I have a dataset of 40 patients which receive 4 different treatments, group 1, is placebo, group 2, drug A, group 3, drug B and group 4, drug A+B. Only one dose from each drug and combination. I would like to use Bliss independence model to test for synergy of drugs A and B. My response variable is tumor growth from baseline, so values are ranging between 20-1500. I first caluculated tumor inhibition rate for each group (1-mean(tumor growth for group i)/(mean growth for placebo)), i = (A,B,A+B). Next I calculated the expected tumor inhibition for drug A+B as: E(AB)=E(A)+E(B)-E(A)*E(B), where E(A) and E(B) are the observed tumor inhibition rates which I calculated earlier. And finally I calculated Bliss independence index as BII=(O(AB)-E(AB))/(1-E(AB)), where O(AB) is the observed tumor inhibition rate for A+B. I get positive value for this BII, so apparently the combination is synergic, but is there any statistical way to test for this? I would like to get some confidence interval or p-value to test how significantly the combination is synergic. Or is it possible if we have only one dose? Would it make sense to use e.g. bootstrap method to create variance for the BII value? Any other method than Bliss also would work.


1 Answer 1


This could be quite a long answer as it's something I've also been working on recently. The Bliss Independence model (Bliss, 1939; Greco et al., 1995, Geary, 2013) is an Effect-Based method (in contrast to Dose–Effect-Based method which I will discuss below) that compares the effect resulting from the combination of two drugs ($E_{AB}$) directly to the effects of its individual components ($E_A$ and $E_B$). If the actual observed effect of the drug combination is greater than the expected effect under the Bliss model, it suggests synergy between the drugs – the combination is more effective than expected from their independent actions. If the observed effect is less than the expected effect, it suggests antagonism – the drugs interfere with each other, reducing their overall effectiveness.

To test for synergy using the Bliss independence model in your dataset, and, given the nature of your study design (single dose for each treatment and combination), a bootstrap method may be a reasonable approach to estimate the variability and confidence intervals for the Bliss Independence Index (BII). However, with this fairly small dataset, there's a risk of overfitting, and the results might not be robust and statistical power to detect an effect might be limited. I will talk about bootstrapping in a bit of detail, then mention some alternative to bootstrapping, and alternatives to the BII method. Here's a general outline of how you could approach this using a bootstrap method in SAS:

  • Perform bootstrap resampling to create multiple simulated datasets. For each bootstrap sample, randomly select patients with replacement from each treatment group (including the combination group) to create a new dataset of the same size as the original.

  • For each bootstrap sample, calculate the tumor inhibition rates for each treatment and the combination, the expected inhibition rate for the combination under the Bliss model ($E_{AB}$), and the Bliss Independence Index (BII).

  • From the distribution of BII values across all bootstrap samples, estimate the standard error, confidence intervals, and potentially a p-value for the BII. The p-value can be approximated based on the proportion of bootstrap BII values that exceed the observed BII from your actual data (or are more extreme in the direction indicating synergy).

In SAS, you can use PROC SURVEYSELECT for bootstrap resampling and PROC MEANS or similar procedures for calculating the necessary statistics within each bootstrap sample. Here is a simplified example of how you can approach this in SAS:

   %do i = 1 %to &n;
      PROC SURVEYSELECT DATA = mydata METHOD = srs SAMPSIZE = size_of_your_data out = bootstrap_sample;
         STRATA treatment_group;

      /* Calculate tumor inhibition rates and BII for the bootstrap sample here and Store the results */
      DATA bootstrap_results;
         SET bootstrap_results;
         BII_samp = calc_BII; /* Replace with BII calculation from previous step */
%mend bootstrap;

%BOOTSTRAP(n = 1000); 

/* Calculate the SE and CIs */
PROC MEANS data = bootstrap_results mean stddev clm;
   var BII_samp;

This just provides a basic framework and would need to be adapted to your specific situation. The main point is to replicate the calculation of the BII across many bootstrap samples and then analyse the distribution of these BII values to assess the significance of the synergy.

As mentioned, the small sample size is a cause for concern for bootstrapping, so I would suggest considering other approaches, such as permutation testing or Monte Carlo Simulation

Any other method than Bliss also would work.

Other Effect-Based Strategies are:

  • Combination Subthresholding Approach:

This is a straightforward statistical significance test method that relies on the idea that combining ineffective dosages of medications produces significant effects. The main criticism of this approach is that it lacks consideration of clinical significance (Foucquier and Guedj, 2015). While this strategy is occasionally employed, the observed effects may lack accuracy and may not necessarily reflect substantial differences if the distinction between what is significant and what is not is not inherently significant (Foucquier and Guedj, 2015). This approach risks over-interpreting marginal differences in p-values.

  • Highest Single Agent Approach:

Here, the combined effect ($E_{AB}$) is compared with the effects of the individual components ($E_A$ and $E_B$). The method focuses on whether the combination effect $(\frac{\max{E_A, E_B}}{E_{AB}})$ is greater than the highest effect produced by the individual components. While this approach offers an improvement by focusing on the magnitude of effects rather than just statistical significance, it fails to assess whether the combination effect is synergistic beyond what would be expected from a simple additive effect of the drugs.

  • Response Additivity Approach:

This approach compares the observed effect of the combination ($E_{AB}$) to the expected additive effect ($E_A+E_B$). It assumes linear dose-effect curves with zero intercepts. The major drawback with this method is the assumption of linearity in dose-effect relationships, which is often not the case in pharmacological studies. This can lead to misleading conclusions about synergism or antagonism, especially when dose-effect curves are curvilinear.

Each of these methods has its conceptual basis and can be useful in specific contexts. However, they also illustrate the complexity of accurately assessing drug interactions and synergy.

Dose–Effect-Based Methods:

In contrast to the above Effect-Based methods, which primarily focus on the overall outcomes or effects of drugs often irrespective of dosage, Dose-Effect-Based methods emphasise the relationship between drug dosage and its therapeutic or adverse effects. This approach is key for determining drug potency, optimal dosages, and understanding how drug interactions - such as synergy or antagonism - vary with different doses. It's particularly useful in scenarios like isobolographic analysis (Greco, 1995), where understanding the dose-dependent nature of drug interactions is crucial. Dose-Effect-Based strategies for analysing drug combinations, especially in the context of your study involving two drugs, focus on comparing the concentrations of drugs that produce the same effect. This approach is particularly relevant when dealing with nonlinear dose-effect curves, common in pharmacological and pharmacokinetic studies. One popular approach to this is the Loewe Additivity framework (Loewe, 1953). Based on the principles of dose equivalence and sham combination. It allows for the calculation of an expected additive effect of drug combinations, providing a basis for defining synergy, additivity, and antagonism. In your case, you would use the dose-effect curves of drugs A and B to calculate the combination index (CI). A CI less than 1 indicates synergy (the combined effect is greater than expected), while a CI greater than 1 indicates antagonism. On limitation of this approach is the assumption that the dose-effect curves of each individual are well-described by a sigmoid function.

Zero interaction potency (ZIP), proposed by Yadav et al (2015), is a recent dose-effect reference model used to evaluate predicted reactions in medication combinations. It combines elements of the Bliss Independence and Loewe Additivity models, making it a hybrid method. This method assesses the interactions of medications by comparing the alterations in the effectiveness of the dose-response curves when the drugs are administered individually versus when they are administered together. This evaluation is not influenced by the combined effects of the compounds on the pharmacodynamics. It posits that medications are autonomous and do not mutually influence one another when combined, leading to negligible alterations in their response curves when combined (Yadav et al., 2015). Therefore, it is essential to precisely fit the dose-response curves in order to determine parameters such as the relative half-maximal effect concentration and the slope. However, this might be a challenging task if the data is of low quality (Vlot et al, 2019).

In practice, a full assessment of drug synergy frequently entails a combination of these methodologies, as well as careful consideration of the medications' pharmacological features. Experimental designs that incorporate several pairings and alter doses can provide more robust data for evaluating synergy. In all circumstances, the interpretation of synergy should be done with caution, taking into account both statistical and biological importance.


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