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Target aggregate data adjustment transportability analysis

Core Clinical Sciences

transportTADA.Rmd

Introduction

In this vignette, we demonstrate how to use TransportHealth for weighting-based transportability analysis when individual participant-level data (IPD) is available for the original population, while only aggregate-level data (AgD) is available for the target sample.

Brief introduction to Target aggregate data adjustment (TADA)

In transportability and generalizability analysis using target aggregate data adjustment (TADA), the process differs from traditional inverse odds of participation weighting (IOPW) or g-computation due to the absence of individual-level data (IPD) in the target population. Instead of using logistic regression to obtain weights that balance the distribution of effect modifiers in the source and target data, TADA uses a method of moments approach with the aggregate-level data (AgD) from the target population to account for effect modifiers. This is combined with the usual propensity score weighting approach to adjust for confounding in the source data to fit marginal structural models.

Example

Suppose we are interested in estimating the causal effect of a medication on systolic blood pressure in a target population, but we were only able to conduct an observational study using samples from the study population. To obtain unbiased causal effect estimates using the study sample, we account for the following covariates: sex, body fat percentage, and stress level.

We know that the effectiveness of the medication depends on two effect modifiers: 1) stress level, and 2) whether patients are taking another medication.

Note that the covariates adjusted for in the study data can also be effect modifiers.

Coded variables:

  • Medication - med1

    • 1 for treated
    • 0 for untreated

  • Systolic blood pressure (SBP) - sysBloodPressure (continuous)

  • Sex - sex

    • 1 for male
    • 0 for female

  • Body fat percentage - percentBodyFat (continuous)

  • Stress level - stress

    • 1 for stressed
    • 0 for not stressed

  • Medication 2 - med2

    • 1 for treated
    • 0 for untreated

Analyses

First, for the implementation of TADA in TransportHealth specifically, the data from the study and target population should be in separate data frames in the R environment. The study data should contain IPS on response, treatment, covariates and effect modifiers, while the target data should contain only aggregate-level summary statistics of covariates and effect modifiers. These summary statistics should be formatted as proportions, counts, means, medians and standard deviations.

Suppose that we have the study and target data separately as follows.

names(testData)
#> [1] "studyData"           "aggregateTargetData"
print("Study data:")
#> [1] "Study data:"
head(testData$studyData)
#>   sysBloodPressure med1 sex stress med2 percentBodyFat
#> 1         105.5173    1   0      0    0       26.84295
#> 2         107.6679    1   1      0    0       14.66178
#> 3         116.3453    0   1      1    0       11.16153
#> 4         116.7964    0   0      1    0       24.93147
#> 5         117.7243    0   0      1    0       25.77048
#> 6         117.1051    0   0      0    0       31.19562
print("Target data:")
#> [1] "Target data:"
head(testData$aggregateTargetData)
#>      N sex_count sex_prop stress_prop med2_PROP percentBodyFat_mean
#> 1 1500       474     0.93   0.6966667     0.302            22.14501
#>   percentBodyFat_sd percentBodyFat_median
#> 1          5.857062               24.6448

Here for the target data, the names of the aggregate level statistics corresponding to each variable should be formatted as follows.

  • For continuous variables, the mean of a given variable should be named variable_mean, the standard deviation should be named variable_sd, and the median should be named variable_median.

  • The proportion or count of a binary variable can be named variable_prop or variable_count, respectively (corresponding to the non-reference category in the study data).

  • The naming method of categorical variables (with more than two levels) is specified as variable_category_AgD. For example, for an ordinal variable grade with three categories (low, medium and high), the proportions of each category recorded in the target data should be named grade_low_prop, grade_medium_prop and grade_high_prop. This is just to demonstrate the naming format for categorical variables; there are no categorical variables in this example.

We can now perform transportability analysis using TADA with the transportTADA functions.

transportTADA(msmFormula, 
              propensityScoreModel,
              matchingCovariates,
              propensityWeights,
              participationWeights,
              treatment,
              response,
              family,
              studyData,
              aggregateTargetData)

Arguments for the transportTADA function

This function requires specification of the following arguments:

  • msmFormula: A formula expressing the marginal structural model (MSM) to be fit.

  • propensityScoreModel: A formula or a glm object expressing the propensity model, i.e. a model of treatment assignment in terms of covariates.

If a formula is provided, logistic regression is used by default. Custom propensity weights from other weighting methods can also be provided to the customPropensity argument instead; in this case, do not set propensityScoreModel because it is NULL by default and will be overridden. Note that as IPD is available for the study data, it is still possible to use logistic regression to estimate propensity scores.

  • matchingCovariates: A vector of user-specified covariates which are used for the data matching to obtain participation weights when no custom weights provided.

  • propensityWeights: A vector of custom weights balancing covariates between treatments. Providing them will override the formula or model provided by propensityScoreModel. This vector should have as any entries as the sample size of the study data.

  • participationWeights: A vector of custom weights balancing effect modifiers between study and target populations. This vector should have as any entries as the sample size of the study data.

  • treatment: The name of the variable indicating treatment.

  • response: The name of the variable indicating response.

  • family: The type of outcome model to be fit. This can be any of the families used in glm, one of "coxph" or "survreg". The "coxph" and "survreg" options are for survival analysis and will use default options of these methods from the survival package.

  • studyData: The individual participant data (IPD) of study population.

  • aggregateTargetData: The aggregate-level data (AgD) of target population.

Specification of transportability analysis

These components are put together as follows.

Recall that: - sysBloodPressure is the response

  • med1 is the treatment

  • sex, percentBodyFat, toxicGrade, and stress are covariates to be controlled in the original study

  • med2 (other medication) and stress are effect modifiers of interest.

result <- transportTADA(
  # MSM formula
  msmFormula = sysBloodPressure ~ med1,
  
  # Propensity model
  propensityScoreModel = med1 ~ sex + percentBodyFat + stress,
  
  # Matching covariates
  matchingCovariates = c("stress", "med2"),
  
  # Name of treatment variable
  treatment = "med1", 
  
  # Name of response variable
  response = "sysBloodPressure",
  
  # Type of MSM
  family = gaussian,
  
  # Study data
  studyData = testData$studyData,
  
  # Target data
  aggregateTargetData = testData$aggregateTargetData)
#> Warning in processedAgD(aggregateTargetData): Following columns are ignored
#> since it does not follow the naming conventions:N

Producing statistical results

To show the results of the analysis, we can use the base::summary function, similar to how one would use the lm function for fitting a linear model.

The base::summary function prints out a table of unweighted and weighted SMDs of covariates between treatment groups, a pre-post weighting table which includes unweighted and weighted aggregate level summary as well as pre- and post- difference values of effect modifiers of interest between study data and target data, and summaries of the fitted outcome model and the final marginal structural model (MSM) fit with the correct standard errors calculated using bootstrap.

Note that scientific conclusions should only be drawn from the MSM.

summary(result)
#> Absolute SMDs of covariates between treatments before and after weighting:
#>                       variable         smd   method
#> sex                        sex 0.294175794 Observed
#> percentBodyFat  percentBodyFat 0.300317047 Observed
#> stress                  stress 0.072111181 Observed
#> sex1                       sex 0.001101437 Weighted
#> percentBodyFat1 percentBodyFat 0.001912183 Weighted
#> stress1                 stress 0.003064457 Weighted
#> Aggregate level data summary of effect modifiers of interests before and after weighting:
#>   Effect Modifiers Study (N = 1000) Target (N = 1500) Study (Post-Weighting)
#> 1        MED2_PROP            0.086             0.302                  0.299
#> 2      STRESS_PROP            0.415             0.697                  0.701
#>   Pre Weighting Difference Post Weighting Difference
#> 1                    0.216                     0.003
#> 2                    0.282                    -0.004
#> MSM results:
#> 
#> Call:
#> stats::glm(formula = msmFormula, family = family, data = toAnalyze, 
#>     weights = finalWeights)
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  115.856      0.192   604.8   <2e-16 ***
#> med11         -0.763      0.696    -1.1     0.27    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for gaussian family taken to be 14.60882)
#> 
#>     Null deviance: 14681  on 999  degrees of freedom
#> Residual deviance: 14580  on 998  degrees of freedom
#> AIC: 6800
#> 
#> Number of Fisher Scoring iterations: 2

Positivity and conditional exchangeability are two key assumptions that can affect the validity of transportability analysis.

Positivity is the assumption that at all observed levels of covariates and effect modifiers, the probabilities of being in the treatment group and being the study are neither 0 nor 1, respectively.

To evaluate this assumption for the treatment assignment, use the plot function with type = "propensityHist". This outputs mirrored histogram of probabilities of being in the treatment group for different treatment groups. Non-overlap of the ranges of the histograms suggest violations of the positivity assumption (Wizard, n.d.b).

base::plot(result, type = "propensityHist")

The plot function with type = "participationHist" provides the histogram of participation weights and the effective sample size (ESS), which can be used to assess the suitability of the TADA method. For example, the presence of extreme weights or a large reduction in effective sample size may indicate that the assumptions for TADA are violated.

base::plot(result, type = "participationHist")

Conditional exchangeability is roughly the assumption that the only possible confounding is due to the controlled covariates and effect modifiers. Under this assumption, TADA estimates will be reliable if the weighted distributions of covariates and effect modifiers are similar between treatment groups and the study data, respectively. This can be (partially) evaluated using standardized mean differences (SMDs), which are shown in table form by the summary function. The plot function with type = "propensitySMD" provides graphical versions of this table. A general guideline is that an SMD of below 0.1 indicates balance, but this threshold is arbitrary and left to the choice of the analyst (Wizard, n.d.a).

base::plot(result, type = "propensitySMD")

Model coefficient plots showing confidence intervals of the effect estimates are provided by plot function with type = "msm". The standard errors are the correct ones calculated by bootstrap.

base::plot(result, type = "msm")

Note that all plot outputs are designed to be generated with ggplot2. They can be customized further.

References

Wizard, Causal. n.d.a. “Covariate Balance.” https://causalwizard.app/inference/article/covariate-balance.
———. n.d.b. “The Positivity Assumption.” https://causalwizard.app/inference/article/positivity.