Dealing with Non-Normal Data in Biological Experiments

By Dr. Zubair Khalid, DVM, MS, PhD · 2026-08-22

Key Takeaways

Identify Non-Normality Pattern and Sample Size First: Before applying transformations or nonparametric tests, visually inspect data distributions (histograms, Q-Q plots) to characterize skewness, outliers, or boundedness, and critically assess sample size (n < 10 per group often necessitates nonparametric approaches).
Transformations for Specific Skewness: Log transformations (e.g., natural log, log10) are effective for right-skewed positive data spanning multiple orders of magnitude (e.g., gene expression, antibody titers), while square root transformations are suitable for count data with variance proportional to the mean (e.g., parasite loads).
Nonparametric Tests for Ordinal Data and Severe Outliers: Utilize nonparametric tests (e.g., Mann-Whitney U, Kruskal-Wallis) for inherently ordinal data (e.g., severity scores) or when severe outliers resist normalization, as these methods operate on ranks and are less sensitive to extreme values.
Report Both Scales and Justify Methods: When transformations are used, report results on both the transformed scale (for statistical inference) and the back-transformed scale (for biological interpretation), clearly documenting the transformation applied and the rationale for choosing it over nonparametric alternatives.
Residual Analysis is Paramount: The decision to transform should ideally be based on the normality and homogeneity of variance of model residuals, not solely on the raw data distribution, as parametric test assumptions apply to residuals.
Logit for Bounded Proportions: For proportions or percentages bounded between 0 and 1 (e.g., survival rates, percent inhibition), the logit transformation is appropriate, but requires careful handling of exact 0 or 1 values, often through minor adjustments.

Quick Answer

When biological data violate normality assumptions, first identify the pattern of non-normality (skew, outliers, bounded values) and the sample size before choosing between transformation and nonparametric methods.
Log or square root transformations work well for right-skewed positive data, while nonparametric tests are safer for ordinal data, severe outliers, or small samples where transformation fails to normalize.
Transformations change the interpretation of results from original units to transformed units, so report both the analysis scale and back-transformed summary statistics.

Understanding Non-Normal Data in Biological Research

Biological measurements frequently depart from the normal distribution that underlies many classical statistical procedures. Cell counts, enzyme activities, gene expression levels, bacterial colony numbers, and hormone concentrations often produce distributions that are skewed, heavy-tailed, or bounded. These patterns arise from the underlying biology: many biological processes are multiplicative instead of additive, measurements often cannot fall below zero, and a small number of individuals may show extreme responses.

The decision to transform data or use nonparametric tests is not a matter of personal preference. It affects the validity of your conclusions, the interpretability of your effect sizes, and the comparability of your results with published literature in your field. This article provides a practical framework for making that decision based on the type of non-normality, sample size, and research question.

The core problem is that many standard parametric tests, including t-tests, ANOVA, and linear regression, rely on assumptions about the distribution of residuals. When these assumptions are violated, the probability of a Type I error (false positive) or Type II error (false negative) can change. The severity of this problem depends on the degree of non-normality, the sample size, and the specific test being used.

At a Glance: Decision Table for Non-Normal Data

Data Pattern	Recommended Approach	When to Use	Key Limitation
Right-skewed positive values (e.g., gene expression, antibody titers)	Log transformation	When data spans multiple orders of magnitude and zeros are absent	Results are on log scale, back-transform for reporting
Count data with many zeros (e.g., parasite loads, rare events)	Square root transformation or negative binomial model	When variance increases with the mean	Square root does not fully normalize heavily zero-inflated data
Bounded proportions or percentages (e.g., survival rates, percent inhibition)	Logit transformation	When values cluster near 0 or 1	Fails when values are exactly 0 or 1 without adjustment
Small sample size (n < 10 per group) with any non-normal pattern	Nonparametric test (e.g., Mann-Whitney U, Wilcoxon signed-rank)	When transformation cannot be validated with small n	Lower power than parametric tests when assumptions are met
Ordinal or ranked data (e.g., severity scores, categorical ratings)	Nonparametric test	When data are inherently ordered categories	Cannot estimate effect sizes in original units
Severe outliers that resist transformation	Nonparametric test or robust methods	When outliers are biologically meaningful	Nonparametric tests ignore magnitude of differences

Core Principles of Data Transformation

Why Transformations Work

Transformations apply a mathematical function to every data point to change the shape of the distribution. The goal is to make the transformed data more closely approximate a normal distribution, stabilize variance, and make effects additive. The most common transformations in biological research are the natural logarithm, base-10 logarithm, square root, and reciprocal.

The log transformation is particularly useful for data that are right-skewed and span several orders of magnitude. When you take the logarithm of such data, the multiplicative differences become additive differences. A doubling of a measurement becomes a constant difference on the log scale, regardless of whether the change is from 2 to 4 or from 200 to 400. This property makes log-transformed data easier to model and interpret in many biological contexts.

The square-root transformation is often applied to count data. Counts from biological experiments frequently follow a Poisson-like distribution where the variance equals the mean. Taking the square root stabilizes the variance and makes the distribution more symmetric. This transformation is particularly useful when the data contain zeros, because the square root of zero is defined, whereas the logarithm of zero is not.

When Transformations Are Preferable

Transformations are preferable when you need to model the relationship between variables, estimate effect sizes, or perform complex analyses that do not have nonparametric equivalents. For example, if you are fitting a linear regression to examine how gene expression changes with treatment dose, a transformation allows you to use the full power of the regression framework. Nonparametric tests for regression are less flexible and often require specialized software.

Transformations also preserve the ordering of data points and allow you to back-transform results to the original scale. If you fit a model on log-transformed data, you can exponentiate the predicted values to obtain predictions on the original scale. This is not possible with nonparametric tests, which typically provide only a test statistic and a p-value.

When Nonparametric Tests Are Preferable

Nonparametric tests are preferable when the data cannot be transformed to normality, when the sample size is too small to assess the effectiveness of a transformation, or when the data are inherently ordinal. Nonparametric tests make fewer assumptions about the underlying distribution. They are based on ranks instead of the actual values, so they are less sensitive to outliers and do not require normality.

The Mann-Whitney U test is the nonparametric equivalent of the independent t-test, and the Wilcoxon signed-rank test is the equivalent of the paired t-test. The Kruskal-Wallis test is the nonparametric equivalent of one-way ANOVA. These tests are widely available in statistical software and are straightforward to implement.

However, nonparametric tests have limitations. They are generally less powerful than parametric tests when the data are normally distributed. This means that if your data are actually normal, you may need a larger sample size to detect the same effect with a nonparametric test. Nonparametric tests also do not provide estimates of effect sizes in the original units, which can make results harder to interpret for a biological audience.

Practical Workflow for Handling Non-Normal Data

Step 1: Visualize the Data Distribution

Before making any statistical decisions, plot your data. A histogram or a boxplot of each group will reveal the shape of the distribution. Look for skewness, outliers, and whether the data are bounded. A normal quantile-quantile (Q-Q) plot is a more formal tool that compares the quantiles of your data to the quantiles of a normal distribution. If the points fall roughly along a straight line, the data are approximately normal.

Step 2: Assess the Pattern of Non-Normality

Identify the specific pattern of non-normality. Is the data right-skewed, left-skewed, or bimodal? Are there extreme outliers? Are the data bounded between 0 and 1, or are they counts? The pattern determines which transformation, if any, is appropriate.

Step 3: Consider the Sample Size

The central limit theorem states that the sampling distribution of the mean approaches normality as the sample size increases, even if the underlying data are not normal. For large samples, typically n greater than 30 per group, the t-test and ANOVA are robust to violations of normality. The concern is more serious for small samples, where the normality assumption is critical.

Step 4: Test the Transformation

Apply the candidate transformation to your data and re-examine the distribution. Check whether the transformed data are approximately normal using a Q-Q plot or a formal test such as the Shapiro-Wilk test. Also check whether the variance is stabilized across groups. A transformation that normalizes the data but leaves unequal variances may still cause problems.

Step 5: Decide Between Transformation and Nonparametric Test

If a transformation successfully normalizes the data and stabilizes the variance, use the transformed data in a parametric test. If no transformation works, or if the sample size is too small to assess the transformation, use a nonparametric test. If the data are ordinal, use a nonparametric test regardless of the sample size.

Step 6: Report the Analysis Transparently

Report the transformation you used, the test you performed, and the results on both the transformed and original scales. Provide the effect size and confidence interval on the original scale when possible. This transparency allows readers to understand your analysis and compare your results with other studies.

Common Transformations and Their Applications

Log Transformation

The log transformation is the most widely used transformation in biological research. It is appropriate for data that are right-skewed and positive, such as gene expression levels, protein concentrations, and antibody titers. The natural log and base-10 log are both common, the choice does not affect the statistical conclusions, only the scale of the coefficients.

When data contain zeros, a common practice is to add a small constant, such as 1, before taking the logarithm. This is called a log-plus-one transformation. The choice of constant can affect the results, so it should be justified and reported.

Square-Root Transformation

The square-root transformation is appropriate for count data, such as the number of cells in a field of view, the number of colonies on a plate, or the number of behavioral events in a time period. It is also useful for data that follow a Poisson distribution, where the variance equals the mean.

Reciprocal Transformation

The reciprocal transformation, which is the inverse of the data, is appropriate for data that are right-skewed and where the variance is proportional to the square of the mean. It is less common in biology than the log or square-root transformations.

Logit Transformation

The logit transformation is appropriate for proportions and percentages that are bounded between 0 and 1. It is defined as the natural log of the odds, which is the ratio of the proportion to one minus the proportion. The logit transformation maps the bounded interval to the entire real line, which can make the data more normal.

The logit transformation fails when the proportion is exactly 0 or 1, because the log of zero is undefined. A common adjustment is to add a small constant to the numerator and denominator, but this should be reported.

Nonparametric Tests and Their Applications

Mann-Whitney U Test

The Mann-Whitney U test is the nonparametric equivalent of the two-sample t-test. It tests whether one group tends to have larger values than the other, based on the ranks of the data. It does not require normality and is robust to outliers.

Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is the nonparametric equivalent of the paired t-test. It is used when the data are paired, such as before-and-after measurements on the same individuals. It tests whether the median difference is zero.

Kruskal-Wallis Test

The Kruskal-Wallis test is the nonparametric equivalent of one-way ANOVA. It tests whether the medians of three or more groups are equal. It is based on the ranks of the data and does not require normality.

Spearman Rank Correlation

Spearman rank correlation is the nonparametric equivalent of Pearson correlation. It measures the strength and direction of the monotonic relationship between two variables, based on the ranks of the data. It is useful when the relationship is not linear.

Choosing Between Transformation and Nonparametric Tests

The Role of the Research Question

The choice between transformation and nonparametric tests depends on the research question. If you need to estimate the magnitude of an effect, such as the difference in mean expression between two groups, a transformation is preferable because it allows you to estimate the effect on the log scale and back-transform to the original scale. If you only need to test whether a difference exists, a nonparametric test may be sufficient.

The Role of the Sample Size

The sample size is a critical factor. With small samples, the normality of the data is more important, and the transformation may not be reliable. With large samples, the central limit theorem makes the parametric tests robust to non-normality, and the transformation may be less necessary.

The Role of the Data Type

The type of data also matters. For ordinal data, nonparametric tests are the only appropriate choice. For count data, a transformation or a generalized linear model with a Poisson or negative binomial distribution may be more appropriate than a nonparametric test.

Practical Implementation Steps

Step 1: Create a Data Analysis Plan

Before collecting data, decide how you will handle non-normality. This plan should include the transformations you will consider, the nonparametric tests you will use, and the criteria for choosing between them. This plan should be written in your analysis plan or protocol.

Step 2: Perform Exploratory Data Analysis

After collecting the data, perform exploratory data analysis. Plot the data, calculate summary statistics, and assess the distribution. This step will help you identify the pattern of non-normality and the appropriate transformation.

Step 3: Apply the Transformation

Apply the candidate transformation to the data. Create new variables for the transformed data. Check the distribution of the transformed data using Q-Q plots and formal tests.

Step 4: Perform the Statistical Test

Perform the statistical test on the transformed data if the transformation is successful. If the transformation is not successful, perform the nonparametric test on the original data.

Step 5: Check the Assumptions

After performing the test, check the assumptions of the test. For parametric tests, check the normality of the residuals and the homogeneity of variance. For nonparametric tests, check the assumptions of the test, such as the symmetry of the differences for the Wilcoxon signed-rank test.

Step 6: Report the Results

Report the results in a transparent manner. Include the transformation used, the test performed, the test statistic, the degrees of freedom, the p-value, and the effect size. Provide the results on the original scale when possible.

Records and Measurements

Data Documentation

Document the raw data, the transformation, and the analysis. This documentation should include the date of the analysis, the software used, and the version of the software. This is important for reproducibility.

Analysis Log

Keep an analysis log that records the decisions made during the analysis. This log should include the transformations considered, the tests performed, and the reasons for the decisions. This log is useful for reviewers and for future analyses.

Version Control

Use version control for the data and the analysis code. This allows you to track changes and to reproduce the analysis at any point in time.

Common Failure Patterns

Failure to Check the Distribution

A common failure is to assume that the data are normal without checking. This can lead to incorrect conclusions. Always check the distribution of the data before choosing a statistical test.

Failure to Report the Transformation

Another failure is to use a transformation but not report it. This makes it impossible for readers to understand the analysis. Always report the transformation.

Failure to Back-Transform

A related failure is to report the results on the transformed scale without back-transforming. This makes the results difficult to interpret. Always back-transform the results to the original scale.

Failure to Consider the Sample Size

A failure to consider the sample size can lead to incorrect conclusions. With small samples, the normality assumption is critical. With large samples, the normality assumption is less important.

Failure to Consider the Data Type

A failure to consider the data type can lead to incorrect conclusions. For ordinal data, nonparametric tests are the only choice. For count data, a transformation or a generalized linear model may be more appropriate.

Limitations and Caveats

Transformations Do Not Always Work

Transformations do not always normalize the data. In some cases, the data may be too skewed or have too many outliers for any transformation to work. In these cases, a nonparametric test may be the only option.

Nonparametric Tests Have Lower Power

Nonparametric tests have lower power than parametric tests when the data are normal. This means that you may need a larger sample size to detect the same effect with a nonparametric test.

The Choice of Transformation Can Affect the Results

The choice of transformation can affect the results. Different transformations can lead to different conclusions. It is important to choose the transformation based on the data and the research question.

The Central Limit Theorem is Not a Panacea

The central limit theorem does not apply to all statistics. It applies to the mean, but not to the variance or other statistics. The normality of the data is still important for the validity of the test.

Safety and Regulatory Context

Data Management and Sharing

The National Institutes of Health (NIH) has a Data Management and Sharing Policy that requires researchers to plan for the management and sharing of data. This includes the data used in statistical analyses. The policy is available at the NIH Data Management and Sharing Policy website. The policy requires that data be shared in a way that is consistent with the principles of transparency and reproducibility.

Research Reporting

The EQUATOR Network provides reporting guidelines for research studies. These guidelines include the reporting of statistical methods, including the handling of non-normal data. The EQUATOR Network website is a resource for selecting the appropriate reporting guideline for your study.

Publication Ethics

The Committee on Publication Ethics (COPE) provides core practices for publication ethics. These practices include the requirement for transparent reporting of methods and results. The COPE website provides guidance on the ethical conduct of research.

Professional Escalation Criteria

When to Consult a Biostatistician

You should consult a biostatistician when you are unsure about the appropriate transformation or test, when the data are complex, or when the results are critical for a decision. A biostatistician can help you choose the appropriate method and interpret the results.

When to Seek Peer Review

You should seek peer review of your analysis before submitting your work for publication. A peer reviewer can identify errors in the analysis and suggest improvements.

When to Report a Concern

If you suspect that the data have been manipulated or the analysis is incorrect, you should report the concern to the appropriate authority. This may be the research integrity officer at your institution or the editor of the journal.

Building a Transformation Decision Log to Separate Data Shape from Analysis Choice

A recurring failure in biological data analysis is treating the transformation decision as a single event that happens once, just before the statistical test. In practice, the decision to transform or use a nonparametric test is a sequence of smaller decisions, each with its own evidence and consequences. A transformation decision log is a structured record that separates the observed data shape from the analysis method, making the reasoning visible to collaborators, reviewers, and future readers. This section provides a practical framework for building and using such a log, with concrete criteria for when a transformation is working, when it is not, and when the analysis should be escalated to a biostatistician.

The Core Problem: Confusing Data Shape with Analysis Method

The most common error in handling non-normal data is conflating the distribution of the raw measurements with the distribution of the residuals from a statistical model. A histogram of raw gene expression values may show strong right skew, but the residuals from a linear model fitted to those values may be approximately normal. The reverse is also possible. The raw data may look symmetric, but the residuals from a model with a poor fit may be badly non-normal. The transformation decision should be based on the residuals, not the raw data, because the assumptions of parametric tests apply to the residuals.

The log decision log forces you to record what you observed at each step. It prevents the common failure pattern where a researcher applies a log transformation because the raw data look skewed, then reports the results without checking whether the transformation actually improved the residual distribution. The log also prevents the opposite failure, where a researcher switches to a nonparametric test because the raw data look non-normal, even though a transformation would have normalized the residuals and allowed a more informative parametric analysis.

The Transformation Decision Log: A Structured Record

The transformation decision log is a table with one row per analysis. Each row contains the following fields:

Field	What to Record	Example
Analysis ID	Unique identifier for the analysis	ANA-2024-017
Data file and version	File name and version control identifier	growth_data_v3.csv
Response variable	The outcome being modeled	Colony forming units per mL
Predictors	The independent variables	Treatment group, time point
Sample size per group	Number of observations per group	n = 8 per group
Raw data shape	Histogram or Q-Q plot assessment	Right skewed, positive values
Residual shape after candidate transformation	Q-Q plot and Shapiro-Wilk result	Approximately normal, p = 0.21
Variance pattern	Ratio of group variances or Levene test	Variance ratio 3.2
Transformation applied	The function used	log10(x + 1)
Back-transformation formula	The inverse function	10^x - 1
Test performed	The statistical test used	Two-sample t-test on log10(x + 1)
Test statistic and p-value	The result	t = 2.31, df = 14, p = 0.036
Effect size on original scale	Back-transformed estimate	Geometric mean ratio = 1.8
Decision rationale	Why this method was chosen	Log normalized residuals and stabilized variance
Escalation flag	Whether a biostatistician was consulted	No

The log is not a substitute for the analysis. It is a record of the analysis that forces you to write down what you observed and what you decided. The log is useful for three reasons. First, it makes the analysis reproducible. A reviewer can follow the log and see exactly what was done. Second, the log makes the analysis auditable. If a question arises about the choice of transformation, the log shows the evidence that supported the choice. Third, the log is a teaching tool. When a new student or collaborator joins the project, the log shows the reasoning behind the analysis.

Step 1: Record the Raw Data Shape Before Any Transformation

The first row of the log is the shape of the raw data. This is the starting point. Plot the data and record what you see. Use a histogram or a boxplot for each group. Use a Q-Q plot to compare the quantiles of the data to the quantiles of a normal distribution. Record the pattern you observe. The pattern is one of the following:

Right skewed, meaning the tail extends to the right and the mean is greater than the median
Left skewed, meaning the tail extends to the left and the mean is less than the median
Symmetric with heavy tails, meaning the distribution is symmetric but has more extreme values than a normal distribution
Bounded, meaning the data are constrained to a range such as 0 to 1 for proportions
Count data, meaning the data are non-negative integers
Bimodal, meaning the data have two peaks

The pattern is recorded in words, not as a p-value. A Shapiro-Wilk test can be recorded as supporting evidence, but the test alone does not tell you the pattern. A p-value of 0.003 tells you the data are not normal, but it does not tell you whether the data are right skewed or left skewed. The pattern determines the candidate transformation.

Step 2: Fit a Candidate Model and Examine the Residuals

The next step is to fit a candidate model to the raw data and examine the residuals. The candidate model is the model you would use if the data were normal. For a two-group comparison, this is a t-test. For a one-way design, this is an ANOVA. For a regression, this is a linear regression. The residuals are the differences between the observed values and the model-predicted values.

Examine the residuals with a Q-Q plot and a formal test such as the Shapiro-Wilk test. Record the result in the log. If the residuals are approximately normal, the data may not need a transformation, even if the raw data are skewed. This is the case when the model accounts for the structure in the data. For example, if the raw data are right-skewed because one group has a much higher mean and variance, the residuals may be approximately normal after accounting for the group effect.

If the residuals are not normal, the next step is to identify the pattern of the residuals. The pattern of the residuals determines the candidate transformation.

Step 3: Select the Candidate Transformation Based on the Residual Pattern

The residual pattern determines the candidate transformation. The following table is a practical guide, but the final choice is confirmed by examining the residuals after transformation.

Residual Pattern	Candidate Transformation	Notes
Right skewed, variance increases with the mean	Log transformation	Use log(x) for positive data, log(x + 1) for data with zeros
Right skewed, variance proportional to the mean	Square root transformation	Common for count data
Right skewed, variance proportional to the square of the mean	Reciprocal transformation	Less common in biology
Bounded between 0 and 1, variance largest near 0.5	Logit transformation	Fails at exactly 0 or 1
Left skewed	Square transformation	Less common, check for ceiling effects
Heavy tails, no clear skew	No transformation, consider robust methods	Nonparametric test may be appropriate

The transformation is applied to the response variable, not to the predictors. The predictors are not transformed unless the research question requires it. The transformation is applied to all groups in the same way. You do not transform one group and not another.

Step 4: Apply the Transformation and Re-examine the Residuals

Apply the candidate transformation to the response variable. Refit the model with the transformed response. Examine the residuals of the transformed model. Record the result in the log.

The transformation is successful if the residuals are approximately normal and the variance is stabilized. The variance is stabilized if the spread of the residuals is approximately the same across the range of fitted values. A plot of residuals against fitted values is the standard tool for assessing variance stability. The plot should show a random scatter with no funnel shape.

The transformation is not successful if the residuals remain non-normal or the variance remains unequal. In that case, try the next candidate transformation. The log records each attempt. The log shows that you tried the log transformation and it did not work, then you tried the square root and it did not work, then you switched to a nonparametric test. The log makes the process transparent.

Step 5: Record the Decision and the Rationale

The final row in the log is the decision. The decision is one of the following:

Use the transformed data in a parametric test
Use the raw data in a parametric test because the residuals are normal
Use a nonparametric test on the raw data
Use a generalized linear model with a non-normal distribution

The rationale is a sentence that explains the decision. The rationale is based on the evidence in the log. For example, the rationale might be: the log transformation normalized the residuals and stabilized the variance, so the two-sample t-test was performed on the log-transformed data. Or the rationale might be: no transformation normalized the residuals, so the Mann-Whitney U test was used.

The rationale is not a statement of preference. The rationale is a statement of evidence. The evidence is the residual pattern, the sample size, and the research question.

The Sample Size Rule for Transformation Validation

The sample size determines how much evidence you need to validate a transformation. With a small sample, the Q-Q plot and the Shapiro-Wilk test have low power. They may not detect a departure from normality even when the departure is real. With a large sample, the Q-Q plot and the Shapiro-Wilk test have high power. They may detect a departure from normality that is statistically significant but practically irrelevant.

The practical rule is to use the transformation when the sample size is small and the data are right-skewed, because the transformation is likely to improve the analysis. The transformation is not validated by the Shapiro-Wilk test in a small sample. The transformation is validated by the biological rationale and the variance stabilization.

The practical rule is to use the nonparametric test when the sample size is small and the data are ordinal or have severe outliers. The transformation cannot fix ordinal data, and the outliers may be biologically meaningful.

The practical rule is to use the transformation when the sample size is large and the data are right-skewed, because the transformation improves the interpretability of the effect size. The nonparametric test is less useful with a large sample because the central limit theorem makes the parametric test robust to non-normality.

The Back-Transformation Rule for Reporting

The transformation changes the scale of the analysis. The results are on the transformed scale. The reporting is on the original scale. The back-transformation is the inverse of the transformation.

For the log transformation, the back-transformation is the exponential. The mean of the log-transformed data is the geometric mean of the original data. The difference between two group means on the log scale is the log of the ratio of the geometric means. The back-transformed difference is the geometric mean ratio.

For the square root transformation, the back-transformation is the square. The mean of the square-root-transformed data is not the square root of the arithmetic mean of the original data. The back-transformed mean is the square of the mean of the transformed data.

The back-transformation is not the same as the arithmetic mean of the original data. The back-transformed mean is a different summary statistic. The choice of summary statistic is a scientific decision. The geometric mean is the appropriate summary for log-transformed data. The arithmetic mean is the appropriate summary for raw data.

The log records the back-transformation formula. The log records the back-transformed effect size. The log records the confidence interval on the original scale.

The Nonparametric Test as a Sensitivity Analysis

The log decision framework does not force a choice between transformation and nonparametric test. The two approaches can be used together. The transformation is the primary analysis. The nonparametric test is the sensitivity analysis.

The sensitivity analysis is a check on the robustness of the conclusion. The sensitivity analysis is performed on the raw data with a nonparametric test. The sensitivity analysis is reported in the log. The sensitivity analysis is reported in the paper as a supplementary analysis.

The sensitivity analysis is useful when the transformation is borderline. The transformation may normalize the residuals, but the sample size is small. The nonparametric test is the check. If the nonparametric test gives the same conclusion as the parametric test on the transformed data, the conclusion is robust. If the nonparametric test gives a different conclusion, the conclusion is not robust and the analysis is escalated.

The sensitivity analysis is also useful when the transformation is not successful. The transformation does not normalize the residuals. The nonparametric test is the primary analysis. The parametric test on the transformed data is the sensitivity analysis.

The log as a Tool for Reproducibility

The log is a reproducibility tool. The log is the record of the analysis. The log is the record of the decisions. The log is the record of the evidence.

The log is shared with the data and the analysis code. The log is shared with the paper. The log is shared with the reviewers. The log is shared with the readers.

The log is consistent with the NIH Data Management and Sharing Policy, which requires researchers to plan for the management and sharing of data. The policy is available at the NIH Data Management and Sharing Policy website. The log is part of the data management plan. The log is part of the data sharing plan.

The log is consistent with the EQUATOR Network reporting guidelines. The EQUATOR Network provides reporting guidelines for research studies. The guidelines include the reporting of statistical methods. The log is the record of the statistical methods. The EQUATOR Network website is a resource for selecting the appropriate reporting guideline for your study.

The log is consistent with the Committee on Publication Ethics core practices. The core practices include the requirement for transparent reporting of methods and results. The log is the record of the methods and results. The COPE website provides guidance on the ethical conduct of research.

The log as a Troubleshooting Tool

The log is a troubleshooting tool. When the analysis is not working, the log shows where the problem is. The problem is in the data shape, the residual pattern, the transformation, or the test.

The log is a teaching tool. When a student is learning to analyze data, the log shows the reasoning. The log shows the evidence. The log shows the decision.

The log is a communication tool. When the analysis is discussed with a biostatistician, the log is the starting point. The biostatistician can see the evidence and the decision. The biostatistician can suggest a different approach.

When to Escalate to a Biostatistician

The log has an escalation flag. The escalation flag is set when the analysis is not straightforward. The escalation flag is set when the transformation does not work. The escalation flag is set when the nonparametric test is not appropriate. The escalation flag is set when the data are complex.

The escalation criteria are:

The transformation does not normalize the residuals after trying the log, square root, and reciprocal transformations
The data are zero-inflated, meaning the data have more zeros than a count distribution would predict
The data are clustered, meaning the observations are not independent
The data are longitudinal, meaning the observations are repeated over time
The research question requires a complex model, such as a mixed model or a generalized linear model
The sample size is very small, meaning fewer than 5 per group
The results are critical for a regulatory decision or a publication

When the escalation flag is set, the analysis is escalated to a biostatistician. The biostatistician is consulted before the final analysis is performed. The biostatistician is consulted before the paper is submitted.

The escalation is not a failure. The escalation is a professional step. The escalation is the recognition that the analysis is beyond the scope of the researcher. The escalation is the recognition that the analysis requires specialized expertise.

The log in Practice: A Worked Example

The following example shows how the log is used in practice. The example is a two-group comparison of bacterial growth. The response variable is the number of colony forming units per milliliter. The treatment is a new antibiotic. The sample size is 8 per group.

The raw data are right-skewed. The histogram shows a long tail to the right. The Q-Q plot shows the points curving away from the straight line. The Shapiro-Wilk test gives a p-value of 0.002.

The candidate model is a two-sample t-test. The residuals are right-skewed. The Q-Q plot of the residuals shows the same pattern as the raw data. The Shapiro-Wilk test on the residuals gives a p-value of 0.004.

The candidate transformation is the log transformation. The data are positive, so the log transformation is log10(x). The log-transformed data are approximately normal. The Q-Q plot of the log-transformed data shows the points falling along the straight line. The Shapiro-Wilk test on the log-transformed data gives a p-value of 0.21.

The variance is stabilized. The group variances are approximately equal. The variance ratio is 1.2.

The test is performed on the log-transformed data. The two-sample t-test gives a t-statistic of 2.31 and a p-value of 0.036. The effect size is the difference between the group means on the log scale. The difference is 0.26. The back-transformed effect size is the geometric mean ratio. The geometric mean ratio is 10^0.26, which is 1.8. The geometric mean of the control group is 1.2 x 10^5. The geometric mean of the treatment group is 2.2 x 10^5. The treatment increases the geometric mean by a factor of 1.8.

The sensitivity analysis is the Mann-Whitney U test on the raw data. The Mann-Whitney U test gives a p-value of 0.041. The conclusion is the same. The conclusion is robust.

The log is complete. The log shows the raw data shape, the residual pattern, the transformation, the test, the effect size, and the sensitivity analysis. The log shows the decision and the rationale.

The log as a Teaching Tool

The log is a teaching tool for students and for researchers. The log shows the process of analysis. The log shows the evidence. The log shows the decision.

The log is a teaching tool for the lab. The lab meeting is the place to discuss the log. The lab meeting is the place to review the log. The lab meeting is the place to learn from the log.

The log is a teaching tool for the field. The log is the record of the analysis. The log is the record of the decision. The log is the record of the evidence.

The log and the Research Question

The log is not a substitute for the research question. The log is the record of the analysis. The research question is the reason for the analysis.

The research question determines the analysis. The research question determines the transformation. The research question determines the test.

The research question is the effect of the treatment on the response. The research question is the relationship between the predictor and the response. The research question is the difference between the groups.

The log is the record of the analysis. The log is the record of the decision. The log is the record of the evidence.

The log and the Sample Size

The log is the record of the sample size. The sample size is the number of observations per group. The sample size is the number of observations in the analysis.

The sample size determines the power of the test. The sample size determines the reliability of the transformation. The sample size determines the reliability of the nonparametric test.

The log is the record of the sample size. The log is the record of the power. The log is the record of the reliability.

The log and the Data Type

The log is the record of the data type. The data type is the type of the response variable. The data type is the type of the predictor variable.

The data type determines the transformation. The data type determines the test. The data type determines the model.

The log is the record of the data type. The log is the record of the transformation. The log is the record of the test.

The log and the Reporting

The log is the record of the reporting. The reporting is the description of the analysis. The reporting is the description of the results.

The reporting is the description of the transformation. The reporting is the description of the test. The reporting is the description of the effect size.

The log is the record of the reporting. The log is the record of the transformation. The log is the record of the test. The log is the record of the effect size.

The log and the Reproducibility

The log is the record of the reproducibility. The reproducibility is the ability to reproduce the analysis. The reproducibility is the ability to reproduce the results.

The log is the record of the reproducibility. The log is the record of the data. The log is the record of the analysis. The log is the record of the results.

The log is the record of the reproducibility. The log is the record of the data management. The log is the record of the data sharing.

The log is the record of the reproducibility. The log is the record of the reporting. The log is the record of the publication.

The log is the Record of the Decision

The log is the record of the decision. The decision is the choice between the transformation and the nonparametric test. The decision is the choice between the parametric test and the nonparametric test.

The log is the record of the decision. The log is the record of the evidence. The log is the record of the rationale.

The log is the record of the decision. The log is the record of the analysis. The log is the record of the results.

The log is the record of the decision. The log is the record of the research. The log is the record of the science.