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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

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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 11th September 2009

1 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Outline

1

Motivation

2

Selection Models Outcome Model Reminder Process Model

3

Results

4

Conclusions & Future Work

2 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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, r = 1 ⇐ ⇒ z ∈ {0, 1, 2, 3, 4} and r = 0 ⇐ ⇒ z = 5.

6 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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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

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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

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Motivation Selection Models Results Conclusions & Future Work

Informative or Non-Ignorable Missingness

yi = (yi0, yi4, yi,12, yi,39)⊤; yi,obs observed part and yi,mis missing part; ri = (ri0, ri4, ri,12, ri,39)⊤ denotes the missingness indicator; xi summarizes the explanatory variables for subject i; Yi ∼ P(θ) and Ri ∼ P(φ).

8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Informative or Non-Ignorable Missingness

yi = (yi0, yi4, yi,12, yi,39)⊤; yi,obs observed part and yi,mis missing part; ri = (ri0, ri4, ri,12, ri,39)⊤ denotes the missingness indicator; xi summarizes the explanatory variables for subject i; Yi ∼ P(θ) and Ri ∼ P(φ). Observed Data Likelihood LYi,obs,Ri(θ, φ) =

  • f(yi, ri|xi, θ, φ) dyi,mis

8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Informative or Non-Ignorable Missingness

yi = (yi0, yi4, yi,12, yi,39)⊤; yi,obs observed part and yi,mis missing part; ri = (ri0, ri4, ri,12, ri,39)⊤ denotes the missingness indicator; xi summarizes the explanatory variables for subject i; Yi ∼ P(θ) and Ri ∼ P(φ). Observed Data Likelihood LYi,obs,Ri(θ, φ) =

  • f(yi, ri|xi, θ, φ) dyi,mis

8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Informative or Non-Ignorable Missingness

yi = (yi0, yi4, yi,12, yi,39)⊤; yi,obs observed part and yi,mis missing part; ri = (ri0, ri4, ri,12, ri,39)⊤ denotes the missingness indicator; xi summarizes the explanatory variables for subject i; Yi ∼ P(θ) and Ri ∼ P(φ). Observed Data Likelihood LYi,obs,Ri(θ, φ) =

  • f(yi, ri|xi, θ, φ) dyi,mis

Selection Model f(yi, ri|xi, θ, φ) = f(yi|xi, θ) f(ri|yi, xi, φ)

8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Informative or Non-Ignorable Missingness

yi = (yi0, yi4, yi,12, yi,39)⊤; yi,obs observed part and yi,mis missing part; ri = (ri0, ri4, ri,12, ri,39)⊤ denotes the missingness indicator; xi summarizes the explanatory variables for subject i; Yi ∼ P(θ) and Ri ∼ P(φ). Observed Data Likelihood LYi,obs,Ri(θ, φ) =

  • f(yi, ri|xi, θ, φ) dyi,mis

Selection Model f(yi, ri|xi, θ, φ) = f(yi|xi, θ) f(ri|yi, xi, φ)

8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Informative or Non-Ignorable Missingness

yi = (yi0, yi4, yi,12, yi,39)⊤; yi,obs observed part and yi,mis missing part; ri = (ri0, ri4, ri,12, ri,39)⊤ denotes the missingness indicator; xi summarizes the explanatory variables for subject i; Yi ∼ P(θ) and Ri ∼ P(φ). Observed Data Likelihood LYi,obs,Ri(θ, φ) =

  • f(yi, ri|xi, θ, φ) dyi,mis

Selection Model f(yi, ri|xi, θ, φ) = f(yi|xi, θ) f(ri|yi, xi, φ)

8 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Selection Model

Likelihood using Ri LYobs,R (θ, φ) =

553

  • i=1
  • f (yi|xi, θ) f (ri|yi, xi, φ) dyi,mis.

9 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Selection Model

Likelihood using Ri LYobs,R (θ, φ) =

553

  • i=1
  • f (yi|xi, θ) f (ri|yi, xi, φ) dyi,mis.

zi = (zi0, zi4, zi,12, zi,39)⊤ denotes the reminder indicator and Zi ∼ P(ψ). Likelihood using Zi LYobs,Z (θ, ψ) =

553

  • i=1
  • f (yi|xi, θ) f (zi|yi, xi, ψ) dyi,mis.

9 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Selection Model

Likelihood using Ri LYobs,R (θ, φ) =

553

  • i=1
  • f (yi|xi, θ) f (ri|yi, xi, φ) dyi,mis.

zi = (zi0, zi4, zi,12, zi,39)⊤ denotes the reminder indicator and Zi ∼ P(ψ). Likelihood using Zi LYobs,Z (θ, ψ) =

553

  • i=1
  • f (yi|xi, θ) f (zi|yi, xi, ψ) dyi,mis.

9 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Selection Model

Likelihood using Ri LYobs,R (θ, φ) =

553

  • i=1
  • f (yi|xi, θ) f (ri|yi, xi, φ) dyi,mis.

zi = (zi0, zi4, zi,12, zi,39)⊤ denotes the reminder indicator and Zi ∼ P(ψ). Likelihood using Zi LYobs,Z (θ, ψ) =

553

  • i=1
  • f (yi|xi, θ) f (zi|yi, xi, ψ) dyi,mis.

9 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Outcome Model – Average Evolution

10 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Outcome Model

Non-Linear Mixed Model Yi|Ui

ind.

∼ N4

  • ηi, σ2I4
  • ;

Ui

iid

∼ N(0, D2); ηij = g(xij, θi) for j ∈ {0, 4, 12, 39}, where g is the model function and the parameter vector θi = (θ, Ui)⊤ varies across subjects.

11 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Outcome Model

Non-Linear Mixed Model Yi|Ui

ind.

∼ N4

  • ηi, σ2I4
  • ;

Ui

iid

∼ N(0, D2); ηij = g(xij, θi) for j ∈ {0, 4, 12, 39}, where g is the model function and the parameter vector θi = (θ, Ui)⊤ varies across subjects.

11 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Outcome Model

Non-Linear Mixed Model Yi|Ui

ind.

∼ N4

  • ηi, σ2I4
  • ;

Ui

iid

∼ N(0, D2); ηij = g(xij, θi) for j ∈ {0, 4, 12, 39}, where g is the model function and the parameter vector θi = (θ, Ui)⊤ varies across subjects.

11 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Non-linear Model for CAST

Assumption g

′(xij, θi) = ktrt g(xij, θi)

  • max − g(xij, θi)
  • 12 (22)

Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Non-linear Model for CAST

Assumption g

′(xij, θi) = ktrt g(xij, θi)

  • max − g(xij, θi)
  • Non-linear Mixed Model

g(xij, θi) = β1 + α1 · age exp( − [β2,trt + α2 · age]tj)(β1+α1·age

β0

− 1) + 1 + Ui, where β2,trt = ktrt · β1.

12 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Non-linear Model for CAST

Assumption g

′(xij, θi) = ktrt g(xij, θi)

  • max − g(xij, θi)
  • Non-linear Mixed Model

g(xij, θi) = β1 + α1 · age exp( − [β2,trt + α2 · age]tj)(β1+α1·age

β0

− 1) + 1 + Ui, where β2,trt = ktrt · β1.

12 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Non-linear Model for CAST

Assumption g

′(xij, θi) = ktrt g(xij, θi)

  • max − g(xij, θi)
  • Non-linear Mixed Model

g(xij, θi) = β1 + α1 · age exp( − [β2,trt + α2 · age]tj)(β1+α1·age

β0

− 1) + 1 + Ui, where β2,trt = ktrt · β1.

12 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Non-linear Model for CAST

Assumption g

′(xij, θi) = ktrt g(xij, θi)

  • max − g(xij, θi)
  • Non-linear Mixed Model

g(xij, θi) = β1 + α1 · age exp( − [β2,trt + α2 · age]tj)(β1+α1·age

β0

− 1) + 1 + Ui, where β2,trt = ktrt · β1.

12 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

zj ∈ {0, 1, 2, 3, 4, 5} for j ∈ {0, 4, 12, 39};

13 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

zj ∈ {0, 1, 2, 3, 4, 5} for j ∈ {0, 4, 12, 39}; let pj0 be the probability to respond at the first attempt, i.e. zj = 0;

13 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

zj ∈ {0, 1, 2, 3, 4, 5} for j ∈ {0, 4, 12, 39}; let pj0 be the probability to respond at the first attempt, i.e. zj = 0; let pjk denote the probability to respond at the k − th attempt, given the subject has not replied earlier;

13 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

zj ∈ {0, 1, 2, 3, 4, 5} for j ∈ {0, 4, 12, 39}; let pj0 be the probability to respond at the first attempt, i.e. zj = 0; let pjk denote the probability to respond at the k − th attempt, given the subject has not replied earlier; the unconditional probabilities to reply at attempt k and time point j are given by µj0 = pj0; µj1 = pj1 (1 − pj0); . . . µj4 = pj4

3

  • k=0

(1 − pjk)

13 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

zj ∈ {0, 1, 2, 3, 4, 5} for j ∈ {0, 4, 12, 39}; let pj0 be the probability to respond at the first attempt, i.e. zj = 0; let pjk denote the probability to respond at the k − th attempt, given the subject has not replied earlier; the unconditional probabilities to reply at attempt k and time point j are given by µj0 = pj0; µj1 = pj1 (1 − pj0); . . . µj4 = pj4

3

  • k=0

(1 − pjk) and thus µj5 = 1 −

4

  • k=0

µjk.

13 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

We redefine Zj in terms of an indicator random vector Vj: zj = 0 ⇐ ⇒ vj = (1, 0, 0, 0, 0, 0)⊤ . . . zj = 5 ⇐ ⇒ vj = (0, 0, 0, 0, 0, 1)⊤.

14 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Reminder Process Model

We redefine Zj in terms of an indicator random vector Vj: zj = 0 ⇐ ⇒ vj = (1, 0, 0, 0, 0, 0)⊤ . . . zj = 5 ⇐ ⇒ vj = (0, 0, 0, 0, 0, 1)⊤. Multinomial distribution Vj ∼ Multinomial(1, µj0, ..., µj5), where µjk is a function of pj0, ..., pjk.

14 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Different Reminder Processes

MCAR: logit(pjk) = ψ0k + ψ1 tj + ψ5 ✶Bledsoe MAR: logit(pjk) = ψ0k + ψ1 tj + ψ3 yj−1 + ψ5 ✶Bledsoe MNAR: logit(pjk) = ψ0k + ψ1tj + ψ3yj−1 + ψ4yj + ψ5✶Bledsoe where k ∈ {0, 1, 2, 3, 4}, j ∈ {4, 12, 39}

15 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Different Reminder Processes

MCAR: logit(pjk) = ψ0k + ψ1 tj + ψ5 ✶Bledsoe MAR: logit(pjk) = ψ0k + ψ1 tj + ψ3 yj−1 + ψ5 ✶Bledsoe MNAR: logit(pjk) = ψ0k + ψ1tj + ψ3yj−1 + ψ4yj + ψ5✶Bledsoe where k ∈ {0, 1, 2, 3, 4}, j ∈ {4, 12, 39}

15 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Different Reminder Processes

MCAR: logit(pjk) = ψ0k + ψ1 tj + ψ5 ✶Bledsoe MAR: logit(pjk) = ψ0k + ψ1 tj + ψ3 yj−1 + ψ5 ✶Bledsoe MNAR: logit(pjk) = ψ0k + ψ1tj + ψ3yj−1 + ψ4yj + ψ5✶Bledsoe where k ∈ {0, 1, 2, 3, 4}, j ∈ {4, 12, 39}

15 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work Outcome Model Reminder Process Model

Different Reminder Processes

MCAR: logit(pjk) = ψ0k + ψ1 tj + ψ5 ✶Bledsoe MAR: logit(pjk) = ψ0k + ψ1 tj + ψ3 yj−1 + ψ5 ✶Bledsoe MNAR: logit(pjk) = ψ0k + ψ1tj + ψ3yj−1 + ψ4yj + ψ5✶Bledsoe where k ∈ {0, 1, 2, 3, 4}, j ∈ {4, 12, 39} and ψ00 = ψ02 = ψ04 = 0.

15 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Parameter Ignorable MCAR MAR MNAR Est. SE p-val. Est. SE p-val. Est. SE p-val. Est. SE p-val. β0 41.04 0.77 ∗ 41.04 0.78 ∗ 41.04 0.77 ∗ 41.04 0.77 ∗ β1 110.20 6.79 ∗ 106.89 6.79 ∗ 106.89 6.89 ∗ 106.82 6.79 ∗ α1

  • 9.292

2.05 ∗

  • 8.276

2.05 ∗

  • 8.276

2.07 ∗

  • 8.261

2.04 ∗ α2

  • 0.208

0.04 ∗

  • 0.213

0.04 ∗

  • 0.213

0.04 ∗

  • 0.213

0.04 ∗ σ2 186.63 7.35 ∗ 186.27 7.33 ∗ 186.27 7.33 ∗ 186.25 7.33 ∗ D2 144.79 12.32 ∗ 147.53 12.65 ∗ 147.52 12.65 ∗ 147.59 12.67 ∗ ψ01 0.124 0.08 0.06 0.194 0.08 0.01 0.205 0.08 0.01 ψ03 0.045 0.12 0.36 0.107 0.13 0.20 0.115 0.13 0.19 ψ1

  • 0.012

0.002 ∗

  • 0.006

0.003 0.04

  • 0.006

0.003 0.04 ψ3

  • 0.004

0.001 0.002

  • 0.002

0.002 0.16 ψ4

  • 0.002

0.002 0.25 ψ5 0.133 0.08 0.06 0.212 0.09 0.01 0.218 0.09 0.01 −2ℓ

  • 19577.16

19568.3 19567.32

Table: Overview of the parameter estimates, standard errors, p-values and deviances under the different reminder processes.

16 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Parameter Ignorable MCAR MAR MNAR Est. SE p-val. Est. SE p-val. Est. SE p-val. Est. SE p-val. β21 − β22

  • 11.89

4.26 0.0054

  • 11.82

4.25 0.0028

  • 11.82

4.25 0.0028

  • 11.83

4.25 0.0028 β21 − β23

  • 5.64

3.80 0.1383

  • 5.51

3.78 0.0720

  • 5.52

3.78 0.0721

  • 5.52

3.77 0.0717 β21 − β24

  • 1.20

3.37 0.7217

  • 1.05

3.36 0.3771

  • 1.05

3.36 0.3772

  • 1.07

3.36 0.3752 ψ5 13.32 8.43 0.0573 21.21 8.83 0.0083 21.85 8.87 0.0070

Table: Overview of the estimates for the treatment differences, standard errors, p-values and deviances under the different reminder processes.

17 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness. Combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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SLIDE 49

Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness. Combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process. Treatment differences are estimated robustly across a range

  • f plausible reminder (i.e. missingness) processes.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

slide-50
SLIDE 50

Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness. Combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process. Treatment differences are estimated robustly across a range

  • f plausible reminder (i.e. missingness) processes.

Results suggest that phone calls are effective in retrieving questionnaires.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

slide-51
SLIDE 51

Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness. Combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process. Treatment differences are estimated robustly across a range

  • f plausible reminder (i.e. missingness) processes.

Results suggest that phone calls are effective in retrieving questionnaires. Patients who reply early tend to have more symptoms than those who reply at a later occasion or not at all.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

slide-52
SLIDE 52

Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness. Combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process. Treatment differences are estimated robustly across a range

  • f plausible reminder (i.e. missingness) processes.

Results suggest that phone calls are effective in retrieving questionnaires. Patients who reply early tend to have more symptoms than those who reply at a later occasion or not at all. As time passes patients tend to need more reminders.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

slide-53
SLIDE 53

Motivation Selection Models Results Conclusions & Future Work

Conclusions

Selection model framework that uses number of contact attempts to adjust for informative or non-ignorable missingness. Combines a non-linear mixed model for the underlying response model with a logistic regression model for the reminder process. Treatment differences are estimated robustly across a range

  • f plausible reminder (i.e. missingness) processes.

Results suggest that phone calls are effective in retrieving questionnaires. Patients who reply early tend to have more symptoms than those who reply at a later occasion or not at all. As time passes patients tend to need more reminders. Patients randomized to Bledsoe treatment tend to return the questionnaire earlier than the other treatment groups.

18 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Further Work

Extension to account for non-monotone missingness patterns. Assumption of the compound symmetry covariance structure needs to be scrutinized. Relax underlying normality assumptions to a probabilistic model that accounts for the skewed distribution of the residuals.

19 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

slide-55
SLIDE 55

Motivation Selection Models Results Conclusions & Future Work

Further Work

Extension to account for non-monotone missingness patterns. Assumption of the compound symmetry covariance structure needs to be scrutinized. Relax underlying normality assumptions to a probabilistic model that accounts for the skewed distribution of the residuals.

19 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

slide-56
SLIDE 56

Motivation Selection Models Results Conclusions & Future Work

Further Work

Extension to account for non-monotone missingness patterns. Assumption of the compound symmetry covariance structure needs to be scrutinized. Relax underlying normality assumptions to a probabilistic model that accounts for the skewed distribution of the residuals.

19 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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SLIDE 57

Motivation Selection Models Results Conclusions & Future Work

Thank you for your attention!

20 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

Acknowledgment

Thanks to Prof. Sallie Lamb (Warwick Medical School) and the CAST team for allowing the authors to use the data and helpful discussions.

21 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data

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Motivation Selection Models Results Conclusions & Future Work

References

J.M. Alho. Adjusting for nonresponse bias using logistic regression. Biometrika, 77:617–624, 1990. R.J.A. Little and D.B. Rubin. Statistical analysis with missing data. Wiley Interscience, 2002. A.M. Wood, I.R. White, and M. Hotopf. Using number of failed contact attempts to adjust for non-ignorable non-response. Journal of the Royal Statistical Society, Series A, 169: 525–542, 2006.

22 (22) Mouna Akacha – M.Akacha@warwick.ac.uk Rate of Change Modelling and Missing Data