What is Regression Analysis? A Practical Introduction

Regression analysis is a statistical method used to examine the relationship between one or more predictor variables and an outcome variable. For farmers and animal scientists, regression analysis helps answer practical questions such as how feed intake affects weight gain, how ambient temperature influences milk production, or how stocking density impacts mortality rates. This article explains the core concepts of regression analysis, describes the main types available, outlines the assumptions that must be checked, and provides a decision framework for selecting the appropriate model for your data.

At a Glance

Regression analysis is a family of statistical techniques instead of a single procedure. The choice of method depends on the nature of your outcome variable, the number of predictors, and the shape of the relationship you expect to find. The table below summarizes the most common regression approaches used in animal agriculture and life sciences.

What Regression Analysis Does

Regression analysis provides a formal framework for quantifying how changes in predictor variables relate to changes in an outcome. In animal science, you might record the weight of pigs at slaughter along with their average daily feed intake, pen temperature, and days on feed. Regression allows you to estimate the average change in slaughter weight for each additional kilogram of daily feed intake while accounting for the other recorded factors.

The method serves two broad purposes. First, it can describe the association between an outcome and its predictors. Second, it can predict outcomes for new observations based on known predictor values. Both purposes require careful attention to data quality and model assumptions.

Proper evaluation of data does not always require advanced statistical methods, but these tools give researchers the freedom to evaluate more complex hypotheses than simple comparisons allow. An overview of regression and multivariate statistics describes general concepts, reviews basic definitions and conventions, and discusses the types of regression analysis including simple regression, multiple regression, multivariate multiple regression, and logistic regression. The steps required to perform these analyses and the advantages and disadvantages of each are detailed in that source.

Core Principles of Regression

The Dependent and Independent Variables

Regression analysis distinguishes between the outcome you want to explain and the factors you use to explain it. The dependent variable, also called the response or outcome, is the measurement you are trying to predict or understand. The independent variables, also called predictors or explanatory variables, are the measurements you believe influence the outcome.

In a dairy herd example, daily milk yield is the dependent variable. Independent variables might include days in milk, parity, body condition score, and ambient temperature. The regression equation estimates how milk yield changes on average for each unit change in a predictor, holding the other predictors constant.

The Regression Equation

Linear regression fits a straight line to the data. The equation takes the form outcome equals the intercept plus the slope multiplied by the predictor. The intercept is the predicted outcome when all predictors are zero. The slope is the average change in the outcome for a one-unit increase in the predictor.

For multiple predictors, the equation extends to include a separate slope for each predictor. Each slope coefficient represents the average change in the outcome associated with a one-unit change in that predictor, assuming the other predictors remain unchanged. This property makes multiple regression useful for separating the effects of correlated factors.

Fitting the Model

Regression models are fitted to data using standard software packages. The fitting process finds the coefficients that minimize the difference between the observed outcomes and the outcomes predicted by the model. Most statistical software performs this calculation automatically once you specify the outcome and predictor variables.

The ease of fitting a model creates a risk. Analysts may rush to perform sophisticated analyses without sufficient knowledge of the basic properties, associations, and errors in their data. This can lead to wrong interpretation and unclear presentation of modeling results. Ignorance about special features of the data such as redundancies or particular distributions may even invalidate the chosen analysis strategy.

Types of Regression Analysis

Simple Linear Regression

Simple linear regression uses one predictor variable to explain a continuous outcome. This is the most basic form of regression and is appropriate when you have a single explanatory factor and a straight-line relationship is plausible.

In animal agriculture, simple linear regression might be used to estimate the relationship between crude protein content in feed and average daily gain in growing lambs. The analysis produces an intercept and a slope. The slope tells you the expected change in daily gain for each percentage point increase in crude protein.

The advantage of simple linear regression is its transparency. You can plot the data and the fitted line on the same graph, inspect the scatter around the line, and identify unusual observations directly. The limitation is that real production systems rarely depend on a single factor.

Multiple Linear Regression

Multiple linear regression extends the simple case to include two or more predictors. This allows you to account for several factors simultaneously and to estimate the effect of each predictor while controlling for the others.

A swine operation might use multiple regression to predict days to market weight from birth weight, weaning age, average daily feed intake, and pen stocking density. The model produces a coefficient for each predictor. The coefficient for stocking density estimates the change in days to market for each additional pig per square meter, holding birth weight, weaning age, and feed intake constant.

Multiple regression offers the advantage of evaluating more complex hypotheses than simple comparisons allow. The main limitation is that the interpretation of each coefficient depends on the other variables included in the model. Adding or removing predictors can change the estimated coefficients for the remaining variables.

Logistic Regression

Logistic regression is used when the outcome is binary, meaning it takes one of two values such as alive or dead, diseased or healthy, or pregnant or open. The method estimates the probability of the event of interest based on one or more predictor variables.

Logistic regression is used to obtain odds ratios in the presence of more than one explanatory variable. The procedure is quite similar to multiple linear regression, with the exception that the response variable is binomial. The result is the impact of each variable on the odds ratio of the observed event of interest. The main advantage is to avoid confounding effects by analyzing the association of all variables together.

In a beef feedlot, logistic regression could estimate the probability of respiratory disease treatment based on arrival weight, transport distance, vaccination history, and season. The model produces odds ratios for each predictor. An odds ratio above one indicates that the predictor increases the odds of the event, while an odds ratio below one indicates a protective effect.

Nonlinear Regression

Not all relationships in biology follow a straight line. Growth curves, enzyme kinetics, and dose-response relationships often show curvature that linear models cannot capture. Nonlinear regression fits models where the relationship between the outcome and predictors follows a specified mathematical form.

Linear and nonlinear regression models represent critically important statistical analyses used to define the relationships between variables of interest in biological systems. Biological assays are essential tools in biomedical and pharmaceutical research. In simplest terms, such an assay is an analytical method used to measure or predict a response in a biological system in the presence of a given stimulus such as a drug. The inherent complexity involved in evaluating a biological system requires the use of rigorous and appropriate tools for data analysis.

For example, the relationship between a drug concentration and its effect on cell growth often follows a sigmoidal curve. Nonlinear regression can estimate the parameters of that curve, including the maximum effect and the concentration producing half the maximum effect. The same approach applies to growth curves in livestock where weight gain slows as animals approach mature size.

Principal Component Regression

Principal component regression addresses a common problem in agricultural data where many predictor variables are correlated with each other. The method first reduces the predictors to a smaller set of uncorrelated components, then regresses the outcome on those components.

This approach has been applied to relate microstructural texture to microhardness in fine-blanked steel. Texture is quantified using Gabor filters that decompose micrographs into orientation and scale specific components. These features are analyzed using principal component analysis and related to microhardness through a linear principal component regression framework. The linear structure of the model enables regression weights to be mapped back to specific image space features, allowing direct interpretation of which microstructural patterns correspond to hardening and softening behavior.

In animal science, principal component regression might be useful when you have many body measurements that are highly correlated. instead of including all measurements directly, the method creates a few composite variables that capture most of the variation in the original measurements.

Regression for Censored Data

Some agricultural and medical data have outcomes that are not fully observed. Interval censoring occurs when the event of interest happens between two known time points instead of at an exact recorded time. This situation arises in studies of disease incubation periods, time to calving, or time to first service.

Interval censored failure time data occur in many medical investigations as well as other studies such as demographical and sociological studies. They include the usual right censored failure time data as a special case but provide much more complex structure and less relevant information than the right censored data. Estimation of a survival function, comparison of several treatments, and regression analysis are discussed in the context of interval censoring.

The incubation period is a key characteristic of an infectious disease. In the outbreak of a novel infectious disease, accurate evaluation of the incubation period distribution is critical for designing effective prevention and control measures. Estimation of the incubation period distribution based on limited information from retrospective inspection of infected cases is highly challenging due to censoring and truncation. A semiparametric regression model for the incubation period has been proposed using a sieve maximum likelihood approach based on symptom onset time, travel history, and basic demographics of reported cases.

Assumptions and Data Screening

The Importance of Initial Data Analysis

Before fitting any regression model, you must understand the basic properties of your data. Initial data analysis is a prerequisite to regression analyses because it provides knowledge about the data needed to confirm the appropriateness of or to refine a chosen model building strategy, to interpret the modeling results correctly, and to guide the presentation of modeling results.

Initial data analysis should be preplanned. An initial data analysis plan should be included in the general statistical analysis plan of a research project, and results should be well documented. Biased statistical inference of the final regression model can be minimized if initial data analysis abstains from evaluating associations of outcome and predictors.

The initial data analysis plan for data screening should cover several aspects. You need to examine the distribution of each variable, check for missing values, identify outliers, and assess whether the relationships between variables are approximately linear. Data visualizations are recommended as part of this screening process.

Checking Linearity

Linear regression assumes that the relationship between each continuous predictor and the outcome is approximately linear. If the true relationship is curved, the model will produce biased estimates.

You can check linearity by plotting the outcome against each predictor and examining the scatter for curvature. A loess smooth line added to the plot can help reveal patterns that are not obvious from the raw points. If curvature is present, you may need to transform the predictor, add a quadratic term, or use a nonlinear regression approach.

Checking Independence

Regression assumes that the observations are independent of each other. In agricultural settings, this assumption is often violated because animals within the same pen, herd, or litter tend to be more similar to each other than to animals in different groups.

If your data have a hierarchical structure, such as animals nested within pens and pens nested within barns, you may need to account for this clustering in your analysis. Ignoring the structure can lead to standard errors that are too small and p-values that are too optimistic.

Checking Constant Variance

Linear regression assumes that the variability of the outcome is the same across all levels of the predictors. This assumption is called homoscedasticity. When the variability changes with the level of the outcome, the data are heteroscedastic.

You can check this assumption by plotting the residuals from the fitted model against the predicted values. The residuals should scatter randomly around zero with roughly constant spread. If the spread widens or narrows as the predicted values increase, the constant variance assumption is violated.

Checking Normality of Residuals

Regression does not require the outcome variable itself to be normally distributed. The assumption is that the residuals, which are the differences between observed and predicted outcomes, follow a normal distribution. This assumption is most important for small samples and for constructing confidence intervals and p-values.

A normal quantile plot of the residuals can help assess this assumption. Points that fall roughly along a straight line suggest the residuals are approximately normal. Severe departures from normality may indicate that a transformation is needed or that a different type of regression model is more appropriate.

Practical Workflow for Regression Analysis

Step 1: Define the Research Question

Start by writing a clear statement of what you want to estimate or predict. Specify the outcome variable and the predictor variables of interest. This step forces you to think about the causal structure of your problem and prevents you from including predictors without a clear rationale.

Step 2: Plan the Data Collection

Decide how many observations you need and how you will select them. Consider whether your sampling scheme will produce independent observations or whether clustering is present. Document your data collection procedures so that others can understand how the data were generated.

The Experimental Design Assistant from the NC3Rs provides a tool for planning experiments and avoiding common design flaws. Using a structured design tool can help you identify potential problems before you collect data.

Step 3: Perform Initial Data Analysis

Before fitting any regression model, screen your data. Examine the distribution of each variable, check for missing values, and identify outliers. Plot the outcome against each predictor to assess linearity. Document your findings so that the choices you make in the modeling stage are transparent.

Step 4: Select the Regression Type

Use the decision table in the At a Glance section to choose the appropriate regression type based on your outcome variable and research question. If your outcome is continuous and you have one predictor, use simple linear regression. If you have multiple predictors, use multiple linear regression. If your outcome is binary, use logistic regression. If the relationship is curved, consider nonlinear regression.

Step 5: Fit the Model

Fit the regression model using your chosen statistical software. Examine the coefficient estimates, their standard errors, and the model fit statistics. Check that the signs of the coefficients make biological sense. A negative coefficient for a predictor that should increase the outcome may indicate a data problem or a misspecified model.

Step 6: Check the Assumptions

Examine the residuals from the fitted model to check the assumptions of linearity, independence, constant variance, and normality. If the assumptions are violated, consider transformations, alternative model forms, or more advanced methods.

Step 7: Interpret the Results

Interpret the coefficients in the context of your research question. For linear regression, each coefficient represents the average change in the outcome for a one-unit change in the predictor, holding other predictors constant. For logistic regression, exponentiate the coefficients to obtain odds ratios.

Step 8: Document and Report

Document your analysis decisions, including the initial data analysis findings, the model selection process, and the assumption checks. Report the model coefficients with their confidence intervals and the measures of model fit. This documentation supports reproducibility and allows others to evaluate the robustness of your conclusions.

Choosing the Right Regression Model

Decision Tree for Model Selection

The choice of regression model depends primarily on the nature of the outcome variable and the structure of the relationship you expect.

Start by identifying the outcome variable. If the outcome is continuous and can take any value within a range, consider linear regression. If the outcome is binary, consider logistic regression. If the outcome is a count, such as the number of disease cases per pen, consider Poisson or negative binomial regression.

Next, consider the number of predictors. With one predictor and a continuous outcome, simple linear regression is appropriate. With multiple predictors and a continuous outcome, multiple linear regression is appropriate. With multiple predictors and a binary outcome, logistic regression is appropriate.

Then consider the shape of the relationship. If plots of the outcome against the predictors suggest curvature, consider nonlinear regression or transformations. If the relationship follows a known biological function, such as exponential growth or a dose-response curve, nonlinear regression with that function may be appropriate.

Finally, consider the structure of the data. If observations are clustered within groups, you may need to account for the clustering. If predictors are highly correlated, principal component regression may be useful.

Comparing Model Performance

When several candidate models are plausible, compare their performance using appropriate criteria. For prediction, cross-validation provides an estimate of how well the model will perform on new data. For explanation, information criteria such as the Akaike information criterion can help balance model fit against model complexity.

The choice between simpler and more complex models deserves careful consideration. A calibration benchmark of a single-cell large language model for cell type annotation found that classical logistic regression on a matched gene set reached high accuracy and low calibration error on every atlas tested, matching or beating every more complex pipeline. The classical baselines were well calibrated on every atlas. This finding illustrates that simpler models can perform as well as or better than more complex alternatives, particularly when the data are well understood.

The Role of Statistical Software

Regression analysis can be performed with a wide range of software packages, from spreadsheet programs to specialized statistical packages. The choice of software depends on your familiarity, the complexity of your analysis, and the need for reproducibility.

Traditional statistical software remains the standard for transparent and reproducible analyses. A comparison of statistical consistency between a traditional statistical package and a large language model found concordance across most descriptive and inferential statistical procedures, although notable discrepancies did occur. Identical or near identical findings were observed for descriptive statistics and several correlation procedures. The discrepancies highlight the importance of using established software for analyses that will be reported or published.

Records and Measurements

What to Record

Good regression analysis depends on good data. Record your measurements with enough detail to support the analysis you plan to conduct. For each observation, record the outcome variable, all predictor variables, and any identifying information such as animal ID, pen number, or date.

Document the units of measurement for each variable. A regression coefficient for feed intake measured in kilograms per day will differ from the coefficient for feed intake measured in grams per day. Consistent units across observations are essential.

Record the date and time of each measurement. This information allows you to check for temporal trends and to identify any changes in measurement procedures over the course of the study.

Data Quality Checks

Before analysis, check your data for errors. Look for values that are impossible, such as negative weights or temperatures outside the plausible range. Check that categorical variables have consistent coding. Verify that dates are in the correct format and order.

Missing data require careful handling. You should document the amount and pattern of missingness. If a large proportion of observations are missing for a particular variable, consider whether that variable should be included in the model. The choice of method for handling missing data should be made before fitting the model and documented in the analysis plan.

Linking Records Across Sources

Agricultural data often come from multiple sources. Weights may come from the scale system, health records from the treatment log, and environmental data from the barn controller. Combining these sources requires a reliable linking variable such as animal ID or pen number.

The Research Data Framework from the National Institute of Standards and Technology provides guidance on managing research data across its lifecycle. Following structured data management practices helps ensure that records can be linked accurately and that the analysis can be reproduced.

Common Failure Patterns

Rushing to Analysis Without Data Screening

The most common failure in regression analysis is fitting a model before understanding the data. Analysts may rush to perform sophisticated analyses without sufficient knowledge of basic properties, associations, and errors in their data. This can lead to wrong interpretation and unclear presentation of modeling results.

The remedy is to perform initial data analysis before fitting any regression model. Screen the data, examine distributions, check for outliers, and plot relationships. Document your findings and use them to guide the modeling strategy.

Ignoring Nonlinearity

Fitting a straight line to curved data produces biased estimates and misleading conclusions. Growth in livestock is rarely linear across the full production cycle. Feed efficiency changes as animals mature. Dose response relationships often show saturation at high doses.

Plot the outcome against each predictor before fitting the model. If curvature is present, consider transformations or nonlinear regression. The choice between linear and nonlinear models should be based on the data and the underlying biology, not on convenience.

Overlooking Clustering

Animals within the same pen or herd are not independent. They share the same environment, feed, and management. Treating them as independent observations produces standard errors that are too small and p-values that are too optimistic.

If your data have a hierarchical structure, account for it in the analysis. Options include including group as a predictor, using mixed models, or using cluster robust standard errors. The choice depends on the research question and the structure of the data.

Including Too Many Predictors

Including many predictors in a regression model can lead to overfitting, where the model fits the observed data well but performs poorly on new data. Overfitting is particularly problematic when the number of predictors is large relative to the number of observations.

The calibration benchmark study found that classical logistic regression on a matched gene set matched or beat every more complex pipeline tested on accuracy and calibration. This result supports the principle that simpler models often perform better than complex alternatives, especially when the number of observations is limited.

Misinterpreting Coefficients

Regression coefficients describe associations, not necessarily causal effects. A coefficient estimates the average change in the outcome for a one-unit change in the predictor, holding other predictors constant. This interpretation assumes that the model is correctly specified and that all relevant confounders are included.

Be cautious about interpreting coefficients as causal effects unless the study design supports causal inference. Observational data from farm records can identify associations that are useful for prediction and hypothesis generation, but they cannot establish causation without additional evidence.

Limitations of Regression Analysis

Correlation Is Not Causation

Regression analysis quantifies the association between variables. It does not by itself establish that one variable causes another. A strong association between two variables may reflect a causal relationship, a common cause, or a spurious correlation.

In agricultural settings, many factors change together. Temperature, humidity, and ventilation rate are correlated in barn environments. Separating their individual effects requires careful study design and often experimental manipulation instead of observational data alone.

Sensitivity to Outliers

Regression estimates can be strongly influenced by unusual observations. A single animal with an extreme weight or a single pen with an unusual mortality rate can change the estimated coefficients substantially.

Examine your data for outliers before fitting the model. Consider whether outliers represent data errors, unusual but real observations, or a different population. The decision to exclude or downweight outliers should be documented and justified.

Limited to the Range of Observed Data

Regression models describe relationships within the range of the observed data. Extrapolating beyond that range is risky because the relationship may change outside the observed values. A linear relationship that holds for feed intakes between 2 and 4 kilograms per day may not hold at 6 kilograms per day.

When using regression for prediction, restrict predictions to the range of predictor values observed in the data. If you need predictions outside that range, collect additional data or use a model form that is justified by biological theory.

Assumption Violations

All regression models make assumptions. When the assumptions are violated, the model estimates may be biased, the standard errors may be incorrect, and the conclusions may be misleading.

The key assumptions for linear regression are linearity, independence, constant variance, and normality of residuals. Logistic regression assumes a linear relationship between the predictors and the log odds of the outcome. Nonlinear regression assumes that the specified functional form is correct.

Checking assumptions requires time and attention. The effort is justified because the validity of your conclusions depends on the validity of the assumptions.

Welfare and Safety Context

Using Regression to Monitor Animal Welfare

Regression analysis can support animal welfare monitoring by identifying factors associated with poor outcomes. For example, a logistic regression model might estimate the probability of lameness based on flooring type, stocking density, and days in milk. The model can identify herds or management practices with elevated risk.

The NC3Rs Experimental Design Assistant provides a tool for planning experiments that minimizes animal use and maximizes the reliability of results. Using structured design tools before conducting animal studies supports both scientific quality and animal welfare.

Reporting Standards

Transparent reporting of statistical methods supports reproducibility and allows readers to evaluate the validity of conclusions. The EQUATOR Network provides reporting guidelines for health research, including guidelines for studies that use regression analysis. Following these guidelines helps ensure that the methods are described clearly and completely.

The National Center for Biotechnology Information provides literature resources that can help you find examples of regression analysis applied to agricultural and biomedical questions. PubMed, maintained by the National Library of Medicine, indexes peer reviewed literature and can be searched for studies using regression methods in your area of interest.

When to Seek Professional Help

Regression analysis can be performed by researchers with appropriate training, but some situations warrant consultation with a statistician. Seek professional help when your data have complex structures such as clustering or censoring, when you are unsure which model form is appropriate, or when the results will be used for regulatory decisions or high stakes predictions.

A statistician can help with study design before data collection, which is more efficient than trying to fix design problems after the data are collected. The Experimental Design Assistant from the NC3Rs is one tool that can help identify design issues early.

Frequently Asked Questions

What is the difference between simple and multiple regression?

Simple regression uses one predictor variable to explain an outcome. Multiple regression uses two or more predictors. Multiple regression allows you to estimate the effect of each predictor while controlling for the other predictors in the model. This is useful when predictors are correlated with each other, as is common in agricultural data.

When should I use logistic regression instead of linear regression?

Use logistic regression when the outcome variable is binary, meaning it takes one of two values such as alive or dead, diseased or healthy, or pregnant or open. Logistic regression estimates the probability of the event of interest and produces odds ratios for each predictor. Linear regression requires a continuous outcome and is not appropriate for binary outcomes.

What are odds ratios in logistic regression?

An odds ratio describes the change in the odds of the event of interest for a one-unit change in the predictor variable. An odds ratio of one indicates no association. An odds ratio above one indicates that the predictor increases the odds of the event. An odds ratio below one indicates that the predictor decreases the odds. In logistic regression with multiple predictors, each odds ratio is adjusted for the other predictors in the model.

How do I check whether my data meet the assumptions of linear regression?

Plot the outcome against each predictor to check linearity. Plot the residuals from the fitted model against the predicted values to check constant variance. Use a normal quantile plot of the residuals to check normality. Consider whether the observations are independent or whether clustering is present. Initial data analysis before fitting the model is the first step in checking assumptions.

What should I do if my data violate the assumptions of linear regression?

Consider transformations of the outcome or predictors to address nonlinearity or non normality. Consider weighted regression or robust standard errors to address non constant variance. Consider mixed models or cluster robust standard errors to address clustering. If the relationship is fundamentally curved, consider nonlinear regression with a biologically justified functional form.

Can regression analysis prove that one factor causes an outcome?

No. Regression analysis quantifies associations between variables. A strong association may reflect causation, but it may also reflect confounding or chance. Establishing causation requires study designs that control for confounders, such as randomized experiments, or additional evidence from multiple studies. Regression results should be interpreted as describing associations unless the study design supports causal inference.

How many observations do I need for regression analysis?

The number of observations needed depends on the number of predictors, the strength of the relationships, and the precision required. As a general rule, more observations are better, and models with many predictors require more observations than models with few predictors. The risk of overfitting increases when the number of predictors is large relative to the number of observations. Consult a statistician for guidance on sample size for your specific study.

What is the difference between univariate and multivariate regression?

Univariate regression has one outcome variable. Multivariate regression has multiple outcome variables analyzed together. The term multivariate is sometimes used incorrectly to describe multiple regression, which has multiple predictors but one outcome. The distinction matters because the statistical methods and interpretations differ between the two approaches.

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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.
• Understanding the Basics of Meta-Analysis and How to Read a Forest Plot: As Simple as It Gets.. The Journal of clinical psychiatry, 2020.
• Regression analysis and multivariate analysis.. Seminars in reproductive endocrinology, 1996.
• Understanding logistic regression analysis.. Biochemia medica, 2014.
• Statistics for Immunologists.. Current protocols in immunology, 2018.
• Regression without regrets -initial data analysis is a prerequisite for multivariable regression.. BMC medical research methodology, 2024.
• Interval censoring.. Statistical methods in medical research, 2010.
• Introduction to the Use of Linear and Nonlinear Regression Analysis in Quantitative Biological Assays.. Current protocols, 2023.
• Semiparametric regression analysis of doubly-censored data with applications to incubation period estimation.. Lifetime data analysis, 2023.
• WHO estimates of the global, regional, and national disease burden of nine foodborne chemicals, 2000-21: an updated data synthesis.. 2027.
• Interpretable mapping between microstructural texture and microhardness in fine-blanked steel.. 2026.
• When Simpler Models Win: A Calibration Benchmark of scGPT for Cell-Type Annotation. 2026.
• Statistical Consistency in Artificial Intelligence-Assisted Computations: A Comparison of SPSS 31.0 and GPT-5.5.. 2026.
• A simple program in BASIC to compare regression lines by analysis of covariance. Computer Methods and Programs in Biomedicine, 1993.
• Principal component regression analysis on drape performance and basic structural parameters of the fabric. Wool Textile Journal, 2015.
• Univariate and multivariate regression analysis: Some basic statistical principles. CME Journal of Gynecologic Oncology, 1999.
• Regression basics: A student's guide to quantitative methods and statistical analysis: Third edition. Regression Basics A Student S Guide to Quantitative Methods and Statistical Analysis Third Edition, 2024.
• How Efficient are Basic Public Health Services Between Urban and Rural in Shandong Province, China? A Data Envelopment Analysis and Panel Tobit Regression Approach. Risk Management and Healthcare Policy, 2022.

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