# An Alternative Sensitivity Approach for Longitudinal Analysis with Dropout

Authors: Amal Almohisen; Robin Henderson; Arwa M. Alshingiti
Journal: Journal of Probability and Statistics
 (2019)

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## Abstract

In any longitudinal study, a dropout before the final timepoint can rarely be avoided. The chosen dropout model is commonly one of these types: Missing Completely at Random (MCAR), Missing at Random (MAR), Missing Not at Random (MNAR), and Shared Parameter (SP). In this paper we estimate the parameters of the longitudinal model for simulated data and real data using the Linear Mixed Effect (LME) method. We investigate the consequences of misspecifying the missingness mechanism by deriving the so-called least false values. These are the values the parameter estimates converge to, when the assumptions may be wrong. The knowledge of the least false values allows us to conduct a sensitivity analysis, which is illustrated. This method provides an alternative to a local misspecification sensitivity procedure, which has been developed for likelihood-based analysis. We compare the results obtained by the method proposed with the results found by using the local misspecification method. We apply the local misspecification and least false methods to estimate the bias and sensitivity of parameter estimates for a clinical trial example.


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## Body

## 1. Introduction

Missing data are common in various settings, including surveys, clinical trials, and longitudinal studies. Methods for handling missing data strongly depend on the mechanism that generated the missing values as well as the distributional and modeling assumptions at various stages. This study focuses only on Missing at Random and Missing Not at Random dropout models, under a Linear Mixed Effect (LME) model.Much of the literature on missing data problems assumes the dropout model is only MAR and not MNAR, but this assumption is clearly limited [1]. The consequences of misspecifying the missingness mechanism are investigated by deriving the so-called least false values, which are the values the parameter estimates converge to when the assumptions may be wrong. Derivation and illustration of theoretical least false values for the LME method are made under Missing at Random (MAR) and Missing Not at Random (MNAR) dropout. The misspecified dropout model MAR is assumed in this study.Copas and Eguchi [2] gave a formula to estimate the bias under such misspecification using a likelihood approach. As the LME is a likelihood-based method, the estimates obtained through the Copas and Eguchi method can be compared with the LME least false estimates. The procedure will be applied by adding a tilt to the MAR dropout model to provide what Copas and Eguchi call local misspecification.The local model uncertainty is elaborated as proposed by Copas and Eguchi [2] and illustrated both when model misspecification is present and when the data is incomplete. Furthermore, we find that the Copas and Eguchi method gives very similar results to the least false method. Misspecification will be dealt with assuming MAR where actually the truth is MNAR. Beside Copas and Eguchi [2], many other authors have developed methods to assess the sensitivity of inference under the MAR assumption [3, 4]. Moreover, Lin et al. [5] extended the Copas and Eguchi method and assumed a doubly misspecified model while having only single misspecification. Also, there has been interest in the Copas and Eguchi method from a Bayes perspective [6–10]. Recently, [11] performed simulation based sensitivity analysis.In Section2, the LME method is presented and we show how to calculate the least false values. A description of the Copas and Eguchi method is provided in Section 3.1, followed by an example in Section 3.2. A simulation study is described in Section 4. The Copas and Eguchi bias estimate results are studied and examined with the least false values derived from the LME method, and we then show the coverage of nominal confidence intervals. A sensitivity analysis is conducted to assess how inference can depend on missing data. In Section 5, the methods are applied to data from a clinical trial with two treatments and two measurement times as introduced and analysed by Matthews et al. [12]. We compared the results obtained by the proposed method with the results found by using the Copas and Eguchi method.


## 2. Linear Mixed Effect (LME) Method

A statistical model containing fixed effects and random effects is called a mixed effect model. These models have been shown to be effective in many disciplines in the biological, physical, and social sciences. Usually a linear form is assumed.Reference [13] gave a definition of the response Y in the LME model which is of the form: (1)measuredresponse=covariateeffects+randomeffects+measurementerror.For example, a simplified version of the Liard and Ware [14] mixed model approach for longitudinal data would include a random effect in the intercept term in a model for responses. If Yij is the response at time j on subject i, the model is (2)Yij=μij+Ui+ϵijwhere μij is the marginal mean, which will usually be a linear function of covariates, ϵij is independent Gaussian noise, and Ui is a realisation of a zero mean scalar Gaussian random variable. Since Ui has zero mean, the marginal mean of Yij remains μij after integrating out Uij. However, since Ui is common to all j, we get dependence between observations on the same subject. For example, if Ui is positive, then all values would tend to be above the marginal mean and so on. In the context of longitudinal data, some reviews of linear mixed models can be found in [15, 16].


### 2.1. Assumptions

Suppose there aren individuals in a study and each provides longitudinal responses Y and dropout information R. Generally, we will assume a linear model for Y (in the absence of dropout) and logistic models for the probability ofcontinuing to the next timepoint t+1 given that a subject is still under observation at time t. At times, we refer to atrue orgenerating model as the way in which data are obtained and to anassumed orfitting model as that chosen by the analyst for estimation.For simplicity in this work, the study assumes that there are just two observations or treatment periods. The methods are of course more general.At time 1, there is a measurement provided for all subjects, denoted byYi1 for subject i. Then at time 2, some subjects are dropped out before measurement. Let Ri=1 indicate that there is a measurement at time 2 and Ri=0 otherwise. Let Yi=(Yi1,Yi2)T and assume E[Yi]=xiβ where β is a parameter vector of dimension p and xi is the design matrix associated with subject i, which is of dimension 2×p. The standard model assumes just one covariate and is(3)Y1i=β1G+β2Gxi+Ui+ϵ1iY2i=β3G+β4Gxi+Ui+ϵ2iYi=XiβG+Ui1+ϵi,where Yi=Y1iY2i, Xi=1xi00001xi, βG=(β1G,β2G,β3G,β4G)T, 1=11, ϵi=ϵ1iϵ2i, and xi~N(0,σx2), Ui~N(0,σU2), ϵ1i~N(0,σϵ12) and ϵ2i~N(0,σϵ22). Let σ12=σU2+σϵ12, σ22=σU2+σϵ22 and ρ=σU2/σ1σ2.Returning to the general case, the influence of missing data depends on the missingness mechanism, that is, the probability model for missingness. Knowing the reason for the missingness is obviously helpful to handle missing data. There are four generalmissingness mechanisms as introduced by Little and Rubin [17] and Wu and Carroll [18]. They are Missing Completely at Random (MCAR), Missing at Random (MAR), Missing Not at Random (MNAR), and Shared Parameter (SP).For simplicity in this investigation, the parameters are assumed to be common between timepoints. Let the dropout parameters beθ=(θ0,θ1). The MAR dropout logistic model is then(4)πiθ=PRi=1∣Y1i,Y2i=eθ0+θ1Y1i1+eθ0+θ1Y1i=expitθ0+θ1Y1i.The missingness is called Missing Not at Random, if it depends on unrecorded information, which predicts the missing values. An example is that a patient was unsatisfied with a particular treatment, and thus this patient is more likely to quit the study. If missingness is not at random, then some bias is expected in inferences.Let the dropout parameters now be(θ,θ2). The MNAR version for the two-timepoint example is the logistic model:(5)πiθ,θ2=PRi=1∣Y1i,Y2i=eθ0+θ1Y1i+θ2Y2i1+eθ0+θ1Y1i+θ2Y2i=expitθ0+θ1Y1i+θ2Y2i.


### 2.2. LME Least False

In this section, the Linear Mixed Effect (LME) method is investigated, which is based on a maximum likelihood estimating approach. The performance of the LME method under MAR and MNAR dropout is examined. Derivation and illustration of theoretical least false values are made. Assuming a Gaussian random intercept model, the score equation of current interest is [19](6)∑i=1nRiXiTV-1Yi-Xiβ^+1-Riσ12xi1Yi1-xi1Tβ^=0where Yi=(Yi1,Yi2), Xi is a 2×4 design matrix associated with subject i which is Xi=1xi00001xi, and we will use xi1T as notation for the first row of Xi; thus xi1T=(1,xi,0,0), β^T=(β^1,β^2,β^3,β^4), and V=σ12ρσ1σ2ρσ1σ2σ22. We can rearrange the terms in (6) to be(7)∑i=1nRiXiTV-1Xi+1-Riσ12xi1xi1Tβ^(8)=∑i=1nRiXiTV-1Yi+1-Riσ12xi1Yi1.These components are in detail(9)V-1=Kσ22-ρσ1σ2-ρσ1σ2σ12where K=1/σ12σ22(1-ρ2), and(10)XiTV-1Xi=Kσ22σ22xi-ρσ1σ2-ρσ1σ2xiσ22xiσ22xi2-ρσ1σ2xi-ρσ1σ2xi2-ρσ1σ2-ρσ1σ2xiσ12σ12xi-ρσ1σ2xi-ρσ1σ2xi2σ12xiσ12xi2.Also(11)xi1xi1T=1xi00xixi20000000000.Similarly for the right hand side of (8)(12)XiTV-1Yi=Kσ22Yi1-ρσ1σ2Yi2σ22Yi1xi-ρσ1σ2Yi2xiσ12Yi2-ρσ1σ2Yi1σ12Yi2xi-ρσ1σ2Yi1xi.Finally(13)xi1Yi1=Yi1xiYi100.We assume independent and identically distributed responses, with finite variance for the covariate and error distributions, and dropout probabilities bounded away from both zero and one. On dividing all sums by n, the weak law of large numbers applies and we can replace the sums with expectations as follows:(14)ERXTV-1X+1-Rσ12x1x1Tβ∗=ERXTV-1Y+1-Rσ12x1Y1.In the left hand side of (14), there will be two parts. First(15)ERXTV-1X(16)=Kσ22ERσ22ERx-ρσ1σ2ER-ρσ1σ2ERxσ22ERxσ22ERx2-ρσ1σ2ERx-ρσ1σ2ERx2-ρσ1σ2ER-ρσ1σ2ERxσ12ERσ12ERx-ρσ1σ2ERx-ρσ1σ2ERx2σ12ERxσ12ERx2,and second(17)1-Rσ12x1x1T=1σ121-EREx-ERx00Ex-ERxEx2-ERx20000000000.Similarly, the right hand side is(18)Kσ22ERY1-ρσ1σ2ERY2σ22ERY1x-ρσ1σ2ERY2xσ12ERY2-ρσ1σ2ERY1σ12ERY2x-ρσ1σ2ERY1x+1σ12EY1-ERY1EY1x-ERY1x00.Expressions for E[R], E[Rx], E[Rx2], E[RY1], E[RY2], E[RY1x], and E[RY2x] have been obtained under different dropout models. For illustration, we show calculation of E[R] under MAR in the Supplementary Materials available at the journal website (available here).Finally to find the least false valueβ∗, the inverse of the matrix has been considered in the left hand side of (14) and we multiply this inverse by the matrix in the right hand side, which will yield the array of the least false values β∗T=(β1∗,β2∗,β3∗,β4∗). In the following section, we present simulations regarding how the LME method performs under MAR and MNAR dropout model.


### 2.3. Numerical Investigation

A scalarN(0,1) variable x is generated, and then the longitudinal means are generated μ1=β1+β2x, μ2=β3+β4x. This was followed by (Y1,Y2) from a bivariate normal distribution with mean (μ1,μ2). Missingness was generated from (4) and (5) for the MAR and MNAR models, respectively. In all of the following simulations, unless it is stated otherwise, the parameters β=(-2,-2,-1,-1), σx=σ1=σ2=1, ρ=0.5 were followed. In the following, we show the effect of dropout on the limiting values β3∗ and β4∗.As LME provides consistent estimates under MAR, the least false valuesβ3∗ and β4∗ are not affected by changing the dropout probabilities under MAR. Therefore, only MNAR concentrations were considered. From a contour plot of β3∗ under MNAR (Figure 1), in order to minimise the bias in β3∗, θ1 should be chosen to be around zero. For negative θ1, the dropout is associated with large U, so Y1 and Y2 both tend to be low if dropout does not occur. Hence β3∗ is lower than it should be. The opposite happens for a positive θ1.Figure 1
Contour plot ofβ3∗ under MNAR.Figure2 shows a contour plot of β4∗ under MNAR. Here, negative bias is obtained as θ1 moves away from zero in either direction. Such an attenuation of regression effect is common when there are errors in variables [20]. It seems that a similar effect is obtained here.Figure 2
Contour plot ofβ4∗ under MNAR.Having obtained least false values, we propose their use in sensitivity analyses. Before doing so, a sensitivity procedure is investigated for local misspecification as proposed by Copas and Eguchi [2].


## 2.1. Assumptions

Suppose there aren individuals in a study and each provides longitudinal responses Y and dropout information R. Generally, we will assume a linear model for Y (in the absence of dropout) and logistic models for the probability ofcontinuing to the next timepoint t+1 given that a subject is still under observation at time t. At times, we refer to atrue orgenerating model as the way in which data are obtained and to anassumed orfitting model as that chosen by the analyst for estimation.For simplicity in this work, the study assumes that there are just two observations or treatment periods. The methods are of course more general.At time 1, there is a measurement provided for all subjects, denoted byYi1 for subject i. Then at time 2, some subjects are dropped out before measurement. Let Ri=1 indicate that there is a measurement at time 2 and Ri=0 otherwise. Let Yi=(Yi1,Yi2)T and assume E[Yi]=xiβ where β is a parameter vector of dimension p and xi is the design matrix associated with subject i, which is of dimension 2×p. The standard model assumes just one covariate and is(3)Y1i=β1G+β2Gxi+Ui+ϵ1iY2i=β3G+β4Gxi+Ui+ϵ2iYi=XiβG+Ui1+ϵi,where Yi=Y1iY2i, Xi=1xi00001xi, βG=(β1G,β2G,β3G,β4G)T, 1=11, ϵi=ϵ1iϵ2i, and xi~N(0,σx2), Ui~N(0,σU2), ϵ1i~N(0,σϵ12) and ϵ2i~N(0,σϵ22). Let σ12=σU2+σϵ12, σ22=σU2+σϵ22 and ρ=σU2/σ1σ2.Returning to the general case, the influence of missing data depends on the missingness mechanism, that is, the probability model for missingness. Knowing the reason for the missingness is obviously helpful to handle missing data. There are four generalmissingness mechanisms as introduced by Little and Rubin [17] and Wu and Carroll [18]. They are Missing Completely at Random (MCAR), Missing at Random (MAR), Missing Not at Random (MNAR), and Shared Parameter (SP).For simplicity in this investigation, the parameters are assumed to be common between timepoints. Let the dropout parameters beθ=(θ0,θ1). The MAR dropout logistic model is then(4)πiθ=PRi=1∣Y1i,Y2i=eθ0+θ1Y1i1+eθ0+θ1Y1i=expitθ0+θ1Y1i.The missingness is called Missing Not at Random, if it depends on unrecorded information, which predicts the missing values. An example is that a patient was unsatisfied with a particular treatment, and thus this patient is more likely to quit the study. If missingness is not at random, then some bias is expected in inferences.Let the dropout parameters now be(θ,θ2). The MNAR version for the two-timepoint example is the logistic model:(5)πiθ,θ2=PRi=1∣Y1i,Y2i=eθ0+θ1Y1i+θ2Y2i1+eθ0+θ1Y1i+θ2Y2i=expitθ0+θ1Y1i+θ2Y2i.


## 2.2. LME Least False

In this section, the Linear Mixed Effect (LME) method is investigated, which is based on a maximum likelihood estimating approach. The performance of the LME method under MAR and MNAR dropout is examined. Derivation and illustration of theoretical least false values are made. Assuming a Gaussian random intercept model, the score equation of current interest is [19](6)∑i=1nRiXiTV-1Yi-Xiβ^+1-Riσ12xi1Yi1-xi1Tβ^=0where Yi=(Yi1,Yi2), Xi is a 2×4 design matrix associated with subject i which is Xi=1xi00001xi, and we will use xi1T as notation for the first row of Xi; thus xi1T=(1,xi,0,0), β^T=(β^1,β^2,β^3,β^4), and V=σ12ρσ1σ2ρσ1σ2σ22. We can rearrange the terms in (6) to be(7)∑i=1nRiXiTV-1Xi+1-Riσ12xi1xi1Tβ^(8)=∑i=1nRiXiTV-1Yi+1-Riσ12xi1Yi1.These components are in detail(9)V-1=Kσ22-ρσ1σ2-ρσ1σ2σ12where K=1/σ12σ22(1-ρ2), and(10)XiTV-1Xi=Kσ22σ22xi-ρσ1σ2-ρσ1σ2xiσ22xiσ22xi2-ρσ1σ2xi-ρσ1σ2xi2-ρσ1σ2-ρσ1σ2xiσ12σ12xi-ρσ1σ2xi-ρσ1σ2xi2σ12xiσ12xi2.Also(11)xi1xi1T=1xi00xixi20000000000.Similarly for the right hand side of (8)(12)XiTV-1Yi=Kσ22Yi1-ρσ1σ2Yi2σ22Yi1xi-ρσ1σ2Yi2xiσ12Yi2-ρσ1σ2Yi1σ12Yi2xi-ρσ1σ2Yi1xi.Finally(13)xi1Yi1=Yi1xiYi100.We assume independent and identically distributed responses, with finite variance for the covariate and error distributions, and dropout probabilities bounded away from both zero and one. On dividing all sums by n, the weak law of large numbers applies and we can replace the sums with expectations as follows:(14)ERXTV-1X+1-Rσ12x1x1Tβ∗=ERXTV-1Y+1-Rσ12x1Y1.In the left hand side of (14), there will be two parts. First(15)ERXTV-1X(16)=Kσ22ERσ22ERx-ρσ1σ2ER-ρσ1σ2ERxσ22ERxσ22ERx2-ρσ1σ2ERx-ρσ1σ2ERx2-ρσ1σ2ER-ρσ1σ2ERxσ12ERσ12ERx-ρσ1σ2ERx-ρσ1σ2ERx2σ12ERxσ12ERx2,and second(17)1-Rσ12x1x1T=1σ121-EREx-ERx00Ex-ERxEx2-ERx20000000000.Similarly, the right hand side is(18)Kσ22ERY1-ρσ1σ2ERY2σ22ERY1x-ρσ1σ2ERY2xσ12ERY2-ρσ1σ2ERY1σ12ERY2x-ρσ1σ2ERY1x+1σ12EY1-ERY1EY1x-ERY1x00.Expressions for E[R], E[Rx], E[Rx2], E[RY1], E[RY2], E[RY1x], and E[RY2x] have been obtained under different dropout models. For illustration, we show calculation of E[R] under MAR in the Supplementary Materials available at the journal website (available here).Finally to find the least false valueβ∗, the inverse of the matrix has been considered in the left hand side of (14) and we multiply this inverse by the matrix in the right hand side, which will yield the array of the least false values β∗T=(β1∗,β2∗,β3∗,β4∗). In the following section, we present simulations regarding how the LME method performs under MAR and MNAR dropout model.


## 2.3. Numerical Investigation

A scalarN(0,1) variable x is generated, and then the longitudinal means are generated μ1=β1+β2x, μ2=β3+β4x. This was followed by (Y1,Y2) from a bivariate normal distribution with mean (μ1,μ2). Missingness was generated from (4) and (5) for the MAR and MNAR models, respectively. In all of the following simulations, unless it is stated otherwise, the parameters β=(-2,-2,-1,-1), σx=σ1=σ2=1, ρ=0.5 were followed. In the following, we show the effect of dropout on the limiting values β3∗ and β4∗.As LME provides consistent estimates under MAR, the least false valuesβ3∗ and β4∗ are not affected by changing the dropout probabilities under MAR. Therefore, only MNAR concentrations were considered. From a contour plot of β3∗ under MNAR (Figure 1), in order to minimise the bias in β3∗, θ1 should be chosen to be around zero. For negative θ1, the dropout is associated with large U, so Y1 and Y2 both tend to be low if dropout does not occur. Hence β3∗ is lower than it should be. The opposite happens for a positive θ1.Figure 1
Contour plot ofβ3∗ under MNAR.Figure2 shows a contour plot of β4∗ under MNAR. Here, negative bias is obtained as θ1 moves away from zero in either direction. Such an attenuation of regression effect is common when there are errors in variables [20]. It seems that a similar effect is obtained here.Figure 2
Contour plot ofβ4∗ under MNAR.Having obtained least false values, we propose their use in sensitivity analyses. Before doing so, a sensitivity procedure is investigated for local misspecification as proposed by Copas and Eguchi [2].


## 3. The Effect of Local Misspecification of the Dropout Model When Using Likelihood-Based Methods under the MAR Assumption

In the previous section, we investigated the consequences of misspecifying the missingness mechanism by deriving the so-called least false values, which are the values the parameter estimates converge to when the assumptions may be wrong.As an alternative, Copas and Eguchi [2] give a formula to estimate the bias under such misspecification using a likelihood approach. As the LME is a likelihood-based method, we can compare the Copas and Eguchi method with the LME least false estimates. The procedure will be applied by adding a tilt to the MAR dropout model to provide what Copas and Eguchi [2] call local misspecification.


### 3.1. Description of Copas and Eguchi Method

We use the notation of Copas and Eguchi [2], denoting by Z complete data and by Y incomplete data. There are two types of model: the true model and the assumed model. The true model is also called the generating model and it means how the data are actually generated or simulated. On the other hand, the assumed model or what is also known as the fitting model is what we fit to data. The true model for complete data is denoted by gZ=gZ(z;ψ) and the corresponding true model for incomplete data is gY=gY(y:ψ) which can be derived from gZ. Here ψ is a generic (vector) parameter. The assumed or working model is a parametric model fZ=fZ(z;ψ) which gives the distribution of Z, and its marginal density is fY=fY(y:ψ).Thus(19)fY=∫yfZdzwhere the notation (y) means integration over all missing values in Z that are consistent with the observed Y.A method is provided to approximate the bias in the estimation of the parameters of the misspecified model following Copas and Eguchi [2]. We consider MAR as the working model and MNAR as the true model. Thus, the misspecification is caused by assuming MAR but the truth is MNAR.Suppose there is a random sample ofn observations, and the true model is given by gZ, which is defined by (16) in Copas and Eguchi [2] as a tilt model:(20)gZ=gZz;ψ,ε,uZ=fZz;ψexp⁡εuZz;ψ.Thus, the misspecification is determined by the quantityεuZ(z;ψ). In this, ε, which is assumed to be small, measures the size of misspecification while uZ(z;ψ) determines its direction. We assume uZ(z;ψ) has zero mean and unit variance under the working model fZ. The misspecification is local because ε is small. Hence, gZ is close to fZ and can be written as(21)gZfZ=exp⁡εuZz;ψ.Now if the model actually used to fit the data is fZ(z;ψ), then the limiting value of the MLE ψ^ as n→∞ is given by equation (18) in Copas and Eguchi [2](22)ψgZ=argψEgsZz;ψ=0=ψ+εIZ-1EfZuZz;ψsZz;ψ,where sZ(·;ψ)=∂{log⁡(fZ)}/∂ψ and IZ=E[-∂2{log⁡(fZ)}/∂ψ∂θT] are the score and information matrix for the model fZ, respectively.However,fY will be considered as the working model for the marginal data. Copas and Eguchi [2] show that if (20) is true and ε is small, then a similar approximation holds for the marginal data Y, i.e.,(23)gY=gYy;ψ,ε,uY=fYy;ψexp⁡εuYy;ψwhere again uY(y;ψ) has zero mean and unit variance. In this case according to (19) in Copas and Eguchi [2] the limiting value is(24)ψgY≈ψ+εIY-1EfuYsY=ψ+IY-1EfεuYsYwhere sY(·;ψ)=∂{log⁡(fY)}/∂ψ and IY=E-∂2{log⁡(fY)}/∂ψ∂ψT are the score and information matrix for the model fY, respectively. To calculate the bias, IY-1Ef[εuYsY], tilt εuY. In the next section, how to calculate this amount under MAR and MNAR in our setting of two timepoints will be determined.


### 3.2. Copas and Eguchi Method for Two-Timepoint Example

The bias consists of, as shown in (24), the score, information matrix, and the tilt. In order to calculate these components, the likelihood model in use is defined. Under MAR, either of the following equivalent formulations can be selected:(25)L=fY1,Y2PR=1∣Y1,Y2RfY1PR=0∣Y11-R(26)=fY2∣Y1fY1PR=1∣Y1,Y2RfY1PR=0∣Y11-R.The conditional distribution of Y2 given Y1 is needed quite a lot in this section. Hence, for simplicity, we use Y21 to denote this quantity. Since f(Y1,Y2) is bivariate normal in this assumed model, Y21~N(μ21,σ21) where μ21=μ2+σ2/σ1ρ(Y1-μ1) and σ21=σ21-ρ2. Also, the complete data is Z=(Y1,Y2,R) and incomplete data is Y=(Y1,Y2(R),R) where(27)Y2R=Y2,R=1undefined,R=0.Therefore, at R=1, Y=Z, but Y will differ from Z at R=0.In addition, the models are defined asfZ, fY, gZ, and gY. MAR is assumed as the working model or misspecified model. Under MAR, there is P(R=1∣Y1,Y2)=P(R=1∣Y1); then from (25) the working model for complete data by assuming R=1 is(28)fZ=fY1,Y2PR=1∣Y1.Similarly, from (26) the working model for incomplete data by assuming R=0 is(29)fY=fY1PR=0∣Y1.Under MNAR, if there is complete data, then we will always setR=1. Thus, from (25), the true model for complete data is(30)gZ=fY1,Y2PR=1∣Y1,Y2.The true model for incomplete data on the other hand is the marginal density: (31)gY=∫ygZdz=∫Y2RfY1,Y2PR=1∣Y1,Y2dY2R.Note that the integral is over the missing values Y2R. Referring to (27), the missing values Y2 are undefined in case that R=0.This means that in order to use Copas and Eguchi’s ideas, we should convert the specificgY in (31) into the general form of (23). To do this, we will redefine the MNAR model in tilt form:(32)PR=1∣Y1,Y2=expitθ0+θ1Y1exp⁡εuY.Here ε=θ2σ21 and uY=uY(y;θ)=(Y2-μ21)/σ21. For small θ2 this is a good approximation to the logistic MNAR model.Calculation of the terms needed for the bias expression (24) is now possible and follows directly. Details are in the Supplementary Materials available at the journal website.


## 3.1. Description of Copas and Eguchi Method

We use the notation of Copas and Eguchi [2], denoting by Z complete data and by Y incomplete data. There are two types of model: the true model and the assumed model. The true model is also called the generating model and it means how the data are actually generated or simulated. On the other hand, the assumed model or what is also known as the fitting model is what we fit to data. The true model for complete data is denoted by gZ=gZ(z;ψ) and the corresponding true model for incomplete data is gY=gY(y:ψ) which can be derived from gZ. Here ψ is a generic (vector) parameter. The assumed or working model is a parametric model fZ=fZ(z;ψ) which gives the distribution of Z, and its marginal density is fY=fY(y:ψ).Thus(19)fY=∫yfZdzwhere the notation (y) means integration over all missing values in Z that are consistent with the observed Y.A method is provided to approximate the bias in the estimation of the parameters of the misspecified model following Copas and Eguchi [2]. We consider MAR as the working model and MNAR as the true model. Thus, the misspecification is caused by assuming MAR but the truth is MNAR.Suppose there is a random sample ofn observations, and the true model is given by gZ, which is defined by (16) in Copas and Eguchi [2] as a tilt model:(20)gZ=gZz;ψ,ε,uZ=fZz;ψexp⁡εuZz;ψ.Thus, the misspecification is determined by the quantityεuZ(z;ψ). In this, ε, which is assumed to be small, measures the size of misspecification while uZ(z;ψ) determines its direction. We assume uZ(z;ψ) has zero mean and unit variance under the working model fZ. The misspecification is local because ε is small. Hence, gZ is close to fZ and can be written as(21)gZfZ=exp⁡εuZz;ψ.Now if the model actually used to fit the data is fZ(z;ψ), then the limiting value of the MLE ψ^ as n→∞ is given by equation (18) in Copas and Eguchi [2](22)ψgZ=argψEgsZz;ψ=0=ψ+εIZ-1EfZuZz;ψsZz;ψ,where sZ(·;ψ)=∂{log⁡(fZ)}/∂ψ and IZ=E[-∂2{log⁡(fZ)}/∂ψ∂θT] are the score and information matrix for the model fZ, respectively.However,fY will be considered as the working model for the marginal data. Copas and Eguchi [2] show that if (20) is true and ε is small, then a similar approximation holds for the marginal data Y, i.e.,(23)gY=gYy;ψ,ε,uY=fYy;ψexp⁡εuYy;ψwhere again uY(y;ψ) has zero mean and unit variance. In this case according to (19) in Copas and Eguchi [2] the limiting value is(24)ψgY≈ψ+εIY-1EfuYsY=ψ+IY-1EfεuYsYwhere sY(·;ψ)=∂{log⁡(fY)}/∂ψ and IY=E-∂2{log⁡(fY)}/∂ψ∂ψT are the score and information matrix for the model fY, respectively. To calculate the bias, IY-1Ef[εuYsY], tilt εuY. In the next section, how to calculate this amount under MAR and MNAR in our setting of two timepoints will be determined.


## 3.2. Copas and Eguchi Method for Two-Timepoint Example

The bias consists of, as shown in (24), the score, information matrix, and the tilt. In order to calculate these components, the likelihood model in use is defined. Under MAR, either of the following equivalent formulations can be selected:(25)L=fY1,Y2PR=1∣Y1,Y2RfY1PR=0∣Y11-R(26)=fY2∣Y1fY1PR=1∣Y1,Y2RfY1PR=0∣Y11-R.The conditional distribution of Y2 given Y1 is needed quite a lot in this section. Hence, for simplicity, we use Y21 to denote this quantity. Since f(Y1,Y2) is bivariate normal in this assumed model, Y21~N(μ21,σ21) where μ21=μ2+σ2/σ1ρ(Y1-μ1) and σ21=σ21-ρ2. Also, the complete data is Z=(Y1,Y2,R) and incomplete data is Y=(Y1,Y2(R),R) where(27)Y2R=Y2,R=1undefined,R=0.Therefore, at R=1, Y=Z, but Y will differ from Z at R=0.In addition, the models are defined asfZ, fY, gZ, and gY. MAR is assumed as the working model or misspecified model. Under MAR, there is P(R=1∣Y1,Y2)=P(R=1∣Y1); then from (25) the working model for complete data by assuming R=1 is(28)fZ=fY1,Y2PR=1∣Y1.Similarly, from (26) the working model for incomplete data by assuming R=0 is(29)fY=fY1PR=0∣Y1.Under MNAR, if there is complete data, then we will always setR=1. Thus, from (25), the true model for complete data is(30)gZ=fY1,Y2PR=1∣Y1,Y2.The true model for incomplete data on the other hand is the marginal density: (31)gY=∫ygZdz=∫Y2RfY1,Y2PR=1∣Y1,Y2dY2R.Note that the integral is over the missing values Y2R. Referring to (27), the missing values Y2 are undefined in case that R=0.This means that in order to use Copas and Eguchi’s ideas, we should convert the specificgY in (31) into the general form of (23). To do this, we will redefine the MNAR model in tilt form:(32)PR=1∣Y1,Y2=expitθ0+θ1Y1exp⁡εuY.Here ε=θ2σ21 and uY=uY(y;θ)=(Y2-μ21)/σ21. For small θ2 this is a good approximation to the logistic MNAR model.Calculation of the terms needed for the bias expression (24) is now possible and follows directly. Details are in the Supplementary Materials available at the journal website.


## 4. Simulation Study

We use the same simulation setup as before. The limiting valuesβ3∗ and β4∗ are compared using different methods under MAR and MNAR dropout models. Next, the local model uncertainty will be elaborated as proposed by Copas, and we illustrate how to apply it both when model misspecification is present and when the data is incomplete. We find that the Copas and Eguchi [2] method gives very similar results to the least false. Misspecification will be dealt with assuming MAR where actually the truth is MNAR.


### 4.1. Comparing the Copas and Eguchi Method with LME Least False Results

In this section, the parameter estimates are affected when a MAR model is fitted to data that are MNAR, and compared with the values that the Copas and Eguchi method predicts.The sample size is 10000, and 10 simulations are used. We used large samples here, as our first task is to check the accuracy of the large-sample approximations underpinning the least false values. The aim is to show the variation in treatment effect estimates asθ2 varies. A grid of θ2 from -0.2 to 0.2 is selected. Figure 3 is produced when β=(-2,-2,-1,-1) and θ=(-0.5,0), which gives dropout rate around 40%. Here the blue lines (dotted lines) are simulation estimates using maximum likelihood, the red lines (solid lines) are Copas and Eguchi estimates, and the light blue lines are the LME least false estimates. These show that the least false, simulations, and Copas and Eguchi [2] results all match well. Therefore, we can use the least false results for bias correction as an alternative to Copas and Eguchi.Figure 3
Comparison 1:β=(-2,-2,-1,-1), θ=(-0.5,0). The blue lines (dotted lines) are simulation estimates using maximum likelihood, the red lines (solid lines) are Copas and Eguchi estimates, and the light blue lines are the LME least false estimates.


### 4.2. CI Coverage for the Estimatedβ3 and β4

The Copas and Eguchi and LME least false values show how estimates are biased by assuming MAR when the data are MNAR. The misspecification parameter isθ2, with θ2=0 meaning no misspecification. If the value of θ2 was known, then the parameter estimates will be adjusted to take into account the misspecification. This idea will be illustrated in this section.For a range of true (generating)θ2, 1000 samples are simulated, each of size 1000. This is a realistic number for applications. In each case, β3 and β4 are estimated using maximum likelihood under a MAR assumption. Afterwards, the estimates are adjusted using either the estimated Copas and Eguchi bias or the bias arising through least false calculations, in both cases taking anassumedθ2. Coverage of the resulting nominal 95% confidence intervals is then recorded. The estimated confidence interval width is not adjusted, just its location.Tables1 and 2 give the results. Here we use θ2T for the true θ2, and θ2A denotes the assumed value used in adjusting the estimates. Also, (β3∗∗,β4∗∗) are used for the Copas and Eguchi adjustment method and (β3∗,β4∗) for the least false adjustment method.Table 1
CI coverage in percent for the estimatedβ3 and β4 at assumed θ2=0. We use θ2T for the true θ2, θ2A for the assumed value in adjusting the estimates, (β3∗∗,β4∗∗) for the Copas and Eguchi adjustment method, and (β3∗,β4∗) for the least false adjustment method. Results based on 1000 samples of size 1000.
θ 2 T θ 2 A β 3 ∗ ∗ β4∗∗ β 3 ∗ β 4 ∗ -0.10 0.00 84.80 95.30 84.90 95.30  -0.09 0.00 85.90 96.70 85.80 96.70  -0.06 0.00 92.00 94.70 91.90 94.70  -0.03 0.00 95.00 95.00 95.10 94.90  0.00 0.00 94.70 95.10 94.70 95.10  0.03 0.00 95.20 94.20 95.00 94.20  0.06 0.00 91.70 94.70 91.70 94.70  0.09 0.00 88.00 95.00 87.80 95.00  0.10 0.00 83.40 95.10 83.60 95.00Table 2
CI coverage for the estimatedβ3 and β4 in percent at assumed θ2=-0.10. We use θ2T for the true θ2, θ2A for the assumed value in adjusting the estimates, (β3∗∗,β4∗∗) for the Copas and Eguchi adjustment method, and (β3∗,β4∗) for the least false adjustment method. Results based on 1000 samples of size 1000.
θ 2 T θ 2 A β 3 ∗ ∗ β4∗∗ β 3 ∗ β 4 ∗ -0.10 -0.10 95.30 95.40 95.70 95.10  -0.09 -0.10 94.80 95.10 95.50 94.80  -0.06 -0.10 95.80 95.20 94.90 95.60  -0.03 -0.10 92.50 94.90 92.00 95.10  0.00 -0.10 89.30 95.30 87.40 95.40  0.03 -0.10 83.70 93.80 81.20 93.80  0.06 -0.10 74.40 95.10 70.50 95.00  0.09 -0.10 62.20 96.00 58.30 95.80  0.10 -0.10 62.70 93.90 57.80 94.20In Table1, the assumed θ2 is zero, meaning no correction. Results at the correct value of θ2A =0 are good. Otherwise, the CI for β3 goes badly wrong. Note that there is no correction here, so the Copas and Eguchi and least false results should be the same. Small differences are just because of the different calculations that are involved. For example, the least false calculation needs an estimate of σx but the Copas and Eguchi one does not. The CI coverage is noted for β4 which is not too much affected at any true θ2 in the range (-0.1,+0.1). For example, at θ2A=-0.1, the CI coverage for β4 is about 95%, whereas there is undercoverage for β3 when θ2T deviates from zero. For example at θ2T=-0.1, the CI coverage for β3 is about 85%. This indicates that β4 is less sensitive to the misspecification than β3 in this scenario.In Table2, the assumed value is taken of θ2=-0.1, which means that dropout is associated with high Y2. Note that, in contrast to the Table 2, there is correction here, so the Copas and Eguchi and least false results will not be the same; for example, at θ2T=+0.1, the CI coverage for β3∗∗ is about 62.7%, but the CI coverage for β3∗ is about 57.8%. However, both estimates β3∗∗ and β3∗ have undercoverage as θ2T goes further from the assumed value -0.1.


### 4.3. Sensitivity Analysis

Of course, in practiceθ2 is not known. For any given data set, a sensible sensitivity procedure would mean plotting bias-corrected estimates and confidence intervals for a range of assumed θ2 values. Here, a grid of assumed θ2 is used from -0.2 to 0.2. We will show that, for each limiting value calculated by the Copas and Eguchi method, the simulated values are within noise of the theoretical values for large sample sizes (n=10000). The noise is estimated from the simulations; that is, a confidence interval is achieved from the simulations with reassurance that the population values are present. A correct MAR model is obtained and after that, under true MNAR, MAR is assumed.Figure4 illustrates the case when MAR is the correct model (θ2=0) and the unadjusted confidence intervals (red lines) include the true parameter values (β3=-1 and β4=-1), as in this case so do the adjusted ones (blue lines). The horizontal lines are at the true values. We note that β3∗∗ decreases as θ2 increases whereas β4∗∗ increases as θ2 increases. Note that β4 has a wider CI than β3.Figure 4
CI under MAR:β=(-2,-2,-1,-1), θ=(-0.5,-0.5), θ2T=0. The blue lines are the adjusted estimates, and red lines are the unadjusted estimates. The horizontal dotted lines are at the true values.Figure5 has the true θ2=0.1, so the study has fitted MAR to data that are really MNAR. The lines cross at θ2=0 because the same MAR model is fitted. The important point is that better estimates of the true β’s are obtained at the correct θ2. Also, as mentioned in Figure 4, β4 has wider CI than β3.Figure 5
CI under MNAR:β=(-2,-2,-1,-1), θ=(-0.5,-0.5), θ2T=0.1. The blue lines are the adjusted estimates, and red lines are the unadjusted estimates. The horizontal lines are at the true values.Note that, both under MAR and MNAR,β3 and β4 have opposite trends; β3 decreases as θ2 increases whereas β4 increases as θ2 increases.


## 4.1. Comparing the Copas and Eguchi Method with LME Least False Results

In this section, the parameter estimates are affected when a MAR model is fitted to data that are MNAR, and compared with the values that the Copas and Eguchi method predicts.The sample size is 10000, and 10 simulations are used. We used large samples here, as our first task is to check the accuracy of the large-sample approximations underpinning the least false values. The aim is to show the variation in treatment effect estimates asθ2 varies. A grid of θ2 from -0.2 to 0.2 is selected. Figure 3 is produced when β=(-2,-2,-1,-1) and θ=(-0.5,0), which gives dropout rate around 40%. Here the blue lines (dotted lines) are simulation estimates using maximum likelihood, the red lines (solid lines) are Copas and Eguchi estimates, and the light blue lines are the LME least false estimates. These show that the least false, simulations, and Copas and Eguchi [2] results all match well. Therefore, we can use the least false results for bias correction as an alternative to Copas and Eguchi.Figure 3
Comparison 1:β=(-2,-2,-1,-1), θ=(-0.5,0). The blue lines (dotted lines) are simulation estimates using maximum likelihood, the red lines (solid lines) are Copas and Eguchi estimates, and the light blue lines are the LME least false estimates.


## 4.2. CI Coverage for the Estimatedβ3 and β4

The Copas and Eguchi and LME least false values show how estimates are biased by assuming MAR when the data are MNAR. The misspecification parameter isθ2, with θ2=0 meaning no misspecification. If the value of θ2 was known, then the parameter estimates will be adjusted to take into account the misspecification. This idea will be illustrated in this section.For a range of true (generating)θ2, 1000 samples are simulated, each of size 1000. This is a realistic number for applications. In each case, β3 and β4 are estimated using maximum likelihood under a MAR assumption. Afterwards, the estimates are adjusted using either the estimated Copas and Eguchi bias or the bias arising through least false calculations, in both cases taking anassumedθ2. Coverage of the resulting nominal 95% confidence intervals is then recorded. The estimated confidence interval width is not adjusted, just its location.Tables1 and 2 give the results. Here we use θ2T for the true θ2, and θ2A denotes the assumed value used in adjusting the estimates. Also, (β3∗∗,β4∗∗) are used for the Copas and Eguchi adjustment method and (β3∗,β4∗) for the least false adjustment method.Table 1
CI coverage in percent for the estimatedβ3 and β4 at assumed θ2=0. We use θ2T for the true θ2, θ2A for the assumed value in adjusting the estimates, (β3∗∗,β4∗∗) for the Copas and Eguchi adjustment method, and (β3∗,β4∗) for the least false adjustment method. Results based on 1000 samples of size 1000.
θ 2 T θ 2 A β 3 ∗ ∗ β4∗∗ β 3 ∗ β 4 ∗ -0.10 0.00 84.80 95.30 84.90 95.30  -0.09 0.00 85.90 96.70 85.80 96.70  -0.06 0.00 92.00 94.70 91.90 94.70  -0.03 0.00 95.00 95.00 95.10 94.90  0.00 0.00 94.70 95.10 94.70 95.10  0.03 0.00 95.20 94.20 95.00 94.20  0.06 0.00 91.70 94.70 91.70 94.70  0.09 0.00 88.00 95.00 87.80 95.00  0.10 0.00 83.40 95.10 83.60 95.00Table 2
CI coverage for the estimatedβ3 and β4 in percent at assumed θ2=-0.10. We use θ2T for the true θ2, θ2A for the assumed value in adjusting the estimates, (β3∗∗,β4∗∗) for the Copas and Eguchi adjustment method, and (β3∗,β4∗) for the least false adjustment method. Results based on 1000 samples of size 1000.
θ 2 T θ 2 A β 3 ∗ ∗ β4∗∗ β 3 ∗ β 4 ∗ -0.10 -0.10 95.30 95.40 95.70 95.10  -0.09 -0.10 94.80 95.10 95.50 94.80  -0.06 -0.10 95.80 95.20 94.90 95.60  -0.03 -0.10 92.50 94.90 92.00 95.10  0.00 -0.10 89.30 95.30 87.40 95.40  0.03 -0.10 83.70 93.80 81.20 93.80  0.06 -0.10 74.40 95.10 70.50 95.00  0.09 -0.10 62.20 96.00 58.30 95.80  0.10 -0.10 62.70 93.90 57.80 94.20In Table1, the assumed θ2 is zero, meaning no correction. Results at the correct value of θ2A =0 are good. Otherwise, the CI for β3 goes badly wrong. Note that there is no correction here, so the Copas and Eguchi and least false results should be the same. Small differences are just because of the different calculations that are involved. For example, the least false calculation needs an estimate of σx but the Copas and Eguchi one does not. The CI coverage is noted for β4 which is not too much affected at any true θ2 in the range (-0.1,+0.1). For example, at θ2A=-0.1, the CI coverage for β4 is about 95%, whereas there is undercoverage for β3 when θ2T deviates from zero. For example at θ2T=-0.1, the CI coverage for β3 is about 85%. This indicates that β4 is less sensitive to the misspecification than β3 in this scenario.In Table2, the assumed value is taken of θ2=-0.1, which means that dropout is associated with high Y2. Note that, in contrast to the Table 2, there is correction here, so the Copas and Eguchi and least false results will not be the same; for example, at θ2T=+0.1, the CI coverage for β3∗∗ is about 62.7%, but the CI coverage for β3∗ is about 57.8%. However, both estimates β3∗∗ and β3∗ have undercoverage as θ2T goes further from the assumed value -0.1.


## 4.3. Sensitivity Analysis

Of course, in practiceθ2 is not known. For any given data set, a sensible sensitivity procedure would mean plotting bias-corrected estimates and confidence intervals for a range of assumed θ2 values. Here, a grid of assumed θ2 is used from -0.2 to 0.2. We will show that, for each limiting value calculated by the Copas and Eguchi method, the simulated values are within noise of the theoretical values for large sample sizes (n=10000). The noise is estimated from the simulations; that is, a confidence interval is achieved from the simulations with reassurance that the population values are present. A correct MAR model is obtained and after that, under true MNAR, MAR is assumed.Figure4 illustrates the case when MAR is the correct model (θ2=0) and the unadjusted confidence intervals (red lines) include the true parameter values (β3=-1 and β4=-1), as in this case so do the adjusted ones (blue lines). The horizontal lines are at the true values. We note that β3∗∗ decreases as θ2 increases whereas β4∗∗ increases as θ2 increases. Note that β4 has a wider CI than β3.Figure 4
CI under MAR:β=(-2,-2,-1,-1), θ=(-0.5,-0.5), θ2T=0. The blue lines are the adjusted estimates, and red lines are the unadjusted estimates. The horizontal dotted lines are at the true values.Figure5 has the true θ2=0.1, so the study has fitted MAR to data that are really MNAR. The lines cross at θ2=0 because the same MAR model is fitted. The important point is that better estimates of the true β’s are obtained at the correct θ2. Also, as mentioned in Figure 4, β4 has wider CI than β3.Figure 5
CI under MNAR:β=(-2,-2,-1,-1), θ=(-0.5,-0.5), θ2T=0.1. The blue lines are the adjusted estimates, and red lines are the unadjusted estimates. The horizontal lines are at the true values.Note that, both under MAR and MNAR,β3 and β4 have opposite trends; β3 decreases as θ2 increases whereas β4 increases as θ2 increases.


## 5. Application: Sensitivity Analysis for Clinical Trial

In this section, the method is illustrated using a real data example. The data is considered from a clinical trial with two treatments and two measurement times as introduced and analysed by Mathews et al. [12]. The covariates are only treatment type and time. The parameter vector is (β1,β2,β3,β4), ignoring any time interaction. There are 422 subjects, assigned to either treatment A or B. Treatment A is associated with treatment effect x=1 and treatment B is when x=0. Then, at time 2, the mean of the group receiving treatment B is β3 and the mean of the group receiving treatment A is β3+β4. At time 1, all subjects provided a response, but 24.4% dropped out by time 2. There are 212 subjects receiving treatment A, but only 126 provided a response at time 2 and the other 86 dropped out. Hence the missingness percentage is about 40%. The dropout reason is not known. For treatment B, there are 210 subjects, of which 193 subjects continued to time 2 and hence there are 17 that did not, and this gave around 8% missingness.A sensitivity analysis approach (over a grid ofθ2) using the Copas and Eguchi and LME methods is shown in Figure 6. The blue lines use the Copas and Eguchi method and the red lines use the least false method. The idea is to adjust the estimate to compensate for bias from a misspecified MAR fit. Consequently, for example, if the least false value is known under MAR to underestimate a parameter, the difference for the estimate is added to back-calculate. Dashes are the CIs, based on the MAR standard errors. The first plot shows confidence intervals for the treatment B mean as the assumed value of θ2 changes. The horizontal line is the estimate under MAR. The second plot shows the confidence intervals for the mean of treatment A. The third plot is the difference in means between treatment A and B, which yields the treatment effect means, i.e., β4 means. In the first plot, the horizontal line is at -0.74 which is the same value for the LME estimate for β3. Again, the LME estimate for β4 is about -0.40 in Figure 2. Also, note that β3+β4 equals -1.15. This supports the finding here and shows better results.Figure 6
Clinical trial example: 95% CI forβ3, β3+β4, and β4. The blue lines use the Copas and Eguchi method, the red lines use the least false method, and the horizontal line is at the MAR estimate.The first thing to note is how close the least false and Copas and Eguchi estimates are. There is almost no difference over this range ofθ2. We take θ2 from -1.5 to +1.5. The value of θ1^ under MAR is -1.66, meaning the range of θ2 allows Y2 to have the same order of effect as Y1. Clearly at large values of θ2, there is concern that the misspecification is not local, which is the assumption of Copas and Eguchi. However, the least false results apply to any misspecification, not necessarily local, and the fact that Copas and Eguchi estimate is so close to the least false one suggests that it can work well even under quite large misspecification.Whenθ2 is negative, the estimates get adjusted upwards, and the opposite is true for positive θ2. This makes sense: at negative θ2, large Y2 values have low probability of staying in the trial. Hence the observed means are lower than they would be in the hypothetical no-dropout situation, so we adjust upwards.The estimates seem to be affected more at the positiveθ2 than at the negative one. At the very largest θ2 shown, there would be a significant change in the value of the estimated true mean. However, there is very little effect of misspecification on the difference between means (third subplot), as the adjustments essentially cancel.


## 6. Conclusion

We considered the Linear Mixed Effect models (maximum likelihood method) for handling missing data. Then, by deriving the so-called least false values, we investigated the consequences of misspecifying the missingness mechanism. The closed form expressions were given to calculate the least false valuesβ3∗ and β4∗. The knowledge of these least false values allowed us to conduct sensitivity analysis, which was illustrated for the LME method.Copas and Eguchi [2] gave a formula to estimate the bias under the misspecification. We derived and explored the Copas and Eguchi approximation for the bias raised by the misspecification of the working model. The results found by using Copas and Eguchi method are compared with the results obtained by the method proposed. Also, we applied the Copas and Eguchi method to estimate the bias for the real data example.Moreover, we explained how to use a sensitivity analysis to see how the methods work under a range ofθ2. We found that the Copas and Eguchi method and LME least false match very well. Both gave very close results over the grid of θ2 considered. This suggests that the least false method can provide a credible alternative to Copas and Eguchi in sensitivity analysis. In fact, it might be preferred since there is no assumption of local misspecification. Finally, we illustrated the results using example data from a clinical trial with two measurement times.



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*Source: 1019303-2019-07-01.xml*