2026-07-09

Effect Measures: RR, HR, and OR

While the risk (or incidence) is an important measure from an individual and public health perspective and for planning health care resources, the biological effect of a beneficial or harmful exposure is measured as the relative risk. With some prospective study designs, the relative risk (RR) can be measured directly from risks or indirectly from incidence density rates.

Other statistical methods produce other effect measures, such as the hazard ratio (HR) or odds ratio (OR), that sometimes can approximate the RR. For example, RR can be estimated from cohort data or a randomised trial using a log-binomial regression model as well as other models (1), HR using a proportional hazards regression model, and OR using a logistic regression model. However, it is a mistake to always interpret odds ratios and hazard ratios as relative risks.

When common outcomes are studied, the odds ratio provides an inflated estimate of the relative risk. This phenomenon is well known but often neglected. As an example, in obesity research, nearly a quarter of articles presenting odds ratios misinterpreted them as relative risk despite common outcomes (2).

Sutradhar and Austin (3) show that for a given hazard ratio, the corresponding relative risk can vary substantially depending on factors like baseline event rate and duration of follow-up. The use of hazard ratios to approximate the relative risk can therefore be questioned.

In both cases, a better alternative may be to estimate the relative risk directly, but the odds ratio may also be corrected (4) to better approximate the relative risk.

References

1. McNutt LA, Wu C, Xue X, Hafner JP. Estimating the relative risk in cohort studies and clinical trials of common outcomes. Am J Epidemiol. 2003 May 15;157(10):940-3. doi: 10.1093/aje/kwg074. PMID: 12746247.

2. Tajeu GS, Sen B, Allison DB, Menachemi N. Misuse of odds ratios in obesity literature: an empirical analysis of published studies. Obesity (Silver Spring). 2012 Aug;20(8):1726-31. doi: 10.1038/oby.2012.71. Epub 2012 Mar 22. PMID: 22436842; PMCID: PMC3399983.

3. Sutradhar R, Austin PC. Relative rates not relative risks: addressing a widespread misinterpretation of hazard ratios. Ann Epidemiol. 2018 Jan;28(1):54-57. doi: 10.1016/j.annepidem.2017.10.014. Epub 2017 Nov 7. PMID: 29239842.

4. Zhang J, Yu KF. What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. 1998 Nov 18;280(19):1690-1. doi: 10.1001/jama.280.19.1690. PMID: 9832001.

2026-07-08

Incidence

The term 'incidence' may seem straightforward, but it can refer to three distinct metrics: the total number of new cases, cumulative incidence, and incidence density. The calculations of these metrics can vary in difficulty (1). As demonstrated by Havers-Borgersen et al. (2) mistakes are published; however, numerous errors remain probably undetected as a result of unclear methodological descriptions.

1. Incident numbers

In the most basic form, incidence begins with the number of new cases that occur in a population over a specified period. It does not convey any information regarding risk on its own: 500 new cases in a village of 5000 is clearly not the same as in a city of 5 million. To be useful, the number of new cases of a disease needs to be compared to the number of people who are at risk.

2. Cumulative Incidence

The cumulative incidence is a classic risk measure, the number of new cases divided by the number of people free of disease at the start of the follow-up period, over a fixed time window. The measure is a proportion, even if it sometimes is presented as per 1,000 persons, and it answers a specific question: What proportion of an initially disease-free group develops the disease during follow up? Because the measure is based on a head count, it is easy to understand. However, it can only be measured if everyone is followed for the same amount of time and no one is lost to follow-up.

3. Incidence Density

In general, real populations are more dynamic. People enter and leave a cohort, die of other causes, emigrate, or are simply followed for different lengths of time. To deal with this, incidence density divides new cases by the total person-time at risk rather than by the number of people: the sum of the time each individual actually spent under observation before developing disease, dying, being lost to follow-up, or study ending.

Person-time at risk

For population-level incidence rates, person-time is typically not calculated from individual follow-up but approximated using official demographic statistics. It is then calculated as the average number of people in the population during the period, multiplied by the length of that period. The average population (mid-year population) is calculated as the average of the population size at the beginning and end of the period.

References

1. Spronk, I., Korevaar, J.C., Poos, R. et al. Calculating incidence rates and prevalence proportions: not as simple as it seems. BMC Public Health 19, 512 (2019). https://doi.org/10.1186/s12889-019-6820-3.

2. Havers-Borgersen E, Butt JH, Johansen M, Petersen OB, Ekelund CK, Rode L, Olesen JB, Køber L, Fosbøl EL. Preeclampsia and Long-Term Risk of Venous Thromboembolism," JAMA Network Open, 2023;6(11):e2343804 — formal correction notice, "Miscalculation of Incidence Rates," JAMA Network Open, published Jan 11, 2024 (DOI: 10.1001/jamanetworkopen.2023.54306).

2026-07-07

A very brief history of statistics in medicine

Medical science is one of the youngest sciences, at least among those based on empirical evidence. From 1665, when the first scientific journals were established, to the mid-20th century, when modern medical research emerged, the publications were primarily about case reports and expert opinions. From the mid-20th century, the focus is on evidence collected using samples of patients.

But first, during the second half of the 19th century, some remarkable events happened. In Vienna, Ignaz Semmelweis showed that the incidence of childbed fever could be drastically reduced by requiring healthcare workers to disinfect their hands. With this procedure, the maternal mortality rate dropped from 18% to less than 2%. The findings were published in 1861 in the book, "Etiology, Concept and Prophylaxis of Childbed Fever". However, at this time, it was believed that diseases were caused by "bad air" (the miasma theory). As Semmelweiss had no theoretical explanation for his findings, they were rejected by the medical community.

Florence Nightingale, a British nurse, managed, during the Crimean War (1853–1856), to reduce the mortality rate at military hospitals in Crimea from around 42 per cent to 2 by improving the hospitals' sanitary conditions. Having received formal training in statistics and mathematics, she created the Polar Area Diagram to illustrate the importance of hygiene. Florence Nightingale was in 1858 the first female fellow elected to the Royal Statistical Society, and in 1874 she became an honorary member of the American Statistical Association. Her work was presented in a book written by Harriet Martineau, "England And Her Soldiers".

The British physician John Snow, now known as "the father of epidemiology", investigated cholera outbreaks in London. His findings indicated that the disease was spread by water contaminated by some biological agent. In 1854, One investigation was named the "Broad Street Pump outbreak", which led to the authorities' removal of the handle from the pump, even though the cause of the outbreak was considered uncertain. The action is commonly credited as ending the outbreak. Like many modern epidemiologists, Snow relied on data compiled from public listings, in his case provided by the Registrar General's Office.

In the 1860s, the germ theory gained widespread acceptance after being developed by the French microbiologist Louis Pasteur, the English surgeon Joseph Lister, and the German physician Robert Koch. However, for many surgeons the consequences were not immediately apparent. As described in the Britannica Academic, "bloodstained frock coats were considered suitable operating-room attire even in the late 1870s, and surgeons operated without masks or head coverings as late as the 1890s." The time lag between the problem's resolution and the acceptance of its important consequences illustrates the importance of scientific communication.

Early in the 20th century, textbooks written by later famous statisticians presented statistics methods for use in medical research. Ronald Fisher published his "Statistical Methods for Research Workers" in 1925, and Austin Bradford Hill "The Principles of Medical Statistics" in 1937. Bradford Hill's book was based on a set of articles that had first been published in the journal Lancet. After the war, in 1948, the report from the British Streptomycin Trial, which was the first modern clinical trial, was published in the British Medical Journal. It was followed by reports from two observational studies on smoking and lung cancer published in the British Medical Journal in the 1950s, one case-control study that included 1,465 lung cancer cases and 1,465 controls, and one cohort study involving 40,000 British doctors. These three reports are the first to use modern statistical inference to develop empirical evidence from patient samples. Austin Bradford Hill was engaged in all of them. The number of publications based on statistical inference increased rapidly after this.

Today, medical research is conducted using the same or more developed statistical approaches. The exceptionally rapid growth of scientific publications implies a parallell increase in the demand for statistical expertise. In spite of this, however, there is a severe shortage of medical statisticians. Mistakes and misunderstandings are not infrequent and accidental. They are the result of inadequate education and the subsequent development of an illusion of knowledge.

2026-07-05

Confirmatory trials and their interpretation

Unfortunately, the findings of confirmatory trials are often misinterpreted. There is no guarantee that a hypothesis is true just because it passes a statistically significant test. The significance level, typically 5%, represents the likelihood of a false positive result. Hence, systematic reviews and meta-analyses, which lessen the uncertainty by integrating the findings from multiple trials of the same endpoint, play a significant role in the pursuit of truth.

Conversely, confirmatory trials that fail to confirm the tested hypothesis because of statistical nonsignificance are frequently presented to the reader as negative trials. See, for example, Que et al. (1) "this was a negative trial as it failed to reach its primary endpoint."

However, a negative trial offers evidence that the intervention does not have any significant impact on the outcome under investigation, and statistical nonsignificance is not such evidence. There are several other reasons why a statistically significant effect might not be present, such as a too small sample size. Showing that a drug's effect is comparable to that of a placebo, requires a successful equivalence trial, not a failed superiority trial. If it cannot be determined in an unsuccessful trial whether the intervention is beneficial, neutral, or harmful, it is inconclusive.

References

1. Que LG, Yang Z, Lugogo NL, Katial RK, Shoemaker SA, Troha JM, Rodman DM, Tighe RM, Kraft M. Effect of the S-nitrosoglutathione reductase inhibitor N6022 on bronchial hyperreactivity in asthma. Immun Inflamm Dis. 2018 Jun;6(2):322-331. doi: 10.1002/iid3.220. Epub 2018 Apr 11. PMID: 29642282; PMCID: PMC5946144.

2026-07-04

Exploratory studies, confirmatory trials, and Bonferroni correction

Medical research is primarily performed using samples of humans, laboratory animals, or cells, but the studied phenomena are rarely limited to what can be observed in samples of these. On the contrary, the aim is almost always to learn about the population from which the sample was drawn. This leads to generalisation problems. Sampling variability makes the results from sample studies uncertain, and non-random sampling may induce bias. However, the uncertainty can, under certain conditions, be quantified. Quantification and reduction of uncertainty are thus essential components of successful scientific research. Statistical inference is a crucial port of modern empirical science.

A fundamental principle in statistical inference is that a hypothesis cannot be generated and confirmed using the same sample. Doing so induces selection bias in the confirmation testing and invalidates standard p-values and confidence intervals. The aim of studying a sample can therefore be either to generate (or re-generate) a new hypothesis or to confirm an old one. The first type of studies are known as exploratory, the second ones confirmatory. While the results from exploratory studies may be more uncertain, perhaps even in an unknown degree, the results data dependent, and relying on subjective assumptions, confirmatory studies are designed to give an objective result with a specified uncertainty level. This typically requires a prospective study with detailed pre-specification of endpoints and statistical analysis, randomised and concealed allocation to study groups, and masking of the study groups. In practice, this can be achieved only with an experimental study design, i.e. a clinical trial. Furthermore, to provide results with objectively specified uncertainty, the pre-specified statistical analysis needs to account for all possible multiplicity problems (simultaneous inference), which may have consequences for the experimental design, not least regarding the sample size.

For practical reasons, multiplicity is usually impossible to correct adequately for in observational studies (1). Nevertheless, many exploratory studies include Bonferroni corrections, typically without a clear and rational motivation. One example is the publication by Lenes et al. (2), which includes Bonferroni correction in order "to avoid false positive results". The consequence of the correction is, in this study, that the significance level for the Bonferroni corrected tests is lowered from 0.05 to 0.008. Not surprisingly, "[a]fter Bonferroni correction, no significant associations were found".

It is possible that multiplicity correction has a place in exploratory studies, but it is vital to understand that just performing a Bonferroni correction does not make an exploratory study confirmatory. Moreover, not providing a clear and rational motivation for the use of the multiplicity correction may make the critical reader question why a Bonferroni correction was implemented, whether the purpose just was to eliminate inopportune significant findings.

References

1. Bender R, Lange S. Adjusting for multiple testing--when and how? J Clin Epidemiol. 2001 Apr;54(4):343-9. doi: 10.1016/s0895-4356(00)00314-0. PMID: 11297884.

2. Lenes A, Klasen M, Bohorquez-Mendoza G, Gecht J, Sopka S, Vogt L. Psychological determinants of successful practical teaching: personality traits, self-efficacy, and subjective perception in a hands-on clinical skills course. BMC Med Educ. 2026 Jul 2;26(1):1069. doi: 10.1186/s12909-026-09788-2. PMID: 42393695.

2026-07-03

Quartiles, interquartile range, and range

Misuse of statistical terminology is unfortunately very common in medical research reports. The misuse not only indicate methodological ignorance, it also threatens the consistency of the statistical terminology. For example, Nahoui et al. (1) state that in their sample of patients, those "in 3rd and 4th quartiles of median PES [esophageal pressure] had increased mortality risk compared to 1st quartile". Given that only three quartiles exist, the statement is remarkable.

The correct definition (2) of quartile is: "There are three values which separate the total frequency of a distribution into four equal parts. The central value is called the median and the other two the lower (first) and upper (third) quartiles respectively."

The authors obviously confuse quartiles with the four quarts of the distribution defined by the three quartiles. Similar misunderstandings of other quantiles, such as tertiles and quintiles, are as frequent. Furthermore, the misunderstanding of the term quartiles have implications for the interpretation of a widely used measure of dispersion: The interquartile range (IQR), defined as the difference between the upper and lower quartiles. This measure shows the spread of half of the total frequency.

In addition, note that the range (the largest minus the smallest of a set of variable values) and the interquartile range are one value each, not the two quartiles or values that are used to calculate them, as is frequently the case in medical research reports.

References

1. Nahoui H, Schirmer H, Einvik G, Øvrebotten T, Berge K, Myhre PL, Husby H, Hrubos-Strøm H. Respiratory effort during sleep predicts mortality in patients with suspected obstructive sleep apnea. Am J Respir Crit Care Med. 2026 Jun 29:aamag344. doi: 10.1093/ajrccm/aamag344. Epub ahead of print. PMID: 42371748.

2. International Statistical Institute. The Oxford Dictionary of Statistical Terms. Sixth Edition. Oxford, New York: Oxford University Press; 2006.

2026-07-02

Univariate, multivariate, univariable, and multivariable

The terms univariate, multivariate, univariable, and multivariable often appear in scientific medical publications. For example, Kanbaş et al., claim that they have evaluated factors prognostic for cervical cancer using multivariate Cox regression. However, Cox regression is a semi-parametric technique that they use to evaluate how a single response variable (survival time) is affected by one or more explanatory (potentially prognostic) variables. This is not a multivariate analysis.

Univariate and multivariate are terms that refer to the probability distribution upon which a statistical model is based. If a model only has one response variable, a univariate probability distribution works. But if the model has more than one response variable, a multivariate probability distribution is needed. In contrast, the model's univariable or multivariable nature is determined by the number of explanatory variables.

As an example, a model for evaluating the relationship between systolic blood pressure and sex is univariate and univariable. If the explanatory variable age is added to the model, the new model is still univariate but has become multivariable. If the two models are further developed for evaluating two response variables, both systolic and diastolic blood pressures, they become multivariate, the first one univariable and the second one multivariable.

References

1. Kanbaş CÖ, Keles E, Mert EÖ, Özdaş CE, Can HK. Prognostic impact of adenomyosis in cervical cancer: insights from machine learning-driven survival analysis. Rev Assoc Med Bras (1992). 2026 Jun 29;72(4):e20251656. doi: 10.1590/1806-9282.20251656. PMID: 42385036.

2026-07-01

Nonparametric data

Using the correct terminology helps to ensure that the same words are used for the same concepts, which is crucial for a clear communication and for avoiding misunderstandings. The term 'nonparametric data' appears often in the statistics section of research reports. For example, Dugan et al. (1) state that the "data from patients in the two groups were compared using Mann-Whitney U tests for nonparametric data."

However, statistical tests are used to evaluate sampling uncertainty, to test a hypothesis about the properties of an unobservable population represented by a sample of studied patients. The hypothesis may or may not be based on an assumption about a specific data distribution (e.g. Normal) for a test of its parameters. If this is the case, the hypothesis is parametric. Such hypotheses are tested using asymptotic methods. Otherwise, the hypothesis is nonparametric, which requires a distribution-free method. "Nonparametric data" is a nonexistent concept.

References

1. Dugan B, Welch CA, Rahman A, Korona MV, Deipolyi AR. Reduction of hemorrhagic complications after non-focal renal biopsy with pre-procedure desmopressin administration. Diagn Interv Radiol. 2026 Mar 23. doi: 10.4274/dir.2026.263734. Epub ahead of print. PMID: 41866967.

Effect Measures: RR, HR, and OR

While the risk (or incidence) is an important measure from an individual and public health perspective and for planning health care resource...