facilitates the understanding of interdependence within a relationship and the design of more efficient prevention programs . to account for the nonignorable dropout 
   , we adopt a  discrete time survival model to link the dropout process with the longitudinal measurement process . 
   we used the data augment method to address the complex missing data problem caused by dropout and interdependence within dyads . 
   the simulation study shows that the proposed method yields consistent estimates with correct coverage rates . 
   we apply our methodology to the longitudinal dyadic data collected from a breast cancer study . 
   our method identifies more `` partner '' effects than the methods that ignore the missing data , thereby providing extra insights into the interdependence of the dyads . 
   for example , the methods that ignore the missing data suggest that the spouse s cesd related only to his own previous cesd scores , whereas the proposed method suggested that the spouse s cesd related not only to his own cesd but also to the patient s cesd and mpi level 
   . this extra information can be useful for the design of more efficient depression prevention programs for breast cancer patients . 
   [ tab5 ]    @lcd2.14d2.14d2.14@ & & & & + & intercept & 4.72  ( 3.32 , 6.11 ) & 5.00  ( 3.48 , 6.47)&5.02  ( 3.57 , 6.48 ) + & patient cesd & 0.87  ( 0.81 , 0.93 ) & 0.86  ( 0.80 , 0.92 ) & 0.88  ( 0.83 , 0.94 ) + & spouse cesd & 0.14  ( 0.09 , 0.19 ) & 0.14  ( 0.08 , 0.19 ) & 0.13  ( 0.08 , 0.18 ) + & patient mpi & 1.27  ( 0.84 , 1.71 ) & 1.12  ( 0.67 , 1.60 ) & 1.20  ( 0.85 , 1.57 ) + & spouse mpi & 0.71  ( 0.49 , 0.91 ) & 0.68  ( 0.46 , 0.87 ) & 0.61  ( 0.39 , 0.82 ) + & cancer stage & -0.03  ( -0.50 , 0.50)&0.18  ( -0.31 , 0.65)&-0.08  ( -0.57 , 0.40 ) + spouses & intercept & 6.40  ( 4.39 , 8.41 ) & 7.56  ( 5.35 , 9.93 ) & 7.52  ( 5.43 , 9.55 ) + & patient cesd & 0.67  ( 0.62 , 0.73 ) & 0.67  ( 0.62 , 0.72 ) & 0.69  ( 0.64 , 0.73 ) + & spouse cesd & 0.76  ( 0.71 , 0.80 ) & 0.75  ( 0.71 , 0.81 ) & 0.75  ( 0.71 , 0.80 ) + & patient mpi & 0.51  ( 0.32 , 0.71 ) & 0.54  ( 0.35 , 0.73 ) & 0.53  ( 0.34 , 0.72 ) + & spouse mpi & 0.79  ( -0.05 , 1.46 ) & 0.54  ( -0.03 , 1.06 ) & 0.45  ( -0.23 , 1.09 ) + & cancer stage & -0.41  ( -0.86 , 0.02 ) & -0.38  ( -0.81 , 0.03 ) & -0.48  ( -0.87 , 0.08 ) +    in the proposed dropout model ( [ dropmodel ] ) , we assume that time - dependent covariates @xmath5 and @xmath159 , @xmath46 , have captured all important time - dependent factors that influence dropout . however , this assumption may not be always true . 
   a more flexible approach is to include in the model a time - dependent random effect @xmath160 to represent all unmeasured time - variant factors that influence dropout . 
   we can further put a hierarchical structure on @xmath160 to shrink it toward a dyad - level time - invariant random effect @xmath59 to account for the effects of unmeasured time - invariance factors on dropout . 
   in addition , in ( [ dropmodel ] ) , in order to allow members in a dyad to drop out at different times , we specify separate dropout models for each dyadic member , linked by a common random effect . although the common random effect makes the members in a dyad more likely to drop out at the same time , it may not be the most effective modeling approach when dropout mostly occurs at the dyad level . in this case 
   , a more effective approach is that , in addition to the dyad - level random effect , we further put hierarchical structure on the coefficients of common covariates ( in the two dropout models ) to shrink toward a common value to reflect that dropout is almost always at the dyad level . 
   we would like to thank the referees , associate editor and editor ( professor susan paddock ) for very helpful comments that substantially improved this paper .
summary: dyadic data are common in the social and behavioral sciences , in which members of dyads are correlated due to the interdependence structure within dyads . 
   the analysis of longitudinal dyadic data becomes complex when nonignorable dropouts occur . 
   we propose a  fully bayesian selection - model - based approach to analyze longitudinal dyadic data with nonignorable dropouts . 
   we model repeated measures on subjects by a transition model and account for within - dyad correlations by random effects . in the model , we allow subject s outcome to depend on his / her own characteristics and measure history , as well as those of the other member in the dyad 
   . we further account for the nonignorable missing data mechanism using a selection model in which the probability of dropout depends on the missing outcome . 
   we propose a gibbs sampler algorithm to fit the model . 
   simulation studies show that the proposed method effectively addresses the problem of nonignorable dropouts . 
   we illustrate our methodology using a longitudinal breast cancer study .    .