Motivation Selection Models Results Conclusions & Future Work Analysing the Rate of Change in a Longitudinal Study with Missing Data, Taking into Account the Number of Contact Attempts Mouna Akacha Prof. Jane L. Hutton Department of Statistics University of Warwick RSS Conference 2009, Edinburgh 11 th September 2009 1 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Outline Motivation 1 Selection Models 2 Outcome Model Reminder Process Model Results 3 Conclusions & Future Work 4 2 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work The Collaborative Ankle Support Trial (CAST) randomized, multicenter study; 553 people with a severe sprain of the ankle; four treatment groups ( Tubigrip , Plaster of Paris , Aircast brace and Bledsoe boot ); four points in time (baseline, 4 weeks, 12 weeks and 39 weeks); clinical status measured via the Foot and Ankle Outcome Score. 3 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work The Collaborative Ankle Support Trial (CAST) randomized, multicenter study; 553 people with a severe sprain of the ankle; four treatment groups ( Tubigrip , Plaster of Paris , Aircast brace and Bledsoe boot ); four points in time (baseline, 4 weeks, 12 weeks and 39 weeks); clinical status measured via the Foot and Ankle Outcome Score. AIM: Estimate the clinical effectiveness of ankle treatments compared to the standard treatment Tubigrip . 3 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Individual Evolution FAOSS-score time 4 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Missing Data FAOSS-score completely observed for 67 . 26 % ; Non-monotone missingness pattern for approx. 10 % ; 5 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Missing Data FAOSS-score completely observed for 67 . 26 % ; Non-monotone missingness pattern for approx. 10 % ; Missingness Processes Missing Completely at Random (MCAR); Missing at Random (MAR); ignorability Missing not at Random (MNAR) 5 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Missing Data FAOSS-score completely observed for 67 . 26 % ; Non-monotone missingness pattern for approx. 10 % ; Missingness Processes Missing Completely at Random (MCAR); Missing at Random (MAR); ignorability patients with low baseline score and high 4 week and 12 week score Missing not at Random (MNAR) 5 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Missing Data FAOSS-score completely observed for 67 . 26 % ; Non-monotone missingness pattern for approx. 10 % ; Missingness Processes Missing Completely at Random (MCAR); Missing at Random (MAR); ignorability patients with low baseline score and high 4 week and 12 week score Missing not at Random (MNAR) patients who considered themselves to have made fully recovery. 5 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Reminder Process System of reminder letters and telephone calls: 6 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Reminder Process System of reminder letters and telephone calls: z = 0: no chasing; z = 1: telephone chase; z = 2: 2nd copy sent with no further telephone chasing; z = 3: 2nd copy sent with further telephone chasing; z = 4: core outcomes obtained over telephone; z = 5: non responder. 6 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Reminder Process System of reminder letters and telephone calls: z = 0: no chasing; z = 1: telephone chase; z = 2: 2nd copy sent with no further telephone chasing; z = 3: 2nd copy sent with further telephone chasing; z = 4: core outcomes obtained over telephone; z = 5: non responder. In particular, ⇐ ⇒ z ∈ { 0 , 1 , 2 , 3 , 4 } r = 1 and ⇐ ⇒ r = 0 z = 5 . 6 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Objective Compare the treatments by modelling the rate of improvement, taking into account the longitudinal and bounded nature of the data, 7 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Objective Compare the treatments by modelling the rate of improvement, taking into account the longitudinal and bounded nature of the data, the reminder process in order to account for informative or non-ignorable missingness. 7 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Informative or Non-Ignorable Missingness y i = ( y i 0 , y i 4 , y i , 12 , y i , 39 ) ⊤ ; y i , obs observed part and y i , mis missing part; r i = ( r i 0 , r i 4 , r i , 12 , r i , 39 ) ⊤ denotes the missingness indicator; x i summarizes the explanatory variables for subject i ; Y i ∼ P ( θ ) and R i ∼ P ( φ ) . 8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Informative or Non-Ignorable Missingness y i = ( y i 0 , y i 4 , y i , 12 , y i , 39 ) ⊤ ; y i , obs observed part and y i , mis missing part; r i = ( r i 0 , r i 4 , r i , 12 , r i , 39 ) ⊤ denotes the missingness indicator; x i summarizes the explanatory variables for subject i ; Y i ∼ P ( θ ) and R i ∼ P ( φ ) . Observed Data Likelihood � L Y i , obs , R i ( θ, φ ) = f ( y i , r i | x i , θ, φ ) d y i , mis 8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Informative or Non-Ignorable Missingness y i = ( y i 0 , y i 4 , y i , 12 , y i , 39 ) ⊤ ; y i , obs observed part and y i , mis missing part; r i = ( r i 0 , r i 4 , r i , 12 , r i , 39 ) ⊤ denotes the missingness indicator; x i summarizes the explanatory variables for subject i ; Y i ∼ P ( θ ) and R i ∼ P ( φ ) . Observed Data Likelihood � L Y i , obs , R i ( θ, φ ) = f ( y i , r i | x i , θ, φ ) d y i , mis 8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Informative or Non-Ignorable Missingness y i = ( y i 0 , y i 4 , y i , 12 , y i , 39 ) ⊤ ; y i , obs observed part and y i , mis missing part; r i = ( r i 0 , r i 4 , r i , 12 , r i , 39 ) ⊤ denotes the missingness indicator; x i summarizes the explanatory variables for subject i ; Y i ∼ P ( θ ) and R i ∼ P ( φ ) . Observed Data Likelihood � L Y i , obs , R i ( θ, φ ) = f ( y i , r i | x i , θ, φ ) d y i , mis Selection Model f ( y i , r i | x i , θ, φ ) = f ( y i | x i , θ ) f ( r i | y i , x i , φ ) 8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Informative or Non-Ignorable Missingness y i = ( y i 0 , y i 4 , y i , 12 , y i , 39 ) ⊤ ; y i , obs observed part and y i , mis missing part; r i = ( r i 0 , r i 4 , r i , 12 , r i , 39 ) ⊤ denotes the missingness indicator; x i summarizes the explanatory variables for subject i ; Y i ∼ P ( θ ) and R i ∼ P ( φ ) . Observed Data Likelihood � L Y i , obs , R i ( θ, φ ) = f ( y i , r i | x i , θ, φ ) d y i , mis Selection Model f ( y i , r i | x i , θ, φ ) = f ( y i | x i , θ ) f ( r i | y i , x i , φ ) 8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
Motivation Selection Models Results Conclusions & Future Work Informative or Non-Ignorable Missingness y i = ( y i 0 , y i 4 , y i , 12 , y i , 39 ) ⊤ ; y i , obs observed part and y i , mis missing part; r i = ( r i 0 , r i 4 , r i , 12 , r i , 39 ) ⊤ denotes the missingness indicator; x i summarizes the explanatory variables for subject i ; Y i ∼ P ( θ ) and R i ∼ P ( φ ) . Observed Data Likelihood � L Y i , obs , R i ( θ, φ ) = f ( y i , r i | x i , θ, φ ) d y i , mis Selection Model f ( y i , r i | x i , θ, φ ) = f ( y i | x i , θ ) f ( r i | y i , x i , φ ) 8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data
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