Smoking hot joint models for attrition bias—XMAR-ks the spot

Abstract

It’s been said that smoking is a leading cause of statistics. Striving to understand relationships between smoking, cognition, dementia and death has long served as a prototypical example of selection bias issues when studying longitudinal change in the presence of likely informative censoring events. The article by Stuckwisch et al. (Am J Epidemiol. 2025: https://doi.org/10.1093/aje/kwaf107) compares results between a standard linear mixed model and an increasingly used Joint Model approach to investigate this long-vexing problem. We describe the nature of the “extended missing at random” (XMAR) assumption of the Joint Model, position the XMAR assumption in an expanded four-level missingness taxonomy, and illustrate a sequential sensitivity analysis using example data on smoking and cognitive decline with attrition by dementia and death events to illuminate approaches.

This article is linked to “The association between smoking status and aging cognitive trajectory accounting for bias in attrition” (https://doi.org/10.1093/aje/kwaf107).

There are two types of people in this world: (1) those who can extrapolate from incomplete data

Stuckwisch et al.¹ examine associations between smoking and cognitive decline in older adults while investigating survival and attrition effects through joint modelling. While experts agree that smoking increases dementia risks,^2,3 demonstrating similar effects on cognitive decline has been more elusive, potentially due to strong relationships between smoking and mortality. Selection bias in this area has long been recognized,^4,5 as Stuckwisch and colleagues nicely describe: “Because smoking status increases an individual’s risk of death (and possibly drop out), the longitudinal cognitive follow-up may be truncated, leaving only a ‘cognitively healthy’ population for analysis in later ages, leading to potentially biased results”. While semi-competing risk (multistate\illness-death) models provide a natural approach to examining non-terminal and terminal events (such as dementia and death),^6-8 dealing with semi-competing selection influences in longitudinal models (e.g. cognitive trajectories) is more challenging. Joint models (JMs) that connect “submodels” for longitudinal and survival/event processes are “hot” choices for exploring potentially informative censoring effects on longitudinal trajectories, and the authors’ work supports other findings for adverse effects of smoking on cognitive decline in the presence of dementia and death using JMs.^9-11 Many excellent reviews, tutorials and extensions of JMs^12-14 provide general guidance on their use, fewer focus on how they can be most useful in the selection bias context. Our aims with this commentary are to (1) review the updated four-level missing data taxonomy that includes “extended missing at random” (XMAR) definitions; (2) clarify the XMAR assumption of JMs and distinguish it from MNAR; and (3) encourage expanded sensitivity analyses.

XMAR-ks the spot: updated missingness taxonomies and joint models

All models rely on underlying assumptions about missing data. The XMAR definition^15,16 and its inclu sion in an updated four-level taxonomy of missing data assumptions (9) are outlined in Table 1.

Table 1.

A four-level taxonomy of missing data assumptions and related models.

Suppose the full data to be collected is Yfull = (Yobs, Ymiss), where Yobs is then observed and Ymiss is unexpectantly missing. Denoting M as missingness indicators that separate Yfull into its components, and the joint distribution of outcomes and missingness as P(Yfull,M), a four-level missingness assumption taxonomy is:
MCAR: Missing Completely at Random; Completely Ignorable Missingness; P(M|Yfull) = P(M)
Assumption:	Missingness is completely random
Common Models:	Generalized Estimating Equations (GEE)
Plausibility:	MCAR is a strong assumption, often difficult to justify
MAR: Missing at Random; Conditionally Ignorable Missingness; P(M|Yfull) = P(M|Yobs)
Assumption:	Missingness only depends on observed data
Common Models:	Mixed models, multiple imputation (MI), weighting (IPW)
Plausibility:	Often reasonable with richly measured longitudinal outcomes and predictors
XMAR: Extended Missing at Random; Partly-Ignorable Missingness; P[M|Yfull] = P[M|Yobs, g(M,Ymiss)]
Assumption:	Missingness depends on observed data and a function of the missing values, g(M,Ymiss)
Common Models:	“Conventional” Joint/Shared-Parameter models JM/SPM, extended-MI
Plausibility:	Commonly plausible with informative data on the reason(s) the outcomes are missing
MNAR: Missing Not at Random; Non-Ignorable Missingness; P(M|Yfull) = P(M|Yobs, Ymiss)
Assumption:	Missingness depends on the unobserved data (“informative missingness”).
Common Models:	pattern mixture (PMM), selection (SEM), and generalized shared parameter models (GSPM).
Plausibility:	Frequently likely, analyses only succeed in eliminating bias under additional unverifiable assumptions; broad MNAR sensitivity analyses are the primary strategy.

severity continuum: MCAR < MAR < XMAR < <<<< MNAR

XMAR designations correspond to partly-ignorable missingness assumptions which are weaker than MAR,¹⁷ where it becomes reasonable to assume part of the missingness may be ignored by incorporating additional observed information related to the missing values, denoted by g(M,Ymiss); choices of which are based on available measures thought to capture important aspects of the missingness. For example, if steeper cognitive declines induce faster dementia event times, then including information linking declines with dementia event times may recover lost information when some cognitive decline measures are missing. JMs use random effects linked to informative censoring events as g(M,Ymiss) structures, forming the basis of (latent) XMAR assumptions. Common practical choices for linking structures are given in Table 2.

Table 2.

Structure of a joint model.

1)	Longitudinal submodel(s): (eg, mixed models; Linear mixed model, LMM, or other)
	LMM:	Outcome over time = fixed effects + random effects + error
	Example:	E(Cognition) = β0 + β1Time + β2Smoking + β3Time·Smoking + b0i
	Generalization *:	Yij = Xijβ + Zijbi + εij with εij ~ N(0,σ2)
2)	Event submodel(s): (eg, survival models; proportional hazards models, PHM, or other)
	PHM:	log(Event hazard) = baseline hazard + predictors + f (random +/- fixed effects)
	Example:	log(Death hazard) = h0(t) + α1Smoking + ρ0·b0i
	Generalization *:	hi(t) = h0(t)·exp(Xaij α + ρi(t))
3)	Linking structure(s): (eg, function of shared effects: f (random +/- fixed effects)
	Random Effects:	random intercepts, slopes, etc.
	Example:	random intercept, b0i ~ N(0,τ2)
	Generalization *:	many forms, generally assumed bi ~ MVN(0,τ), examples include:
		“current value”: ρi(t) = ρ ·mi(t) = ρ ·(Xijβ + Zijbi) (fitted value from mixed model) “current slope”: ρi(t) = ρ ·mi′(t) = ρ ·d(Xijβ + Zijbi)/dt (derivative from mixed model) random intercept: ρi(t) = ρ0·b0i (subject-specific index deviation) random slopes: ρi(t) = ρ0·b0i + ρ1·b1i (subject-specific index and slope deviation)

^* Conventional Specifications: The above are for subjects i = 1..N, measured at timepoints j = 1..n_i, with predictor sets X_ij, Z_ij, X^*_ij, Z^*_ij, random effects vector b_i, parameter vectors β, α, ρ, and covariance matrix τ, with time T_i = min(T^a_i,C_i), where T^a_i is the true time to a censoring event potentially informative about Y_ij and C_i an independent stochastic censoring mechanism.

Notes: Longitudinal submodel(s): most often linear mixed models (LMM) but can be generalized.

Event submodels(s): most often PHM but can be generalized.

Linking structure(s): Most often random effects are assumed MVN but can be generalized. The term “current value” and for m_i(t) pertains to the estimated value at time t and not the observed outcome; the term “current true value” is sometimes used to distinguish this, but m_i(t) is just an estimated value based on an assumed model and not truly “true.”

The severity of missingness assumption hierarchy is:

That is, MCAR assumes there is no problem, MAR assumes problems can be handled using observed information that drives outcome processes, XMAR assumes problems can be handled by including additional external observed information related to why the outcomes are missing, and MNAR assumes the problem is so severe that only additional, unverifiable outside assumptions can help. The multiple “<” signs designate that MNAR is fundamentally distinct and highly separated from the others. For estimation, the direction is similar:

with simple GEE (MCAR) being easiest and fastest, mixed-models/IPW/basic-MI (MAR) being straightforward and widely available, conventional JMs (such as those in Stuckwisch et al.) and extended-MI (both XMAR) being moderately difficult but becoming more accessible, and for PMM/SEM/GSPM (MNAR)^16,18 results one generally needs to find a helpful expert. The XMAR addition to the classic three-level missingness taxonomy¹⁹ helps distinguish between standard mixed models (ignorable under MAR), “conventional” JMs (partly/latently ignorable under XMAR)—which still rely on observed data (measured time to dementia, death, etc.), and more nefarious, non-ignorable, full MNAR situations. It is well known that it is difficult or impossible to tell whether a situation is truly MNAR because the data needed to test the MNAR assumption is unobserved by definition. Many investigations using JMs state that they are operating under MNAR assumptions, but conventional JMs use additional observed event data to help account for potential missingness influences on the longitudinal trajectories, which is not fully MNAR. However, JMs incorporate observed informative event information above and beyond a standard mixed-model, so they improve upon MAR. The XMAR designation clarifies the position of approaches like JMs that use additionally observed data to address missing data problems which operate in-between MAR and MNAR. An added benefit of examining XMAR approaches is that observed data can be used to support an XMAR assumption over MAR with standard diagnostic criteria,²⁰ demonstrated below.

What kind of joint model are we smoking?

“Joint Models“, also known as shared parameter models (SPMs), are multivariate/multiple-outcome models which are commonly defined as having three components: (1) one or more longitudinal outcome submodels, (2) one or more survival/event submodels, and (3) linking structures between the submodels¹³; Table 2 gives an outline.

For JMs, all the myriad modeling decisions for longitudinal and survival models need to be made, plus additional decisions on how to connect the submodels. As in any modeling effort, specific choices for JM components (1)-(3) lead to a specific JM with specific results. For example, the JM in Stuckwisch et al. used (1) a LMM with fixed and random (Gaussian) quadratic effects on time (years since baseline) for cognitive trajectories, (2) a single parametric PHM for a composite censoring event (death and/or drop-out), and (3) a “current-value” plus “current-slope” linking structure. The authors compared results between the JM (XMAR) and the separately fit LMM (MAR) and found them similar; they state this could “suggest minimal to no effect from non-ignorable data.” However, they also reported that a local sensitivity analysis showed that “the current smoker variable suggested sensitivity to non-random missingness” and advised that “further analysis…is recommended.” We appreciate this conundrum and explore further analyses.

“If we have data, let’s look at data. (if all we have are opinions, let’s go with mine.)”—Jim Barksdale

The Atherosclerosis Risk in Communities (ARIC) Study²¹ generously provided data for illustration (38 086 visits from n = 13 180 participants over 32 years (1990-2022), with 382 133 total observed person-years). Briefly, we follow Stuckwisch et al. using a composite cognitive z-score (GlobalZ) as the longitudinal outcome, a baseline three-level (never, past, current) smoking-status variable as the primary exposure, and baseline adjustors including age (centered at 65-years), indicators for male sex, Black participant race, and high school education, three-level alcohol drinking status (never, past, current) and “practice effect” (number of prior cognitive assessments). Studies on cognitive decline (a broad term for memory/thinking deterioration) generally also study dementia (a severe outcome related to significant cognitive decline that substantially impairs daily life). ARIC utilizes in-person cognitive exams with dementia adjudication and has active surveillance outside of visits for death and dementia,²² both of these events censor longitudinal cognitive trajectory measurement. Appendix S1 gives study flow, variable definitions, participant characteristics, and other details from our ARIC sample. Briefly, participants classified as never smokers at baseline were more likely to have higher baseline cognitive function, were more likely to be female and alive without dementia (Tables S1-S3). Figure 1 shows simple explorations of observed cognitive trajectories (Fig. 1A-1B), dementia rates (Fig. 1C), and death rates (Fig. 1D), stratified by smoking status. As in numerous other studies, while global cognitive z-scores appear lower at baseline for current smokers, they show a potentially attenuated observed decline over time. However, overall dementia rates appear potentially higher and death rates are dramatically higher in current smokers, two main suspects of potential informative censoring.

Figure 1.

Simple explorations of (A) observed cognition with lowess smoothers, (B) simple linear cognitive trajectories, (C) dementia incidence, and (D) death incidence, all by baseline smoking status.

We note that our intentions are to illuminate our commentary aims above, and not to make substance inferences nor to necessarily replicate/compare findings; hence, while we generally follow the impressive work of Stuckwisch and colleagues, we employ some alternative modelling choices to ease exposition (such as using linear trajectories and random intercept+slope JM linking structures). Table 3 compares adjusted estimates for separate (MAR) versus JM (XMAR) approaches using LMMs for Cognition (GlobalZ), PHMs with a single composite censoring event (death and/or attrition), and linking structures, comparable in spirit to Stuckwisch. Appendix S2 gives full model descriptions, additional results and parameter interpretations. Table 3 shows (1) while the model with the most parameters (B3) had the best fit (lowest AIC,BIC), all the models give comparable estimates; (2) strong baseline associations were apparent for current-smokers (-0.15 SD lower than never-smokers), (3) never-smokers’ cognition declined at about -0.36 SDs each decade (4) steeper declines were not supported for either smoking group across these models, (5) current-smokers had about a 40% higher rate of the composite event (HR ~ 1.43), (6) participants with higher than expected cognition at baseline (represented by b0i), showed 20% lower composite event rates (HR ~ 0.83). While the JM (B3) had the best fit and the linking function estimate (ρ₀) was highly supported (indicating support for XMAR over MAR), the JM did not move the needle on the primary smoking-by-time interaction estimates of interest. Similar to the Stuckwisch results, the conundrum persists (some suggested non-ignorable missingness but no real movement in the parameters).

Table 3.

Separate (MAR) and joint (XMAR) model estimates of smoking associations with cognition and a composite censoring event (death and/or attrition) using (A) random intercepts, or (B) random intercepts plus slopes linking structures.

	(A) Random Intercepts						(B) Random Intercepts & Slopes
	Separate Models (MAR)			Joint Model (XMAR)			Separate Models (MAR)			Joint Model (XMAR)
	(A1) LMM			(A3) JM: LMM			(B1) LMM			(B3) JM: LMM
Longitudinal GlobalZ	β	CI	p	β	CI	p	β	CI	p	β	CI	p
current-smoker (baseline)	−0.15	(−0.18, −0.13)	<0.001	−0.15	(−0.18, −0.12)	<0.001	−0.15	(−0.18, −0.13)	<0.001	−0.15	(−0.18, −0.12)	<0.001
former-smoker (baseline)	0.00	(−0.03, 0.03)	0.991	0.00	(−0.02, 0.03)	0.974	0.00	(−0.03, 0.02)	0.819	0.00	(−0.03, 0.02)	0.898
time (10-years)	−0.36	(−0.37, −0.35)	<0.001	−0.36	(−0.37, −0.35)	<0.001	−0.37	(−0.38, −0.36)	<0.001	−0.38	(−0.40, −0.36)	<0.001
current-smoker x  time	0.01	(−0.01, 0.03)	0.443	0.00	(−0.02, 0.02)	0.902	0.01	(−0.01, 0.03)	0.176	−0.01	(−0.03, 0.02)	0.743
former-smoker x  time	0.00	(−0.02, 0.01)	0.702	−0.01	(−0.02, 0.01)	0.526	0.00	(−0.02, 0.01)	0.861	−0.01	(−0.02, 0.01)	0.411
RE Var: τ0 = sd(b0i)	0.51	(0.50, 0.52)	<0.001	0.51	(0.50, 0.52)	<0.001	0.52	(0.51, 0.53)		0.52	(0.51, 0.53)
RE Var: τ1 = sd(b1i)							0.22	(0.21, 0.23)		0.22	(0.21, 0.23)
RE Cov(b0i,b1i)							−0.02	(−0.02, −0.01)		−0.01	(−0.02, −0.01)
	(A2) PHM			(A3) JM: PHM			(B2) PHM			(B3) JM: PHM
Composite Event	HR	CI	p	HR	CI	p	HR	CI	p	HR	CI	p
current-smoker  (baseline)	1.42	(1.34, 1.50)	<0.001	1.43	(1.35, 1.50)	<0.001	1.42	(1.34, 1.5)	<0.001	1.43	(1.35, 1.51)	<0.001
former-smoker  (baseline)	1.03	(0.99, 1.07)	0.167	1.03	(0.99, 1.07)	0.134	1.03	(0.99, 1.07)	0.167	1.03	(0.99, 1.08)	0.119
link: ρ0∙(b0i)				0.80	(0.77, 0.84)	<0.001				0.83	(0.76, 0.90)	<0.001
link: ρ1∙(b1i)										0.59	(0.29, 1.18)	0.134
Akaike Info Criterion  (AIC)	121 211			120 982			118 118			117 884
Bayes Info Criterion  (BIC)	121 416			121 196			118 340			118 124
Primary Objectives: LMM cognitive trajectory associations with smoking – differences in trajectories vs. never-smokers
Related Objectives: PHM composite event associations with smoking – Hazard Ratios (HRs)
Linking Structures: random effect variance(s), HR associations for subject-specific cognitive intercepts & slopes

AIC, BIC: lower is better.

HR: Hazard Ratio.

Adjusted for baseline age, sex, race, education, alcohol and “practice effect”.

Heavy outlines and numbering, eg, (A1), (B3), etc., indicate individual models.

GlobalZ units are standard deviations (SD).

Never-Smokers were used as the reference group.

Appendix S2 gives additional details on methods and interpretations.

Guidelines for missing data centrally recommend exploring specific reasons for the missingness and how those specific reasons might be informative. As mentioned, differential deaths may induce survival bias, but general attrition may be less informative about cognitive decline than a specific causal agent such as dementia. In sensitivity analyses then, increasing the specificity of the missingness mechanisms by separating death and dementia from other potentially non-informative attrition sources may be useful. Fortunately, a good amount of work has gone into expanding JMs to handle multiple longitudinal outcomes and multiple events, including competing and semi-competing risk approaches.^23-25 For dementia and death events, a semi-competing risk framework (multi-state, illness-death model) is examined below, since deaths censor dementia, but dementia events do not censor death. We also examined a “marginal-hazards” approach, simply connecting the multiple events with the random effects; see Appendix S3 for details. Table 4 compares the MAR and XMAR estimates and shows (1) supported steeper cognitive declines in current compared to never smokers, (current-smoker x time), after inclusion of dementia and death censoring events using either JM approach, (2) substantial increases in the estimated HRs of dementia and death with both current and former smoking in the JMs, (3) improved AIC, BIC model fit criteria from the JMs, and (4) comparable estimates from either JM.

Table 4.

Comparison of separate (MAR) and joint (XMAR) models using dementia and death in (A) semi-competing events and (B) marginal hazards approaches, both with random intercept and slope linking structures.

	(A) Semi-Competing Risks Approach						(B) Marginal Hazards Approach
	Separate Models (MAR)			Joint Model (XMAR)			Separate Models (MAR)			Joint Model (XMAR)
	(A1) LMM			(A3) JM: LMM			(B1) LMM			(B4) JM: LMM
Longitudinal GlobalZ	β	CI	p	β	CI	p	β	CI	p	β	CI	p
current-smoker  (baseline)	−0.15	(−0.18, −0.13)	<0.001	−0.15	(−0.18, −0.12)	<0.001	−0.15	(−0.18, −0.13)	<0.001	−0.15	(−0.18, −0.12)	<0.001
former-smoker  (baseline)	0.00	(−0.03, 0.02)	0.819	0.00	(−0.03, 0.02)	0.854	0.00	(−0.03, 0.02)	0.819	0.00	(−0.03, 0.02)	0.882
time (10-years)	−0.37	(−0.38, −0.36)	<0.001	−0.43	(−0.44, −0.42)	<0.001	−0.37	(−0.38, −0.36)	<0.001	−0.44	(−0.45, −0.43)	<0.001
current-smoker x  time	0.01	(−0.01, 0.03)	0.176	−0.02	(−0.04, −0.01)	0.013	0.01	(−0.01, 0.03)	0.176	−0.03	(−0.05, −0.01)	0.006
former-smoker x  time	0.00	(−0.02, 0.01)	0.861	−0.01	(−0.03, 0)	0.102	0.00	(−0.02, 0.01)	0.861	−0.01	(−0.03, 0)	0.088
RE Var: τ0 = sd(b0i)	0.52	(0.51, 0.53)		0.50	(0.49, 0.51)		0.52	(0.51, 0.53)		0.51	(0.50, 0.52)
RE Var: τ1 = sd(b1i)	0.22	(0.21, 0.23)		0.19	(0.18, 0.21)		0.22	(0.21, 0.23)		0.22	(0.21, 0.24)
RE Cov(b0i,b1i)	−0.01	(−0.02, −0.01)		0.01	(0.01, 0.02)		−0.01	(−0.02, −0.01)		0.00	(−0.01, 0.01)
	(A2) PHM			(A3) JM: PHM			(B2) PHM			(B4) JM: PHM
Dementia Event	HR	CI	p	HR	CI	p	HR	CI	p	HR	CI	p
current-smoker  (baseline)	1.30	(1.18, 1.44)	<0.001	3.05	(2.55, 3.65)	<0.001	1.30	(1.18, 1.44)	<0.001	2.84	(2.00, 4.03)	<0.001
former-smoker  (baseline)	0.98	(0.90, 1.07)	0.678	1.18	(1.04, 1.35)	0.013	0.98	(0.90, 1.07)	0.678	1.18	(1.02, 1.36)	0.028
link: ρ0∙(b0i)				0.77	(0.69, 0.86)	<0.001				0.63	(0.57, 0.69)	<0.001
link: ρ1∙(b1i)				0.10	(0.09, 0.13)	<0.001				0.11	(0.06, 0.23)	<0.001
	(A2) PHM			(A3) JM: PHM			(B3) PHM			(B4) JM: PHM
Death Event	HR	CI	p	HR	CI	p	HR	CI	p	HR	CI	p
current-smoker  (baseline)	1.99	(1.88, 2.11)	<0.001	2.57	(2.37, 2.78)	<0.001	1.99	(1.88, 2.11)	<0.001	2.41	(2.21, 2.64)	<0.001
former-smoker  (baseline)	1.14	(1.09, 1.21)	<0.001	1.23	(1.15, 1.31)	<0.001	1.14	(1.09, 1.21)	<0.001	1.21	(1.14, 1.29)	<0.001
link: ρ0∙(b0i)				0.94	(0.89, 0.99)	0.027				0.86	(0.81, 0.91)	<0.001
link: ρ1∙(b1i)				0.39	(0.35, 0.43)	<0.001				0.45	(0.38, 0.55)	<0.001
	(A2) PHM			(A3) JM: PHM
Dementia- > Death  Event	HR	CI	p	HR	CI	p
current-smoker  (baseline)	1.18	(1.06, 1.31)	0.003	1.88	(1.60, 2.21)	<0.001
former-smoker  (baseline)	1.01	(0.92, 1.1)	0.888	1.17	(1.03, 1.33)	0.014
link: ρ0∙(b0i)				1.29	(1.15, 1.44)	<0.001
link: ρ1∙(b1i)				0.21	(0.17, 0.25)	<0.001
Akaike Info Criterion  (AIC)	110 153			104 271			104 469			101 403
Bayes Info Criterion  (BIC)	110 547			104 715			104 777			101 745
Primary Objectives: LMM cognitive trajectory associations with smoking – differences in trajectories vs. never-smokers
Related Objectives: PHM composite event associations with smoking – Hazard Ratios (HRs)
Linking Structures: random effect variance(s), HR associations for subject-specific cognitive intercepts & slopes

AIC, BIC: lower is better; here, only compare within approach (A) or approach (B), not (A)-vs-(B).

HR: Hazard Ratio.

Adjusted for baseline age, sex, race, education, alcohol and “practice effect”.

Heavy outlines indicate individual models.

GlobalZ units are standard deviations (SD).

Never-Smokers were used as the reference group.

Appendix S3 gives additional details on methods and interpretations.

Why all the different XMAR formulations? Wonderful work by Stuckwisch’s co-author Rouanet²⁶ has shown via simulation that JM estimates are generally attenuated whenever the JM submodels (1)-(3) are not exactly correct, eg, “well-specified.” While no model is ever likely to be exactly correct (“All models are wrong, but some are useful”—George Box), the additional specificity in moving from the single composite event JM to a dual event JM appears worth the added complexity, using either the semi-competing or marginal approach; AIC/BIC criterion favor the JM\XMAR models over the LMM\MAR models throughout.

The world has become so sensitive I can’t even make fun of myself without upsetting someone

Table 4 supports Stuckwisch“s and others” findings that “smoking is likely to have negative impacts on cognition”. Our goal was not to find significant estimates though, but to highlight additional avenues for missing data sensitivity analyses. The 4-level hierarchy in Table 1, (MCAR<MAR < XMAR<<<<<MNAR), suggests a sequential strategy. Figure 2 adds GEE (MCAR) estimates to the LMM and JM results from Table 4, as well as MNAR constructs under a Pattern Mixture Model (PMM) to show movement in the estimates across the missingness spectrum. Figure 2 shows potentials of additional declines for current and former smokers under additionally informative (unverifiable) MNAR assumptions, where smokers who had dementia prior to death were assumed to decline 3, 5, and 9 times faster than never smokers within six-months of a dementia diagnosis (using PMMs). We don’t necessarily believe that any of these specific steeper rates are the precise truth, instead, they are chosen to illustrate potential results from increasingly stronger, unverifiable MNAR assumptions in a sensitivity analysis. Overall, the role of a good sensitivity analyses is to explore an assumed space of potentially relevant and probabilistically possible worlds that may affect conclusions. Importantly, as shown in Figure 2B, similarity between MAR and XMAR estimates does not necessarily indicate a lack of egregious MNAR biases; “Absence of evidence is not evidence of absence”—Carl Sagan. As a side-note, Rouanet²⁶ astutely investigates population-alive versus immortal-cohort estimands and parameter interpretations for GEE, LMM and JM, providing insights on appropriateness of subject-specific parameters from JMs and making valuable recommendations on additional sensitivity analyses for population-average parameters following Dufoil.²⁷

Figure 2.

Estimates of additional cognitive decline per decade across MCAR, MAR and XMAR assumptions from GEE, LMM, joint models using dementia and death with random intercept plus slope linking structures and patter mixture models (PMM) examining sensitivity to assumptions of additional (unobserved) declines in (A) current smokers and (B) former smokers who were censored by dementia events. Current smoker estimates show the most sensitivity to potential informative missingness effects from death and dementia.

We congratulate Stuckwish and colleagues on their informative work and for stimulating readers to dig deeper. In doing so we learned that in the ARIC data, current smokers appear to have steeper cognitive declines, with estimated additional steepness being related to informative dementia and death events (Table 2, JM\XMAR); importantly, relatively similar conclusions are found even if there is further unmeasured informative missingness up to 3 times as steep as observed declines (Figure 2A, JM\XMAR vs. PMM). Former smokers have less supported additional declines, unless there exist stronger informative missingness structures than the included XMAR dementia and death components cover (Figure 2B, PMM results).

Good statistics is about uncertainty, not certainty. Embrace the unknown—John Tukey

Sensitivity analyses for missing data necessarily embraces the unknown. We’ve sought to clarify the position of XMAR assumptions in the updated four-level missingness hierarchy and illuminate strategies for pursuing sensitivity analyses. We ardently agree that JMs are powerful tools in the modern analyst’s arsenal and show that JMs come in a wide variety of shapes, sizes and assumptions. Examining multiple JM formulations by varying event types and linking structures can illuminate the evidence base. Nicely, JMs operate under an XMAR assumption that missingness is latently ignorable after including information from additional observed events outside of the outcome trajectories, enabling JM\XMAR models to be based on extended measured data and compared directly to LMM\MAR models using standard methods to determine support for XMAR versus MAR assumptions in terms of model fit. MNAR models are based on higher degrees of unverifiable assumptions which are untestable. When XMAR results reveal useful inferences (such as for cognitive declines among current smokers above), additional MNAR investigations may be less necessary, though they might still be examined. However, even when MCAR, MAR, and XMAR results are all similar, missingness mechanisms may still be nonignorable, and additional MNAR sensitivities may need to be pursued (such as for cognitive declines among former smokers above). Importantly, for any study, the more information collected concerning the reasons for missingness, the better one can understand and compensate for potential missingness effects.

Howard Wainer said: “The best that can be managed is to estimate the extent to which missing data have influenced the inferences we wish to draw.” We hope that our commentary has shed additional light on the usefulness of XMAR assumptions, and on how JMs can be used to help assess missing data influences on inferences through informative, well-rounded sensitivity analyses. Links to example code and complementary synthesized data are provided in supplementary materials.

Supplementary material

Supplementary material is available at the American Journal of Epidemiology online.

Conflicts of interest

None declared.

Funding

The Atherosclerosis Risk in Communities Study (ARIC) is supported by National Heart, Lung, and Blood Institute contracts (75N92022D00001, 75N92022D00002, 75N92022D00003, 75N92022D00004, 75N92022D00005). The ARIC Neurocognitive Study is supported by grants U01HL096812, U01HL096814, U01HL096899, U01HL096902, and U01HL096917 from the NIH (NHLBI, NINDS, NIA and NIDCD). The authors thank the investigators, staff and participants of the ARIC study for their important contributions.

Data availability

ARIC data can be made available to interested researchers through established study protocols (https://sites.cscc.unc.edu/aric/). Example code and complementary synthetic data are available at https://github.com/MichaelGriswold/XMAR-JMs.