Medical Statistics

Statistical inference focuses on quantifying uncertainty. It is primarily used to generalise information from samples by estimating unobservable parameters and testing hypotheses about the properties of a population that cannot be observed in its entirety. Data science, which many statisticians consider to be part of statistical science (machine learning), is mainly about exploring patterns in data and developing predictions. For example, to answer the question, "How well will this treatment work for the next patient?" a machine learning approach may be useful, and the answer should include the risk of an erroneous classification. If the question is instead, "What is the average effect of this treatment?" statistical inference is necessary, and the answer should include an assessment of the uncertainty of the presented treatment effect estimate. The two approaches often use the same statistical methods, such as logistic regression analysis, but the methods are typic...

As treatment is not assigned randomly in observational studies, patients receiving a particular exposure (or treatment) may differ systematically from patients receiving other exposures or no exposure at all. Consequentially, when investigating the effects of an exposure, the effect of the systematic differences (confounders) needs to be taken into account, preferably be adjusted away in the statistical analysis. Traditionally, this has been made by stratifying on the confounders or including them in a regression model. More recently, adjustment using propensity scores has become increasingly popular (1). A propensity score is a subject’s estimated probability of receiving a particular exposure or treatment, given their observed baseline characteristics. The propensity score thus condenses several measured baseline covariates into a single score. The purpose of this score is to balance the groups receiving different exposures or treatments. Given the score, the covariates included in t...

  Confounding is a distortion of the estimated effect of an exposure on an outcome. The distortion is caused by a confounder that impacts both exposure and outcome. Exposure <- Confounder -> Outcome An observed exposure effect may then reflect the effect of the confounder rather than exposure. For example, when studying the effect of alcohol consumption on cardiovascular disease risk, prior health status, is a confounder if it affects both alcohol consumption and the risk of a cardiovascular event. Confounding can be dealt with in two ways. The first is to perform an experiment with randomised assignment of subjects to exposure and a control group. The advantage with this strategy is that it prevents against confounding from all confounders, known as well as unknown. However, the disadvantage is that it cannot be used to investigate effects of harmful exposures as that would be unethical and in many cases practically impossible. Other disadvantages are that trials are expensi...

  To produce valid results, both observational studies and randomised trials depend on correct classifications of exposure (or treatment) and follow-up. Immortal time bias is a time-related misclassification that can seriously bias the outcome of an investigation. The phenomenon occurs when either the exposed group is assigned a follow-up period that could not have included the events under study (like time on a treatment waiting list). Or when exposed subjects with early events are excluded from the evaluation, ensuring corresponding survival among the remaining. For example, it was reported (1) from a randomised trial of the effect of a radiation sensitiser on the survival of inoperable lung cancer patients, that the subgroup of patients (36%) who completed the course of radiation therapy augmented by the sensitiser survived a median of 22 months, twice as long as the patients in the control group. However, the median survival for the entire treatment group was only 13 ...

  One of the terms most frequently misused in medical publications is ‘correlation’. It is often used to describe any kind of relationship or interaction between variables. However, the term is very specific and using it arbitrarily may be misleading. Apart from non-parametric alternatives and minor variants, two major types of correlation coefficients have been developed: Pearson's correlation coefficient and the intraclass correlation coefficient. The first one measures the linear association between two variables, the second one the resemblance of measurements made within the same unit (repeated measurements) or cluster. Pearson's correlation coefficient Pearson's correlation coefficient ( ρ ) for the two variables X and Y is the covariance of the two variables divided by the product of their standard deviations: ρ  = cov(X,Y)/(σ X σ Y ) Pearson's correlation coefficient ranges from −1 to 1. The value 1 implies that a linear equation describes the relationship bet...

One common misunderstanding demonstrated in medical publications is the testing of normal distribution as part of a decision process: If Shapiro-Wilk Test or Kolmogorov-Smirnov Test indicates that a variable has a statistically significant departure from normal distribution, group mean differences are tested using the Mann-Whitney Test instead of Student's t-test. This decision rule may sound rock solid, but it may be a serious mistake. First, the practice treats absence of evidence as evidence of absence. A statistically nonsignificant normality test is simply not evidence of a normal distribution. Second, for a two-sample t-test or a linear statistical model, the relevant distributional issue concerns the difference or model residual conditional on covariates, not whether each observed treatment group's raw outcome values are exactly normal. Third, with small sample sizes, where the normality assumption may be important, the normality test has low power to detect non-normalit...

  In early clinical trials patients were allocated to treatment and control groups by alternation, i.e. patients were assigned to the treatment group and control group in a strict order. The advantage with this method was that it produced groups with equal size. The two groups would also be fairly compared if the order of the patients included in the trial was random. However, the group allocation for a patient was predictable, which could potentially introduce selection bias into the trial's outcome through differential invitations to participate. The only way to avoid this issue was, of course, to make the group assignment concealed, which necessitated a non-systematic group assignment sequence. D'Arcy Hart (1) describes the development with two trials from the 1940s. The first one investigating the effects of patulin (a mycotoxin) on the common cold in 1943-1944 with strict alternation. The second one, having a randomised group assignment, investigating the effects of strept...

The most important guideline for writing a manuscript to be submitted to a medical scientific journal is the Manuscript Preparation and Submission recommendations from the International Committee of Medical Journal Editors (ICMJE). It can be found here (1). From a statistical viewpoint, two of the recommendations are exceptionally useful. The first is: "Describe statistical methods with enough detail to enable a knowledgeable reader with access to the original data to judge its appropriateness for the study and to verify the reported results." To be able to judge appropriateness and verify reported results is not only about naming the methods that have been used, it is also about why a particular method has been used, the investigators' intention. It may therefore be necessary to motivate the method choice by explaining the analysis strategy. Vague or otherwise unclear statements such as "independent samples t-test" and "as appropriate" should be avoi...

  Statistical analyses of related data need to take the relations into account to avoid misleading results. Two main types of statistical models are used, conditional and marginal models (1). A conditional model can be fitted within the framework of generalized linear mixed models (GLMM) and a marginal model using generalized estimating equations (GEE). Analyses based on conditional and marginal models give answers to different questions. While a conditional model can be used to estimate the outcome within a subject or cluster after conditioning on covariates, a marginal model can estimate the average outcome for the population accounting group-specific effects. For example, in a longitudinal study comparing patients receiving diet recommendations (treated) with patients receiving exercise recommendations (controls), uncontrolled blood pressure at repeated visits can be modelled using both a conditional and marginal model. Neither of these two models is automatically more correct ...

  In statistical inference, degrees of freedom is a measure of the available amount of independent information. For example, when estimating a mean value of a population, the degrees of freedom, df , are the same as the number of observations, n . When estimating the variance of the same population, one degree of freedom is first lost when estimating the mean value because variance is defined relative to the mean value. The population variance is thus estimated with n - 1 degrees of freedom. More generally, with independent observations the degrees of freedom can be calculated as the number of observations minus the number of estimated parameters, df = n - p , where p is the number of estimated parameters. However, with repeated measurements, things become more complicated because observations from the same person are correlated and do not provide as much independent information as the same number of measurements from different people. Ignoring this problem is a common mistake, wh...

  While classical statistical methods are based on an assumption of independent observations, many currently used statistical models include observations that are related instead of independent, such as repeated measurements from the same patient and patients randomised at the same centre. Fixed effects models, random effects models, and mixed models provide different ways to deal with independent and related, and a mix of independent and related observations (1). Fixed effects estimate a population-average association or a specific comparison that is central to the research question. For example, in a clinical study, treatment group, follow-up time, baseline age, and sex may be modelled as fixed effects. The estimated treatment effect then represents the expected difference between treatment groups, conditional on the other predictors. Random effects represent the variation among units drawn from a wider population. Instead of estimating a separate, unrelated parameter for each ho...

  The concepts of efficacy, effectiveness, and efficiency are frequently misunderstood. They all pertain to the outcome of a medical intervention, but it is crucial to comprehend their differences accurately. Efficacy refers to whether an intervention produces more benefit than harm when tested under tightly controlled, ideal circumstances, such as strict inclusion/exclusion criteria, close monitoring, and high adherence to the treatment protocol. This effect is typically investigated in a classical randomised trial designed for high internal validity, an explanatory trial (1). Effectiveness refers to how well an intervention works in everyday clinical practice with a more diverse patient population including comorbidities, inconsistent adherence, and varying follow-up care. This effect is often investigated in a pragmatic trial, prioritizing external validity (representativity) instead of internal validity. The distinction between efficacy and effectiveness is important because an...

  Blinding randomised patients and doctors by masking treatment, where this is possible, is an established approach in clinical trials, aiming to prevent bias. It is not uncommon that published trial reports state that the statistician was also blinded, not only during the planning of the final analysis, which has been routine for a long time, but also during the analysis. Professional clinical trials units seem to have variable approaches to the blinding of trial statisticians (1). You might argue that knowing which arm is which affects the outcome of the statistical analysis, and that blinding the statistician results in a more objective result. However, this misses the fact that statistical comparisons of treatment groups, masked or not, in itself are likely to unblind the data. For example, adverse-event profiles typically show side effect patterns that reveal the identity of treatment and placebo groups. A reasonable explanation for the statisticians' influence on the analysis...

Followers of today's two main traditions of statistical inference are known as frequentist and Bayesian. Ronald Fisher attempted during the 1930s to develop a third school called fiducial inference (1), but this was broadly considered controversial and do not play a major role today. The fundamental difference between frequentists and Bayesians is that they define probability in different ways. For a frequentist, a probability is an objective measure, a long-run relative frequency. For example, a fair coin has probability 0.5 of heads because, across a very large number of comparable tosses, half are heads, and a drug lowering blood pressure has an objectively measurable average effect among the patients taking the drug. The central limit theorem, which states that as the sample size increases, the distribution of sample means tends to approximate a normal distribution, is essential in frequentist inference. A Bayesian, on the other hand, sees probability as a subjective measure, a...

A search in PubMed shows that the use of the statistical term estimand has increased markedly during the past ten years. Yet, the term remains unfamiliar to many readers. In statistics, a parameter is a numerical characteristic of a population, probability distribution, or statistical model. Examples include a mean, proportion, variance, regression coefficient, hazard ratio, or risk. Because the population is rarely observed in its entirety, such quantities are usually inferred from sample data, for example, from participants in a clinical trial. As sampling uncertainty affects the resulting estimates and should be quantified, statistical inference is necessary. An estimand is different. It is a precise statement of the quantity that a study is intended to estimate in order to answer its scientific or clinical question. It should be defined before selecting a statistical model or analysis method. This distinction matters because different analyses can answer different questions. For ex...

  Statistical inference is used to evaluate sampling uncertainty in medical research. The two most commonly used uncertainty measures are confidence intervals and p-values. However, it is often useful to distinguish between the uncertainty resulting from random variation (aleatoric uncertainty) and the uncertainty caused by incomplete knowledge (epistemic uncertainty). A small p-value indicates disagreement between observed data and a tested null hypothesis, but it is not in itself a direct measure of either aleatoric or epistemic uncertainty. P-values do not, by themselves, separate random variability from the uncertainty about whether the statistical model and its assumptions are appropriate. A confidence interval quantifies sampling uncertainty about an estimate under the assumed statistical model and study design. Its width is influenced by outcome variability and sample size, but it does not generally capture uncertainty from model misspecification, unmeasured confounding, or ...

While the risk (or incidence) in absolute numbers is an important measure from an individual and a public health perspective and for planning health care resources, the biological effect of a beneficial or harmful exposure is always measured and medically interpreted in terms of 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 be used to 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 t...

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

  Medical science is one of the youngest sciences, at least as we define science today. From 1665, when the first scientific journals were established, to the mid-20th century, when modern medical research emerged, medical research publications were primarily descriptive (case reports) or presenting subjective comments (expert opinions). It lasted until the mid-20th century, until medical research focused on empirical evidence collected from samples of patients and evaluating the uncertainty of the findings objectively, using statistical science. 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 finding was published in 1861 in the book, "Etiology, Concept and Prophylaxis of Childbed Fever". However, ...

  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 nothing more than 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 w...

  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. However, this leads to generalisation problems. Sampling variability makes the results from sample studies uncertain, and the consequences of non-random sampling may induce an uncertain amount of bias. The uncertainty can, under certain conditions, be quantified, and a sound study design may reduce it. 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 a...

  Misuse of statistical terminology is 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, this statement is remarkable. A quartile is defined (2) like this: "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 quarti...

  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, in this case of 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. The term "Nonparametric data" is a...

  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 associated to one or more explanatory (potentially prognostic) variables. That is not a multivariate analysis. Univariate and multivariate 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 a...