In a traditional clinical trial, there is normally one variable under investigation.

Are there any existing mathematical foundations for testing two or more variables in a clinical trial setting?

The reason I ask? There are many diseases that require multiple biomarkers to be pushed back into the optimum range before the symptoms of the disease will recede.

The answer not only has to be mathematically correct, it must be simple enough to persuade non-mathematical people in a healthcare regulatory setting to accept a multivariate clinical trial.

The perfect answer would be a simple, compelling paragraph, with appropriate references to well respected peer-reviewed papers. As existing clinical trials usually operate under a frequentist assumption, the ideal answer would make minimal changes to the traditional methods.

p.s. This answer could save millions of lives, and improve the health of millions more, by allowing clinical trials that can deal with health interventions that involve more than one variable.

  • $\begingroup$ It's not really up to me. One of my associates had a proposal for a clinical trial rejected as it used more than one variable. If he ends up using this answer in any papers, letters or correspondence, I will recommend that he cites the author (due to the obvious risks of plagiarism). $\endgroup$
    – Contango
    Commented Sep 26, 2016 at 9:41
  • 1
    $\begingroup$ If you really mean to treat this as a multi-variate (ie vector-valued) outcome then there should be no problem as it is only one primary analysis. Do you want to treat them all separately? $\endgroup$
    – mdewey
    Commented Sep 26, 2016 at 10:36
  • $\begingroup$ I probably should have phrased the question better. In order to cure a single disease, the trial involves making five interventions to drive five different biomarkers into the "green" zone. Thus the goal for every patient in the treatment group is identical, but the exact amount of drugs or supplements given to each patient may vary depending on the biomarker results of their blood tests. $\endgroup$
    – Contango
    Commented Sep 27, 2016 at 15:59
  • $\begingroup$ @Contango more than one variable or more than one outcome? If you use an appropriate intent-to-treat analysis, there is really only one treatment: treat the 5 biomarkers. I have seen many studies along these lines approved. I have also seen some rejected, but not because of the complexity of the intervention. The biggest issue is that biomarkers are surrogate endpoints and those don't matter for squat. $\endgroup$
    – AdamO
    Commented Jan 22, 2018 at 21:22
  • $\begingroup$ (-1) for lack of clarity and incendiary approach around phrasing the question. $\endgroup$
    – AdamO
    Commented Jan 22, 2018 at 21:25

2 Answers 2


the scenario you describe is not unusual. For example you could reference Butler et al. Circ Heart Fail 2017 "It is important to move away from over-reliance on a single primary end point, declaring an entire trial neutral or negative based on it. Thoughtful assessment of numerous important data should allow for the totality of the information to inform subsequent development decisions.”

or you could refer to the Medical Research Council guideline on Developing and Evaluating Complex Interventions states: “A single primary outcome, and a small number of secondary outcomes, is the most straightforward from the point of view of statistical analysis. However, this may not represent the best use of the data, and may not provide an adequate assessment of the success or otherwise of an intervention which may have effects across a range of domains”

there are a few approaches you could take: 1) analyse endpoints separately requiring control of alpha, 2) derive a univariate measure that is a function of the endpoints ie a composite endpoint, or 3) model the endpoints simultaneously allowing for correlations between outcomes ie multivariate modelling

for 1) you could consider eg Hochberg which is a common approach. For 2) you could consider the average z-score as per Sun et al. And for 3) you could see eg Mascha et al. or Brown et al. for the multivariate modelling option. Re the most cogent analysis to persuade non-mathematical people, this will likely push you towards a composite outcome. Re the need for a 'traditional' method, all three approaches are tenable, the modelling approach may offer superior power however but comes with additional assumptions. References follow

Craig P, Dieppe P, Macintyre S, Michie S, Nazareth I, M. P. Developing and evaluating complex interventions: new guidance: Medical Research Council; 2006 [Available from: https://www.mrc.ac.uk/documents/pdf/complex-interventions-guidance/

Hochberg. A sharper bonferroni proecedure for multiple tests of significance. Biometrika 1988

Sun et al. Evaluating Treatment Effcacy by Multiple End Points in Phase II Acute Heart Failure Clinical Trials. Circ HF 2012

Brown et al. Multitype Events and the Analysis of Heart Failure Readmissions Illustration of a New Modeling Approach and Comparison With Familiar Composite End Points. Circ Cardiovasc Qual Outcomes 2017

Mashca et al. Design and Analysis of Studies with Binary Event Composite Endpoints: Guidelines for Anesthesia Research. Anesth Analg 2011


This issue is not new, but a unified approach is lacking. A nice, brief review with good references is discussed in Analysis of Longitudinal Data by Diggle Heagerty Liang Zeger ch 14 section Multivariate longitudinal data. Ronald Brookmeyer at UCLA has been particularly interested in latent variable models where there is one or two latent variables affected by a process which in turn feed hierarchically into a process for each outcome.

Single-exposure-multiple-outcome models using scaled outcomes and treatment/test interactions for long-transformed data can be estimated using just fixed effects.

Growth models and their analogues were discussed by Bryan, Heagerty in a recent Statistics in Medicine publication: Bryan M, Heagerty P: Multivariate Analysis of Longitudinal Rates of Change. Statistics in Medicine 35(28): 5117-5134, December 2016.

It's certainly possible to conduct inference for each outcome using a mixture of time-to-event, logistic, and linear analyses, then transposing the standardized coefficients and their confidence intervals onto a forest plot, then using meta-analytic methods to evaluate whether the general trend is homogeneous, and inferring the effect's general direction.


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