Why Historical Data Format Matters
A historical control data set may arrive as participant-level records or as a report containing only totals, means, and standard deviations. Either format can contribute information through a power prior, but the available likelihood depends on what was retained. Overall event counts, for example, can support a binomial model for an overall event probability. They do not, by themselves, reveal how event risk varies with age, disease severity, or another covariate.
The key question is not simply whether the data are called “aggregate” or “individual.” It is whether the information supplied is sufficient for the particular historical model being used. Some aggregates preserve the full likelihood for a model; others discard associations needed for covariate adjustment. Individual records can support more detailed adjustment, provided the relevant variables were measured and are appropriately harmonized.
For fixed \(a_0\), a factor in the likelihood that does not depend on \(\theta\) does not affect the posterior distribution of \(\theta\). This is why a sufficient summary can replace individual records for a specified model. But a summary sufficient for one model need not be sufficient for a more detailed model that adjusts for covariates.
When Aggregates Preserve the Likelihood
A summary preserves the likelihood for a model when the likelihood based on the full records can be expressed using that summary, up to factors that do not depend on the model parameters. In that case, using the summary loses no information about those parameters under that model. The qualification “under that model” is essential: sufficient information for an overall event probability may not be sufficient for a regression coefficient.
Worked Example: Overall Binomial Data
Suppose a hypothetical historical control group has 8 events among 20 participants. Assume outcomes are independent Bernoulli observations with a common event probability \(p\), with no covariates in the model. Let \(S=8\) be the event count and \(n=20\). The likelihood is
For these data, this is proportional to \(p^8(1-p)^{12}\). With fixed \(a_0=0.5\), the powered likelihood is proportional to
If the base prior is \(p\sim\operatorname{Beta}(1,1)\), the resulting conditional power prior for \(p\) is \(\operatorname{Beta}(5,7)\): the beta shape parameters are \(1+4\) and \(1+6\). The calculation uses the event count and total, not the identities of the 20 participants. Under the common-probability Bernoulli model, those totals retain all the information needed for this likelihood.
This does not establish that adjustment is unnecessary. It establishes only that the totals are sufficient for the stated model. If risk differs by covariate and the historical covariate distribution differs from the current trial’s distribution, the overall event count cannot recover that relationship.
Why Overall Summaries Cannot Usually Adjust for Covariates
Suppose a continuous historical control outcome \(Y_i\) is modeled as a function of a covariate \(x_i\). An overall outcome mean and an overall covariate mean do not generally determine the relationship between \(Y_i\) and \(x_i\). In particular, they do not tell us whether higher outcomes occurred among participants with higher or lower covariate values.
Worked Example: Same Means, Different Covariate Relationships
Consider two hypothetical historical data sets, each with four control participants and a binary covariate \(x\). Both have two participants with \(x=0\), two with \(x=1\), \(\bar{x}=0.5\), and \(\bar{y}=4\).
In data set A, the \((x,y)\) pairs are \((0,1),(0,3),(1,5),(1,7)\). The group means are 2 for \(x=0\) and 6 for \(x=1\). In data set B, the pairs are \((0,5),(0,7),(1,1),(1,3)\). The group means are 6 for \(x=0\) and 2 for \(x=1\). Thus, the same overall outcome mean and the same covariate distribution are compatible with opposite covariate-outcome patterns.
For the linear regression model \(Y_i=\beta_0+\beta_1x_i+\epsilon_i\), the least-squares slope for a binary \(x\) is the difference in group means. In data set A, \(\hat{\beta}_1=6-2=4\), and \(\hat{\beta}_0=2\). In data set B, \(\hat{\beta}_1=2-6=-4\), and \(\hat{\beta}_0=6\). The overall means alone cannot distinguish these fitted relationships. A power prior based only on those means therefore cannot represent the historical regression likelihood for \((\beta_0,\beta_1)\).
With individual records, the historical model can include measured covariates directly. The likelihood then reflects the joint pattern of outcomes and covariates rather than only their marginal summaries. Adjustment is still conditional on the model assumptions: the covariates must be defined consistently across historical and current data, measured with adequate quality, and included in a suitable outcome model. Adjustment for measured variables cannot correct differences due to unmeasured variables or make incompatible endpoints comparable.
A Justification Using the Normal Linear Model
The distinction can be made precise for a normal linear regression. Suppose the historical outcomes satisfy \(Y\mid X\sim N(X\beta,\sigma^2 I)\), where \(X\) is the design matrix, \(\beta\) is the regression coefficient vector, and \(\sigma^2\) is known. This assumes independent outcomes, a correctly specified linear mean, and a common known variance. The likelihood is
Expanding the quadratic gives
Consequently, for this model the likelihood’s dependence on \(\beta\) is fully determined by \(X^\mathsf{T}X\), \(X^\mathsf{T}y\), and \(y^\mathsf{T}y\). These cross-products are sufficient summaries for the regression likelihood when \(\sigma^2\) is known. Raising the likelihood to \(a_0\) does not change which summaries appear: it multiplies the exponent by \(a_0\).
This derivation explains why the contrast is not simply “aggregates cannot adjust.” If a report supplies the required cross-products, or equivalent sufficient statistics, the specified regression likelihood can be recovered without the original rows. But an overall mean and standard deviation ordinarily do not supply the cross-products needed to estimate covariate effects. If the variance is unknown, additional information about the residual variation is required; summaries should not be assumed sufficient without checking them against the exact model.
Worked Example: Stratified Summaries Support Adjustment
Suppose a hypothetical report gives historical control outcomes separately for two levels of a binary severity covariate. There are 10 participants with \(x=0\), whose outcomes sum to 20, and 10 with \(x=1\), whose outcomes sum to 50. Assume a model with a separate normal mean in each stratum and a common known variance. The stratum means are
The overall mean is \((20+50)/(10+10)=3.5\). It hides the difference between strata, whereas the stratified summaries retain the stratum-specific outcome information. For this model, each stratum’s count and outcome sum determine the likelihood’s dependence on its mean parameter, apart from parameter-independent factors.
For illustration, if a target population is 80% \(x=1\) and 20% \(x=0\), its mean under these stratum means is \(0.8(5)+0.2(2)=4.4\). This standardization uses the stratum-specific means and the stated target covariate proportions; it is not the historical overall mean of 3.5. The calculation assumes the two stratum means are applicable to the target population and that the covariate and outcome definitions are compatible. The reported summaries permit this particular stratified adjustment; a report containing only the overall mean would not.
Choosing and Checking the Data Representation
Before constructing the historical contribution, specify the historical outcome model and then ask whether the available data identify its likelihood. The answer depends on both the data format and the parameters to be estimated. A report may provide enough information for an unadjusted binomial model, but not for a logistic regression. A table of outcomes by prespecified strata may support stratified borrowing, while an overall total does not.
| Historical data supplied | What it can support | What it does not generally recover |
|---|---|---|
| Overall event count and total | Common-probability binomial likelihood | Covariate-specific event risks or regression effects |
| Overall mean and sample size | Some specified models for an overall mean, with required variance information | Outcome-covariate associations or subgroup means |
| Counts and outcome summaries by covariate stratum | Adjustment using the reported strata and a compatible stratified model | Adjustment for unreported covariates or finer groupings |
| Individual outcomes and covariates | Models using recorded covariates, subject to model and data quality assumptions | Adjustment for unmeasured covariates or unsupported extrapolation |
| Sufficient cross-products for a specified regression | The corresponding regression likelihood without retaining each row | Models requiring information beyond the supplied cross-products |
A useful diagnostic is to write the likelihood before deciding that the summaries are adequate. Identify every term involving the model parameters, then check whether the reported quantities determine those terms. For covariate models, specifically check whether the supplied data retain the joint outcome-covariate information, not merely each variable’s separate marginal summary. If the answer is unclear, the summary should not be treated as if it were the full likelihood for an adjusted model.
Individual records are not automatically superior in every practical respect. They can enable adjustment and more flexible checks, but require access, harmonized variable definitions, and a defensible model. Aggregates may be easier to obtain and can be entirely adequate when they are sufficient for the prespecified model. Conversely, an aggregate analysis may force an unadjusted model because the necessary covariate information was never reported. The appropriate comparison is therefore between likelihoods supported by the available information, not between formats in the abstract.
Common Mistakes and Limitations
- Assuming every aggregate is sufficient. An event count is sufficient for a common-probability binomial model, not for every model that could be fitted to the participants.
- Assuming aggregates can never support adjustment. Covariate-stratified summaries or model-specific sufficient statistics can preserve the likelihood needed for a stated adjustment.
- Adjusting using marginal summaries alone. Separate overall summaries of outcome and covariate do not generally contain their association. The worked regression example shows that opposite slopes can produce the same marginal means.
- Treating adjustment as a cure for noncomparability. Adjustment requires measured, harmonized covariates and an adequate model. It does not remove bias from unmeasured differences or incompatible outcome definitions.
- Powering an incomplete likelihood as though it were complete. Raising an unadjusted likelihood to a discount does not restore covariate information that was omitted from the historical summaries.
The power prior’s role is to discount a specified historical likelihood, not to determine whether that likelihood represents the historical data appropriately. Choosing the data representation and model therefore comes before applying the discount. The principal limitation of aggregate data is information loss relative to models that require joint participant-level patterns; the principal limitation of individual records is that more modeling options do not guarantee valid adjustment.
Check Your Understanding
Answer each question, then compare your reasoning with the short solutions.
- A report gives 12 events among 30 historical controls. Under what model can these totals supply the historical likelihood, and what information would be missing for covariate adjustment?
- Two data sets have the same overall outcome mean and the same distribution of a binary covariate. Can their fitted regression slopes differ? Explain.
- A report supplies outcome counts and means separately for each of three harmonized severity groups. What kind of adjustment may these summaries support, and what limitation remains?
- For a normal linear model with known variance, name the cross-products that determine the likelihood for the regression coefficients.
Solutions. (1) The totals supply the likelihood under an independent Bernoulli model with a common event probability. They do not generally give event patterns by covariate or covariate-outcome associations. (2) Yes. The slope depends on how outcomes vary across covariate groups, which the two marginal summaries do not specify. (3) They may support a stratified model using the three reported groups, assuming the model and variable definitions are suitable; they cannot support adjustment for unreported covariates or finer distinctions. (4) \(X^\mathsf{T}X\), \(X^\mathsf{T}y\), and \(y^\mathsf{T}y\) determine the likelihood’s parameter-dependent quadratic form.