Missing data and multiple imputation in clinical epidemiological research

Figures & data

Table 1 An example of a situation when data are MAR rather than MCAR

Observed data	Patients with BMI value (%)	Patients with missing BMI value (%)
Smoking
Yes	80	20
No	60	40
Comorbidity prior diagnosis
Yes	85	15
No	25	75

Abbreviations: BMI, body mass index; MAR, missing at random; MCAR, missing completely at random.

Table 2 Proposed methods for dealing with missing data in the analytic phase

Methods	Brief description	Assumption to achieve unbiased estimates	Advantages	Limitation(s)
Complete-case analysis	Include only individuals with complete information on all variables in the dataset	MCAR	• Simplicity • Comparability across analyses	• Data may not be representative. Reduction of sample size and thereby of statistical power • Too large standard error (lack of precision of the results) • Discarding valuable data
Missing indicator method	For categorical variables, missing values are grouped into a “missing” category. For continuous variables, missing values are set to a fixed value (usually zero), and an extra indicator or dummy (1/0) variable is added to the main analytic model to indicate whether the value for that variable is missing	None	• Uses all available information about missing observation and retains the full dataset	• The magnitude and direction of bias difficult to predict • Too small standard error • The results may be meaningless since method is not theoretically driven • Bias due to residual confounding
Single value imputation	Replace missing values by a single value (eg, mean score of the observed values or the most recently observed value for a given variable if data are measured longitudinally)	MCAR, only when estimating mean	• Run analyses as if data are complete • Retains full dataset	• Too small standard error (overestimation of precision of the results) • Potentially biased results • Weakens covariance and correlation estimates in the data (ignores relationship between variables)
Sensitivity analyses with worst- and best-case scenarios	Missing data values are replaced with the highest or lowest value observed in the dataset	MCAR	• Simplicity • Retains full dataset	• Too small standard error and thereby overestimation of precision of the results • Analyses yielding opposite results may be difficult to interpret
Multiple imputation	Missing data values are imputed based on the distribution of other variables in the dataset	MAR (but can handle both MCAR and MNAR)	• Variability more accurate for each missing value since it considers variability due to sampling and due to imputation (standard error close to that of having full dataset with true values)	• Room for error when specifying models

Abbreviations: MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random.

Figure 1 Distribution of BMI values by outcome in full dataset (A) and in a dataset with 35% missing values (B) for BMI handled by creating a missing BMI category.

Abbreviation: BMI, body mass index.

Figure 2 Normal distribution of observed BMI in a full dataset of 10,000 observations.

Abbreviation: BMI, body mass index.

Figure 3 Distribution of BMI in a dataset of 10,000 observations, where 35% of BMI values are missing and replaced by the observed mean BMI value.

Abbreviation: BMI, body mass index.

Figure 4 Selection of variables in order to create multiple imputed datasets when looking into the association between body mass index and transfusion risk.

Figure 5 The three main stages of implementing multiple imputation.

Table 3 An example of the imputed missing BMI values generated with five imputed datasets

Patient number	Imputed data set 1 (BMI 1)	Imputed data set 2 (BMI 2)	Imputed data set 3 (BMI 3)	Imputed data set 4 (BMI 4)	Imputed data set 5 (BMI 5)
10	25.3	26.4	27.0	24.8	29.7
25	19.7	21.3	22.3	20.5	23.8
23	22.1	27.6	22.9	28.1	25.8
150	20.1	22.5	23.4	21.7	23.0
175	19.7	20.2	21.2	22.4	21.9

Abbreviation: BMI, body mass index.

Table 4 Association between BMI and risk of blood transfusion adjusted for age and gender

Patient characteristics	Full data (n=3,500)			Complete case analysis (n=2,733)			Multiple imputation (n=3500, m=5)			Multiple imputation (n=3500, m=30)
	OR	SE	95% CI	OR	SE	95% CI	OR	SE	95% CI	OR	SE	95% CI
BMI	0.980	0.0085	(0.963, 0.997)	0.978	0.0098	(0.959, 0.997)	0.976	0.0087	(0.959, 0.994)	0.978	0.0098	(0.959, 0.997)
Age (years)
<75	Baseline
≥75	2.100	0.1928	(1.754, 2.514)	2.244	0.2421	(1.816, 2.772)	2.097	0.1927	(1.752, 2.511)	2.098	0.1928	(1.752, 2.511)
Gender
Female	Baseline
Male	0.815	0.0630	(0.700, 0.948)	0.906	0.0779	(0.765, 1.072)	0.818	0.0633	(0.702, 0.952)	0.817	0.0634	(0.702, 0.951)

Note: Results are presented for full-observed data, complete-case analysis, and multiple imputation and contain point estimates for ORs, SEs, and 95% CIs.

Abbreviations: BMI, body mass index; CI, confidence interval; OR, odds ratio; SE, standard error.