Descriptive Statistics for Clinical Trial Data

Introduction

Descriptive statistics are the foundation of clinical trial data analysis. Before investigators perform hypothesis tests, estimate treatment effects, fit statistical models, or interpret clinical endpoints, they need to understand what the observed data look like.

In a clinical trial, descriptive statistics are used throughout the analysis process. They summarize demographic characteristics, baseline disease characteristics, laboratory measurements, vital signs, treatment exposure, adverse events, efficacy endpoints, pharmacokinetic measurements, and many other variables.

The objective is not merely to calculate a mean or median. Good descriptive statistics should provide a clear and clinically meaningful representation of the study population while preserving important information about variability, distribution, missingness, and extreme observations.

What Are Descriptive Statistics?

Descriptive statistics are numerical or graphical summaries used to describe the distribution and characteristics of a dataset.

For clinical trial data, they commonly include:

• Sample size: the number of observations contributing to a summary.
• Mean: the arithmetic average.
• Standard deviation: a measure of variability around the mean.
• Median: the middle observation after ordering the data.
• Minimum and maximum: the observed range.
• Quartiles: values describing the distribution at selected percentiles.
• Interquartile range: the distance between the first and third quartiles.
• Frequencies: counts within categorical levels.
• Percentages: proportions expressed as percentages.

Continuous Versus Categorical Variables

One of the first decisions in a descriptive analysis is determining whether a variable is continuous or categorical.

The variable type does not completely determine the summary statistics. The clinical meaning and distribution of the variable also matter.

Sample Size: The First Statistic to Report

The number of observations contributing to a descriptive statistic is often more important than users initially realize.

Consider a laboratory result summarized as:

The number is difficult to interpret without knowing how many patients contributed to it.

A clinically useful summary might therefore be:

In clinical trial reporting, the analysis population and the number of patients contributing to each summary should be clear.

The Mean

The arithmetic mean is calculated by summing all observed values and dividing by the number of observations.

Suppose five patients have baseline systolic blood pressures of:

The mean is:

The mean is particularly useful when the distribution is reasonably symmetric and extreme observations are not dominating the result.

The Standard Deviation

The standard deviation describes the amount of variability in observations around the mean.

For a sample, the usual sample standard deviation is:

A small standard deviation indicates that observations tend to cluster around the mean. A larger standard deviation indicates greater variability.

The Median

The median is the middle observation after the values are sorted.

For example:

has a median of:

For an even number of observations, the median is commonly calculated as the average of the two central observations.

Why the Median Is Important in Clinical Trials

Many clinical variables are not normally distributed. Examples can include hospital length of stay, time to an event, certain laboratory values, biomarkers, pharmacokinetic parameters, and some measures of healthcare utilization.

A highly skewed distribution can cause the mean to be strongly influenced by a small number of extreme observations.

For example, consider:

The median is 4, while the mean is substantially larger because of the observation of 30.

Mean and Standard Deviation Versus Median and Quartiles

A common clinical-trial convention is:

There is no universal requirement that every continuous variable use the same summary. The statistical analysis plan should specify the intended reporting approach.

Quartiles

Quartiles divide ordered data into portions of the distribution.

The first quartile, Q1, represents a lower point in the distribution, while the third quartile, Q3, represents an upper point.

The interquartile range is:

For example, if:

then:

The IQR describes the spread of the middle portion of the observations and is less sensitive to extreme observations than the full range.

Minimum and Maximum

The minimum and maximum provide the observed range:

For example:

Ranges are especially useful for detecting unusual values and communicating the span of the observed population.

Categorical Data: Frequencies and Percentages

Categorical variables are generally summarized using counts and percentages.

If 42 of 100 patients are female, the percentage is:

A typical clinical trial table might therefore report:

How Should Percentages Be Calculated?

The denominator must be defined carefully.

For a simple baseline characteristic, the denominator may be the number of patients in the treatment group.

For an adverse-event table, the denominator may depend on the specific analysis population and table convention.

Baseline Characteristics and Table 1

One of the most familiar applications of descriptive statistics is the baseline characteristics table, often called Table 1.

Typical variables include:

• Age
• Sex
• Race and ethnicity where collected and appropriate
• Body weight
• Disease duration
• Disease stage
• Prior treatments
• Baseline disease severity
• Relevant laboratory measurements
• Performance status

Continuous variables are commonly summarized using statistics such as mean, SD, median, quartiles, and range. Categorical variables are commonly shown as n and %.

Example Baseline Table

The table provides a descriptive overview of the randomized population without requiring formal hypothesis tests for every baseline characteristic.

Should Baseline Characteristics Be Tested?

In randomized trials, baseline descriptive tables are generally intended to describe the study populations rather than establish that randomization "worked" through a series of significance tests.

A small p-value for a baseline characteristic does not necessarily indicate a problem with randomization, particularly when many characteristics are examined.

Descriptive Statistics for Laboratory Data

Laboratory variables often require several descriptive summaries because both central tendency and abnormal values can be clinically relevant.

Examples include:

• Hemoglobin
• Platelet count
• White blood cell count
• Creatinine
• ALT and AST
• Bilirubin
• Glucose

A laboratory summary might include:

Descriptive Statistics for Change From Baseline

Clinical trials frequently summarize not only absolute measurements but also changes from baseline.

For a continuous endpoint:

where AVAL is the analysis value and BASE is the baseline value.

For example, if baseline systolic blood pressure is 145 mmHg and the post-baseline value is 132 mmHg:

The patient therefore has a 13 mmHg decrease from baseline.

Percentage Change From Baseline

Some endpoints are summarized using percentage change:

For example, if baseline tumor burden is 100 units and the follow-up value is 70:

A negative percentage indicates a decrease relative to baseline.

Change From Baseline Tables

Clinical trial efficacy tables often present descriptive statistics at each scheduled assessment.

The decline in the mean provides descriptive information about the observed change, but interpretation requires consideration of the endpoint definition, analysis population, missing data, and treatment comparison.

Standard Error Versus Standard Deviation

A common mistake is to confuse SD and SE.

The standard error of the sample mean is:

As sample size increases, the standard error generally decreases even when the underlying variability of individual patients remains unchanged.

Confidence Intervals Are Not Purely Descriptive

Confidence intervals are frequently displayed alongside descriptive summaries, but they are inferential quantities rather than purely descriptive statistics.

For a simple mean under standard assumptions, a confidence interval can be written as:

Whether a confidence interval is appropriate depends on the estimand, analysis method, and assumptions.

Adverse Events and Descriptive Statistics

Safety analysis relies heavily on descriptive statistics.

Common safety summaries include:

• Number and percentage of patients with at least one treatment-emergent adverse event
• Number and percentage with serious adverse events
• Adverse events leading to treatment discontinuation
• Adverse events leading to death
• Events by system organ class and preferred term
• Laboratory abnormalities
• Vital-sign changes
• Exposure duration

For adverse-event tables, percentages are commonly accompanied by counts, because the count communicates the actual number of affected patients.

Why "n (%)" Is So Common

Clinical trial reports frequently display categorical safety and efficacy results in the form:

For example:

Patients Versus Events

A particularly important issue in safety reporting is distinguishing the number of patients from the number of events.

Suppose one patient experiences three episodes of the same adverse event. The patient count is one, but the event count is three.

Missing Data

Missing observations are an unavoidable feature of many clinical trials. Patients may miss visits, discontinue treatment, withdraw consent, or simply fail to provide a particular measurement.

A descriptive statistic should therefore be interpreted together with the number of available observations.

For example:

The decreasing number of observations can affect how the later descriptive statistics should be interpreted.

Do Not Hide the Denominator

Reporting a percentage without its denominator can be misleading.

For example:

does not communicate whether 25 patients out of 100 were affected or 1 patient out of 4.

Clinical trial tables should therefore make the analysis population and denominators transparent.

Outliers

Outliers are observations that are unusually distant from the rest of the data.

An outlier may be:

• A legitimate extreme biological value
• A data-entry error
• A measurement error
• A protocol-related observation
• A genuinely rare clinical observation

An extreme value should not automatically be removed merely because it makes the mean look unusual.

Mean Versus Median: A Practical Example

Consider a hypothetical clinical variable with the following values:

The median is the average of the fourth and fifth observations:

The mean is larger because of the extreme observation of 30.

This example demonstrates why reporting only the mean can conceal important distributional characteristics.

Descriptive Statistics by Treatment Arm

Randomized clinical trials frequently display descriptive statistics separately for each treatment group.

The descriptive table allows readers to understand the treatment groups before considering formal treatment-effect estimates.

Descriptive Statistics and the Statistical Analysis Plan

The Statistical Analysis Plan, or SAP, should define important descriptive analysis conventions.

Examples include:

• Analysis populations
• Baseline definitions
• Visit windows
• Handling of unscheduled assessments
• Rules for derived variables
• Missing-data conventions
• Rounding conventions
• Denominator definitions
• Rules for repeated observations
• Handling of treatment-emergent adverse events

The purpose is to ensure that the reported summaries are reproducible and consistent across tables, listings, figures, and analyses.

Rounding Descriptive Statistics

Clinical trial reporting usually applies prespecified rounding rules.

For example:

The exact number of decimal places should be determined by the analysis specification and reporting standards rather than chosen arbitrarily for each table.

Descriptive Statistics in SAS

SAS provides several procedures for producing clinical-trial descriptive statistics. PROC MEANS and PROC SUMMARY are particularly useful for continuous variables.

proc means data=adsl n mean std median q1 q3 min max;
    var age weight;
run;

The resulting output can provide the basic descriptive statistics needed for many continuous clinical variables.

Descriptive Statistics by Treatment

A common requirement is to produce summaries separately by treatment group.

proc means data=adsl n mean std median q1 q3 min max;
    class trt01p;
    var age weight;
run;

For production clinical programming, the exact implementation depends on the analysis dataset, treatment variables, output requirements, and validated programming conventions.

Categorical Summaries in SAS

PROC FREQ is commonly used to summarize categorical variables.

proc freq data=adsl;
    tables sex race;
run;

Treatment-group summaries can be generated using a two-way table:

proc freq data=adsl;
    tables trt01p*sex;
run;

Descriptive Statistics in R

In R, base functions can provide simple descriptive summaries.

mean(adsl$AGE, na.rm = TRUE)
sd(adsl$AGE, na.rm = TRUE)
median(adsl$AGE, na.rm = TRUE)
quantile(adsl$AGE, probs = c(0.25, 0.75), na.rm = TRUE)
range(adsl$AGE, na.rm = TRUE)

For clinical-trial programming, packages such as dplyr and gtsummary can be used to build reproducible summary tables.

library(dplyr)

adsl %>%
  group_by(TRT01P) %>%
  summarise(
    N = n(),
    mean_age = mean(AGE, na.rm = TRUE),
    sd_age = sd(AGE, na.rm = TRUE),
    median_age = median(AGE, na.rm = TRUE)
  )

Creating a Clinical Summary Table in R

A compact descriptive table can be created using gtsummary.

library(gtsummary)

adsl %>%
  select(TRT01P, AGE, SEX, RACE) %>%
  tbl_summary(
    by = TRT01P,
    statistic = list(
      all_continuous() ~ "{mean} ({sd})",
      all_categorical() ~ "{n} ({p}%)"
    )
  )

In a production environment, the exact formatting and statistical specifications should match the intended table shell and reporting standards.

Descriptive Statistics and CDISC Analysis Datasets

Clinical-trial descriptive statistics are often derived from standardized analysis datasets.

Common datasets include:

The exact dataset structure depends on the study and implementation, but the central principle is the same: descriptive summaries should be derived from well-defined analysis populations and analysis variables.

Descriptive Statistics for Longitudinal Data

Repeated clinical measurements create an additional issue: the same patient may contribute multiple observations.

For example, blood pressure might be measured at:

A descriptive table may summarize the distribution at each visit.

However, observations from the same patient are not independent. This matters when moving from descriptive summaries to inferential modeling.

Descriptive Statistics for Time-to-Event Data

Time-to-event variables require special treatment because some patients may not experience the event during follow-up.

For example, overall survival is usually not appropriately summarized simply by taking the arithmetic mean of observed survival times.

Kaplan-Meier methods are commonly used to describe the distribution of time-to-event outcomes in the presence of censoring.

Common descriptive quantities include:

• Number of events
• Number censored
• Kaplan-Meier survival estimates at specified time points
• Median time to event when estimable
• Confidence intervals for time-to-event estimates

Descriptive Statistics for Pharmacokinetic Data

Pharmacokinetic parameters such as C_max, AUC, half-life, and clearance often have skewed distributions.

For such variables, arithmetic and geometric summaries may both be useful.

The geometric mean is:

Geometric means are particularly relevant for positive, multiplicative variables such as many pharmacokinetic exposure measures.

Graphical Descriptive Statistics

Numerical summaries should often be complemented by graphical displays.

Useful clinical-trial displays include:

• Histograms
• Box plots
• Scatter plots
• Line plots
• Spaghetti or trajectory plots
• Waterfall plots
• Forest plots
• Kaplan-Meier plots
• Shift plots and shift tables

A mean and standard deviation can summarize a distribution, but a graph can reveal skewness, multimodality, outliers, or unusual longitudinal patterns that the numerical summaries do not make obvious.

Box Plots and Descriptive Statistics

A box plot provides a visual representation of the median, quartiles, spread, and potential extreme observations.

Its basic components include:

• Median
• First quartile
• Third quartile
• Interquartile range
• Whiskers according to the plotting convention
• Potentially flagged observations

Box plots are particularly useful when comparing the distribution of a continuous variable across treatment groups or study visits.

Descriptive Statistics Are Not Inferential Statistics

This distinction is fundamental.

Descriptive statistics are often the first step in an analysis, but they do not replace the prespecified inferential analysis.

A Practical Clinical Trial Descriptive-Statistics Workflow

Common Mistakes

1. Reporting Only the Mean

A mean alone can hide substantial skewness or extreme values.

2. Reporting Percentages Without Counts

A percentage is difficult to interpret without its numerator and denominator.

3. Ignoring Missing Data

A mean calculated from 40 patients should not be interpreted identically to a mean calculated from 400 patients.

4. Automatically Removing Outliers

Extreme values may be legitimate clinical observations.

5. Confusing SD and SE

SD describes variability among observations; SE describes precision of an estimated quantity.

6. Using the Wrong Denominator

Percentages must correspond to the relevant analysis population and table definition.

7. Ignoring Distributional Shape

The mean and SD may not adequately describe strongly skewed data.

8. Treating Descriptive Differences as Treatment Effects

A difference in observed means between treatment groups is descriptive until it is evaluated using the prespecified inferential framework.

How Descriptive Statistics Fit Into a Clinical Study Report

Descriptive statistics appear throughout the Clinical Study Report (CSR).

They may be used in:

• Demographic and baseline-characteristic tables
• Patient disposition summaries
• Treatment exposure summaries
• Concomitant medication tables
• Adverse-event tables
• Laboratory summaries
• Vital-sign summaries
• ECG summaries
• Efficacy endpoint tables
• Pharmacokinetic summaries
• Patient-reported outcome tables

A well-designed descriptive analysis therefore supports almost every major section of a clinical-trial report.

Quality Control of Descriptive Statistics

Descriptive statistics can appear simple, but errors in denominators, population selection, derivations, visit selection, or missing-data handling can materially change a clinical-trial result.

Quality-control checks should therefore consider:

• Does the patient count agree with the analysis population?
• Do treatment-group totals reconcile?
• Do categorical percentages use the correct denominator?
• Are missing values handled as specified?
• Are minimum and maximum values plausible?
• Does the median agree with the underlying ordered data?
• Are derived variables calculated correctly?
• Are the correct visits included?
• Does the output match the table shell?
• Are rounding and formatting consistent?

Example: A Complete Descriptive Summary

Suppose 120 patients have baseline body weight measurements.

A useful summary might be:

This communicates substantially more information than simply stating that the mean body weight was 78.6 kg.

What Good Descriptive Statistics Should Accomplish

A strong descriptive analysis should allow the reader to answer several basic questions quickly:

• How many patients are represented?
• What is the typical value?
• How variable are the observations?
• Is the distribution skewed?
• Are there extreme observations?
• How much missing data are present?
• How do treatment groups compare descriptively?
• How do values change over time?
• What proportion of patients experience a categorical outcome?

Summary

Descriptive statistics are among the most important tools in clinical biostatistics because they provide the first quantitative description of a clinical trial population and its observed outcomes.

For continuous variables, mean and standard deviation are useful when the distribution is reasonably symmetric, while median and quartiles are often more informative for skewed variables. Minimum and maximum values provide additional information about the observed range.

Categorical variables are generally summarized using counts and percentages, with careful attention to the appropriate denominator.

In clinical trials, descriptive statistics are used extensively for demographics, baseline characteristics, efficacy, safety, laboratory measurements, vital signs, treatment exposure, pharmacokinetics, and other analysis domains.

References

ICH. E9: Statistical Principles for Clinical Trials.

ICH. E3: Structure and Content of Clinical Study Reports.

U.S. Food and Drug Administration. Statistical Principles for Clinical Trials. ICH E9 guidance for industry.

U.S. Food and Drug Administration. Guidance for Industry: Patient-Reported Outcome Measures: Use in Medical Product Development to Support Labeling Claims.

Fleiss, J.L., Levin, B., Paik, M.C. Statistical Methods for Rates and Proportions.

Altman, D.G. Practical Statistics for Medical Research.