P-Value Formula and Interpretation: A Researcher's Guide

A p-value is the probability of obtaining test results at least as extreme as those observed, assuming the null hypothesis is true. It quantifies how compatible your data are with the null hypothesis, not the probability that your hypothesis is correct. This article explains the formulas for common tests, correct interpretation, frequent misconceptions, and why effect size and confidence intervals must accompany any p-value report.

At a Glance

What Is a P-Value

The p-value is the maximum probability of getting the observed outcome by chance when the null hypothesis is true. Biomedical research relies on proving or disproving a research hypothesis, and the p-value becomes a cornerstone of null hypothesis significance testing. For a statistical test to achieve significance, the error by chance must be less than 5 percent. The p-value gives the strength of evidence against the null hypothesis, and researchers can reject a null hypothesis depending on a small p-value. However, the value of p is a function of sample size. When the sample size is large, the p-value is destined to be small or significant. This relationship between sample size and p-value is a critical limitation that researchers must understand before designing studies or interpreting results.

The p-value does not tell you the probability that your hypothesis is correct. It does not tell you the probability that the results occurred by chance alone. It tells you how compatible your data are with the null hypothesis. A small p-value indicates that the observed data would be unusual if the null hypothesis were true, which leads researchers to question the null hypothesis. A large p-value indicates that the data are consistent with the null hypothesis, but it does not prove the null hypothesis is true.

The P-Value Formula for Common Tests

Z-Test Formula

The z-test is used when the population standard deviation is known and the sample size is large. The test statistic is calculated as:

z = (x̄ - μ₀) / (σ / √n)

Where x̄ is the sample mean, μ₀ is the hypothesized population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size.

The p-value is then derived from the standard normal distribution. For a one-tailed test, the p-value is the area in the tail beyond the calculated z-value. For a two-tailed test, the p-value is twice the area in the tail. The p-value calculation depends on the direction of the alternative hypothesis. If the alternative hypothesis states that the mean is greater than μ₀, the p-value is the area to the right of the z-value. If the alternative hypothesis states that the mean is less than μ₀, the p-value is the area to the left. If the alternative hypothesis is two-sided, the p-value includes both tails.

T-Test Formula

The t-test is used when the population standard deviation is unknown and must be estimated from the sample. The test statistic is calculated as:

t = (x̄ - μ₀) / (s / √n)

Where x̄ is the sample mean, μ₀ is the hypothesized population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size.

The p-value is derived from the t-distribution with n - 1 degrees of freedom. For a two-sample t-test comparing two independent groups, the formula becomes:

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

The degrees of freedom for this test depend on whether the variances are assumed equal. The Welch test uses a more complex degrees of freedom calculation that does not assume equal variances.

Chi-Square Test Formula

The chi-square test is used for categorical data. The test statistic is calculated as:

χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ

Where Oᵢ is the observed frequency in category i and Eᵢ is the expected frequency under the null hypothesis. The p-value is derived from the chi-square distribution with degrees of freedom equal to (rows - 1) × (columns - 1) for a contingency table.

Correlation Coefficient P-Value

For correlation coefficients, the p-value calculation methods for partial and semi-partial correlation coefficients were proposed by Cohen et al. in 2003 and Kim in 2015. The method of Cohen et al. depends on a single and identical statistic for both partial and semi-partial correlations, while Kim's method uses two separate statistics for each type of correlation. The mathematical expression of the p-value calculation for the semi-partial correlation coefficient differs between the two methods. Comparisons through theoretical aspects and simulations show that the semi-partial correlation coefficients by Kim are monotonic to the corresponding p-values, while the Cohen method does not maintain monotonicity. This means that for the same data, the ranking of p-values can differ depending on which method is used, which has practical implications for researchers testing multiple correlations.

Step-by-Step Guide to Calculating P-Values

Step 1: State the Hypotheses

Define the null hypothesis and the alternative hypothesis before collecting data. The null hypothesis typically states that there is no effect or no difference. The alternative hypothesis states what you expect to find. These hypotheses must be stated in advance to avoid post hoc reasoning.

Step 2: Choose the Appropriate Test

Choosing the correct test for assessing the significance of the difference is profoundly important. The choice can be arrived by asking three questions: the type of data, whether the data is paired or not, and the number of study groups. Continuous data with two groups typically use a t-test. Categorical data use a chi-square test. Paired data use a paired t-test or Wilcoxon signed-rank test. More than two groups use ANOVA or Kruskal-Wallis tests. Using the wrong test can produce misleading p-values.

Step 3: Calculate the Test Statistic

Apply the appropriate formula for your chosen test. This calculation uses your sample data and the assumptions of the null hypothesis. The test statistic measures how far your sample result is from what the null hypothesis predicts.

Step 4: Determine the Degrees of Freedom

Each test has specific degrees of freedom that determine the shape of the sampling distribution. For a one-sample t-test, degrees of freedom equal n - 1. For a chi-square test, degrees of freedom depend on the number of categories. Using the correct degrees of freedom is essential for accurate p-value calculation.

Step 5: Find the P-Value

Use statistical software, a calculator, or a distribution table to find the p-value corresponding to your test statistic and degrees of freedom. Most modern research uses statistical software for this step. The p-value is the area in the tail or tails of the distribution beyond your test statistic.

Step 6: Interpret in Context

Compare your p-value to your pre-specified significance level. Report the actual p-value instead of only stating whether it is below or above a threshold. Interpret the result in the context of your study design, sample size, and prior evidence.

Correct Interpretation of P-Values

The correct interpretation of p < 0.05 is that the probability of obtaining the observed results or more extreme results is less than 5 percent if the null hypothesis is true. This interpretation is conditional on the null hypothesis being true. It is not the probability that the null hypothesis is false. It is not the probability that the results occurred by chance. It is not the probability that the alternative hypothesis is true.

The p-value provides a measure of the degree of data compatibility with the null hypothesis. A smaller p-value indicates less compatibility. When reporting p-values, authors should always provide the actual value, also statements of p < 0.05 or p ≥ 0.05, because p-values give a measure of the degree of data compatibility with the null hypothesis. Reporting the actual value allows readers to assess the strength of evidence for themselves.

The p-value is a function of sample size. When the sample size is large, the p-value is destined to be small or significant. This means that a very large study can produce a small p-value for a trivial effect, while a small study can produce a non-significant p-value for a clinically important effect. Researchers must consider the sample size when interpreting p-values.

Common Misconceptions About P-Values

Misconception: P-Value Is the Probability That the Null Hypothesis Is True

The p-value is calculated under the assumption that the null hypothesis is true. It cannot be used to assign a probability to the null hypothesis itself. The null hypothesis is either true or false in reality, and the p-value does not quantify this. This misconception leads researchers to incorrectly state that a non-significant result proves the null hypothesis.

Misconception: P-Value Is the Probability That Results Occurred by Chance

The p-value is the probability of the data given the null hypothesis, not the probability that the results are due to chance. These are different quantities. The p-value conditions on the null hypothesis being true and asks how unusual the data are under that condition.

Misconception: P < 0.05 Means the Effect Is Clinically Important

Statistical significance does not equal clinical importance. Clinical importance is best inferred by looking at the effect size, that is how much is the actual change or difference. Statistical significance in terms of p only suggests whether there is any difference in probability terms. A statistically significant result can be clinically trivial, and a clinically important effect can fail to reach statistical significance in a small study.

Misconception: P > 0.05 Means There Is No Effect

A non-significant p-value means the data do not provide strong evidence against the null hypothesis. It does not prove that the null hypothesis is true. The study may have been underpowered to detect a real effect. When a study concludes with a negative result, there are two possibilities: there is no treatment effect, or the study made a mistake. The problem is usually that the null hypothesis was not rejected because there were too few subjects.

Misconception: Post Hoc Power Calculations Explain Non-Significant Results

Observed power should not be used to interpret results of a negative study. Observed power should not be calculated after a study is completed and analyzed. The power of the study to reject or not reject the null hypothesis is already incorporated in the calculation of the p-value. Calculating observed power after the fact and claiming it supports the null hypothesis is a logical error.

The Role of Effect Size

Effect size measures the magnitude of the difference or association, independent of sample size. Common effect size measures include Cohen's d for mean differences, Pearson's r for correlations, and odds ratios for categorical outcomes. Effect sizes allow researchers to assess whether a statistically significant result is practically meaningful.

The p-value alone cannot distinguish between a large effect in a small study and a small effect in a large study. Both can produce the same p-value. Effect size provides the additional information needed to interpret the practical importance of findings. Researchers should report effect sizes alongside p-values to give readers the full picture.

The scientific community is inclined toward knowing the effect size, confidence interval, and descriptive statistics, and researchers need to highlight them along with the p-value. This shift reflects the recognition that p-values alone are insufficient for meaningful interpretation of research findings.

The Role of Confidence Intervals

A confidence interval provides a range of plausible values for the population parameter. The confidence interval is estimated on the basis of a desired confidence level. Calculation of the confidence interval of a sample statistic takes the general form: point estimate plus or minus the margin of error, where the margin of error is given by the product of a critical value derived from the standard normal curve and the standard error of the point estimate.

The factors affecting the width of the confidence interval include the desired confidence level, the sample size, and the variability in the sample. Although the 95 percent confidence interval is most often used in biomedical research, a confidence interval can be calculated for any level of confidence. A 99 percent confidence interval will be wider than a 95 percent confidence interval for the same sample.

Use of the confidence interval supplements the p-value by providing an estimate of actual clinical effect. A confidence interval that excludes the null value is consistent with a statistically significant result. The width of the confidence interval indicates the precision of the estimate. A narrow interval indicates precise estimation, while a wide interval indicates uncertainty.

Choosing the Correct Statistical Test

The choice of statistical test depends on the type of data, whether the data is paired or not, and the number of study groups. These three questions guide the selection of an appropriate test. Using the wrong test can produce misleading p-values and incorrect conclusions.

For continuous data with two independent groups, the independent samples t-test is appropriate when assumptions are met. For paired continuous data, the paired t-test is appropriate. For continuous data with more than two groups, ANOVA is appropriate. For categorical data, the chi-square test or Fisher's exact test is appropriate. For ranked or non-normal data, non-parametric alternatives such as the Mann-Whitney U test or Wilcoxon signed-rank test are appropriate.

Association between variables, agreement between assessments, and time trends cannot be arrived at by calculating the p-value alone but need to highlight the correlation and regression coefficients, odds ratio, relative risk, and other appropriate statistics. Researchers must match the test to the research question and data type.

The Debate Over Statistical Significance Thresholds

The calculation of a p-value in research and especially the use of a threshold to declare the statistical significance of the p-value have both been challenged in recent years. There are at least two important reasons for this challenge: research data contain much more meaning than is summarized in a p-value and its statistical significance, and these two concepts are frequently misunderstood and consequently inappropriately interpreted.

The American Statistical Association has issued statements against the misuse of statistical significance. Some researchers have proposed abandoning the use of p-values entirely, while others have proposed reducing the significance threshold from 0.05 to 0.005. The debate continues in the scientific literature, with some arguing for a conservative approach because clinical research needs dichotomous answers to guide decision-making, particularly in diagnostic imaging and interventional radiology.

With a reduced p-value threshold, the cost of research could increase while spontaneous research could be reduced. Secondary evidence from systematic reviews and meta-analyses, data sharing, and cost-effective analyses are better ways to mitigate the false discovery rate and lack of reproducibility associated with the use of the 0.05 threshold. Radiomics and big data applications involve hundreds or thousands of tested features, similar to genomics, where a reduction in the significance threshold based on well-known corrections for multiple testing has already been adopted.

Advanced P-Value Calculation Methods

Higher Criticism Tests

In modern data science, the higher criticism method is effective for detecting rare and weak signals. The computation has long been an issue when the number of p-values combined and the number of repeated higher criticism tests are large. Some computing methods have been developed, but they all have significant shortcomings, especially when a stringent significance level is required.

An accurate and highly efficient computing strategy has been proposed for four variations of higher criticism. This includes an unbiased cross-entropy-based importance sampling method to benchmark all existing computing methods and a modified SetTest method that resolves numerical issues of the existing approach. An ultra-fast approach combines pre-calculated statistical tables and cubic spline interpolation. Following extensive simulations, a computing strategy integrates these methods with an R package for virtually any number of p-values and small p-values down to approximately 10 to the power of negative 20. The method has been applied to disease surveillance for spatio-temporal outbreak detection from case numbers across multiple counties.

Generalized Fisher Combination

Combining dependent tests of significance has broad applications, but the related p-value calculation is challenging. For Fisher's combination test, current p-value calculation methods tend to inflate the type I error rate when the desired significance level is substantially less than 0.05. The problem could lead to significant false discoveries in big data analyses.

A general family of Fisher type statistics, referred to as the GFisher, covers many classic statistics such as Fisher's combination, Good's statistic, Lancaster's statistic, and weighted Z-score combination. The GFisher allows a flexible weighting scheme as well as an omnibus procedure that automatically adapts proper weights and the statistic-defining parameters to a given data set. New p-value calculation methods based on moment-ratio matching and joint-distribution surrogating are more accurate under multivariate Gaussian distributions and more robust under generalized linear models and multivariate t-distributions.

Gaussian Approximation Methods

It is of fundamental interest in statistics to test the significance of a set of covariates. In genome-wide association studies, a joint null hypothesis of no genetic effect is tested for a set of multiple genetic variants. The minimum p-value method, higher criticism, and Berk-Jones tests are particularly effective when the covariates with nonzero effects are sparse. However, the correlations among covariates and the non-Gaussian distribution of the response pose a great challenge toward the p-value calculation of the three tests.

Permutation is commonly used to obtain accurate p-values, but it is computationally very intensive, especially when a large amount of hypothesis testing is needed. A Gaussian approximation method based on a Monte Carlo scheme is computationally more efficient than permutation while still achieving similar accuracy. Non-asymptotic approximation error bounds can vanish in the limit even if the number of covariates is much larger than the sample size.

Alternative Approaches to P-Value Interpretation

Delta Likelihood Ratio

Interpretation of clinical research findings using the paradigm of null hypothesis significance testing has a number of limitations. These include arbitrary dichotomization of results, lack of incorporation of study power and prior probability, and the confusing use of conditional probability. A novel method of p-value interpretation addresses these limitations.

The delta likelihood ratio is an application of Bayes' rule incorporating the p-value and study power. Calculation of the delta likelihood ratio allows the determination of the most likely effect size using the maximum likelihood principle. The delta likelihood ratio is easily calculated and produces plausible results using examples of previously published research studies. The delta likelihood ratio incorporates study power and provides a continuous measure of the probability that the research result is a true effect. The most likely effect size is an easily understood metric that should aid the interpretation of research.

Bayesian Approaches

Bayesian methods provide an alternative framework that incorporates prior information and produces probability statements about hypotheses directly. These methods do not rely on p-values and instead produce posterior probabilities. While Bayesian methods have advantages, they require specification of prior distributions, which can be subjective. Researchers should be aware of both frequentist and Bayesian approaches and choose the framework appropriate for their research question.

Practical Workflow for P-Value Reporting

Before Data Collection

Define the research question and hypotheses in advance. Determine the appropriate sample size based on the expected effect size and desired power. Pre-specify the significance level and the statistical tests to be used. Register the study protocol if appropriate. These steps prevent post hoc decisions that can bias results.

During Analysis

Check the assumptions of your chosen statistical test. Calculate the test statistic and p-value using appropriate software. Calculate the effect size and confidence interval. Consider whether multiple testing corrections are needed. Document all analysis decisions.

When Reporting Results

Report the actual p-value instead of only stating whether it is below a threshold. Report the effect size with its confidence interval. Report descriptive statistics for each group. Describe the statistical methods used, including the software and version. Follow reporting guidelines appropriate for your study design. The EQUATOR Network provides reporting guidelines for various study types, and researchers should consult these resources before submitting their work.

When Interpreting Results

Interpret the p-value in the context of the study design, sample size, and prior evidence. Do not overinterpret non-significant results as evidence of no effect. Do not overinterpret significant results as evidence of large or important effects. Consider the confidence interval to assess the precision of the estimate. Consider whether the effect size is clinically or practically meaningful.

Common Failure Patterns in P-Value Use

P-Hacking

P-hacking refers to analyzing data in multiple ways until a significant p-value is found. This practice inflates the false positive rate and undermines the validity of research findings. Researchers should pre-specify their analysis plan and report all analyses conducted, also those that produced significant results.

HARKing

Hypothesizing After the Results are Known involves presenting post hoc hypotheses as if they were pre-specified. This practice misleads readers about the strength of evidence. Researchers should clearly distinguish between confirmatory and exploratory analyses.

Dichotomania

Dichotomania refers to the rigid classification of results as significant or non-significant based solely on whether the p-value falls below 0.05. This practice ignores the continuous nature of evidence and can lead to different conclusions for p-values of 0.049 and 0.051, which represent nearly identical evidence.

Ignoring Effect Size

Reporting only p-values without effect sizes prevents readers from assessing the practical importance of findings. A statistically significant result with a trivial effect size may not be worth acting on. Researchers should always report effect sizes alongside p-values.

Misinterpreting Non-Significance

Concluding that a non-significant result proves the absence of an effect is a common error. Non-significance can result from a true absence of effect, insufficient sample size, or high variability. Researchers should report confidence intervals to help readers understand what effects are compatible with the data.

Limitations of P-Values

The p-value is a function of sample size. When the sample size is large, the p-value is destined to be small or significant. This means that large studies can detect trivial effects, while small studies may miss important effects. The p-value does not distinguish between these scenarios.

The p-value is frequently misunderstood and consequently inappropriately interpreted. The arbitrary dichotomization of results based on a threshold creates a false sense of certainty. Research data contain much more meaning than is summarized in a p-value and its statistical significance.

The p-value does not incorporate prior probability or study power. The delta likelihood ratio approach addresses these limitations by incorporating the p-value and study power into a single metric. Researchers should be aware of these limitations and use additional metrics to support their conclusions.

Quality Controls for Statistical Analysis

Pre-Registration

Pre-registering the study design, hypotheses, and analysis plan before data collection reduces the risk of p-hacking and HARKing. Pre-registration makes the distinction between confirmatory and exploratory analyses clear.

Reporting Guidelines

The EQUATOR Network provides reporting guidelines for various study types, including randomized trials, observational studies, and diagnostic accuracy studies. Following these guidelines ensures that statistical methods and results are reported completely and transparently.

Data Sharing

Sharing data and analysis code allows other researchers to verify results and conduct secondary analyses. Data sharing increases transparency and reduces the risk of errors going undetected.

Independent Verification

Having a second statistician review the analysis plan and results can catch errors and reduce bias. Independent verification is particularly important for complex analyses.

Multiple Testing Corrections

When conducting many statistical tests, the probability of false positives increases. Corrections such as the Bonferroni correction or false discovery rate control should be applied when appropriate. In genomics and radiomics, where hundreds or thousands of features are tested, a reduction in the significance threshold based on well-known corrections for multiple testing has been adopted.

Professional Escalation Criteria

Researchers should seek additional statistical expertise when they encounter any of the following situations:

The study design is complex, such as cluster-randomized trials, longitudinal studies, or adaptive designs. The data violate assumptions of standard tests, such as severe non-normality, heteroscedasticity, or missing data. The analysis involves multiple endpoints, subgroup analyses, or interim analyses. The results are borderline, with p-values close to the significance threshold. The study findings will inform clinical decisions, regulatory submissions, or policy decisions. The analysis requires advanced methods such as mixed models, survival analysis, or machine learning.

In these situations, consulting a statistician before data collection and during analysis can prevent errors and improve the quality of the research.

Frequently Asked Questions

What is the difference between a p-value and a confidence interval?

A p-value is the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. A confidence interval provides a range of plausible values for the population parameter. The confidence interval supplements the p-value by providing an estimate of the actual effect. A confidence interval that excludes the null value is consistent with a statistically significant result, and the width of the interval indicates the precision of the estimate.

How do I calculate a p-value for a z-test?

Calculate the z-statistic using the formula z = (x̄ - μ₀) / (σ / √n), where x̄ is the sample mean, μ₀ is the hypothesized mean, σ is the population standard deviation, and n is the sample size. Then find the area in the tail or tails of the standard normal distribution beyond the calculated z-value. For a two-tailed test, multiply the one-tailed p-value by 2.

How do I calculate a p-value for a t-test?

Calculate the t-statistic using the formula t = (x̄ - μ₀) / (s / √n), where s is the sample standard deviation. Then find the p-value from the t-distribution with n - 1 degrees of freedom. For a two-sample t-test, use the appropriate formula for the test statistic and degrees of freedom based on whether variances are assumed equal.

Why is my p-value significant but my effect size small?

A significant p-value with a small effect size typically occurs in large studies. The p-value is a function of sample size, and when the sample size is large, the p-value is destined to be small or significant. The effect size tells you the magnitude of the difference, which may be too small to be practically meaningful. Report both the p-value and the effect size so readers can assess the practical importance of the finding.

Why is my p-value not significant even though I see a difference?

A non-significant p-value can result from insufficient sample size, high variability, or a true absence of effect. When the conclusion of a study is negative, there are two possibilities: there is no treatment effect, or the study made a mistake. The problem is usually that the null hypothesis was not rejected because there were too few subjects. Do not calculate post hoc power to explain the result, as observed power should not be used to interpret results of a negative study.

Should I use p < 0.05 or p < 0.005 as my significance threshold?

The choice of significance threshold depends on your field, the consequences of false positives, and the number of tests being conducted. The traditional 0.05 threshold has been challenged, and some have proposed reducing it to 0.005. However, with a reduced p-value threshold, the cost of research could increase while spontaneous research could be reduced. Consider the context of your research and consult field-specific guidelines.

What should I report alongside my p-value?

Report the actual p-value, the effect size, the confidence interval, and descriptive statistics for each group. The scientific community is inclined toward knowing the effect size, confidence interval, and descriptive statistics, and researchers need to highlight them along with the p-value. Also report the statistical methods used, including the software and version, and follow reporting guidelines appropriate for your study design.

How do I choose the correct statistical test?

Ask three questions: what type of data do you have, whether the data is paired or not, and how many study groups are involved. Continuous data with two independent groups use a t-test. Paired continuous data use a paired t-test. More than two groups use ANOVA. Categorical data use a chi-square test. Non-normal data may require non-parametric alternatives. Choosing the correct test for assessing the significance of the difference is profoundly important.

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References and Further Reading

• Research Data Framework. National Institute of Standards and Technology.
• EQUATOR Network. EQUATOR Network.
• Experimental Design Assistant. NC3Rs.
• NCBI Literature Resources. National Center for Biotechnology Information.
• PubMed. National Library of Medicine.
• P-value calculation methods for semi-partial correlation coefficients.. Communications for statistical applications and methods, 2022.
• P-Value Demystified.. Indian dermatology online journal, 2019.
• Accurate and Ultra-Efficient p-Value Calculation for Higher Criticism Tests.. Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America, 2024.
• The generalized Fisher's combination and accurate p-value calculation under dependence.. Biometrics, 2023.
• The P Value and Statistical Significance: Misunderstandings, Explanations, Challenges, and Alternatives.. Indian journal of psychological medicine, 2019.
• Accurate and Efficient P-value Calculation via Gaussian Approximation: a Novel Monte-Carlo Method.. Journal of the American Statistical Association, 2019.
• Using the confidence interval confidently.. Journal of thoracic disease, 2017.
• Inappropriate use of statistical power.. Bone marrow transplantation, 2023.
• A likelihood-based approach to P-value interpretation provided a novel, plausible, and clinically useful research study metric.. 2017.
• Surgeon interpretation of margin status utilizing intraoperative 3D tomosynthesis during breast-conserving surgery.. 2026.
• Diagnostic Accuracy and Time Efficiency of Artificial Intelligence for Intracranial Hemorrhage Detection on Emergency Brain CT.. 2026.
• From C1 Exit Standards to Enacted Proficiency: A Mixed-Methods Study of Score Interpretation and Assessment Alignment in an EFL Higher Education Programme. 2026.
• Statistical Reporting in Plastic Surgery Research. 2026.
• Effect of AI-assisted caries annotation on dental students' performance in caries detection on panoramic radiographs.. 2026.
• Scientists rise up against statistical significance. Nature, 2019.
• Clinical significance: a statistical approach to defining meaningful change in psychotherapy research.. Journal of Consulting and Clinical Psychology, 1991.
• Statistical significance: p value, 0.05 threshold, and applications to radiomics-reasons for a conservative approach. European Radiology Experimental, 2020.
• Title: Redefine Statistical Significance. 2017.
• Should bootstrapping be used in pls-sem? Toward stable p-value calculation methods. Journal of Applied Structural Equation Modeling, 2018.

This article is educational and does not replace institutional policy, professional advice, or applicable safety and regulatory requirements.