Statistical Analysis & Data Interpretation
Every assay described in the preceding phases ultimately produces a set of numbers, and the scientific validity of a pharmacological research programme rests as much on the correctness of its statistical analysis as on the quality of its experimental technique. Phase 6 addresses the selection of appropriate statistical tests, the assumptions underlying parametric analysis, dose-response modelling, sample-size and power calculation, and the practical workflow of statistical software such as GraphPad Prism.
Biostatistics is the branch of applied statistics concerned specifically with the design, analysis, and interpretation of data arising from biological and health-related research. It differs from general statistics chiefly in its emphasis on the particular data structures and design challenges characteristic of biological experimentation: small sample sizes dictated by ethical and cost constraints, biological variability that is often considerably larger than measurement error, hierarchical or repeated-measures data structures (multiple measurements from the same animal over time), and outcome variables that are frequently non-normally distributed, bounded (such as percentage inhibition), or censored (such as survival time). A firm grounding in biostatistical principles is what allows a pharmacologist to move from a raw dataset to a defensible scientific conclusion, and, conversely, its absence is what most commonly separates a technically well-executed experiment from a publishable, reproducible one.
The biostatistical workflow in pharmacological research typically proceeds through four stages: study design (specifying the hypothesis, the experimental units, and the randomisation and blocking scheme, as introduced in Phase 4), data collection (ensuring that the planned statistical analysis is compatible with the data structure actually being generated), data analysis (selecting and correctly applying the appropriate statistical test, the principal focus of the remainder of this phase), and data interpretation (translating a statistical result — a p-value, a confidence interval, an effect size — into a scientifically meaningful conclusion, resisting the common temptation to conflate statistical significance with biological or clinical importance).
Statistical analysis is not an optional add-on performed after data collection; it is the scientific foundation that determines whether an observed pharmacological effect reflects a genuine biological phenomenon or is merely the product of random variation. Regulatory agencies including the FDA, EMA, and CDSCO require rigorous, pre-specified statistical analysis for all preclinical data submitted in support of a drug application, and incorrect statistical test selection remains among the most frequent causes of manuscript rejection and, in more serious cases, published-article retraction.
The appropriate statistical test is determined jointly by the number of groups being compared, whether the data meet the assumptions of parametric analysis, whether observations are paired or independent, and the nature of the outcome variable itself.
For two groups of normally distributed (parametric) data, the independent-samples (unpaired) t-test is appropriate when the two groups consist of different animals or subjects, while the paired t-test is used when the same animals are measured before and after treatment. When more than two groups are compared along a single factor — for example, several dose levels of the same drug — one-way ANOVA followed by an appropriate post-hoc test (Tukey's Honestly Significant Difference test when comparing all groups pairwise, or Dunnett's test when comparing each treatment group only against a single control) is the correct approach. When two factors are varied simultaneously, such as dose and time, two-way ANOVA is required, using a repeated-measures design if the same animals are measured at each time point, with Bonferroni or Šídák correction applied to post-hoc comparisons.
When data are non-parametric — small sample size or a distribution that departs from normality — the Mann-Whitney U test is the non-parametric analogue of the unpaired t-test for two independent groups, while the Wilcoxon signed-rank test is the non-parametric analogue for paired data. For more than two non-parametric groups, the Kruskal-Wallis test followed by Dunn's post-hoc test with Bonferroni correction is appropriate. Categorical data, such as proportions or mortality rates, are analysed using the chi-square test, or Fisher's exact test when expected cell frequencies are small. Dose-response relationships (IC50, EC50, LD50) are properly analysed by non-linear regression of the log-transformed dose against response, typically using a four-parameter logistic (4PL) model, described in detail below. Survival data are analysed using Kaplan-Meier curves, compared statistically using the log-rank (Mantel-Cox) test or the Gehan-Breslow-Wilcoxon test, the latter giving greater weight to early events. Finally, the association between two continuous variables is quantified using the Pearson correlation coefficient for parametric data, the Spearman rank correlation for non-parametric data, or Bland-Altman analysis when the goal is to assess agreement between two measurement methods rather than correlation per se.
- Normality: The data must be approximately normally distributed, formally assessed using the Shapiro-Wilk test for sample sizes below approximately 50, or the Kolmogorov-Smirnov test for larger samples.
- Homogeneity of variance: The groups being compared should have approximately equal variance, assessed using Levene's test or the F-test for two groups, or Bartlett's test across multiple groups.
- Independence: Individual observations must be independent of one another, except in explicitly paired or repeated-measures designs.
- Scale of measurement: The outcome variable must be measured on an interval or ratio scale, not an ordinal or nominal scale.
- Violated assumptions: If any of the above assumptions is violated, the appropriate non-parametric equivalent should be used, or the data may be mathematically transformed (commonly by logarithmic or square-root transformation) before re-testing for normality.
A standard one-way ANOVA analysis proceeds through five sequential steps. First, data are entered with each experimental group in its own column and each row representing an individual animal's value, with results conventionally reported as mean ± standard error of the mean (SEM). Second, normality is tested using Column Statistics → Shapiro-Wilk normality test, with a resulting p-value greater than 0.05 taken as consistent with a normal distribution. Third, equality of variance is tested using Bartlett's test, again with p greater than 0.05 taken as evidence of equal variance across groups. Fourth, the ANOVA itself is run, with the appropriate post-hoc comparison selected — Tukey's test when comparing every group against every other group, or Dunnett's test when comparing each treatment only against a single control group. Fifth, results are interpreted through the F-statistic and its associated p-value, together with the adjusted p-values generated by the chosen post-hoc test, conventionally denoted using asterisk notation (for example, single, double, and triple asterisks representing progressively smaller p-value thresholds) on the resulting graph.
Non-parametric tests make substantially weaker assumptions about the underlying data distribution than their parametric counterparts, analysing the rank order of observations rather than their raw numerical values, and are the appropriate choice whenever the normality or equal-variance assumptions of parametric testing are violated, or whenever sample sizes are too small to assess normality reliably (broadly, fewer than about ten observations per group).
The Mann-Whitney U test (also known as the Wilcoxon rank-sum test) is the non-parametric analogue of the unpaired t-test, comparing the rank distributions of two independent groups rather than their means; it is appropriate when comparing, for example, two independent groups of animals with a non-normally distributed or ordinal outcome. The Wilcoxon signed-rank test serves the analogous role for paired data, comparing the ranked magnitude and direction of within-subject differences between two paired conditions, such as a physiological measurement taken in the same animals before and after treatment.
The Kruskal-Wallis test extends the logic of the Mann-Whitney U test to more than two independent groups, serving as the non-parametric analogue of one-way ANOVA. A significant Kruskal-Wallis result indicates that at least one group's rank distribution differs from the others, and must be followed by an appropriate post-hoc test — most commonly Dunn's test with Bonferroni correction — to identify which specific pairs of groups differ, exactly paralleling the logic of ANOVA post-hoc testing described above.
Correlation and regression analyses quantify the relationship between two (or more) continuous variables, addressing a fundamentally different scientific question from the group-comparison tests discussed above: not whether groups differ, but how strongly, and in what functional form, two variables are related.
The Pearson correlation coefficient (r) quantifies the strength and direction of a linear relationship between two normally distributed continuous variables, ranging from −1 (perfect negative relationship) through 0 (no linear relationship) to +1 (perfect positive relationship); the Spearman rank correlation coefficient provides the non-parametric analogue, based on ranked rather than raw values, and is appropriate for non-normally distributed data or genuinely non-linear but monotonic relationships. It is important to note that correlation quantifies association, not causation, and a statistically significant correlation between two pharmacological variables (for example, plasma drug concentration and a biomarker level) does not by itself establish that one causes the other.
Whereas correlation simply quantifies the strength of association, regression analysis models the functional relationship between an independent (predictor) variable and a dependent (outcome) variable, allowing the outcome to be predicted from the predictor. Simple linear regression fits a straight-line relationship, commonly used in pharmacology to construct calibration curves during analytical method validation (relating instrument response to known analyte concentration) and to derive pharmacokinetic parameters from log-transformed concentration-time data. Non-linear regression, most prominently the four-parameter logistic model introduced below, is required whenever the underlying biological relationship is not linear, as is almost universally the case for dose-response pharmacology.
Dose-response relationships in pharmacology are almost universally sigmoidal when plotted against the logarithm of concentration, and are conventionally fitted using a four-parameter logistic model:
where X represents the logarithm of concentration and Y represents the percentage inhibition or percentage response. The 'Bottom' parameter represents the minimum response (typically constrained to 0%), the 'Top' parameter represents the maximum response (typically constrained to 100%), 'LogIC50' represents the log-concentration at which the response equals 50% of maximum, and the 'Hill slope' describes the steepness of the curve — a Hill slope near 1 is consistent with simple, non-cooperative receptor occupancy, while steeper or shallower slopes suggest cooperative binding or a heterogeneous population of binding sites. In GraphPad Prism, this analysis is performed via Analyze → Nonlinear regression → Dose-response → log(inhibitor) vs. normalised response (variable slope), and results are conventionally reported as IC50 = X.XX µM (with its 95% confidence interval), the goodness-of-fit statistic R², and the number of independent experiments (n), each performed in technical triplicate, contributing to the analysis.
The IC50 is the concentration of a compound that inhibits 50% of a defined enzyme or receptor activity, derived from an inhibition dose-response curve by non-linear regression. The EC50 is the concentration producing 50% of the maximal possible effect, derived analogously from an activation or agonist dose-response curve using the same four-parameter logistic model. The LD50 is the dose that is lethal to 50% of a treated animal population, conventionally derived by probit analysis (the Finney method) or using the AOT425 software associated with OECD Guideline 425, and is expressed in mg/kg. The ED50 is the dose producing the desired therapeutic effect in 50% of a treated population, derived from quantal (all-or-nothing) dose-response data using probit or logit transformation.
Two derived indices are of particular importance in candidate selection. The Selectivity Index (SI) is calculated as the IC50 for cytotoxicity on normal, non-target cells divided by the IC50 for activity on the intended target; an SI greater than 10 is generally regarded as indicating acceptable selectivity for further development. The Therapeutic Index (TI) is calculated as the LD50 divided by the ED50; a higher TI indicates a wider, and therefore safer, margin between the effective and lethal dose, with a TI greater than 10 generally desirable for a new drug candidate, while drugs with a narrow therapeutic index (for example, warfarin or digoxin) require therapeutic drug monitoring in clinical use.
Survival analysis encompasses the statistical methods used to analyse time-to-event data — most commonly time to death, but equally applicable to time to relapse, tumour recurrence, or any other clearly defined biological event — and is distinguished from ordinary continuous-data analysis by its ability to correctly handle censored observations, in which the event of interest has not yet occurred (or was not observed) by the end of the study period for some subjects. The Kaplan-Meier method estimates the survival function — the probability of survival beyond a given time point — non-parametrically from the observed event and censoring times, and is conventionally displayed as a step-function survival curve. Comparison of survival curves between two or more treatment groups is performed using the log-rank (Mantel-Cox) test, which weights each observed death equally across the follow-up period, or the Gehan-Breslow-Wilcoxon test, which instead gives greater statistical weight to deaths occurring earlier in the follow-up period and may therefore be more sensitive in studies where a treatment effect is expected to diminish over time. Survival analysis is central to carcinogenicity studies, chronic toxicity studies, and infectious disease models (such as the sepsis and tuberculosis models described in Phase 4), in all of which overall survival is frequently the most clinically and regulatorily meaningful endpoint.
6.9.1 Why Power Analysis is Mandatory
Sample size must be calculated prospectively, before a study begins — a power analysis performed retrospectively, after data collection, is statistically invalid and does not satisfy regulatory or ethical review requirements. A proper power calculation requires four components: the significance level, alpha (conventionally set at 0.05), the desired statistical power (1 − beta, conventionally set at 0.80 or 0.90), the anticipated effect size (commonly expressed as Cohen's d for two-group comparisons, or Cohen's f for ANOVA designs), and an estimate of the outcome variable's standard deviation, drawn from pilot data or the published literature.
For a two-sample t-test, the required sample size per group is given approximately by n = 2 × [(Zα/2 + Zβ)² × σ²] / δ², where δ represents the minimum clinically or biologically meaningful difference to be detected. Commonly used software for this calculation includes the free G*Power package, the R 'pwr' package, GraphPad StateMate, and SAS PROC POWER. As a worked example, for an anticipated large effect size (Cohen's d = 0.8), a significance level of 0.05, and a desired power of 0.80, the two-sample t-test formula yields a requirement of approximately 26 animals per group. CPCSEA explicitly expects that the minimum number of animals used in any protocol be justified by a formal power analysis, directly operationalising the Reduction principle of the 3Rs discussed in Phase 4.
A typical Prism-based analysis proceeds through seven stages: importing data in the appropriate table format (XY, Column, or Grouped, depending on the experimental design); generating descriptive statistics, conventionally reported as mean ± SEM; testing normality using the Shapiro-Wilk test; selecting the statistically appropriate test (ANOVA, t-test, or a non-parametric alternative) based on the outcome of the normality assessment; applying the appropriate post-hoc test; where relevant, fitting a dose-response curve using non-linear regression; and finally exporting publication-quality figures, conventionally at a minimum resolution of 300 dpi in TIFF or PDF format.
6.10.1 Prism Table Types and Their Applications
The XY table format is used for dose-response curves, time-course data, scatter plots with regression lines, and pharmacokinetic concentration-time profiles. The Column table format is used for comparisons between groups by ANOVA or t-test, including box plots and bar graphs with error bars, and supports built-in normality testing. The Grouped table format is used for two-way ANOVA designs involving multiple treatments across multiple time points, generating clustered bar graphs. The Survival table format supports Kaplan-Meier survival curve analysis with the log-rank test, appropriately handling censored data. The Contingency table format supports chi-square and Fisher's exact tests for categorical data such as mortality rate or disease incidence, typically arranged in a 2×2 format. The Parts-of-whole table format supports pie charts and stacked bar charts, useful for representing percentage composition data such as flow-cytometry (FACS) cell-population results.
How statistical results are presented is nearly as important as how they are calculated, since a reader or reviewer must be able to independently assess the strength and reliability of the evidence being presented. Continuous data are conventionally presented as mean ± SEM in bar or scatter graphs when the sample size is small, with individual data points overlaid wherever practicable so that the reader can directly assess the underlying data spread rather than relying solely on the summary statistic. Box-and-whisker plots, which display the median, interquartile range, and full data range without assuming a particular distribution, are increasingly preferred for non-parametric or skewed datasets over simple bar-and-error-bar graphs, which can visually misrepresent asymmetric data distributions. Dose-response and time-course data are conventionally presented as XY line or scatter graphs with a fitted regression curve and its 95% confidence band, allowing the reader to assess both the central estimate and its uncertainty at a glance. Figure legends must state, at minimum, the sample size (n) and what it represents (biological versus technical replicates), the summary statistic displayed (mean ± SEM or median ± IQR), the statistical test applied, and the meaning of any significance annotation (asterisks or exact p-values) — omitting any of these elements is a common and easily avoidable source of reviewer criticism.
Several recurring statistical errors merit brief mention here, with a comprehensive treatment reserved for the troubleshooting discussion in Phase 7: reporting standard deviation rather than SEM for small samples, running multiple t-tests instead of ANOVA, omitting a post-hoc test following a significant ANOVA, excluding outliers without a pre-specified objective criterion, and applying a parametric test to data that has not been formally assessed for normality. Beyond these mechanical errors, the single most important interpretive principle in pharmacological statistics is that statistical significance (a p-value below the chosen alpha threshold) is not equivalent to biological or clinical importance: a very large sample size can render a trivially small, biologically meaningless difference statistically significant, while a genuinely important biological effect can fail to reach statistical significance in an underpowered study. For this reason, contemporary pharmacological reporting increasingly emphasises effect sizes and their confidence intervals alongside, or in place of, bare p-values, since an effect size directly communicates the magnitude of a finding in a way that a p-value alone cannot.
Phase 6 has established the statistical foundation required to interpret pharmacological data correctly: the logic of test selection according to group number, data distribution, and design; the practical assumptions and workflow of parametric testing; the four-parameter logistic model underlying all dose-response analysis; and the mandatory, prospective calculation of sample size. Even the most carefully designed experiment yields misleading conclusions if analysed incorrectly, which is why the pitfalls associated with each stage of this pipeline — including the statistical errors discussed here — are drawn together and addressed systematically in Phase 7.