Principled Approaches to Missing Data in Epidemiologic Studies

Principled Approaches to Missing Data in Epidemiologic Studies Abstract Principled methods with which to appropriately analyze missing data have long existed; however, broad implementation of these methods remains challenging. In this and 2 companion papers (Am J Epidemiol. 2018;187(3):576–584 and Am J Epidemiol. 2018;187(3):585–591), we discuss issues pertaining to missing data in the epidemiologic literature. We provide details regarding missing-data mechanisms and nomenclature and encourage the conduct of principled analyses through a detailed comparison of multiple imputation and inverse probability weighting. Data from the Collaborative Perinatal Project, a multisite US study conducted from 1959 to 1974, are used to create a masked data-analytical challenge with missing data induced by known mechanisms. We illustrate the deleterious effects of missing data with naive methods and show how principled methods can sometimes mitigate such effects. For example, when data were missing at random, naive methods showed a spurious protective effect of smoking on the risk of spontaneous abortion (odds ratio (OR) = 0.43, 95% confidence interval (CI): 0.19, 0.93), while implementation of principled methods multiple imputation (OR = 1.30, 95% CI: 0.95, 1.77) or augmented inverse probability weighting (OR = 1.40, 95% CI: 1.00, 1.97) provided estimates closer to the “true” full-data effect (OR = 1.31, 95% CI: 1.05, 1.64). We call for greater acknowledgement of and attention to missing data and for the broad use of principled missing-data methods in epidemiologic research. bias (epidemiology), complete-case analysis, inverse probability weighting, missing data, multiple imputation Missing data are a pervasive challenge in biomedical research (1). For example, of 262 studies published in the 2010 volumes of the American Journal of Epidemiology, Epidemiology, and the International Journal of Epidemiology, the amount of missing data was not sufficiently reported to be quantified by reviewers in 68% (2). When quantifiable, the extent of missing data can be substantial. For example, Eekhout et al. (2) reported a prevalence of 26% missing data in 84 epidemiologic studies. Although missing data are nearly ubiquitous in epidemiologic research, the impact of missing data on inference can vary greatly. Despite a rich body of literature on statistical methods for the analysis of missing data, the most widely used technique in epidemiology remains the most basic: “complete-case” analysis (2–10), which ignores potentially valuable observed information by excluding participants with only partially available data on the variables of interest (11, 12). To demonstrate the impact of missing data and illustrate principled methods that account for missingness, we constructed a challenge with 2 independent teams of missing-data experts. The goal of their work was to illustrate the use of methods of their choice to analyze epidemiologic studies with missing data and to evaluate and relax (to the extent possible) the assumptions made about missing data. The motivating example is taken from the Collaborative Perinatal Project (CPP) (13), a prototypical epidemiologic study, and focuses on estimating the association of smoking during pregnancy with risk of spontaneous abortion. Example data sets were created from a completely observed subset of the CPP data, where missingness was introduced using 3 different mechanisms. Two teams then analyzed the example data sets, one using multiple imputation (MI) to account for the missing data (14) and the other using inverse probability weighting (IPW) (15). While in practice researchers may have substantive information regarding the missingness process, the teams here were blinded to the mechanisms that generated the missing data, as well as to the full data from which the 3 incomplete data sets were drawn. In this paper, we review existing nomenclature for missing-data mechanisms and introduce the CPP, along with the series of 3 derived data sets with missingness. We close by revealing the underlying true missing-data-generating mechanisms, summarizing the teams’ findings in the context of the unmasked missing-data mechanisms, and discussing best practices and future directions. In 2 companion papers in this issue of the Journal, each team describes, in turn, the application of a principled approach—one parametric (14) and one semiparametric (15)—to account for the missing data. TYPES OF MISSING DATA Throughout this paper we focus on explicit missing data, characterized by missing values for the exposure, outcome, or covariates in an analytical data set. Observational studies are particularly prone to such missing data, because even in the rare settings where there are no missing data for the exposure and outcome, there are invariably missing values for the covariates necessary to obtain adjusted estimates. Data may be categorized as missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) (16, 17). Data are MCAR when the probability of having a variable with missing data does not depend on any observed or missing variables. Missing data are MAR if the probability that a given subset of variables (i.e., a “pattern”) is observed depends only on the values of observed variables. Data are MNAR if the missingness pattern depends on the values of unobserved variables. MCAR is the strongest assumption, and it is unrealistic in typical epidemiologic studies. MAR is a weaker assumption, and it is generally more likely than MCAR to hold in epidemiologic studies, while MNAR is the weakest assumption of the three. Using observed data, one can test the MCAR assumption by empirically refuting it. Given observed data alone, however, the MAR and MNAR mechanisms are indistinguishable. While complete-case analysis of MCAR data will generally yield asymptotically unbiased (henceforth unbiased) estimates, efficiency is often lost because this technique discards information on incomplete cases. If data are MCAR, application of the principled methods in the companion papers (14, 15) will also yield unbiased estimates, but it can also improve efficiency by recovering information from incomplete cases. A complete-case analysis of MAR data may or may not yield biased results. When a complete-case analysis of MAR data yields valid inference, such inference can be inefficient because data on incomplete cases are discarded. However, the principled methods applied and discussed in the companion papers can provide unbiased results and recover efficiency when data are MAR, as demonstrated later in this paper. MNAR cannot be ruled out empirically and typically implies that aspects of the missing-data mechanism need to be properly modeled to identify the parameter of interest. Intuitively, the observed data are of little help, because the information necessary to model the missing-data mechanism is unobserved by definition (18, 19). Therefore, modeling missing data that are MNAR often requires additional assumptions either in the form of external information or fairly strong parametric assumptions when such information is not available (20, 21). Other approaches to handling MNAR data include conducting a sensitivity analysis, which in extreme scenarios will produce bounds for the impact of missing data. If data are MNAR, then both complete-case analysis and the principled methods discussed in the companion papers may provide biased estimates. Notably, MNAR does not imply or guarantee biased estimates, which depends on the particular causal structure; it only guarantees that we cannot correct for it if present (22). Finally, ignorability is a criterion that is often used interchangeably with MAR but requires that the missing data are MAR (or MCAR) and that the parameters governing the missing-data mechanism are distinct from the parameters governing the full-data model (16, 23). Little and Zanganeh (24) have provided a variety of examples where a data set is MNAR and how ignorability can arise and be used in a subset to perform valid inference on that subset of data. Distinct parameters are those for which, in the likelihood setting, the domain of the parameters is the Cartesian product of their individual domains (values taken by one parameter do not restrict values taken by another parameter); and in the Bayesian setting, distinct parameters require independent priors for the parameters. Ignorability (e.g., MAR with distinct parameters) is the weakest general assumption that allows the likelihood to factor such that one can identify the parameter of interest. In settings where ignorability does not appear to hold, methods for nonignorable models (e.g., pattern mixture models) can be useful (25). MOTIVATING EXAMPLE Data for this challenge comprised information on a subsample of participants from the CPP. The CPP was a multisite US study of pregnancy and pediatric outcomes conducted from 1959 to 1974 (13). The CPP investigators recruited and enrolled 48,197 participants who were seeking prenatal care. Data on demographic factors and medical history were collected at entry into the study. For illustrative purposes, we selected 11,373 women entering the cohort prior to 20 weeks’ gestation who had complete data on the variables birth outcome (spontaneous abortion (<20 weeks’ gestation) or live birth), maternal smoking, maternal age (years), maternal race, and maternal body mass index (BMI; weight (kg)/height (m)2). Spontaneous abortion or live birth and smoking status were binary variables; race was categorized as white, black, or other; and age and BMI were continuous. This subsample is referred to as the “full” data set. The characteristics of the full data set are displayed in Table 1, overall and by birth outcome, with 411 spontaneous abortions and 10,962 live births. Table 1. Distribution of Maternal Characteristics in the Full Subset of Data From the Collaborative Perinatal Project, 1959–1974 Variable  Birth Outcome  Overall (n = 11,373)  Live Birth (n = 10,962)  Spontaneous Abortion (n = 411)  No. of Women  %  No. of Women  %  No. of Women  %  Smoking               No  8,458  74.4  8,169  74.5  289  70.3   Yes  2,915  25.6  2,793  25.5  122  29.7  Race               White  7,192  63.2  6,955  63.5  237  57.7   Black  3,475  30.6  3,333  30.4  142  34.5   Other  706  6.2  674  6.1  32  7.8  Age, yearsa  25.2 (5.9)  25.1 (5.8)  27.2 (6.9)  Body mass indexa,b  22.5 (4.1)  22.5 (4.1)  22.8 (4.1)  Variable  Birth Outcome  Overall (n = 11,373)  Live Birth (n = 10,962)  Spontaneous Abortion (n = 411)  No. of Women  %  No. of Women  %  No. of Women  %  Smoking               No  8,458  74.4  8,169  74.5  289  70.3   Yes  2,915  25.6  2,793  25.5  122  29.7  Race               White  7,192  63.2  6,955  63.5  237  57.7   Black  3,475  30.6  3,333  30.4  142  34.5   Other  706  6.2  674  6.1  32  7.8  Age, yearsa  25.2 (5.9)  25.1 (5.8)  27.2 (6.9)  Body mass indexa,b  22.5 (4.1)  22.5 (4.1)  22.8 (4.1)  a Values are expressed as mean (standard deviation). b Weight (kg)/height (m)2. The missing-data analytical teams were asked to estimate the relationship between smoking exposure measured during early pregnancy and the risk of spontaneous abortion (i.e., <20 weeks’ gestation), adjusted for race, age, and BMI. Linear logistic regression was employed to estimate the odds ratio quantifying this relationship between smoking and spontaneous abortion. Missing data can occur in a variety of ways. Therefore, to broaden our demonstration, we applied 3 missing-data mechanisms. From the full data, we constructed 3 data sets with missing data for the variables spontaneous abortion, smoking, and BMI, for 8 possible missing-data patterns. The patterns of missingness for each data set are shown in Table 2. We attempted to hold the proportion and pattern of missing data approximately constant across the 3 data sets, with some departures as seen in Table 2. One constructed data set had missing data generated under MCAR, one under MAR, and one under MNAR. The parameters governing the missing-data mechanism (see the Web Appendix, available at https://academic.oup.com/aje) were distinct from the parameters governing the substantive model, implying ignorability in the MCAR and MAR data sets. The characteristics of the data sets, arbitrarily numbered as 1, 2, and 3, are displayed in Table 3. Table 2. Mechanism-Specific Missing-Data Patterns and Percentages Induced in the Full Subset of Data From the Collaborative Perinatal Project, 1959–1974 Missing-Data Patterna  Missing Pattern Percentage  Pattern  SA  Smoking  Black Race  Other Race  Age  BMIb  Fixed  Data Set  1 (MAR)  2 (MCAR)  3 (MNAR)  1  X  X  X  X  X  X  60  61.09  62.32  52.86  2  X  X  X  X  X  M  5  5.32  5.17  5.33  3  X  M  X  X  X  X  5  5.48  5.25  7.16  4  X  M  X  X  X  M  5  5.20  4.95  7.59  5  M  X  X  X  X  X  1  1.15  0.92  1.16  6  M  X  X  X  X  M  10  11.07  10.56  11.25  7  M  M  X  X  X  X  10  9.82  9.84  13.80  8  M  M  X  X  X  M  1  0.87  0.98  0.86  Missing-Data Patterna  Missing Pattern Percentage  Pattern  SA  Smoking  Black Race  Other Race  Age  BMIb  Fixed  Data Set  1 (MAR)  2 (MCAR)  3 (MNAR)  1  X  X  X  X  X  X  60  61.09  62.32  52.86  2  X  X  X  X  X  M  5  5.32  5.17  5.33  3  X  M  X  X  X  X  5  5.48  5.25  7.16  4  X  M  X  X  X  M  5  5.20  4.95  7.59  5  M  X  X  X  X  X  1  1.15  0.92  1.16  6  M  X  X  X  X  M  10  11.07  10.56  11.25  7  M  M  X  X  X  X  10  9.82  9.84  13.80  8  M  M  X  X  X  M  1  0.87  0.98  0.86  Abbreviations: BMI, body mass index; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random; SA, spontaneous abortion. a “X” denotes that data are observed in that pattern, and “M” denotes that data are missing in that pattern. b Weight (kg)/height (m)2. Table 3. Characteristics of 3 Observed Data Sets With Constructed Missing Data From the Full Subset of Data, Collaborative Perinatal Project, 1959–1974 Data Set and Variablea  Birth Outcome  Overall  Live Birth  Spontaneous Abortion  No. of Women  %  No. of Women  %  No. of Women  %  Data set 1 (MAR)               Total  8,767    8,464    303     Smoking                No  6,670  74.6  5,828  78.6  108  76.6    Yes  2,273  25.4  1,584  21.4  33  23.4   Race                White  7,192  63.2  5,308  62.7  176  58.1    Black  3,475  30.6  2,627  31.0  104  34.3    Other  706  6.2  529  6.3  23  7.6   Age, yearsb  25.1 (5.8)  25 (5.8)  26.9 (6.8)   BMIc  22.5 (4.0)  22.5 (4.0)  23 (4.5)  Data set 2 (MCAR)               Total  8,836    8,517    319     Smoking                No  6,684  74.4  5,500  74.4  192  68.6    Yes  2,298  25.6  1,896  25.6  88  31.4   Race                White  7,192  63.2  5,416  63.6  184  57.7    Black  3,475  30.6  2,584  30.3  113  35.4    Other  706  6.2  517  6.1  22  6.9   Age, years  25.2 (5.8)  25.1 (5.8)  27.5 (6.9)   BMI  22.6 (4.1)  22.6 (4.1)  22.9 (4.2)  Data set 3 (MNAR)               Total  8,295    8,033    262     Smoking                No  5,961  74.2  5,111  79.1  121  76.1    Yes  2,068  25.8  1,348  20.9  38  23.9   Race                White  7,192  63.2  5,032  62.6  153  58.4    Black  3,475  30.6  2,480  30.9  88  33.6    Other  706  6.2  521  6.5  21  8.0   Age, years  25.2 (5.9)  25.1 (5.8)  27.2 (7.1)   BMI  22.5 (4.1)  22.5 (4.1)  22.8 (3.9)  Data Set and Variablea  Birth Outcome  Overall  Live Birth  Spontaneous Abortion  No. of Women  %  No. of Women  %  No. of Women  %  Data set 1 (MAR)               Total  8,767    8,464    303     Smoking                No  6,670  74.6  5,828  78.6  108  76.6    Yes  2,273  25.4  1,584  21.4  33  23.4   Race                White  7,192  63.2  5,308  62.7  176  58.1    Black  3,475  30.6  2,627  31.0  104  34.3    Other  706  6.2  529  6.3  23  7.6   Age, yearsb  25.1 (5.8)  25 (5.8)  26.9 (6.8)   BMIc  22.5 (4.0)  22.5 (4.0)  23 (4.5)  Data set 2 (MCAR)               Total  8,836    8,517    319     Smoking                No  6,684  74.4  5,500  74.4  192  68.6    Yes  2,298  25.6  1,896  25.6  88  31.4   Race                White  7,192  63.2  5,416  63.6  184  57.7    Black  3,475  30.6  2,584  30.3  113  35.4    Other  706  6.2  517  6.1  22  6.9   Age, years  25.2 (5.8)  25.1 (5.8)  27.5 (6.9)   BMI  22.6 (4.1)  22.6 (4.1)  22.9 (4.2)  Data set 3 (MNAR)               Total  8,295    8,033    262     Smoking                No  5,961  74.2  5,111  79.1  121  76.1    Yes  2,068  25.8  1,348  20.9  38  23.9   Race                White  7,192  63.2  5,032  62.6  153  58.4    Black  3,475  30.6  2,480  30.9  88  33.6    Other  706  6.2  521  6.5  21  8.0   Age, years  25.2 (5.9)  25.1 (5.8)  27.2 (7.1)   BMI  22.5 (4.1)  22.5 (4.1)  22.8 (3.9)  Abbreviations: BMI, body mass index; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random. a Subgroup counts for smoking and race do not sum to the totals because of missingness patterns. b Values for age and BMI are expressed as mean (standard deviation). c Weight (kg)/height (m)2. Missing data were generated under the MAR, MCAR, and MNAR missingness mechanisms for data sets 1, 2, and 3, respectively. For all 3 data sets, missing data for the variables BMI, smoking, and spontaneous abortion were generated using the following multinomial model:   P(R=r|L)=exp(α0r+βr'V+γr'L(r)+ηr'W(r)+θrU)1+∑k=22Kexp(α0k+βk'V+γk'L(k)+ηk'W(k)+θkU),where K=3 denotes the number of variables with missingness and R denotes the missing-data pattern with possible patterns r=1,…,2K. Explicitly, r=1 was the pattern of complete data and r=8 was the pattern with missing data on BMI, smoking, and spontaneous abortion. L is the set of measured variables. The set V denotes the subset of L that is always observed; in this example, V = (age, race). The set L(r) is the subset of L observed under pattern r but missing in at least one pattern (else the variable would be found in V), while W(r) is the complement of L(r), or the subset of L not observed under pattern r. For example, in pattern r=7, L(r) is the observed BMI and W(r) is the missing data on smoking and spontaneous abortion. Finally, U is an unmeasured variable. For data set 1, MAR was created by setting βr≠0,γr≠0, and ηr=θr=0, so that the missingness pattern depended on the variables that were always observed (age and race) and the observed values of spontaneous abortion, BMI, and smoking. For data set 2, MCAR was achieved by setting βr=γr=ηr=θr=0, so that the missingness pattern was a function of a constant and thus was completely random. For data set 3, βr≠0,γr≠0, and ηr≠0 or θr≠0, corresponding to an MNAR mechanism because the set of unobserved values (W(r)) as well as a completely unobserved variable (U) defined the missingness mechanism. See the Web Appendix for detailed SAS code (SAS Institute, Inc., Cary, North Carolina), as well as the parameter values used to induce missingness in the 3 data sets. Notably, the MNAR mechanism of data set 3 was introduced via W(r) (Table 2, pattern 6, specifically) or U (Table 2, patterns 3, 4, and 7) but could be introduced via W(r) and U simultaneously. Across the 3 data sets, the intercepts α0r were chosen to maintain approximately 60% complete data (see Table 2). The analysis teams were provided with the 3 observed data sets, with instructions to estimate the association of smoking with risk of spontaneous abortion, adjusted for the list of potential confounders provided. The teams were not given any indication or instructions regarding the role or use of these variables in the missingness mechanisms or in their analysis, respectively. The analysts applied the methodology they deemed appropriate. Again, the teams were masked to the underlying missing-data mechanisms for each observed data set and did not have access to the full data set. SUMMARY OF FINDINGS In each of the 3 data sets, both teams conducted complete-case analysis as well as used a principled method (i.e., MI or IPW) to estimate the association of smoking with spontaneous abortion (14, 15). The goal in conducting a complete-case analysis was to triangulate results and highlight potential pitfalls of this common technique, particularly in comparison with the principled methods. In the corresponding companion papers (14, 15), the analytical teams closely mimicked the real-world application of these missing-data methods because the underlying mechanism used to create the missing data is rarely known in practice and was not known to the investigators in this exercise. The results from the analyses are unmasked here and consolidated in Table 4, along with the (“true”) results from the full data. The “Full Data” column represents the association of smoking with spontaneous abortion after adjustment for race, age, and BMI in the complete subsample from the CPP cohort, showing an odds ratio of 1.31 (95% confidence interval: 1.05, 1.64) for all 3 data sets. The rows of Table 4 correspond to the data sets in which a given missing-data mechanism was imposed. Results are provided for complete-case analyses, MI, and augmented IPW, along with the results from the full data. Of course, some variation in the point estimates will be expected even for unbiased scenarios due to sampling variability, as these are single realizations of the data. Table 4. Estimated Odds Ratiosa for the Association Between Smoking and Spontaneous Abortion According to Data Analysis Approach, Collaborative Perinatal Project, 1959–1965 Missing-Data Process  Approach  Full Data  Complete-Case  Multiple Imputation  Augmented IPW  OR  95% CI  OR  95% CI  OR  95% CI  OR  95% CI  MCAR (data set 2)  1.31  1.05, 1.64  1.39  1.06, 1.82  1.36  1.05, 1.77  1.42  1.09, 1.86  MAR (data set 1)  1.31  1.05, 1.64  0.43  0.19, 0.93  1.30  0.95, 1.77  1.40  1.00, 1.97  MNAR (data set 3)  1.31  1.05, 1.64  0.93  0.55, 1.57  1.14  0.81, 1.60  1.57  1.09, 2.23  Missing-Data Process  Approach  Full Data  Complete-Case  Multiple Imputation  Augmented IPW  OR  95% CI  OR  95% CI  OR  95% CI  OR  95% CI  MCAR (data set 2)  1.31  1.05, 1.64  1.39  1.06, 1.82  1.36  1.05, 1.77  1.42  1.09, 1.86  MAR (data set 1)  1.31  1.05, 1.64  0.43  0.19, 0.93  1.30  0.95, 1.77  1.40  1.00, 1.97  MNAR (data set 3)  1.31  1.05, 1.64  0.93  0.55, 1.57  1.14  0.81, 1.60  1.57  1.09, 2.23  Abbreviations: CI, confidence interval; IPW, inverse probability weighting; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random; OR, odds ratio. a Adjusted for age, race, and body mass index. For MCAR (i.e., data set 2 in the companion papers), complete-case analyses and use of both principled methods resulted in similar point estimates and confidence intervals. For this realization of the data, all 3 methods estimated odds ratios that were close to those from the full data, with slightly wider confidence intervals, reflecting the loss of information due to the missing data, essentially a consequence of a reduced sample size. For the MAR mechanism (i.e., data set 1 in the companion papers), the complete-case analysis estimate of 0.43 shows a notable and spurious protective effect of smoking on spontaneous abortion. This effect reversal is primarily due to the violated assumption that the missingness is MCAR, when in fact it is a function of the observed variables. In some settings, complete-case analysis can be valid under MAR and even MNAR—for example, if the missingness process depends only on covariates in the regression, even if some of them are not fully observed (26). However, this assumption is not met in our example, which serves to highlight the potential consequences of applying complete-case analysis and subsequently failing to address the impact of the missing data. On the other hand, employing either MI or augmented IPW resulted in point estimates of 1.30 and 1.40, respectively, with similar confidence intervals. Both estimates showed a drastic shift from the (naive) complete-case analyses estimate toward the effect observed in the full data. For the MNAR case (i.e., data set 3 in the companion papers), complete-case analyses resulted in an estimated odds ratio of 0.93 (95% confidence interval: 0.55, 1.57). While both MI and IPW resulted in point estimates closer to the full-data effect estimate, such a finding cannot be expected generally. PRINCIPLED METHODS: PROS, CONS, AND ASSUMPTIONS In general, standard complete-case analyses rely on an assumption that the missing data are MCAR or an equivalent assumption to yield unbiased estimates (26). This assumption may often be unrealistic in epidemiologic settings, in which complete-case analyses will likely result in bias (e.g., data set 2). Furthermore, even if data are MCAR, such a complete-case analysis will typically be inefficient, as it ignores valuable information in incomplete cases, which can be particularly deleterious when a necessary covariate is the predominantly missing variable. When data are MAR, both MI and IPW may still return unbiased estimates when appropriate assumptions are met. In particular, standard MI and IPW as implemented in the companion papers (14, 15) rely on specific modeling assumptions beyond assuming an ignorable nonresponse process (27). MI is formally a Bayesian approach which, as implemented in the companion paper by Harel et al. (14), assumes that the parameters indexing models of interest have a normal prior distribution. The imputation model and the analysis model are also assumed to be correctly specified, which includes correct specification of the conditional distribution of incomplete variables given the observed variables. This also implies that the imputation model for the distribution of covariates given the outcome is compatible with the underlying model of substantive interest for the density of the outcome given covariates. This is guaranteed to occur when the joint distribution of the covariates and outcome is joint normal and for certain model choices within the natural exponential family, but not in general. There are different types of imputation models; some require parametric assumptions (e.g., joint normal distribution in MI) and some do not (e.g., hot-deck). While misspecified models should not be expected to produce unbiased results, simulations have shown that MI is somewhat robust to the choice of imputation model for moderate rates of missing information. All MI implemented by Harel et al. in the companion paper (14) used a “proper” parametric imputation model, which respects the distributional properties of the imputation draws, typically by drawing values of parameters before drawing imputations (12, 28). The goal of MI is not to estimate the missing values themselves but rather to produce unbiased and efficient estimates for the population parameters of interest, by essentially averaging over the (unknown) distribution of the missing data. Thus, a joint normal model might be an appropriate choice, even for nonnormal variables, because it is more important to preserve the relationship between the variables than to impute feasible values for all of the missing variables. In addition, the purpose of the imputation model in MI is to predict, not establish causality, and therefore it does not need to preserve temporality. Rather, all available and relevant data should be used to impute or model the missingness mechanism, regardless of its temporal relationship to the exposure or outcome. For instance, there may be observed variables that are highly predictive of the missing variables or the missingness mechanism but do not enter into the analysis model, either because they are not predictive of the outcome or because they are not confounders. These auxiliary variables are beneficial and easy to use in MI and are considered one of the advantages of the method. In this exercise, only variables chosen a priori to be included in the analysis model were available, implying that the variable sets for the imputation and analysis models were the same. In contrast to MI, IPW assumes that a model for the nonresponse process given the observed data is correctly specified. Because the model for the missing-data process does not restrict the model of substantive interest and vice versa, IPW (and augmented IPW) is not limited by a compatibility of conditional densities under the MAR assumption. However, IPW requires that for all possible realizations of the full data, there is a nonzero probability of observing a person with complete data (i.e., the positivity assumption) (29). IPW as implemented here relies on large-sample theory for valid inference. In practice, MI-based inference also often relies on large-sample inference (30–32). However, when proper imputation is applied under a strict Bayesian framework (implying no P values or confidence intervals), MI can in principle be applied in small samples as well, so long as additional assumptions concerning the fully conditional distribution hold. In actuality, MI and IPW make somewhat complementary modeling assumptions, as they rely on parametric models for distinct components of the joint likelihood for the full data and the missingness process. As stated above, MI relies on a model for the underlying full data but allows the missingness mechanism to remain completely unrestricted. Conversely, IPW models the missingness mechanism, but the full-data model is unrestricted beyond the model of substantive interest. As with a complete-case analysis, IPW does not efficiently use information available in incomplete cases, although these data are used to estimate the nonresponse rates, therefore recovering some information from incomplete cases. This issue does not arise with MI because available information among incomplete cases serves as a basis for imputing missing information. In the other companion paper, Sun et al. (15) considered an augmented IPW, an extension of IPW attributable to Robins et al. (29) and recently implemented for nonmonotone MAR patterns by Sun and Tchetgen Tchetgen (33), which largely resolves the efficiency limitation of standard IPW by allowing the analyst to recover information in incomplete cases. DISCUSSION Based on the above considerations, complete-case analysis should be used with the same caution we ascribe to unadjusted estimates, as its validity relies on strong, often unrealistic assumptions. In contrast, principled methods such as MI or IPW may account for bias due to missing data under weaker assumptions. The most standard application of these methods relies on the MAR assumption. In the absence of model misspecification, we expect that MI will be more efficient than IPW, because the latter assumes that the full-data model is completely unrestricted other than by the model of substantive interest, while the former uses a restricted parametric formulation for the full data (29, 30). While IPW fails to efficiently recover all available information from incomplete cases, augmented IPW recovers such information, at least the portion recoverable without relying on a finite-dimension full-data model. Now, given that we wish to use principled methods, which are we to choose? We can choose a version of MI, a version of IPW, or another formal approach not considered in this set of articles (e.g., direct maximum likelihood, Bayesian analysis) that equally applies under MAR. We believe foremost that we should prioritize a consistent estimator under ignorable MAR settings. Then, to the extent possible, we should minimize variance, although reasonable tradeoffs in allowing some bias in exchange for increased precision may be prudent (e.g., to minimize mean squared error). We remain without consensus as to the extent to which flexible parametric models may be overly restrictive and hide some amount of estimation error, but we do have consensus that low-dimension parametric models can be overly restrictive and hinder our ability to see the world more clearly: In such cases, in our results we might see more of our assumptions than we see of our world (1). Indeed, nuisance models required for controlling selection bias (e.g., the imputation model, the missing-data mechanism model) do not need to return interpretable finite-dimension parameters, and therefore they might be restricted only by the requirement to achieve a consistent estimator of the parameter(s) of interest from the substantive model. This leaves us the ability to pick from several candidate methods. Some of these candidate methods have nonoverlapping assumptions, as mentioned above. Therefore, there should be advantages to conducting more than 1 analysis and comparing results, as we have done with IPW and MI. When results with nonoverlapping assumptions agree, then our confidence in the results should be higher—emboldened, but never certain, because even nonoverlapping assumptions may jointly fail. When results disagree, we should pause and reconsider our assumptions and methods. Current research on missing data is producing more flexible procedures, such as doubly robust estimators, that combine a model for the full data with a model for the missing-data process, such that only 1 of the models need be correctly specified to produce unbiased inferences. While the advent of accessible software packages for these methods may be on the horizon, moving away from complete-case analysis towards a principled method like MI or IPW as demonstrated in the companion papers (14, 15) is vital to ensuring proper analysis of current epidemiologic studies. When analyzing observational data, an epidemiologist presented with a crude odds ratio of 0.43 (analogous to the complete-case analysis estimate) and an adjusted odds ratio of 1.35 (analogous to MI or IPW estimate) might posit that confounding bias is the culprit and report both, while interpreting the adjusted result as probably closer to the underlying truth (barring collider-stratification bias and assuming a rare outcome such that the odds ratio is collapsible). Reporting only the results from a complete-case analysis in the presence of nontrivial missing data is commensurate with reporting only crude associations from nonrandomized studies. We therefore advocate that the same reasoning be followed when dealing with missing data. If we are principled, we are more likely to get closer to the truth. We urge researchers, reviewers, and journal editors to think about the missing-data problem prior to making decisions about a plan of action. We would be wise to plan for missing values, minimize nonresponse, determine the missing-data assumptions, and report them appropriately. Of course, there may be reasonable extenuating circumstances that support reporting a crude association, or an equivalent rationale for reporting results from a complete-case analysis. But we surmise that such cases are rare. In closing, we ask that researchers join us in reevaluating the context of our work, to think more carefully about the assumptions that underlie the claims we make. ACKNOWLEDGMENTS Author affiliations: Division of Intramural Population Health Research, Eunice Kennedy Shriver National Institute of Child Health and Human Development, Rockville, Maryland (Neil J. Perkins, Enrique F. Schisterman); Department of Epidemiology, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina (Stephen R. Cole); Department of Statistics, College of Liberal Arts and Sciences, University of Connecticut, Storrs, Connecticut (Ofer Harel); Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, Massachusetts (Eric J. Tchetgen Tchetgen, BaoLuo Sun); and Agency for Healthcare Research and Quality, Rockville, Maryland (Emily M. Mitchell). This research was partially supported by the Long-Range Research Initiative of the American Chemistry Council (Washington, DC) and the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health. This work was also partially supported by award K01MH087219 from the National Institute of Mental Health. The content of this article is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Mental Health or the National Institutes of Health. Conflict of interest: none declared. Abbreviations BMI body mass index CPP Collaborative Perinatal Project IPW inverse probability weighting MAR missing at random MCAR missing completely at random MI multiple imputation MNAR missing not at random REFERENCES 1 Little RJ, D’Agostino R, Cohen ML, et al.  . The prevention and treatment of missing data in clinical trials. N Engl J Med . 2012; 367( 14): 1355– 1360. Google Scholar CrossRef Search ADS PubMed  2 Eekhout I, de Boer RM, Twisk JW, et al.  . Missing data: a systematic review of how they are reported and handled. Epidemiology . 2012; 23( 5): 729– 732. Google Scholar CrossRef Search ADS PubMed  3 Harel O, Boyko J. Mi??ing data: should we c?re? Am J Public Health . 2013; 103( 2): 200– 201. Google Scholar CrossRef Search ADS PubMed  4 Klebanoff MA, Cole SR. Use of multiple imputation in the epidemiologic literature. Am J Epidemiol . 2008; 168( 4): 355– 357. Google Scholar CrossRef Search ADS PubMed  5 Sterne JA, White IR, Carlin JB, et al.  . Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls. BMJ . 2009; 338: b2393. Google Scholar CrossRef Search ADS PubMed  6 Stuart EA, Azur M, Frangakis C, et al.  . Multiple imputation with large data sets: a case study of the Children’s Mental Health Initiative. Am J Epidemiol . 2009; 169( 9): 1133– 1139. 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On inverse probability weighting for nonmonotone missing at random data. arXiv.org . https://arxiv.org/abs/1411.5310. Accessed December 15, 2015. Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of Public Health 2018. This work is written by (a) US Government employee(s) and is in the public domain in the US. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png American Journal of Epidemiology Oxford University Press

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Abstract

Abstract Principled methods with which to appropriately analyze missing data have long existed; however, broad implementation of these methods remains challenging. In this and 2 companion papers (Am J Epidemiol. 2018;187(3):576–584 and Am J Epidemiol. 2018;187(3):585–591), we discuss issues pertaining to missing data in the epidemiologic literature. We provide details regarding missing-data mechanisms and nomenclature and encourage the conduct of principled analyses through a detailed comparison of multiple imputation and inverse probability weighting. Data from the Collaborative Perinatal Project, a multisite US study conducted from 1959 to 1974, are used to create a masked data-analytical challenge with missing data induced by known mechanisms. We illustrate the deleterious effects of missing data with naive methods and show how principled methods can sometimes mitigate such effects. For example, when data were missing at random, naive methods showed a spurious protective effect of smoking on the risk of spontaneous abortion (odds ratio (OR) = 0.43, 95% confidence interval (CI): 0.19, 0.93), while implementation of principled methods multiple imputation (OR = 1.30, 95% CI: 0.95, 1.77) or augmented inverse probability weighting (OR = 1.40, 95% CI: 1.00, 1.97) provided estimates closer to the “true” full-data effect (OR = 1.31, 95% CI: 1.05, 1.64). We call for greater acknowledgement of and attention to missing data and for the broad use of principled missing-data methods in epidemiologic research. bias (epidemiology), complete-case analysis, inverse probability weighting, missing data, multiple imputation Missing data are a pervasive challenge in biomedical research (1). For example, of 262 studies published in the 2010 volumes of the American Journal of Epidemiology, Epidemiology, and the International Journal of Epidemiology, the amount of missing data was not sufficiently reported to be quantified by reviewers in 68% (2). When quantifiable, the extent of missing data can be substantial. For example, Eekhout et al. (2) reported a prevalence of 26% missing data in 84 epidemiologic studies. Although missing data are nearly ubiquitous in epidemiologic research, the impact of missing data on inference can vary greatly. Despite a rich body of literature on statistical methods for the analysis of missing data, the most widely used technique in epidemiology remains the most basic: “complete-case” analysis (2–10), which ignores potentially valuable observed information by excluding participants with only partially available data on the variables of interest (11, 12). To demonstrate the impact of missing data and illustrate principled methods that account for missingness, we constructed a challenge with 2 independent teams of missing-data experts. The goal of their work was to illustrate the use of methods of their choice to analyze epidemiologic studies with missing data and to evaluate and relax (to the extent possible) the assumptions made about missing data. The motivating example is taken from the Collaborative Perinatal Project (CPP) (13), a prototypical epidemiologic study, and focuses on estimating the association of smoking during pregnancy with risk of spontaneous abortion. Example data sets were created from a completely observed subset of the CPP data, where missingness was introduced using 3 different mechanisms. Two teams then analyzed the example data sets, one using multiple imputation (MI) to account for the missing data (14) and the other using inverse probability weighting (IPW) (15). While in practice researchers may have substantive information regarding the missingness process, the teams here were blinded to the mechanisms that generated the missing data, as well as to the full data from which the 3 incomplete data sets were drawn. In this paper, we review existing nomenclature for missing-data mechanisms and introduce the CPP, along with the series of 3 derived data sets with missingness. We close by revealing the underlying true missing-data-generating mechanisms, summarizing the teams’ findings in the context of the unmasked missing-data mechanisms, and discussing best practices and future directions. In 2 companion papers in this issue of the Journal, each team describes, in turn, the application of a principled approach—one parametric (14) and one semiparametric (15)—to account for the missing data. TYPES OF MISSING DATA Throughout this paper we focus on explicit missing data, characterized by missing values for the exposure, outcome, or covariates in an analytical data set. Observational studies are particularly prone to such missing data, because even in the rare settings where there are no missing data for the exposure and outcome, there are invariably missing values for the covariates necessary to obtain adjusted estimates. Data may be categorized as missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) (16, 17). Data are MCAR when the probability of having a variable with missing data does not depend on any observed or missing variables. Missing data are MAR if the probability that a given subset of variables (i.e., a “pattern”) is observed depends only on the values of observed variables. Data are MNAR if the missingness pattern depends on the values of unobserved variables. MCAR is the strongest assumption, and it is unrealistic in typical epidemiologic studies. MAR is a weaker assumption, and it is generally more likely than MCAR to hold in epidemiologic studies, while MNAR is the weakest assumption of the three. Using observed data, one can test the MCAR assumption by empirically refuting it. Given observed data alone, however, the MAR and MNAR mechanisms are indistinguishable. While complete-case analysis of MCAR data will generally yield asymptotically unbiased (henceforth unbiased) estimates, efficiency is often lost because this technique discards information on incomplete cases. If data are MCAR, application of the principled methods in the companion papers (14, 15) will also yield unbiased estimates, but it can also improve efficiency by recovering information from incomplete cases. A complete-case analysis of MAR data may or may not yield biased results. When a complete-case analysis of MAR data yields valid inference, such inference can be inefficient because data on incomplete cases are discarded. However, the principled methods applied and discussed in the companion papers can provide unbiased results and recover efficiency when data are MAR, as demonstrated later in this paper. MNAR cannot be ruled out empirically and typically implies that aspects of the missing-data mechanism need to be properly modeled to identify the parameter of interest. Intuitively, the observed data are of little help, because the information necessary to model the missing-data mechanism is unobserved by definition (18, 19). Therefore, modeling missing data that are MNAR often requires additional assumptions either in the form of external information or fairly strong parametric assumptions when such information is not available (20, 21). Other approaches to handling MNAR data include conducting a sensitivity analysis, which in extreme scenarios will produce bounds for the impact of missing data. If data are MNAR, then both complete-case analysis and the principled methods discussed in the companion papers may provide biased estimates. Notably, MNAR does not imply or guarantee biased estimates, which depends on the particular causal structure; it only guarantees that we cannot correct for it if present (22). Finally, ignorability is a criterion that is often used interchangeably with MAR but requires that the missing data are MAR (or MCAR) and that the parameters governing the missing-data mechanism are distinct from the parameters governing the full-data model (16, 23). Little and Zanganeh (24) have provided a variety of examples where a data set is MNAR and how ignorability can arise and be used in a subset to perform valid inference on that subset of data. Distinct parameters are those for which, in the likelihood setting, the domain of the parameters is the Cartesian product of their individual domains (values taken by one parameter do not restrict values taken by another parameter); and in the Bayesian setting, distinct parameters require independent priors for the parameters. Ignorability (e.g., MAR with distinct parameters) is the weakest general assumption that allows the likelihood to factor such that one can identify the parameter of interest. In settings where ignorability does not appear to hold, methods for nonignorable models (e.g., pattern mixture models) can be useful (25). MOTIVATING EXAMPLE Data for this challenge comprised information on a subsample of participants from the CPP. The CPP was a multisite US study of pregnancy and pediatric outcomes conducted from 1959 to 1974 (13). The CPP investigators recruited and enrolled 48,197 participants who were seeking prenatal care. Data on demographic factors and medical history were collected at entry into the study. For illustrative purposes, we selected 11,373 women entering the cohort prior to 20 weeks’ gestation who had complete data on the variables birth outcome (spontaneous abortion (<20 weeks’ gestation) or live birth), maternal smoking, maternal age (years), maternal race, and maternal body mass index (BMI; weight (kg)/height (m)2). Spontaneous abortion or live birth and smoking status were binary variables; race was categorized as white, black, or other; and age and BMI were continuous. This subsample is referred to as the “full” data set. The characteristics of the full data set are displayed in Table 1, overall and by birth outcome, with 411 spontaneous abortions and 10,962 live births. Table 1. Distribution of Maternal Characteristics in the Full Subset of Data From the Collaborative Perinatal Project, 1959–1974 Variable  Birth Outcome  Overall (n = 11,373)  Live Birth (n = 10,962)  Spontaneous Abortion (n = 411)  No. of Women  %  No. of Women  %  No. of Women  %  Smoking               No  8,458  74.4  8,169  74.5  289  70.3   Yes  2,915  25.6  2,793  25.5  122  29.7  Race               White  7,192  63.2  6,955  63.5  237  57.7   Black  3,475  30.6  3,333  30.4  142  34.5   Other  706  6.2  674  6.1  32  7.8  Age, yearsa  25.2 (5.9)  25.1 (5.8)  27.2 (6.9)  Body mass indexa,b  22.5 (4.1)  22.5 (4.1)  22.8 (4.1)  Variable  Birth Outcome  Overall (n = 11,373)  Live Birth (n = 10,962)  Spontaneous Abortion (n = 411)  No. of Women  %  No. of Women  %  No. of Women  %  Smoking               No  8,458  74.4  8,169  74.5  289  70.3   Yes  2,915  25.6  2,793  25.5  122  29.7  Race               White  7,192  63.2  6,955  63.5  237  57.7   Black  3,475  30.6  3,333  30.4  142  34.5   Other  706  6.2  674  6.1  32  7.8  Age, yearsa  25.2 (5.9)  25.1 (5.8)  27.2 (6.9)  Body mass indexa,b  22.5 (4.1)  22.5 (4.1)  22.8 (4.1)  a Values are expressed as mean (standard deviation). b Weight (kg)/height (m)2. The missing-data analytical teams were asked to estimate the relationship between smoking exposure measured during early pregnancy and the risk of spontaneous abortion (i.e., <20 weeks’ gestation), adjusted for race, age, and BMI. Linear logistic regression was employed to estimate the odds ratio quantifying this relationship between smoking and spontaneous abortion. Missing data can occur in a variety of ways. Therefore, to broaden our demonstration, we applied 3 missing-data mechanisms. From the full data, we constructed 3 data sets with missing data for the variables spontaneous abortion, smoking, and BMI, for 8 possible missing-data patterns. The patterns of missingness for each data set are shown in Table 2. We attempted to hold the proportion and pattern of missing data approximately constant across the 3 data sets, with some departures as seen in Table 2. One constructed data set had missing data generated under MCAR, one under MAR, and one under MNAR. The parameters governing the missing-data mechanism (see the Web Appendix, available at https://academic.oup.com/aje) were distinct from the parameters governing the substantive model, implying ignorability in the MCAR and MAR data sets. The characteristics of the data sets, arbitrarily numbered as 1, 2, and 3, are displayed in Table 3. Table 2. Mechanism-Specific Missing-Data Patterns and Percentages Induced in the Full Subset of Data From the Collaborative Perinatal Project, 1959–1974 Missing-Data Patterna  Missing Pattern Percentage  Pattern  SA  Smoking  Black Race  Other Race  Age  BMIb  Fixed  Data Set  1 (MAR)  2 (MCAR)  3 (MNAR)  1  X  X  X  X  X  X  60  61.09  62.32  52.86  2  X  X  X  X  X  M  5  5.32  5.17  5.33  3  X  M  X  X  X  X  5  5.48  5.25  7.16  4  X  M  X  X  X  M  5  5.20  4.95  7.59  5  M  X  X  X  X  X  1  1.15  0.92  1.16  6  M  X  X  X  X  M  10  11.07  10.56  11.25  7  M  M  X  X  X  X  10  9.82  9.84  13.80  8  M  M  X  X  X  M  1  0.87  0.98  0.86  Missing-Data Patterna  Missing Pattern Percentage  Pattern  SA  Smoking  Black Race  Other Race  Age  BMIb  Fixed  Data Set  1 (MAR)  2 (MCAR)  3 (MNAR)  1  X  X  X  X  X  X  60  61.09  62.32  52.86  2  X  X  X  X  X  M  5  5.32  5.17  5.33  3  X  M  X  X  X  X  5  5.48  5.25  7.16  4  X  M  X  X  X  M  5  5.20  4.95  7.59  5  M  X  X  X  X  X  1  1.15  0.92  1.16  6  M  X  X  X  X  M  10  11.07  10.56  11.25  7  M  M  X  X  X  X  10  9.82  9.84  13.80  8  M  M  X  X  X  M  1  0.87  0.98  0.86  Abbreviations: BMI, body mass index; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random; SA, spontaneous abortion. a “X” denotes that data are observed in that pattern, and “M” denotes that data are missing in that pattern. b Weight (kg)/height (m)2. Table 3. Characteristics of 3 Observed Data Sets With Constructed Missing Data From the Full Subset of Data, Collaborative Perinatal Project, 1959–1974 Data Set and Variablea  Birth Outcome  Overall  Live Birth  Spontaneous Abortion  No. of Women  %  No. of Women  %  No. of Women  %  Data set 1 (MAR)               Total  8,767    8,464    303     Smoking                No  6,670  74.6  5,828  78.6  108  76.6    Yes  2,273  25.4  1,584  21.4  33  23.4   Race                White  7,192  63.2  5,308  62.7  176  58.1    Black  3,475  30.6  2,627  31.0  104  34.3    Other  706  6.2  529  6.3  23  7.6   Age, yearsb  25.1 (5.8)  25 (5.8)  26.9 (6.8)   BMIc  22.5 (4.0)  22.5 (4.0)  23 (4.5)  Data set 2 (MCAR)               Total  8,836    8,517    319     Smoking                No  6,684  74.4  5,500  74.4  192  68.6    Yes  2,298  25.6  1,896  25.6  88  31.4   Race                White  7,192  63.2  5,416  63.6  184  57.7    Black  3,475  30.6  2,584  30.3  113  35.4    Other  706  6.2  517  6.1  22  6.9   Age, years  25.2 (5.8)  25.1 (5.8)  27.5 (6.9)   BMI  22.6 (4.1)  22.6 (4.1)  22.9 (4.2)  Data set 3 (MNAR)               Total  8,295    8,033    262     Smoking                No  5,961  74.2  5,111  79.1  121  76.1    Yes  2,068  25.8  1,348  20.9  38  23.9   Race                White  7,192  63.2  5,032  62.6  153  58.4    Black  3,475  30.6  2,480  30.9  88  33.6    Other  706  6.2  521  6.5  21  8.0   Age, years  25.2 (5.9)  25.1 (5.8)  27.2 (7.1)   BMI  22.5 (4.1)  22.5 (4.1)  22.8 (3.9)  Data Set and Variablea  Birth Outcome  Overall  Live Birth  Spontaneous Abortion  No. of Women  %  No. of Women  %  No. of Women  %  Data set 1 (MAR)               Total  8,767    8,464    303     Smoking                No  6,670  74.6  5,828  78.6  108  76.6    Yes  2,273  25.4  1,584  21.4  33  23.4   Race                White  7,192  63.2  5,308  62.7  176  58.1    Black  3,475  30.6  2,627  31.0  104  34.3    Other  706  6.2  529  6.3  23  7.6   Age, yearsb  25.1 (5.8)  25 (5.8)  26.9 (6.8)   BMIc  22.5 (4.0)  22.5 (4.0)  23 (4.5)  Data set 2 (MCAR)               Total  8,836    8,517    319     Smoking                No  6,684  74.4  5,500  74.4  192  68.6    Yes  2,298  25.6  1,896  25.6  88  31.4   Race                White  7,192  63.2  5,416  63.6  184  57.7    Black  3,475  30.6  2,584  30.3  113  35.4    Other  706  6.2  517  6.1  22  6.9   Age, years  25.2 (5.8)  25.1 (5.8)  27.5 (6.9)   BMI  22.6 (4.1)  22.6 (4.1)  22.9 (4.2)  Data set 3 (MNAR)               Total  8,295    8,033    262     Smoking                No  5,961  74.2  5,111  79.1  121  76.1    Yes  2,068  25.8  1,348  20.9  38  23.9   Race                White  7,192  63.2  5,032  62.6  153  58.4    Black  3,475  30.6  2,480  30.9  88  33.6    Other  706  6.2  521  6.5  21  8.0   Age, years  25.2 (5.9)  25.1 (5.8)  27.2 (7.1)   BMI  22.5 (4.1)  22.5 (4.1)  22.8 (3.9)  Abbreviations: BMI, body mass index; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random. a Subgroup counts for smoking and race do not sum to the totals because of missingness patterns. b Values for age and BMI are expressed as mean (standard deviation). c Weight (kg)/height (m)2. Missing data were generated under the MAR, MCAR, and MNAR missingness mechanisms for data sets 1, 2, and 3, respectively. For all 3 data sets, missing data for the variables BMI, smoking, and spontaneous abortion were generated using the following multinomial model:   P(R=r|L)=exp(α0r+βr'V+γr'L(r)+ηr'W(r)+θrU)1+∑k=22Kexp(α0k+βk'V+γk'L(k)+ηk'W(k)+θkU),where K=3 denotes the number of variables with missingness and R denotes the missing-data pattern with possible patterns r=1,…,2K. Explicitly, r=1 was the pattern of complete data and r=8 was the pattern with missing data on BMI, smoking, and spontaneous abortion. L is the set of measured variables. The set V denotes the subset of L that is always observed; in this example, V = (age, race). The set L(r) is the subset of L observed under pattern r but missing in at least one pattern (else the variable would be found in V), while W(r) is the complement of L(r), or the subset of L not observed under pattern r. For example, in pattern r=7, L(r) is the observed BMI and W(r) is the missing data on smoking and spontaneous abortion. Finally, U is an unmeasured variable. For data set 1, MAR was created by setting βr≠0,γr≠0, and ηr=θr=0, so that the missingness pattern depended on the variables that were always observed (age and race) and the observed values of spontaneous abortion, BMI, and smoking. For data set 2, MCAR was achieved by setting βr=γr=ηr=θr=0, so that the missingness pattern was a function of a constant and thus was completely random. For data set 3, βr≠0,γr≠0, and ηr≠0 or θr≠0, corresponding to an MNAR mechanism because the set of unobserved values (W(r)) as well as a completely unobserved variable (U) defined the missingness mechanism. See the Web Appendix for detailed SAS code (SAS Institute, Inc., Cary, North Carolina), as well as the parameter values used to induce missingness in the 3 data sets. Notably, the MNAR mechanism of data set 3 was introduced via W(r) (Table 2, pattern 6, specifically) or U (Table 2, patterns 3, 4, and 7) but could be introduced via W(r) and U simultaneously. Across the 3 data sets, the intercepts α0r were chosen to maintain approximately 60% complete data (see Table 2). The analysis teams were provided with the 3 observed data sets, with instructions to estimate the association of smoking with risk of spontaneous abortion, adjusted for the list of potential confounders provided. The teams were not given any indication or instructions regarding the role or use of these variables in the missingness mechanisms or in their analysis, respectively. The analysts applied the methodology they deemed appropriate. Again, the teams were masked to the underlying missing-data mechanisms for each observed data set and did not have access to the full data set. SUMMARY OF FINDINGS In each of the 3 data sets, both teams conducted complete-case analysis as well as used a principled method (i.e., MI or IPW) to estimate the association of smoking with spontaneous abortion (14, 15). The goal in conducting a complete-case analysis was to triangulate results and highlight potential pitfalls of this common technique, particularly in comparison with the principled methods. In the corresponding companion papers (14, 15), the analytical teams closely mimicked the real-world application of these missing-data methods because the underlying mechanism used to create the missing data is rarely known in practice and was not known to the investigators in this exercise. The results from the analyses are unmasked here and consolidated in Table 4, along with the (“true”) results from the full data. The “Full Data” column represents the association of smoking with spontaneous abortion after adjustment for race, age, and BMI in the complete subsample from the CPP cohort, showing an odds ratio of 1.31 (95% confidence interval: 1.05, 1.64) for all 3 data sets. The rows of Table 4 correspond to the data sets in which a given missing-data mechanism was imposed. Results are provided for complete-case analyses, MI, and augmented IPW, along with the results from the full data. Of course, some variation in the point estimates will be expected even for unbiased scenarios due to sampling variability, as these are single realizations of the data. Table 4. Estimated Odds Ratiosa for the Association Between Smoking and Spontaneous Abortion According to Data Analysis Approach, Collaborative Perinatal Project, 1959–1965 Missing-Data Process  Approach  Full Data  Complete-Case  Multiple Imputation  Augmented IPW  OR  95% CI  OR  95% CI  OR  95% CI  OR  95% CI  MCAR (data set 2)  1.31  1.05, 1.64  1.39  1.06, 1.82  1.36  1.05, 1.77  1.42  1.09, 1.86  MAR (data set 1)  1.31  1.05, 1.64  0.43  0.19, 0.93  1.30  0.95, 1.77  1.40  1.00, 1.97  MNAR (data set 3)  1.31  1.05, 1.64  0.93  0.55, 1.57  1.14  0.81, 1.60  1.57  1.09, 2.23  Missing-Data Process  Approach  Full Data  Complete-Case  Multiple Imputation  Augmented IPW  OR  95% CI  OR  95% CI  OR  95% CI  OR  95% CI  MCAR (data set 2)  1.31  1.05, 1.64  1.39  1.06, 1.82  1.36  1.05, 1.77  1.42  1.09, 1.86  MAR (data set 1)  1.31  1.05, 1.64  0.43  0.19, 0.93  1.30  0.95, 1.77  1.40  1.00, 1.97  MNAR (data set 3)  1.31  1.05, 1.64  0.93  0.55, 1.57  1.14  0.81, 1.60  1.57  1.09, 2.23  Abbreviations: CI, confidence interval; IPW, inverse probability weighting; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random; OR, odds ratio. a Adjusted for age, race, and body mass index. For MCAR (i.e., data set 2 in the companion papers), complete-case analyses and use of both principled methods resulted in similar point estimates and confidence intervals. For this realization of the data, all 3 methods estimated odds ratios that were close to those from the full data, with slightly wider confidence intervals, reflecting the loss of information due to the missing data, essentially a consequence of a reduced sample size. For the MAR mechanism (i.e., data set 1 in the companion papers), the complete-case analysis estimate of 0.43 shows a notable and spurious protective effect of smoking on spontaneous abortion. This effect reversal is primarily due to the violated assumption that the missingness is MCAR, when in fact it is a function of the observed variables. In some settings, complete-case analysis can be valid under MAR and even MNAR—for example, if the missingness process depends only on covariates in the regression, even if some of them are not fully observed (26). However, this assumption is not met in our example, which serves to highlight the potential consequences of applying complete-case analysis and subsequently failing to address the impact of the missing data. On the other hand, employing either MI or augmented IPW resulted in point estimates of 1.30 and 1.40, respectively, with similar confidence intervals. Both estimates showed a drastic shift from the (naive) complete-case analyses estimate toward the effect observed in the full data. For the MNAR case (i.e., data set 3 in the companion papers), complete-case analyses resulted in an estimated odds ratio of 0.93 (95% confidence interval: 0.55, 1.57). While both MI and IPW resulted in point estimates closer to the full-data effect estimate, such a finding cannot be expected generally. PRINCIPLED METHODS: PROS, CONS, AND ASSUMPTIONS In general, standard complete-case analyses rely on an assumption that the missing data are MCAR or an equivalent assumption to yield unbiased estimates (26). This assumption may often be unrealistic in epidemiologic settings, in which complete-case analyses will likely result in bias (e.g., data set 2). Furthermore, even if data are MCAR, such a complete-case analysis will typically be inefficient, as it ignores valuable information in incomplete cases, which can be particularly deleterious when a necessary covariate is the predominantly missing variable. When data are MAR, both MI and IPW may still return unbiased estimates when appropriate assumptions are met. In particular, standard MI and IPW as implemented in the companion papers (14, 15) rely on specific modeling assumptions beyond assuming an ignorable nonresponse process (27). MI is formally a Bayesian approach which, as implemented in the companion paper by Harel et al. (14), assumes that the parameters indexing models of interest have a normal prior distribution. The imputation model and the analysis model are also assumed to be correctly specified, which includes correct specification of the conditional distribution of incomplete variables given the observed variables. This also implies that the imputation model for the distribution of covariates given the outcome is compatible with the underlying model of substantive interest for the density of the outcome given covariates. This is guaranteed to occur when the joint distribution of the covariates and outcome is joint normal and for certain model choices within the natural exponential family, but not in general. There are different types of imputation models; some require parametric assumptions (e.g., joint normal distribution in MI) and some do not (e.g., hot-deck). While misspecified models should not be expected to produce unbiased results, simulations have shown that MI is somewhat robust to the choice of imputation model for moderate rates of missing information. All MI implemented by Harel et al. in the companion paper (14) used a “proper” parametric imputation model, which respects the distributional properties of the imputation draws, typically by drawing values of parameters before drawing imputations (12, 28). The goal of MI is not to estimate the missing values themselves but rather to produce unbiased and efficient estimates for the population parameters of interest, by essentially averaging over the (unknown) distribution of the missing data. Thus, a joint normal model might be an appropriate choice, even for nonnormal variables, because it is more important to preserve the relationship between the variables than to impute feasible values for all of the missing variables. In addition, the purpose of the imputation model in MI is to predict, not establish causality, and therefore it does not need to preserve temporality. Rather, all available and relevant data should be used to impute or model the missingness mechanism, regardless of its temporal relationship to the exposure or outcome. For instance, there may be observed variables that are highly predictive of the missing variables or the missingness mechanism but do not enter into the analysis model, either because they are not predictive of the outcome or because they are not confounders. These auxiliary variables are beneficial and easy to use in MI and are considered one of the advantages of the method. In this exercise, only variables chosen a priori to be included in the analysis model were available, implying that the variable sets for the imputation and analysis models were the same. In contrast to MI, IPW assumes that a model for the nonresponse process given the observed data is correctly specified. Because the model for the missing-data process does not restrict the model of substantive interest and vice versa, IPW (and augmented IPW) is not limited by a compatibility of conditional densities under the MAR assumption. However, IPW requires that for all possible realizations of the full data, there is a nonzero probability of observing a person with complete data (i.e., the positivity assumption) (29). IPW as implemented here relies on large-sample theory for valid inference. In practice, MI-based inference also often relies on large-sample inference (30–32). However, when proper imputation is applied under a strict Bayesian framework (implying no P values or confidence intervals), MI can in principle be applied in small samples as well, so long as additional assumptions concerning the fully conditional distribution hold. In actuality, MI and IPW make somewhat complementary modeling assumptions, as they rely on parametric models for distinct components of the joint likelihood for the full data and the missingness process. As stated above, MI relies on a model for the underlying full data but allows the missingness mechanism to remain completely unrestricted. Conversely, IPW models the missingness mechanism, but the full-data model is unrestricted beyond the model of substantive interest. As with a complete-case analysis, IPW does not efficiently use information available in incomplete cases, although these data are used to estimate the nonresponse rates, therefore recovering some information from incomplete cases. This issue does not arise with MI because available information among incomplete cases serves as a basis for imputing missing information. In the other companion paper, Sun et al. (15) considered an augmented IPW, an extension of IPW attributable to Robins et al. (29) and recently implemented for nonmonotone MAR patterns by Sun and Tchetgen Tchetgen (33), which largely resolves the efficiency limitation of standard IPW by allowing the analyst to recover information in incomplete cases. DISCUSSION Based on the above considerations, complete-case analysis should be used with the same caution we ascribe to unadjusted estimates, as its validity relies on strong, often unrealistic assumptions. In contrast, principled methods such as MI or IPW may account for bias due to missing data under weaker assumptions. The most standard application of these methods relies on the MAR assumption. In the absence of model misspecification, we expect that MI will be more efficient than IPW, because the latter assumes that the full-data model is completely unrestricted other than by the model of substantive interest, while the former uses a restricted parametric formulation for the full data (29, 30). While IPW fails to efficiently recover all available information from incomplete cases, augmented IPW recovers such information, at least the portion recoverable without relying on a finite-dimension full-data model. Now, given that we wish to use principled methods, which are we to choose? We can choose a version of MI, a version of IPW, or another formal approach not considered in this set of articles (e.g., direct maximum likelihood, Bayesian analysis) that equally applies under MAR. We believe foremost that we should prioritize a consistent estimator under ignorable MAR settings. Then, to the extent possible, we should minimize variance, although reasonable tradeoffs in allowing some bias in exchange for increased precision may be prudent (e.g., to minimize mean squared error). We remain without consensus as to the extent to which flexible parametric models may be overly restrictive and hide some amount of estimation error, but we do have consensus that low-dimension parametric models can be overly restrictive and hinder our ability to see the world more clearly: In such cases, in our results we might see more of our assumptions than we see of our world (1). Indeed, nuisance models required for controlling selection bias (e.g., the imputation model, the missing-data mechanism model) do not need to return interpretable finite-dimension parameters, and therefore they might be restricted only by the requirement to achieve a consistent estimator of the parameter(s) of interest from the substantive model. This leaves us the ability to pick from several candidate methods. Some of these candidate methods have nonoverlapping assumptions, as mentioned above. Therefore, there should be advantages to conducting more than 1 analysis and comparing results, as we have done with IPW and MI. When results with nonoverlapping assumptions agree, then our confidence in the results should be higher—emboldened, but never certain, because even nonoverlapping assumptions may jointly fail. When results disagree, we should pause and reconsider our assumptions and methods. Current research on missing data is producing more flexible procedures, such as doubly robust estimators, that combine a model for the full data with a model for the missing-data process, such that only 1 of the models need be correctly specified to produce unbiased inferences. While the advent of accessible software packages for these methods may be on the horizon, moving away from complete-case analysis towards a principled method like MI or IPW as demonstrated in the companion papers (14, 15) is vital to ensuring proper analysis of current epidemiologic studies. When analyzing observational data, an epidemiologist presented with a crude odds ratio of 0.43 (analogous to the complete-case analysis estimate) and an adjusted odds ratio of 1.35 (analogous to MI or IPW estimate) might posit that confounding bias is the culprit and report both, while interpreting the adjusted result as probably closer to the underlying truth (barring collider-stratification bias and assuming a rare outcome such that the odds ratio is collapsible). Reporting only the results from a complete-case analysis in the presence of nontrivial missing data is commensurate with reporting only crude associations from nonrandomized studies. We therefore advocate that the same reasoning be followed when dealing with missing data. If we are principled, we are more likely to get closer to the truth. We urge researchers, reviewers, and journal editors to think about the missing-data problem prior to making decisions about a plan of action. We would be wise to plan for missing values, minimize nonresponse, determine the missing-data assumptions, and report them appropriately. Of course, there may be reasonable extenuating circumstances that support reporting a crude association, or an equivalent rationale for reporting results from a complete-case analysis. But we surmise that such cases are rare. In closing, we ask that researchers join us in reevaluating the context of our work, to think more carefully about the assumptions that underlie the claims we make. ACKNOWLEDGMENTS Author affiliations: Division of Intramural Population Health Research, Eunice Kennedy Shriver National Institute of Child Health and Human Development, Rockville, Maryland (Neil J. Perkins, Enrique F. Schisterman); Department of Epidemiology, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina (Stephen R. Cole); Department of Statistics, College of Liberal Arts and Sciences, University of Connecticut, Storrs, Connecticut (Ofer Harel); Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, Massachusetts (Eric J. Tchetgen Tchetgen, BaoLuo Sun); and Agency for Healthcare Research and Quality, Rockville, Maryland (Emily M. Mitchell). This research was partially supported by the Long-Range Research Initiative of the American Chemistry Council (Washington, DC) and the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health. This work was also partially supported by award K01MH087219 from the National Institute of Mental Health. The content of this article is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Mental Health or the National Institutes of Health. Conflict of interest: none declared. Abbreviations BMI body mass index CPP Collaborative Perinatal Project IPW inverse probability weighting MAR missing at random MCAR missing completely at random MI multiple imputation MNAR missing not at random REFERENCES 1 Little RJ, D’Agostino R, Cohen ML, et al.  . The prevention and treatment of missing data in clinical trials. 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Journal

American Journal of EpidemiologyOxford University Press

Published: Mar 1, 2018

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