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Abstract

Missing data are a common challenge across scientific disciplines. Current imputation methods require the availability of individual data to impute missing values. However, missingness often requires using external data for the imputation, particularly in multisite settings and federated analyses. We introduce a new command, mi impute from, designed to impute missing values using linear predictors and their related covariance matrix from imputation models fit in one or multiple external studies. This allows for the imputation of any missing values without sharing individual data between studies. We describe the underlying method and present the syntax of mi impute from alongside practical examples of missing data in collaborative research projects.

Keywords

st0792 mi impute from mi_impute_from_get univariate imputation missing data systematically missing data met a-analysis quantile regression losgistic regression multiple imputation pooling projects data network

1 Introduction

Missing data are a common challenge to statistical inference in various scientific disciplines (Carpenter et al. 2023). Commonly, sporadic or partially missing data occur in single studies and can be handled with conventional imputation approaches available with the mi suite of commands in Stata. First, mi impute is used to create imputed datasets by fitting an imputation model for missing variables. Second, mi estimate (see [Ml] mi estimate) is used to fit a substantive model across imputed datasets and combine estimates with Rubin’s rules. Current imputation approaches require individual- level data to be used as the basis for imputation. However, borrowing information from other studies is often necessary to impute missing values in a single study. For example, many collaborative research efforts involving multiple studies contain completely or systematically missing data (100% missing values) for key variables in one or more studies.

While there have been extensive efforts to address systematically missing data in individual participant data meta-analysis with multiple imputation (Burgess et al. 2013; Resche-Rigon et al. 2013; Jolani et al. 2015; Quartagno and Carpenter 2016; Audigier et al. 2018; Resche-Rigon and White 2018; Jolani 2025), a crucial prerequisite for current multiple-imputation approaches is the availability of individual data. However, sharing individual data is often not possible or desired because of logistic, legal, and administrative barriers. Consequently, standard imputation procedures become infeasible because there is no basis for imputation because of the lack of observed data on missing values at certain study sites. While it is possible to restrict the analysis only to studies with complete data, this approach is undesirable because of the potential loss of efficiency and validity (Debray et al. 2013).

In previous work, we described the substantive idea of imputing missing values without sharing individual participant data across study sites (Thiesmeier et al. 2025). This approach can be particularly useful in multisite settings and federated analyses where covariates across study sites are not consistently collected (Popovic 2017). However, software facilitating the imputation without individual data is currently not available. Therefore, we introduce a new command, mi impute from, that facilitates univariate imputation of missing values using external linear predictors and their covariance structure. The key feature of this approach is to fit a prediction model in an external dataset that has some information on the missing variable. Instead of sharing individual data, the regression coefficients and covariance structure of a prediction model are shared between studies and used as the basis of imputation at the study site with missing data.

The underlying theoretical considerations of imputing missing values without the need of sharing individual participant data have been described in detail in previous work. Resche-Rigon and White (2018) described the fundamental basis of this process as a two-stage imputation process. Chang et al. (2020) presented and evaluated multiple algorithms specifically for distributed health-data networks and aspects of multivariate imputation techniques in federated settings. We further described and evaluated the imputation of systematically missing data for continuous variables using quantile regression (Bottai and Zhen 2013; Thiesmeier, Bottai, and Orsini 2024). An extension of this approach was presented for categorical variables using multinomial logistic regression (Thiesmeier, Hofer, and Orsini 2025).

The remaining parts of this article are structured as follows. First, the methods underlying mi impute from are presented with different imputation options, including quantile, multinomial, and binary logistic imputation. Second, the structure and options of the command are shown. Third, mi impute from is illustrated with different simulated examples, including imputing missing values for a confounding variable, an effect modifier, and a predictor in collaborative research projects. We conclude by highlighting the potential of the new command and pointing out future developments.

2 Methods

In this section, we outline the general steps ol imputing missing values using external data. We follow notations similar to those presented in Thiesmeier, Bottai, and Orsini (2024) and Thiesmeier, Hofer, and Orsini (2025). Here we consider systematically missing data because they are a common missing-data challenge involving multiple studies. Such missing values may affect important covariates that are not measured or collected. Although the focus is on systematically missing data, mi impute from can be used for any type of missing data and currently allows imputation of continuous, discrete, and binary variables with quantile, multinomial, and binary logistic regression, respectively.

2.1 Imputation of missing values using external prediction equations

Let z_ij denote observations from independent random variables Z_ij in the jth study. The index i, identifying an individual, ranges from 1 to n_j, and the index j, identifying a study, ranges from 1 to J. In addition, let us denote with c _ij = (c_1i, c_2i, · · ·, c_ki) a k-dimensional set of predictors of z_ij. Suppose the vector c _ijs is completely observed for all i ∈ 1,…, n_j and all j ∈ 1,…, J, while z_ij is missing in the jth study. Let M_j ⸦ {1,…, n_ij} be the set of indexes corresponding to the individuals with missing values for z_ij in the jth study. The cardinality of M_j divided by the sample size n_i sdenotes the proportion of missing data for the jth study, that is, FM_j = |M_j|n_j, ranging from 0 (no missing) to 1 (all missing, that is, systematically missing). Systematically missing data in a single study are present if FM_j = 1. For the remainder of this article, let J = A U B, with A = j ∈ J : FM_j = 1, denote the set of studies with systematically missing data, whereas the set of studies B = j ∈ J : FM_j < 1 is used as the basis of imputation.

First, in each study J ∈ B, fit an imputation model on z_i dependent on a set of predictors c _i . An imputation model is fit in the form of a quantile, multinomial logistic, or logistic regression model. The resulting vector of imputation regression coefficients ${\hat{γ}}_{i} = (γ_{0 i}, γ_{1 i} γ_{2 i} \cdot \cdot \cdot γ_{k i})$ and the related covariance matrix ${\hat{Φ}}_{i}$ are saved to be shared with other studies in J ∈ A.

Second, the matrices including the regression coefficients ${\hat{γ}}_{i}$ and covariances ${\hat{Φ}}_{i}$ are exported into files (for example, Excel or delimited text files) that can be shared with other studies. In J ∈ A, the files are imported and transformed into matrices to be used for the imputation of missing data with mi_impute_from. The command mi_impute_from_get facilitates this step. If mi_impute_from_get recognizes multiple input files, a series of inverse-variance-weighted least-squares models are fit to compute ${\hat{γ}}_{i} (c_{i})$ and ${\bar{Φ}}_{i}$ . Regardless of the number of files, a random number from a normal distribution is drawn and further used as the regression coefficient, $γ_{i} * (c_{i}) \sim N [{\hat{γ}}_{i} (c_{i}), \hat{SE} {{\hat{γ}}_{i} (c_{i})}]$ , to be used for the mth imputation of the missing value $z_{i}^{(m)}$ .

Third, mi_impute_from_get returns formatted matrices that are used to impute missing values on z_i in J ∈ A. Next, it executes m = 1, 2,…, M imputations of z_i conditional on the predictors c _i . The key feature of mi impute from is that missing values are imputed without individual data as the basis for the imputation. One can impute continuous, discrete, and binary missing variables with quantile, multinomial, and binary logistic regression, respectively. The individual assignment of imputed values to the missing data for each of the three imputation methods is explained in section 2.2.

Finally, a substantive model can be fit in each imputed dataset, and estimates are pooled with Rubin’s rules, provided by mi estimate.

2.2 Imputation methods

We have implemented three imputation methods for continuous, discrete, and binary variables in the current version that can be handled with qreg, mlogit, and logit, respectively. The methods can be extended to any other regression method.

2.2.1 Continuous variables

A continuous missing variable can be imputed by sharing the estimates of linear predictors of quantile regression models; this has been previously described (Bottai and Zhen 2013; Thiesmeier, Bottai, and Orsini 2024).

In J ∈ B, the cumulative distribution function of z_i conditionally on a set of predictors cs _i can be derived by estimating the p-quantile of z_i swith a quantile regression model where p ranges from 0.01 to 0.99 in steps of 0.01,

{\hat{Q}}_{z i | c i} (p) = c_{i} γ_{i} (p) p \in {0.01, 0.02, \dots, 0.99}

There are 99 sets of regression coefficients $\hat{γ_{i}} (p)$ defining the cumulative distribution function for any linear combination of predictors. Each missing value $z_{i}^{(m)}$ is replaced by M independent imputed values. In J ∈ A, a random draw U_i from a continuous uniform distribution U(0,1) is taken. The mth imputation $z_{i}^{(m)}$ for the ith individual is computed as the weighted average of the F and F + 1 conditional predicted quantiles,

z_{i}^{(m)} = (1 - mod) \times {\hat{Q}}_{z_{i} | c_{i}} (F) + mod \times {\hat{Q}}_{z_{i} | c_{i}} (F + 1)

where F = [Ui%] and mod = U_i% — [Us%].

2.2.2 Categorical variables

A categorical missing variable z_i with K levels can be imputed by sharing k linear predictors of multinomial logistic regression models (Thiesmeier, Hofer, and Orsini 2025). We denote θ_ik as the probability that z_i is equal to level k, given a set of predictors c _i , $θ_{i k} | c_{i} = P (z_{i} = k | c_{i})$ with $\sum_{k = 1}^{K} θ_{i k} = 1$ In J∈B, a multinomial logistic regression imputation model is specified as

\ln (\frac{θ_{i k}}{θ_{i r}}) = γ_{i k} (c_{i})

where r is the reference level. The estimated predicted conditional probabilities of falling into the levels of the categorical variable z_i are denoted as ${\hat{θ}}_{i k} = \hat{P} (z = i | c_{i})$ for any individual in J ∈ B. In the set of studies $J \in A, {\hat{θ}}_{k} = \hat{P} (z = i | c_{i})$ is used to compute the conditional predicted probabilities of the missing variable

{\hat{θ}}_{i k} = {\begin{matrix} \frac{1}{1 + \sum_{k = 2}^{K} e^{{\hat{Υ}}_{i k} (c_{i})}} & if k = r \\ \frac{e^{{\hat{Υ}}_{i k} (c_{i})}}{1 + \sum_{k = 2}^{K} e^{\hat{Υ} i k (c_{i})}} & if k > r \end{matrix}

The conditional predicted cumulative distribution function, ${\hat{Θ}}_{i k}$ is equal to the sum of the probabilities of the missing variable being less than or equal to k:

{\hat{Θ}}_{i k} = \hat{P} (z_{i} \leq l | c_{i}) = \sum_{k = 1}^{K} {\hat{θ}}_{i k}

In J ∈ A, a random draw U_i from a continuous uniform distribution U(0,1) is taken.

The mth imputation $z_{i}^{(m)}$ for the ith individual is obtained by mapping U_i to the predicted cumulative probabilities ${\hat{Θ}}_{i k}$ .

2.2.3 Binary variables

A binary (coded as 0 or 1) missing variable can be imputed by sharing the linear predictors of logistic regression models (Thiesmeier, Hofer, and Orsini 2025). When z_i is binary, we denote θ_i as the probability that z_i = 1, given a set of predictors c _i , θ _i |c _i = P(z_i = 1|c _i ). In J ∈ B, we specify a logistic regression imputation model in the form of

\ln (\frac{θ_{i}}{1 - θ_{i}}) = γ_{i} (c_{i})

Here ${\hat{γ}}_{i} (c_{i})$ is the estimated linear predictor. The predicted conditional probability can be expressed as

\hat{θ_{i}} = \frac{e {\hat{γ}}_{i} (c_{i})}{1 + e {\hat{γ}}_{i} (c_{i})}

In the set of studies J ∈ A, we take a random draw U_i from a continuous uniform distribution U(0,1) and assign $z_{i}^{(m)}$ if $U_{i} > \hat{θ_{i}}$ and 0 otherwise.

3 The syntax

3.1 mi impute from

mi impute from fills in missing values using an imputation model fit in one or multiple external studies. The command assumes that variables used in the imputation model are available in the current data.

After mi impute from, mi estimate can be used to compute estimates of regression coefficients by fitting an estimation command to M imputed datasets. You must mi set your data before using mi impute from; see [Ml] mi set.

3.1.1 Options

b (matname) specifies a vector of regression coefficients of an imputation model for ivar.

v (matname) specifies a matrix of covariances for the imputation model to be used for ivar. The covariance structure v() can either come from the imputation model of a single external study or represent the pooled variance-covariance matrix after a meta-regression of multiple-imputation models.

imodel (string) specifies the imputation model, where string can be either qreg, mlogit, or logit.

qreg specifies that the matrix b() contain 99 (q = 0.01(0.01)0.99) linear predictors of the quantitative variable ivar to be imputed.

mlogit specifies that the imputation model be a multinomial logistic regression model for the categorical variable ivar.

logit specifies that the imputation model be a logistic regression model for the binary variable ivar.

Additionally, the options add(), replace, rseed(), and double are available, similarly to [Ml] mi impute.

3.1.2 Stored results

mi impute from stores the following in r():

3.2 mi_impute_from_get

The command mi_impute_from_get facilitates the use of external imputation models by reading the files and formatting matrices to be passed to mi_impute_from. If multiple files are specified, mi_impute_from_get combines regression coefficients across files using an inverse-variance-weighted least-squares model. Users can implement more different meta-analytical models—such as random-effects models that allow for between- study heterogeneity—before passing the resulting pooled estimates to mi impute from.

3.2.1 Options

b(filename [filename [···]]) specifies a list of files containing the estimated regresssion coefficients for the imputation model for ivar. If more than one file is input, the default is to combine regression coefficients across files using an inverse-variance-weighted least-squares model.

v(filename [filename [···]]) specifies a list of files containing the estimated covariances of the regression coefficients for the imputation model for ivar. If more than one file is input, the default is to combine regression coefficients across files using an inverse-variance-weighted least-squares model.

colnames (string) specifies a list of variable names (including the constant _cons) included in the linear predictor of the imputation model.

tf (string) specifies the format of the files (either tab-delimited .txt or Excel . xlsx) containing the fitted imputation model. Adding the format (.txt, .xlsx) to each filename in b() and v() is not needed. The default format for all the files is tab- delimited (.txt).

imodel(string) specifies the type of imputation model (qreg, mlogit, logit) passed in b(). Of note, for a quantitative variable, one must pass the linear predictor for 99 (q = 0.01(0.01)0.99) quantiles of the imputation model. A loop can be used to compute all the estimated linear predictors in a single file. For a categorical variable with k levels, k linear predictors of multinomial logistic regression models must be passed. The reference level is recognized by the fact that all regression coefficients are equal to zero.

values(numlist) specifies all the numerical values of the categorical variable to be imputed. This is needed when using imodel (mlogit) as the imputation model.

path(string) specifies the location of the filenames specified in options b() and v().

3.2.2 Stored results

mi_impute_from_get stores the following in r():

4 Examples

In the following section, we provide multiple examples to impute missing values with external data. The individual data used for the examples are available as supplementary material.

4.1 Example 1: Missing continuous confounder

Considering multiple studies to answer the same research question can improve the gen- eralizability of findings and is increasingly becoming the norm in international research collaborations (Toh et al. 2013). However, data between studies often cannot be shared (Casaletto et al. 2023). In this example, we consider a collaborative research project involving five observational studies. The aim is to estimate the adjusted odds ratio (or) for the exposure in all five studies. All studies have collected data on the outcome (Y), the exposure (X), and two confounding variables (C and Z). However, in study 1, the confounding variable Z is 100% missing, and it is not possible to estimate the fully adjusted OR for the exposure. Either study 1 has to be excluded from the fully adjusted analysis or the adjustment for the confounder Z has to be disregarded. Both options are undesirable. Therefore, we borrow information from studies that have collected data on the confounder Z to impute the missing values in study 1. Table 1 shows descriptive characteristics and estimated s of the five studies.

Table 1.
Descriptive statistics and estimated ORs (95% confidence interval [Cl]) with a different degree of adjustment. Q25 = 25th percentile, Q95 = 95th percentile.

Study 1 Study 2 Study 3 Study 4 Study 5

N = 6437 N = 1334 N = 8623 N = 3603 N = 6673s

Outcome (%) 12.0 24.1 30.1 15.3 39.8

Exposure (%) 20.5 9.8 18.8 18.5 18.5

Confounder C (%) 39.9 37.5 28.3 25.6 23.4

Confounder Zs (median) NA 0.4 0.5 0.4 0.5

(Q25,Q95) NA (0.1, 4.1) (0.1, 3.9) (0.1, 3.7) (0.1, 3.9)

Crude OR 2.1 (1.8, 2.5) 2.1 (1.4, 3.1) 2.4 (2.1, 2.7) 1.8 (1.4, 2.2) 2.0 (1.8, 2.3)

C adjusted OR 1.6 (1.4, 1.9) 1.7 (1.2, 2.6) 1.8 (1.6, 2.1) 1.5 (1.2, 1.9) 1.7 (1.5, 1.9)

Z and Cs adjusted OR NA 1.6 (1.1, 2.4) 1.6 (1.5, 1.8) 1.4 (1.2, 1.8) 1.4 (1.2, 1.6)

	Study 1	Study 2	Study 3	Study 4	Study 5
	N = 6437	N = 1334	N = 8623	N = 3603	N = 6673s
Outcome (%)	12.0	24.1	30.1	15.3	39.8
Exposure (%)	20.5	9.8	18.8	18.5	18.5
Confounder C (%)	39.9	37.5	28.3	25.6	23.4
Confounder Zs (median)	NA	0.4	0.5	0.4	0.5
(Q25,Q95)	NA	(0.1, 4.1)	(0.1, 3.9)	(0.1, 3.7)	(0.1, 3.9)
Crude OR	2.1 (1.8, 2.5)	2.1 (1.4, 3.1)	2.4 (2.1, 2.7)	1.8 (1.4, 2.2)	2.0 (1.8, 2.3)
C adjusted OR	1.6 (1.4, 1.9)	1.7 (1.2, 2.6)	1.8 (1.6, 2.1)	1.5 (1.2, 1.9)	1.7 (1.5, 1.9)
Z and Cs adjusted OR	NA	1.6 (1.1, 2.4)	1.6 (1.5, 1.8)	1.4 (1.2, 1.8)	1.4 (1.2, 1.6)

4.1.1 Using a single external study

We can choose a single study to specify an imputation model for the confounder Z. For example, let us estimate 99 conditional quantiles for Z as a function of Y, X, and C in study 2. The imputation regression coefficients are exported into a text file that is easy to share between studies. In study 1, the text files of the regression coefficients (b_study2) and their covariances (v_study2) from the imputation model are imported with the command mi_impute_from_get. The resulting matrices for the regression coefficients (get_ib) and the covariance matrix (get_iV) are now used for the imputation of the missing values for the confounder Z with mi_impute_from. The correct order and number of variables used in the imputation model is specified in the option colnames. The missing values for Z are imputed 10 times.

The total number of imputed values equals the total sample size in study 1. The mi impute from command has imputed 100% missing values for the confounder Z using external regression coefficients. Finally, mi estimate is used to fit a multivariate logistic regression model across imputed datasets and combine the results with Rubin’s rules to obtain an estimate for the fully adjusted OR in study 1.

The estimated fully adjusted OR in study 1 is OR = 1.48 (95% Cl = [1.23,1.77]), and we are able to include study 1 in the collaborative research project considering all five studies.

4.1.2 Using multiple external studies

Alternatively to using a single study, we can derive an imputation model from all four studies. After specifying a prediction model on Z conditional on Y, X, and C in studies 2, 3, 4, and 5, we import the list of text files of the regression coefficients (b_study2 b_study3 b_study4 b_study5) and covariance matrices (v_study2 v_study3 v_study4 v_study5) from all four studies with mi_impute_from_get. The command recognizes multiple files and computes a weighted average of the imputation regression coefficients resulting in the average regression coefficients and variances in matrices ib and iV.

A multivariate logistic regression model is fit in the imputed datasets, and results are combined with Rubin’s rules to obtain a final estimate of the fully adjusted OR in study 1.

The estimated imputed fully adjusted OR is 1.32 with a 95% Cl from 1.09 to 1.59. The next step in a collaborative effort would be a meta-analytical approach to combine the fully adjusted ORs across studies.

All studies were simulated under a heterogeneous data-generating mechanism. The fully adjusted OR in study 1 before the confounder Z was set to missing is OR =1.3 (95% Cl = [1.1,1.5]). The underlying value of the fully adjusted confounding effect in all studies was set to OR = 1.4. Using multiple studies as the basis for imputation resulted in an OR close to the estimated value before systematically missing data were introduced. This example illustrates how mi impute from can lead to the inclusion of the fully adjusted OR in study 1 to the overall collaborative effort.

4.2 Example 2: Missing confounder in a federated analysis with 10 large studies

In this section, we extend the challenge of systematically missing confounders and illustrate a more complex example on how to impute missing variables in a federated framework using mi impute from. We consider a federated analysis involving 10 observational studies aiming to estimate the pooled adjusted OR. Similarly to the mechanisms described above, all studies have collected data on the outcome (Y), the exposure (X), and one confounding variable (C). However, 3 out of 10 studies do not have any information on the confounding variable C. For simplicity, all variables in this example are binary.

Two approaches to deal with systematically missing data are common in federated analyses. First, only studies with available data on C are used in the meta-analysis. However, a complete case analysis can introduce bias in the analysis because the included studies might not be representative of the overall population any more. Second, only variables that have not been measured are omitted from the statistical model at study sites with systematically missing data. This approach, too, increases the risks of bias in the analysis if, for example, the omitted variables are important confounders. Therefore, both approaches are not desirable. Alternatively, we wish to impute data on the missing variable C in the three studies that have not measured it by using information from the seven studies in which C was fully observed.

We implement the following approaches as described in more detail in Thiesmeier et al. (2025).

Control analysis: Ten studies are analyzed with complete data. This approach serves as a reference (that is, control) scenario.

Complete case analysis: Seven studies are analyzed with complete data. Three studies with completely missing data on C are excluded.

Available data analysis: Ten studies are analyzed, but the confounding variable C is omitted from the statistical model in studies 8, 9, and 10.

Analysis after multiple imputation: Ten studies are analyzed. In studies 8, 9, and 10, mi impute from is used to ismpute C with information from studies in which data on C were collected.

We conduct a two-stage meta-analysis in all approaches. First, a logistic regression model on the outcome Y is fit at each study site, including A’ and, where available, C. Second, the effect estimates are pooled with a meta-analytical model with random effects using restricted maximum likelihood.

Figure 1 presents a forest plot showing the individual study effect estimates and the pooled estimate for the control approach without missing data including all 10 studies. The pooled OR is 1.4 with a 95% Cl between 1.2 and 1.7. Cochran’s Q test indicates evidence against the null hypothesis of homogeneity, suggesting that the observed differences between studies are unlikely to be due to sampling error alone.

Figure 1.

Two-stage meta-analysis of 10 observational studies without missing data

To include all studies in the analysis and adjust for the confounding variable C at study sites where data have not been collected, we used mi impute from to impute C in studies 8, 9, and 10. All studies with complete values on C are used as the basis for imputation. First, a logistic regression model is fit on C with the covariates Y and X in studies 1-7.

Second, the seven prediction models are pooled with a multivariate random-effects meta-analysis using restricted maximum likelihood. We use meta mvregress for this step. The extended do-file is available in the supplementary materials. To initiate the federated analysis, we first create a data frame, metadata, and execute the analysis in the seven studies without missing data on C. We then use the final set of regression coefficients and covariance structure from the multivariate random-effects meta-analysis with mi impute from to execute the analysis after multiple imputation. The set of imputation regression coefficients and covariance structure is saved in the matrices ib and iV, respectively. We specify logit in the imodel option for mi impute from to indicate that we used a logistic regression model as a regression method for the imputation of C. The missing confounder C is imputed 10 times. Finally, mi estimate is used to fit the substantive model in each imputed dataset and pool the results. The final estimates are then placed in the new data frame, metadata, for the subsequent meta-analysis.

Figure 2 shows the results of all implemented approaches (panel A: Complete case analysis; panel B: Available data analysis; and panel G: Analysis after multiple imputation). The complete case analysis underestimates the overall effect estimate compared with the control approach (OR: 1.18, 95% GI: [1.03,1.35]). The available data estimate, on the other hand, overestimates the effect of the exposure on the outcome (OR: 1.56, 95% GI: [1.18,2.06]). After using mi impute from, the individual effect estimates in study 8, 9, and 10 were adjusted for the imputed confounder C. The estimates were similar to the ones in the analysis without missing data (control approach). The pooled OR (OR: 1.40, 95% GI: [1.16,1.69]) is comparable with the one of the control approach (OR: 1.41, 95% GI: [1.16,1.72]).

Figure 2.

Two-stage meta-analysis of A) 7 studies excluding those with systematically missing data; B) 10 studies excluding the systematically missing variable from the regression model in studies 8, 9, and 10; and G) 10 studies after using multiple imputation to impute the systematically missing confounder in studies 8, 9, and 10. The black line represents the pooled estimate from the control approach (OR: 1.41).

4.3 Example 3: Multisite trial missing an effect modifier

Multisite trials are often considered to study differential treatment effects (Riley, Lambert, and Abo-Zaid 2010). In practice, the effect of a treatment commonly depends on different levels of a third factor, such as disease stage (Riley et al. 2020). The rationale for a collaborative effort involving multiple sites is then to increase the possibility to detect such differential treatment effects and increase the generalizability of the findings and the diversity of the participants. In this example, we consider two large randomized controlled trial sites with a randomly allocated treatment, X, and an important discrete-effect modifier, Z, with three levels (low, medium, high). The outcome was defined as time to death from all causes. It is hypothesized that the treatment has beneficial, null, and harmful effects at the low, medium, and high levels of the effect modifier, respectively. Therefore, the target of statistical inference is the effect of the treatment at the three levels of Z. In trial site 1, however, the categorical variable Z has not been measured. Trial site 2 has observed data on Z that can be used to specify a prediction model for the imputation of Z. Table 2 shows the characteristics of the two sites.

Table 2.
Characteristics of trial sites 1 and 2, including the number of cases, the baseline failure rate per 1,000 person-years, and the estimated hazard ratio (HR.) (95% Cl) of the treatment X at the low, medium, and high levels of the effect modifier Z.

Trial site 1 Trial site 2

N = 4000 N = 3175

Number of cases 1,893 2,074

Rate (per 1,000) 64.9 107.5

HR _x _| _z = o (95% Cl) NA 0.5 (0.4, 0.6)

HR _x _| _z = i (95% Cl) NA 0.8 (0.7, 0.9)

HR _x _| _z = 2 (95% Cl) NA 1.5 (1.3, 1.7)

	Trial site 1	Trial site 2
Number of cases	1,893	2,074
Rate (per 1,000)	64.9	107.5
HR _x _\| _z = o (95% Cl)	NA	0.5 (0.4, 0.6)
HR _x _\| _z = i (95% Cl)	NA	0.8 (0.7, 0.9)
HR _x _\| _z = 2 (95% Cl)	NA	1.5 (1.3, 1.7)

To be able to include both sites in the analysis, we specify a multinomial logistic regression model in trial site 2 and export the regression coefficients to trial site 1. The imputation model must be specified congenial to the outcome model (Meng 1994; White, Royston, and Wood 2011). Thus, interaction terms between 1) the treatment and the cumulative hazard (x_cumh) using the Nelson-Aalen estimator (White and Royston 2009) and 2) the treatment and indicator variables for death (x_d) are included in the imputation model. In mi_impute_from_get, the three levels of the effect modifier Z are specified in values (0 1 2). In addition, we model the interaction effects of the treatment X and the effect modifier Z with indicator variables. Therefore, the survival model fitting the effect of the treatment X at the low, medium, and high levels of the effect modifier Z is obtained with the main effect (x) and two product terms (x_zil and x_zi2). Before missing values for Z at trial site 2 are imputed, all indicator variables are registered as passive imputation variables. Finally, the missing values of the effect modifier Z are imputed 10 tsimes.

After we impute the missing values for the effect modifier Z, a Cox regression model is fit in each imputed dataset, and the estimates are pooled with Rubin’s rules.

At trial site 1, the effect of the treatment is estimated to reduce the mortality rate by 46% (HR = 0.54; 95% Cl = [0.28,1.02]) and 22% (HR = 0.78, 95% Cl = [0.55,1.11]) in the low and medium levels of the effect modifier Z, respectively. On the contrary, the effect of the treatment in the high level of the effect modifier Z conferred a 62% higher mortality rate (HR = 1.62; 95% Cl = [1.10, 2.40]).

Of note, the set values of the trial sites were ${HR}_{X | Z = 0} = 0.5, {HR}_{X | Z = 1} = 1$ , and ${HR}_{X | Z = 2} = 1.5$ . Additionally, before Z was set to 100% missing in trial 1, the estimated HRs were ${HR}_{X | Z = 0} = 0.49 (95 % CI = 0.40, 0.61)$ , ${HR}_{X | Z = 1} = 0.98 (95 % CI = [0.85, 1.12])$ , and ${HR}_{X | Z = 2} = 1.67 (95 % Cl = [1.44, 1.94])$ . Therefore, we know that we have not introduced substantial bias in the imputed estimates of the HR in trial site 1 and were able to include both sites in the overall analysis. The postestimation after following mi estimate was done using mi predictnl.

4.4 Example 4: Using mi impute from to evaluate potential predictors

Developing prediction models based on large observational studies is common in many health-related research fields (Debray et al. 2017). In this example, we consider a large study in which a prediction model for the outcome Y is specified with all available predictors, say, X and C. The predictive capacity of the model is quantified with the area under the curve (AUC) and is estimated at AUC = 0.662. In another external study, a similar prediction model has been fit. However, additionally, this study has collected data on a strong binary predictor, Z, and the discriminative ability of the model is fit at AUC = 0.821. The investigator of study 1 would like to evaluate to what extent the predictor Z from study 2 improves the AUC before potentially collecting individual data on Z for study 1. To understand the potential impact of including the predictor Z, let us use information on Z from study 2 and use mi impute from. In this example, mi impute from is not used in the context of multisites studies or federated analyses but to inform the potential impact of adding a variable to a model that has not been collected previously. Table 3 shows the sample size and AUC of both studies.

Table 3.
Sample size and AUC of the prediction model in study 1 and the external study 2

Study 1 External study 2

N = 24087 N = 23614

AUC without predictor Z 0.662 0.715

AUC with predictor Z NA 0.821

	Study 1	External study 2
AUC without predictor Z	0.662	0.715
AUC with predictor Z	NA	0.821

To evaluate the impact of Z on the predictive performance of the model in study 1, we first need to get information on Z in relation to other observed variables in study 2. In study 2, a logistic regression model is specified on Z conditional on Y, X, and C. The regression coefficients and covariance matrix are exported as text files to study 1. The text files are then used for mi_impute_from_get. The missing predictor Z is imputed 10 times.

A multivariate logistic regression model is fit in each imputed dataset to obtain 10 estimates for the AUC. A final estimate of the AUC can be obtained as an average of the estimates across imputed datasets. With 10 imputations, the AUC of the prediction model in study 1 increased from 0.662 to 0.796 after including the predictor Z. in figure 3 , the distribution of the AUC is shown after adding 1.000 imputations. Across imputations, the AUC is centered around the average value of 0.795, ranging from 0.780 to 0.805.

Figure 3.

Distribution of the AUC of 1,000 imputations of the predictor Z in study 1

This example illustrated how mi impute from can be used for the design of a prediction model by borrowing information from another study to evaluate the improvement of the discriminative ability in the presence of an additional predictor.

4.5 Utility of mi impute from

The main use of mi impute from is in imputation settings for multisite studies and federated analyses, where some studies have completely (that is, systematically) missing values on key covariates such as confounding or effect-modifying variables. We illustrated its use in multiple applied examples. The command is particularly useful in the context of growing interest in federated learning and distributed data analysis, where individual participant data cannot be shared because of privacy or regulatory constraints. This is increasingly relevant in fields such as genetics (McCaw et al. 2024; D’Altri et al. 2025) or pharmacoepidemiology (Secrest et al. 2020), where large-scale collaborative analyses are essential and systematically missing data present major methodological challenges.

Additionally, we showed that mi impute from can be used beyond standard imputation frameworks—for example, to evaluate the potential impact of predictors that are not observed in the data. This use is conceptually like quantitative bias analysis, where the robustness of results is assessed under assumptions about unmeasured or missing variables (Lash et al. 2014). In both approaches, users specify plausible distributions or associations for variables that are not observed, allowing sensitivity analyses that assess how inferences might change.

Related concepts also appear in econometrics, where methods such as two-sample two-stage least-squares estimate associations between variables that are never jointly observed, by combining information from structurally related but nonoverlapping datasets (Bjorklund and Jantti 1997; Inoue and Solon 2010). While primarily developed for causal inference, these methods share important features with the described imputation- based strategy for systematically missing values: both rely on shared variables and structural assumptions to recover missing or unobserved information. These parallels illustrate how mi impute from aligns with a broader class of methods designed to synthesize fragmented or incomplete data across studies and domains.

5 Conclusions

We introduced and illustrated a new command, mi impute from, that allows users to impute missing values using regression coefficients derived from external data, without requiring access to individual-level data. This approach supports both sporadically and systematically missing variables—a feature not currently available in other Stata imputation commands.

By enabling users to share model parameters instead of individual data, mi impute from provides a practical solution for data protection challenges, particularly in multisite or federated settings. The command supports a range of variable types and can be extended to accommodate additional imputation models.

While designed for use in the multiple-imputation framework, the method also lends itself to structured sensitivity analysis, allowing researchers to explore the impact of unmeasured or unavailable variables. Future development will focus on improved diagnostics, potential integration with the mi impute chained framework, and use in federated learning workflows.

6 Acknowledgments

This work was supported by the National Infrastructure NEAR, supported by the Swedish Research Council [grant numbers Dnrs 2017-00639 and 2021-00178].

7 Programs and supplemental materials

To install the software files as they existed at the time of publication of this article, type

A more recent version of the command may be available from the Statistical Software Components Archive by typing

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Footnotes

About the authors

Robert Thiesmeier is a doctoral student in medical science (biostatistics) at the Department of Global Public Health, Karolinska Institutet, Stockholm, Sweden.

Matteo Bottai is a professor of biostatistics at the Division of Biostatistics, Institute of Environmental Medicine, Karolinska Institutet, Stockholm, Sweden.

Nicola Orsini is an associate professor of medical statistics, a senior researcher, and the head of the Biostatistics Team at the Department of Global Public Health, Karolinska Institutet, Stockholm, Sweden.

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Supplementary Material

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	Trial site 1	Trial site 2
	N = 4000	N = 3175
Number of cases	1,893	2,074
Rate (per 1,000)	64.9	107.5
HR _x _\| _z = o (95% Cl)	NA	0.5 (0.4, 0.6)
HR _x _\| _z = i (95% Cl)	NA	0.8 (0.7, 0.9)
HR _x _\| _z = 2 (95% Cl)	NA	1.5 (1.3, 1.7)

	Study 1	External study 2
	N = 24087	N = 23614
AUC without predictor Z	0.662	0.715
AUC with predictor Z	NA	0.821

Imputing missing values with external data: Applications for multisite settings and federated analyses

Abstract

Keywords

1 Introduction

2 Methods

2.1 Imputation of missing values using external prediction equations

2.2 Imputation methods

2.2.1 Continuous variables

2.2.2 Categorical variables

2.2.3 Binary variables

3 The syntax

3.1 mi impute from

3.1.1 Options

3.1.2 Stored results

3.2 mi_impute_from_get

3.2.1 Options

3.2.2 Stored results

4 Examples

4.1 Example 1: Missing continuous confounder

4.1.2 Using multiple external studies

4.2 Example 2: Missing confounder in a federated analysis with 10 large studies

Table 3. Sample size and AUC of the prediction model in study 1 and the external study 2 Study 1 External study 2 N = 24087 N = 23614 AUC without predictor Z 0.662 0.715 AUC with predictor Z NA 0.821

5 Conclusions

6 Acknowledgments

7 Programs and supplemental materials

Supplemental Material

sj-dta-1-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

Supplemental Material

sj-dta-2-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-3-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-4-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-5-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-6-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-7-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-8-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-dta-9-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

Supplemental Material

sj-dta-10-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-11-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-12-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-13-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-14-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-15-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-16-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-17-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-18-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-19-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-20-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-21-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-22-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-23-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-24-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-25-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-26-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

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sj-txt-27-stj-10.1177_1536867X251398605 - Supplemental material for Imputing missing values with external data: Applications for multisite settings and federated analyses

Supplemental Material

Table 3.
Sample size and AUC of the prediction model in study 1 and the external study 2

Study 1 External study 2

N = 24087 N = 23614

AUC without predictor Z 0.662 0.715

AUC with predictor Z NA 0.821