In clinical research, mathematical analyses are essential for interpreting study results. These calculations help measure and compare risks, as well as evaluate the effectiveness of treatments. Imagine a researcher studying the effect of a new drug on Disease X by comparing two groups of 100 people suffering from this disease. After randomization, the first group receives the treatment under investigation (treatment group), while the second group receives a placebo (control group). The hypothesis is that the drug is superior to the placebo in preventing the onset of the disease. These two groups are followed over a predetermined period of one year to identify individuals who develop the disease in both groups.
Based on this randomized controlled trial design, let’s explore five key mathematical analysis concepts used to interpret the data collected in this fictional study. These data are frequently presented in the media when the results of scientific studies make headlines.
Absolute Risk (AR)
Absolute risk represents the probability that an event will occur in a given population over a specified time period.
Example: In our fictional study, let’s consider the control group. Out of the 100 people followed for one year, 10 develop the disease under study. For this group, the absolute risk is calculated as follows:
absolute risk = number of events (developing the disease) / total number of individuals in the group
AR = 10 / 100
AR = 0.1
Thus, the absolute risk of developing this disease is 0.1 or 10% over one year.
Relative Risk Ratio (RR)
Relative risk compares the absolute risk between two groups (e.g., those exposed to a treatment and those who are not).
Example: In our fictional study spanning one year, among 100 people receiving preventive treatment, 4 develop the disease (absolute risk of 4/100 or 4%). In the group of 100 untreated individuals, 10 out of 100 fall ill (absolute risk of 10/100 or 10%).
The relative risk is calculated as follows:
relative risk ratio = AR of the treatment group / AR of the control group
RR = 0.04 / 0.1
RR = 0.4
A relative risk of 0.4 means that the risk of developing the disease in the treatment group is 2.5 times lower than in the control group over one year.
The following general rules apply depending on the RR value:
If RR = 1, there is no difference in risk between the two groups.
If RR > 1, the risk is higher in the exposed group (e.g., the treatment might be harmful).
If RR < 1, the risk is lower in the exposed group (e.g., the treatment is protective).
Absolute Risk Reduction (ARR)
Absolute risk reduction measures the difference between the absolute risks of the two groups. This value indicates the treatment’s effectiveness.
Example: Using the results from our study:
absolute risk reduction = AR of the control group - AR of the treatment group
ARR = 0.1 - 0.04
ARR = 0.06
The absolute risk reduction is 6%, meaning 6 out of 100 people benefit directly from the treatment over one year.
In this context, a positive result indicates that the treatment group benefits from protection, while a negative result indicates a harmful effect of the treatment.
Relative Risk Reduction (RRR)
Relative risk reduction expresses, as a percentage, how much the risk is reduced compared to the initial relative risk. This value is also an indicator of the treatment’s effect compared to the control.
Example: Using the data from our study:
relative risk reduction = 1 - RR
RRR = 1 - 0.04
RRR = 0.06
The relative risk reduction is 6%, indicating that the treatment reduces the risk of developing the disease by 60% compared to the untreated group.
Number Needed to Treat (NNT)
The number needed to treat represents the number of individuals who need to be treated to prevent one case of the disease. This metric highlights that treatments are rarely 100% effective and that some treated individuals will still develop the disease.
Example: With an absolute risk reduction of 6% (0.06), the NNT is calculated as follows:
NNT = 1 / ARR
NNT = 1 / 0.06
NNT = 16.6
This result means that 16 people need to receive the treatment to prevent one case of the disease.
Odds Ratio (OR)
The odds ratio is a measure used to evaluate the strength of the association between an exposure (treatment) and an outcome (disease). It compares the odds, or "chances," of the event occurring in two groups.
Example: In our fictional study:
Odds in the treatment group = (4 diseased / 96 non-diseased) = 0.0417
Odds in the control group = (10 diseased / 90 non-diseased) = 0.1111
The odds ratio is calculated as follows:
OR= (4/96) / (10/90)
OR=0.0417 / 0.1111
OR=0.375
An OR of 0.375 indicates that patients in the treatment group are approximately 62.5% less likely to develop the disease compared to the control group. In this context:
OR = 1 means the likelihood of the outcome is the same in both groups.
OR > 1 means the probability of the outcome is higher in the treatment group (an undesirable result!).
OR < 1 means the probability of the outcome is lower in the treatment group (indicating a protective effect).
Summary
These analyses provide powerful tools for understanding the risks and benefits of health interventions. Here is a summary table:
Analysis | Formula | Interpretation |
Absolute Risk (AR) | number of events / total number of individuals | Probability of an event occurring |
Relative Risk (RR) | AR of the treatment group / AR of the control group | Risk ratio between two groups |
Absolute Risk Reduction (ARR) | AR control group - AR treatment group | Difference in risk between groups |
Relative Risk Reduction (RRR) | 1 - RR | Proportional risk reduction |
Number Needed to Treat (NNT) | 1 / ARR | Number of treatments needed to prevent one case |
Odds Ratio (OR) | (cases in treatment group / non-cases in treatment group) / (cases in control group / non-cases in control group) | Strength of the association between an exposure and an event |
Testing the Hypothesis
In our fictional study, researchers initially hypothesized that the tested drug is more effective than the placebo in reducing the onset of a disease. To confirm this hypothesis, a statistical test must be conducted on the results. One of the most commonly used tests for analyzing categorical variables (e.g., absence or presence of disease) is the Chi-squared test (or Pearson’s test). This test determines whether there is a significant association between two variables.
Inserting our study data into the analysis table yields:
Study Groups | Presence of Disease | Absence of Disease | Total |
Treatment Group | 4 | 96 | 100 |
Control Group | 10 | 90 | 100 |
Total | 14 | 186 | 200 |
The Chi-squared test calculates a p-value indicating whether the observed difference between groups is due to chance. For this, the result of the test produces an indicator called the “p-value.” A p-value below 0.05 is generally considered statistically significant.
If p < 0.05 (commonly used threshold), there is less than a 5% chance that the differences are due to chance. The results are considered statistically significant.
If p > 0.05, it is likely that the observed differences are due to chance, and the results are not statistically significant.
In our fictional study, the calculated p-value is 0.166, which is higher than 0.05. This indicates that, despite an analysis initially suggesting benefits of the treatment, the results are insufficient to eliminate the possibility that the observed effect is due to chance.
The p-value does not prove causation, but it helps assess whether study results are robust enough to warrant in-depth interpretation. Several factors can explain non-significant results, including insufficient sample size, high variability in data, or a genuinely small or non-existent effect of the treatment being studied.
While these notions barely scratch the surface of the vast topic of mathematical analysis of scientific study data, it is already evident that data always undergo a series of statistical analyses aimed not only at making them “speak” but also at validating the probability that the observed relationship is not due to chance. Proper interpretation is essential to understanding the information conveyed by clinical studies, as it underpins health-related recommendations.