Robust Inference for Causal Mediation Analysis of Recurrent Event Data

Recurrent events, including cardiovascular events, are commonly observed in biomedical studies. Researchers must understand the effects of various treatments on recurrent events and investigate the underlying mediation mechanisms by which treatments …

Authors: Yan-Lin Chen, Yan-Hong Chen, Pei-Fang Su

Robust Inference for Causal Mediation Analysis of Recurrent Event Data
1 Robust Infer ence for Causal Mediation Analysis of Recurr ent Event Dat a   1      2  3   4     3  1 Institute of Statistics, National Yang Ming Chiao Tung Unive rsity, Hsin -Chu, Taiwan 2 Institute of Statistical Science , Academia Sinica, Taipei, Taiwan 3 Department of Statistics, National Cheng Kung University, Tainan, Ta iw an.         *Corre spon ding author           S UMMAR Y :                                                                                                                                                        2 1. Introduction  Mediation pr oblem with r ecurr ent event data                                                                                                                                                                                                                                                                                                                  3                                                                                                                                                                                      Related work s                                                                                                                                          4                                                                                                                                                                                                                                             Contribution s of the pr esent study                                                                          5                                                                                                                                                                                               2. M ediation an aly sis with outcome s of a counting process outcome  Notations and definitions                               󰇛       󰇜                                                          6      󰇛  󰇜                                               󰇛  󰇜               󰇛  󰇜              󰇛  󰇜     󰇛  󰇜    󰇛  󰇜                                                                                                                   󰇛  󰇜                󰇛     󰇜                     󰇛   󰇜                                            Figure 1.      7  Dir ect and indir ect effect s                                              󰇛  󰇜   󰇛  󰇛   󰇛  󰇜 󰇜󰇜   󰇛  󰇛   󰇛  󰇜 󰇜󰇜   󰇛    󰇜   󰇛   󰇜   󰇛  󰇜  󰇛  󰇛   󰇛  󰇜 󰇜󰇜  󰇛  󰇛    󰇛  󰇜 󰇜󰇜   󰇛   󰇜   󰇛   󰇜   󰇛      󰇜   󰇛  󰇛     󰇛   󰇜 󰇜󰇜                                                                                    󰇛    󰇜    󰇛     󰇛  󰇜 󰇜                                                      󰇛    󰇜   󰇛  󰇛    󰇜 󰇜    󰇛  󰇛   󰇜 󰇜   󰇛   󰇜                     󰇛  󰇜     󰇛  󰇜     󰇛  󰇛    󰇜 󰇜   󰇛  󰇛   󰇜 󰇜                Assump tions and identifica tion                                               8   󰇛       󰇜      (A1)    󰇛     󰇜 󰀓   (A2)    󰇛      󰇜 󰀓    (A3)    󰇛  󰇜 󰀓   (A4)      󰇛      󰇜 󰀓  󰇛   󰇜  (A5)                       󰇛  󰇜      (A6)                                    󰇛     󰇜    󰇛  󰇜                                                                                           󰇛 󰇜       󰇛󰇜                                                                  9                                                               󰇛      󰇜         󰇛      󰇜           󰇛      󰇜       󰇛  󰇜              󰇛           󰇜     󰇛  󰇜   󰇛      󰇜     󰇛       󰇜           󰇛     󰇜     󰇛  󰇜                                                                           󰇛  󰇜   󰇛   󰇜   󰇛   󰇜    󰇡   󰇛  󰇜                 󰇛  󰇜          󰇢   󰇛          󰇜   󰇛  󰇜    󰇛  󰇜   󰇛   󰇜   󰇛   󰇜   󰇧    󰇛  󰇜            󰇛         󰇜        󰇛  󰇜            󰇛         󰇜  󰇨   󰇛  󰇜  3. Estimation usi ng a proportional mean model 10  Regr ession-based estimati on                          󰇛       󰇜                󰇛      󰇜                                 󰇛  󰇜                 󰇛     󰇜                                                       󰇛      󰇜           󰇛  󰇜        󰇛  󰇜                        p             󰇛  󰇜       p                                                      ( CA. 1 )                     󰇛  󰇜 󰀓   󰇛  󰇜                     11  󰇛  󰇜    󰇫      󰇛  󰇜    󰇛     󰇜       󰇛  󰇜  󰇛     󰇜   󰇬   󰇛  󰇜                        󰇛  󰇜                   󰇛  󰇜      󰇛  󰇜            󰇛       󰇜        󰇛      󰇜             󰆹     󰆹            󰇛           󰇜      󰇛  󰇜                                                                                                            󰇛             󰇜      󰇛                󰇜                    󰇛      󰇜         󰇛      󰇜    󰇛  󰇜   󰆹     󰆹           󰇛  󰇜            󰇧         󰆹      󰆹            󰇨    󰇛  󰇜                󰇛  󰇜    󰇛  󰇜   󰆹        󰇛  󰇜    󰇛  󰇜   󰆹    󰆹        󰆹          󰇛  󰇜      󰇛  󰇜           󰇛      󰇜                    󰇛  󰇜   12                           󰇛      󰇜              󰇛      󰇜                                                                         󰇛      󰇜                                                                                  T riply r obust estim ation                            󰇛      󰇜      󰇛      󰇜     󰇫 󰆹  󰇛   󰇜  󰆹  󰇛         󰇜  󰆹  󰇛        󰇜      󰇛  󰇜        󰇛  󰇜             󰆹  󰇛    󰇜       󰇛  󰇜            󰇛          󰇜      󰇛          󰇜 󰇬    󰇝  󰇞     󰇟  󰇠       󰆹  󰇛    󰇜   󰇛     󰇜      󰇛  󰇜              󰇛  󰇜        󰇛        󰇜        󰇛  󰇜            󰇛         󰇜    13         󰇛     󰇜          󰇛     󰇜            󰇛     󰇜      󰆹  󰇛   󰇜     󰇛  󰇜 󰇞       󰇥 󰇡   󰆹  󰇛   󰇜 󰇢    󰇛          󰇜 󰇦                                                󰇛      󰇜      󰆹  󰇛     󰇜       󰇛  󰇜                                      󰇛      󰇜     󰇫 󰆹  󰇛   󰇜  󰆹  󰇛         󰇜  󰆹  󰇛        󰇜     󰇛  󰇜 󰇬                               󰇛      󰇜                                                                                                                                  14                                                                                                                                     (CA.2)                 󰇛  󰇜 󰀓     (CA.3)               󰇛  󰇜 󰀓                                                                                                                            15 T able 1.                            Estimator Censoring Assumption  󰇛  󰇜 󰀓   󰇛  󰇜      󰇛  󰇜 󰀓     󰇛  󰇜 󰀓        ◯    ◯ ◯     ◯ ◯     ◯ ◯ ◯    ◯ ◯ ◯ Abbr eviations: RB, regr ession -base d; TR, triply r obust.                                                                󰇛    󰇜      󰇛   󰇜                                     󰇡                       󰇢     󰇫 󰆹  󰇛      󰇜  󰆹              󰆹            󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                  󰇢󰇤     󰆹  󰇛       󰇜 󰇣  󰇡    󰇛  󰇜 󰇻                  󰇢     󰇡      󰇻               󰇢󰇤     󰇡      󰇻                󰇢󰇬            󰇛       󰇜                  16 T HEOREM 1 : Consistency and asymptotic pr operties Suppose that identification assumptions (A1)–(A6) and independent c ensoring assumptions (CA.1)–(CA.3) hold. (a)    󰇡                      󰇢 is consistent in   . (b)  󰇛    󰇡                        󰇢   󰇛      󰇜 󰇜 is asymptotically normal with mean zer o and varian ce  󰇛 󰇛  󰇛      󰇜 󰇜  󰇜 in   , wher e   󰇛      󰇜                       󰇛  󰇜    󰇧           󰇛    󰇛     󰇜󰇜  󰇨    󰇧                 󰇨                        󰇛  󰇜    󰇛   󰇜     󰇛   󰇜                󰇛  󰇜     󰇛   󰇜      󰇛     󰇜     󰇛          󰇜           󰇛  󰇜                          󰇛  󰇜     󰇛  󰇜                󰇛  󰇜                                                󰇛  󰇜                          󰇛  󰇜    󰇛  󰇜                                                                            󰇛   󰇜     󰇛   󰇜      󰇛   󰇜     󰇛   󰇜                        󰇡  󰇛   󰇜    󰇛    󰇜   󰇢    󰇡  󰇛   󰇜    󰇛   󰇜   󰇢                                           󰇛      󰇜           17    4. Numerical studies                                                                                                 󰇡       󰇢                         󰇛  󰇜                                                            Experiment 1: Robustness of the TR estimator under differ ent scenarios                                                              18                                                          T able 2.                                                        󰇛    󰇜   󰇛    󰇜   󰇛    󰇜   󰇛    󰇜                                                                                             󰇛    󰇜   󰇛    󰇜   󰇛    󰇜   󰇛    󰇜                                                                                     Abbr eviations: ESE, empirical standar d e rror; RB, regr ession -base d ; TR, trip ly r obust ; NDE, natural direc t effect; NIE, natural indir ect effect. 19 Experiment 2: Performance of the TR estimator with a low occurr ence rate                        󰇡   󰇛  󰇜 󰇢      󰇛  󰇜                                                                                    T able 3.                                                                 󰇛   󰇜                                                                                    20                                                                                                 Abbr eviations: ESE, empirical standar d err or; NDE, natural dire ct effect; NIE, natural indire ct effect. Experiment 3: Behavior of the TR estimator under early censoring                                                                                                󰇛  󰇜                                                                         21 T able 4.                                                  󰇛   󰇜   󰇛   󰇜   󰇛   󰇜   󰇛   󰇜                                                  󰇛   󰇜   󰇛   󰇜   󰇛   󰇜   󰇛   󰇜                                        Abbr eviations: ES E, empirical st andar d err or; PC , mean pr opo rtion of c ensored time; NDE, na tural dir ect effec t; NIE, na tural indire ct effect. Experiment 4: Effect of the violation of independent censoring assumptions                                                                                                                             22                                                                                                                                                                       23 Figure 2.                                       24 5. Real data analysis                                                                                                                     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                                                                                                                                                                                                            —            —                                                                26                                             •   󰇛      󰇜    󰇛                    󰇜  •   󰇛      󰇜                         •      󰇛  󰇜            󰇛  󰇜  󰇛                         󰇜    󰇛  󰇜                                                                                                                                                                                  27                                                                                                                      Figure 3.                          28 Figure 4.                 T able 5.                                        Abbr eviations: TE, total effect; NDE, natural dir ect effect; NIE, natural indir ect effect. 6. Discussion                                                   29                                                                                                                                                                                                                         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                         󰇛 󰇜                                                                                                                    31 Figure 5.                Acknowledgemen t                                Re fer enc e   Statistical models based on counting pr ocesses         Diabetes Car e  41        , et al.     Cir culation  129    The statist ic al analysis of re curr ent events    32    , et al.            Diabetes Car e  41                       BMJ open  11       Expert Opin Investig Drugs  12          , et al.      Diabetes Car e  46               , et al.     Lancet  380    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 Int J Car diol  220              , et al.      J Hypertens  32     36   Hypertension  37      Car dio vasc Diabetol  17     Statistics in medicine  39      Biostatistics  23       The international journal of biostatistics  7               The Annals of Statistics  40       Annu Rev Med  66           Epidemiology  22            Statistics and its Interface  2              , et al.        Curr 37 V asc Pharmacol  12     , et al.     N Engl J Med  375              J Am Coll C ar diol  72            Stat Med  40   38 Appendix Appendix 1: Procedure of identification      Assumptions (A1)    󰇛     󰇜 󰀓   (A2)    󰇛      󰇜 󰀓    (A3)    󰇛  󰇜 󰀓   (A4)       󰇛      󰇜 󰀓  󰇛   󰇜  (A5)                    󰇛  󰇜      (A6)                                            󰇛     󰇜    󰇛  󰇜          󰇛      󰇜    󰇛       󰇜  󰇛      󰇜   󰇡       󰇛   󰇜 󰇢          󰇛   󰇜     󰇛  󰇜             󰇛   󰇜  󰇛   󰇜        󰇛   󰇜 󰇛      󰇜    󰇛  󰇜         󰇛      󰇜     󰇛   󰇜 󰇛      󰇜    󰇛  󰇜         󰇛      󰇜         󰇛   󰇜 󰇛      󰇜    󰇛  󰇜         󰇛      󰇜              󰇛   󰇜 󰇛      󰇜    󰇛  󰇜         󰇛      󰇜              󰇛   󰇜 󰇛           󰇜    󰇛  󰇜         󰇛  󰇜             󰇛           󰇜    󰇛  󰇜  39   󰇛       󰇜 Appendix 2: Resu lts for th e binary mediator    󰇧  󰇛        󰇜    󰇛        󰇜 󰇨               󰇛     󰇜    󰇛      󰇜    󰇛  󰇜  󰇛         󰇜           󰇛  󰇜     󰇛  󰇜   󰇛         󰇜 󰇧     󰇛             󰇜    󰇛             󰇜    󰇛             󰇜    󰇨   󰇛  󰇜     󰇛  󰇜     󰇛  󰇜  󰆹    󰆹      󰆹    󰇧                                       󰇨    󰇛  󰇜    󰇛  󰇜     󰇛  󰇜    󰆹    󰆹    󰇭 󰇧                                                   󰇨  󰇧                                      󰇨 󰇮    󰇛  󰇜 Appendix 3: Consistency and asymptotic normality Appendix 3.1 Con sistency         󰇡                      󰇢            󰇛      󰇜      󰇛   󰇜        󰇡                       󰇢       󰇡                       󰇢          󰇛      󰇜    40                                                          󰇛  󰇜                        󰇛  󰇜    󰇛  󰇜             (CA.1)                      󰇛  󰇜 󰀓   󰇛  󰇜     (CA.2)                      󰇛  󰇜 󰀓     (CA.3)                  󰇛  󰇜 󰀓         󰇡                  󰇢     󰇱  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜           󰇛   󰇜   󰇛  󰇜    󰆹              󰆹                 󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                   󰇢󰇤   󰇛     󰇜    󰇛   󰇜   󰇛  󰇜     󰇛     󰇜    󰇛   󰇜   󰇛  󰇜   󰇣  󰇡    󰇛  󰇜 󰇻                  󰇢     󰇡  󰇻                󰇢󰇤     󰇡  󰇻                󰇢󰇲 41    󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤    󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢                  󰇛  󰇜 󰇤                   󰇛  󰇜 󰇬   󰇛      󰇜    󰇛    󰇜    󰇛   󰇜  󰇛  󰇜    󰇝  󰇛  󰇛    󰇜    󰇛   󰇜  󰇛  󰇜  󰇜󰇞   󰇝  󰇟  󰇛    󰇜  󰇛  󰇜  󰇠     󰇛   󰇜󰇞    󰇟  󰇛    󰇜  󰇛  󰇜      󰇠                󰇛      󰇜             󰇛      󰇜    󰇛  󰇛  󰇜      󰇜    󰇛  󰇛  󰇜    󰇜 Case 1: In        󰇡                  󰇢    󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢󰇤    󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢                 󰇛  󰇜 󰇤                 󰇛  󰇜 󰇬   󰇫 󰇫    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇤 󰈅         󰇛  󰇜    󰇬   󰇛           󰇛  󰇜    󰇜 󰇬  󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢                  󰇛  󰇜 󰇤󰇬                   󰇛  󰇜        󰇛   󰇜   󰇛  󰇜  󰇛   󰇜  󰆹                󰆹                󰇣     󰇛  󰇜         󰇛  󰇜        󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢󰇤   󰇛          󰇛  󰇜    󰇜   42  󰇥  󰇛    󰇜    󰇛   󰇜   󰇛  󰇜  󰇛   󰇜 󰇣  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢                 󰇛  󰇜 󰇤 󰇦                   󰇛  󰇜      󰇫    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜 󰇣     󰇛  󰇜              󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢󰇤   󰇛          󰇛  󰇜    󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢                 󰇛  󰇜 󰇤󰇬                   󰇛  󰇜 󰇛󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢                    󰇛  󰇜 󰇤󰇬                      󰇛  󰇜  󰇛        󰇜        󰇛   󰇜   󰇛  󰇜  󰇛     󰇜 󰇣   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢     󰇡   󰇻            󰇛  󰇜 󰇢󰇤          󰇛       󰇜    󰇥    󰇡   󰇻            󰇛  󰇜 󰇢 󰇦   󰇫  󰇛   󰇛  󰇜       󰇜   󰇛           󰇜  󰇛      󰇜  󰇥󰇣  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢                  󰇛  󰇜 󰇤 󰇻      󰇦   󰇛       󰇜 󰇬                    󰇛  󰇜 󰇛󰇜   󰇫  󰇛  󰇛  󰇜       󰇜   󰇛           󰇜  󰇛      󰇜 󰇣  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻      󰇢                    󰇛  󰇜 󰇤   󰇛       󰇜 󰇬                       󰇛  󰇜                    󰇛  󰇜 󰇛     󰇜   󰇛   󰇜  󰇛        󰇜 Case 2: In        󰇡                  󰇢        󰇛   󰇜   󰇛  󰇜  󰇛   󰇜  󰆹                󰆹                󰇣     󰇛  󰇜         󰇛  󰇜         󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢󰇤   󰇛          󰇛  󰇜    󰇜    󰇥  󰇛    󰇜    󰇛   󰇜   󰇛  󰇜  󰇛   󰇜 󰇣  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢                 󰇛  󰇜 󰇤 󰇦  43                  󰇛  󰇜      󰇫    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜 󰇣     󰇛  󰇜              󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇤   󰇛          󰇛  󰇜    󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢                   󰇛  󰇜 󰇤󰇬                    󰇛  󰇜 󰇛󰇜   󰇥  󰇛    󰇜    󰇛   󰇜   󰇛  󰇜  󰇛   󰇜 󰇣   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢                     󰇛  󰇜 󰇤󰇦                       󰇛  󰇜           󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇬  󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜   󰇛     󰇜  󰇛      󰇜                󰇛  󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇬       󰇛   󰇜   󰇛  󰇜   󰇛      󰇜  󰇛     󰇜                󰇛  󰇜         󰇛        󰇜    󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇬  󰇫  󰇛   󰇛  󰇜       󰇜   󰇛     󰇜   󰇛          󰇜  󰇛     󰇜                 󰇛  󰇜  󰇛       󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇬  󰇫  󰇛   󰇛  󰇜       󰇜   󰇛  󰇛  󰇜    󰇜   󰇛          󰇜  󰇛      󰇜                 󰇛  󰇜  󰇛       󰇜 󰇬 󰇛       󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢󰇬 󰇛  󰇜   󰇫    󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢         󰇛        󰇜   󰇛       󰇜 󰇬       󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢         󰇛     󰇜   󰇫 󰇡  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢 󰇻      󰇢  󰇛   󰇛  󰇜       󰇜  󰇛     󰇜 󰇬  󰇛󰇜 44   󰇫 󰇡  󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢 󰇻      󰇢  󰇛   󰇛  󰇜     󰇜  󰇛     󰇜 󰇬 󰇛󰇜   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦  󰇛      󰇜   󰇛   󰇜 󰇛       󰇜 Case 3: In       󰇡                  󰇢    󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤    󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜 󰇣  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢                  󰇛  󰇜 󰇤                   󰇛  󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇬   󰇫󰇧  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜   󰇛      󰇜  󰇛     󰇜 󰇨                 󰇛  󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇬        󰇛   󰇜   󰇛  󰇜   󰇛      󰇜  󰇛     󰇜                  󰇛  󰇜         󰇛        󰇜  45   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇬   󰇫  󰇛   󰇛  󰇜       󰇜   󰇛     󰇜   󰇛          󰇜  󰇛      󰇜                 󰇛  󰇜  󰇛        󰇜 󰇬  󰇛     󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇬   󰇫  󰇛   󰇛  󰇜       󰇜   󰇛  󰇛  󰇜    󰇜   󰇛          󰇜  󰇛     󰇜                 󰇛  󰇜  󰇛        󰇜 󰇬 󰇛       󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜   󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇬 󰇛󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫    󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢         󰇛        󰇜   󰇛       󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬       󰇛  󰇜  󰇛     󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢         󰇛      󰇜 46   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫 󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻      󰇢  󰇛   󰇛  󰇜       󰇜  󰇛     󰇜 󰇬  󰇛     󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇫 󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻      󰇢  󰇛   󰇛  󰇜     󰇜  󰇛      󰇜 󰇬  󰇛      󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦 󰇛     󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜  󰇣     󰇛  󰇜    󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢󰇤󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫  󰇛   󰇛  󰇜       󰇜    󰇛           󰇜  󰇛      󰇜   󰇧  󰆹  󰇛             󰇜  󰆹  󰇛             󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰈅      󰇨  󰇛        󰇜 󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦  󰇛     󰇜 47   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫  󰇛   󰇛  󰇜       󰇜   󰇛     󰇜   󰇧  󰆹  󰇛             󰇜  󰆹  󰇛             󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰈅      󰇨󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦 󰇛     󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫  󰇛   󰇛  󰇜     󰇜   󰇛     󰇜   󰇧  󰆹  󰇛             󰇜  󰆹  󰇛             󰇜   󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰈅      󰇨󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦  󰇛     󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫 󰇧  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜   󰇡    󰇛  󰇜 󰇻                󰇛  󰇜 󰇢 󰈅      󰇨󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦 󰇛     󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜   󰇡    󰇛  󰇜 󰇻             󰇛  󰇜 󰇢   󰇛         󰇜   󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬   󰇫  󰆹  󰇛           󰇜   󰇡    󰇛  󰇜 󰇻              󰇛  󰇜 󰇢   󰇬   󰇥  󰇡  󰇡    󰇛  󰇜 󰇻               󰇛  󰇜 󰇢 󰇻     󰇢 󰇦 󰇛      󰇜   󰇫  󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰇬 󰇛      󰇜   󰇫 󰇧   󰇛  󰇜   󰇛           󰇜  󰇛      󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰈅      󰇨  󰇛        󰇜 󰇬 48   󰇫 󰇧   󰇛  󰇜  󰇛      󰇜  󰆹  󰇛             󰇜  󰆹  󰇛             󰇜     󰇛  󰇜 󰈅      󰇨󰇬 󰇛     󰇜   󰇫 󰇧 󰇧   󰇛  󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜     󰇛  󰇜 󰈅         󰇨 󰈅      󰇨󰇬   󰇫 󰇧  󰇛   󰇛  󰇜          󰇜  󰇛     󰇜  󰆹  󰇛             󰇜  󰆹  󰇛             󰇜       󰇛  󰇜          󰈅      󰇨󰇬 󰇛     󰇜   󰇫 󰇧  󰇛   󰇛  󰇜       󰇜  󰇛     󰇜  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜       󰇛  󰇜          󰈅      󰇨󰇬 󰇛     󰇜   󰇫 󰇧  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜       󰇛  󰇜          󰈅      󰇨󰇬 󰇛      󰇜   󰇫  󰆹  󰇛            󰇜  󰆹  󰇛            󰇜       󰇛  󰇜            󰇛         󰇜  󰇬   󰇫  󰆹  󰇛            󰇜       󰇛  󰇜            󰇬  󰇛       󰇜          󰇛  󰇜                  󰇛      󰇜   󰇛   󰇜 Appendix 3. 2 Asymptotic distr ibution            󰇡                      󰇢   󰇛       󰇜                                  󰇡                      󰇢   󰇛      󰇜         󰇡                       󰇢     󰇡                       󰇢         󰇡                      󰇢     󰇡                      󰇢         󰇡                       󰇢                       󰇛  󰇜      󰇡                     󰇛    󰇜    󰇛      󰇜 󰇢         󰇡                      󰇢     󰇡                      󰇢  49     󰇫    󰆹  󰇛     󰇜  󰆹 󰇛           󰇜  󰆹 󰇛          󰇜     󰇛  󰇜        󰇛  󰇜               󰆹  󰇛      󰇜       󰇛  󰇜              󰇛             󰇜 󰇬󰈅         󰇛  󰇜    󰇛  󰇜   󰇛      󰇜    󰇛  󰇜      󰇒 󰇏        󰆹  󰇛      󰇜     󰇛     󰇜    󰇛  󰇜   󰇛  󰇜            󰇛  󰇜   󰇛  󰇜    󰇟  󰇛     󰇜   󰇛  󰇜 󰇠  󰇡              󰇛  󰇜   󰇛  󰇜    󰇢      󰇛  󰇜     󰇛  󰇜 󰇡                󰇛  󰇜   󰇛  󰇜   󰇢  󰇡              󰇛  󰇜   󰇛  󰇜    󰇢        󰇛  󰇜    󰇧     󰇛        󰇜          󰇛        󰇜 󰇨  󰇭     󰇡    󰇛        󰇜 󰇢     󰇡  󰇛        󰇜 󰇢  󰇮     󰇛        󰇜  󰇭     󰇡    󰇛        󰇜 󰇢     󰇡  󰇛        󰇜 󰇢  󰇮                   󰇛      󰇜   󰇩           󰇛    󰇛     󰇜󰇜  󰇪         󰇧                 󰇨       󰇛  󰇜                                󰇡                      󰇢     󰇡                      󰇢      󰇫    󰆹  󰇛     󰇜  󰇧     󰇡           󰇢 󰇛     󰇜             󰇨        󰇛  󰇜    󰇛   󰇜                50     󰆹  󰇛      󰇜   󰇛   󰇜                    󰇛             󰇜          󰇛  󰇜    󰇛  󰇜   󰇛      󰇜    󰇛  󰇜                       󰇛  󰇜    󰇩           󰇛    󰇛     󰇜󰇜  󰇪       󰇧                 󰇨       󰇛  󰇜                        󰇛  󰇜     󰇣 󰇛     󰇜   󰇛  󰇜 󰇡 󰇛      󰇜  󰇛   󰇜     󰇛   󰇜   󰇛   󰇜  󰇛     󰇜   󰇛  󰇜 󰇢󰇤  󰇛      󰇜          󰇛       󰇜     󰇛  󰇜     󰇛  󰇜                     󰇣 󰇛      󰇜   󰇛  󰇜  󰇛       󰇜  󰇛   󰇜     󰇛   󰇜   󰇛   󰇜  󰇛      󰇜   󰇛  󰇜 󰇤  󰇛       󰇜     󰇩   󰇛  󰇜                        󰇛  󰇜                    󰇛      󰇜   󰇪󰇬  󰇛       󰇜    󰇛     󰇜    󰇛   󰇜   󰇛  󰇜   󰇛   󰇜       󰇛  󰇜     󰇭     󰇡    󰇛        󰇜 󰇢     󰇡  󰇛        󰇜 󰇢  󰇮                  󰇛      󰇜                                       󰇛       󰇜   󰆹 󰇛            󰇜  󰆹 󰇛           󰇜   󰇧      󰇡              󰇢 󰇛     󰇜              󰇨        󰇡                       󰇢     󰇡                       󰇢  51    󰇡                󰇛   󰇜                     󰇛   󰇜 󰇢󰇻       󰇛  󰇜    󰇛  󰇜                    󰇛   󰇜        󰇛  󰇜    󰇛  󰇜           󰇛  󰇜     󰇥  󰆹  󰇛      󰇜     󰇛  󰇜     󰇛  󰇜                     󰇛       󰇜      󰇛       󰇜  󰇡   󰆹  󰇛       󰇜 󰇢                    󰇛  󰇜  󰇦            󰇛  󰇜    󰇛        󰇜              󰇛     󰇜                              󰇛     󰇜          󰇛     󰇜                      󰇛     󰇜                                               󰇛  󰇜     󰇡   󰇛             󰇜 󰇢       󰇛  󰇜    󰇛  󰇜      󰇛  󰇜                     󰇛      󰇜    󰇛      󰇜     󰇛      󰇜                                                                                                                                    52   󰇛  󰇜        󰇛    󰇜      󰇛        󰇜 󰆒 󰇛        󰇜              󰇛        󰇜          󰇛        󰇜  󰇛        󰇜            󰇛        󰇜                                                 󰇛   󰇜        󰇛  󰇜                                   󰇒 󰇏                󰇛   󰇜      󰇛   󰇜    󰇛  󰇜     󰇛   󰇜          󰇛  󰇜 󰇻           󰇡                       󰇢     󰇡                       󰇢                        󰇛  󰇜    󰇛   󰇜        󰇛   󰇜       󰇛  󰇜                        󰇛  󰇜       󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛      󰇜     󰇛  󰇜       󰇛  󰇜                       󰇛       󰇜   󰇛       󰇜  󰇧   󰇛     󰇜    󰇛   󰇜   󰇛  󰇜  󰇛        󰇜 󰇨                   󰇛  󰇜      󰇛   󰇜                                                   53        󰇡                       󰇢                       󰇛  󰇜      󰇡                  󰇛   󰇜                        󰇛    󰇜 󰇢󰇻     󰇛  󰇜    󰇛  󰇜                      󰇛   󰇜      󰇛  󰇜    󰇛  󰇜              󰇛   󰇜                   󰇛   󰇜      󰇛  󰇜    󰇛  󰇜      󰇡            󰇛    󰇜 󰇢    󰇛  󰇜     󰇫󰇧 󰆹  󰇛       󰇜   󰆹  󰇛       󰇜  󰆹 󰇛            󰇜  󰆹 󰇛           󰇜 󰇨    󰇣 󰇡             󰇢              󰇛  󰇜 󰇤  󰇡   󰆹  󰇛        󰇜 󰇢      󰇛  󰇜  󰇡            󰇢           󰇛      󰇜   󰇬            󰇫󰇧 󰆹  󰇛        󰇜   󰆹  󰇛       󰇜  󰆹 󰇛            󰇜  󰆹 󰇛           󰇜 󰇨    󰇣 󰇡             󰇢󰇤  󰇡   󰆹  󰇛        󰇜 󰇢    󰇡            󰇢   󰇛      󰇜   󰇬      󰇡            󰇛  󰇜 󰇢     󰇛  󰇜           󰇛   󰇜                󰇛   󰇜                                                   󰇛     󰇜                                󰆹              󰆹                        54                          󰇛  󰇜                    󰇡                       󰇢                       󰇛  󰇜                  󰇛  󰇜             󰇛          󰇜     󰇡            󰇛  󰇜 󰇢       󰇛  󰇜                󰇛  󰇜    󰇧   󰇛      󰇜  󰇛       󰇜   󰇛     󰇜  󰇛       󰇜    󰇛       󰇜     󰇛   󰇜   󰇛  󰇜       󰇛         󰇜   󰇛        󰇜     󰇛  󰇜  󰇧   󰇛      󰇜    󰇛   󰇜   󰇛  󰇜  󰇛        󰇜 󰇨      󰇛  󰇜    󰇛        󰇜   󰇛       󰇜    󰇛      󰇜   󰇨   󰇛           󰇜   󰇭  󰇛      󰇜  󰇛       󰇜   󰇛     󰇜  󰇛      󰇜    󰇛       󰇜     󰇛   󰇜   󰇛  󰇜     󰇛         󰇜    󰇧   󰇛      󰇜    󰇛   󰇜   󰇛  󰇜  󰇛        󰇜 󰇨      󰇛      󰇜     󰇛      󰇜   󰇮      󰇒 󰇏              󰇒 󰇏    󰇛  󰇜                 󰇛   󰇜         󰇛     󰇜       󰇛  󰇜   󰇡            󰇛  󰇜 󰇢               󰇛  󰇜        󰇛  󰇜  55          󰇛  󰇜      󰇛  󰇜  󰇛  󰇜 󰇛     󰇜      󰇛  󰇜 󰇛     󰇜  󰇛   󰇛  󰇜    󰇜   󰇛  󰇜 󰇛     󰇜         󰇛   󰇜   󰇧    󰇛  󰇜 󰇛     󰇜  󰇛  󰇜 󰇛     󰇜 󰇨   󰇛  󰇜       󰇛   󰇜           󰇛     󰇜        󰇛     󰇜   󰇧    󰇛  󰇜 󰇛     󰇜  󰇛  󰇜 󰇛     󰇜 󰇨   󰇛  󰇜    󰇛  󰇜 󰇛     󰇜       󰇛  󰇜  󰇛     󰇜     󰇛  󰇜 󰇛     󰇜       󰇛  󰇜    󰇛     󰇜      󰇛  󰇜    󰇛  󰇜     󰇛  󰇜            󰇛  󰇜        󰇡                       󰇢                       󰇛  󰇜                  󰇛  󰇜     󰇛   󰇜         󰇛     󰇜        󰇛           󰇜             󰇛  󰇜          󰇛  󰇜       󰇡                     󰇛    󰇜    󰇛      󰇜 󰇢    󰇫    󰇫 󰆹  󰇛       󰇜  󰆹 󰇛            󰇜  󰆹 󰇛           󰇜     󰇛  󰇜        󰇛  󰇜                󰆹  󰇛       󰇜       󰇛  󰇜                󰇛                󰇜      󰇛                󰇜 󰇬   󰇛      󰇜 󰇬 56       󰇫  󰇛     󰇜  󰇛      󰇜   󰇛      󰇜   󰇛     󰇜    󰇛  󰇜    󰇧  󰇛      󰇜  󰇛       󰇜   󰇛     󰇜  󰇛       󰇜   󰇛      󰇜   󰇛     󰇜 󰇨     󰇛   󰇜   󰇛  󰇜    󰇛  󰇜                    󰇧   󰇛      󰇜    󰇛   󰇜   󰇛  󰇜  󰇛       󰇜 󰇨      󰇛  󰇜                    󰇛      󰇜     󰇛      󰇜 󰇬    󰇛  󰇜                          󰇛  󰇜        󰇛  󰇜                         󰇛  󰇜    󰇛     󰇜  󰇛      󰇜   󰇛      󰇜   󰇛    󰇜    󰇛  󰇜  󰇧  󰇛      󰇜  󰇛       󰇜   󰇛     󰇜  󰇛       󰇜   󰇛      󰇜   󰇛     󰇜 󰇨     󰇛   󰇜   󰇛  󰇜    󰇛  󰇜                    󰇧   󰇛      󰇜    󰇛   󰇜   󰇛  󰇜  󰇛       󰇜 󰇨      󰇛  󰇜                    󰇛      󰇜     󰇛      󰇜        󰇡                       󰇢   󰇛       󰇜           󰇡                      󰇢   󰇛      󰇜        󰇛       󰇜       󰇛  󰇜    󰇛      󰇜                       󰇛  󰇜    󰇩           󰇛    󰇛     󰇜󰇜  󰇪    󰇧                 󰇨                        󰇛  󰇜    󰇛   󰇜     󰇛   󰇜                󰇛  󰇜     󰇛   󰇜      󰇛     󰇜     󰇛          󰇜           󰇛  󰇜                          󰇛  󰇜     󰇛  󰇜 57    󰇛      󰇜                    󰇡                      󰇢   󰇛      󰇜               󰇡                      󰇢   󰇛      󰇜          󰇛   󰇛      󰇜  󰇜  Appendix 4: Procedure for misspecif ied mo dels                                                                                                                                   󰇛  󰇜           󰇛     󰇜    󰆒    󰆒      󰆒       󰆒       󰇛  󰇜        󰆒 󰇛  󰇜   󰆒     󰆒      󰆒       󰆒       󰆒       󰆒        󰆒         󰆒        󰆒 󰇛  󰇜     󰇛  󰇜 58   󰆒       󰆒        󰆒         󰆒                󰇛  󰇜                󰆒       󰆒             󰆒             󰆒        󰇛  󰇜    󰆒 󰇛  󰇜   󰆹     󰆒      󰆹     󰆒      󰆹       󰆒   󰆹       󰆒      󰇛       󰇜     󰇛        󰇜     󰇛        󰇜     󰇛           󰇜     󰇛          󰇜                        59 Appendix 5 : Estimated regr essio n parameters     󰇛     󰇜   󰇛               󰇜                                       󰇛     󰇜                                                                           󰇛  󰇜         󰇛  󰇜  󰇛                         󰇜        󰆹         󰆹         󰆹          󰆹         󰆹         󰆹            󰇛  󰇜  60 61 Re fer enc e            The Ame rican Statistician  67            Journal of the Royal Statistical Society: Series B (Statistical Methodology)  62      Statistics in medicine  39 

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