How to write statistical analysis section in medical research

Step 1: specify study objective type and outcomes (overall approach)

The study objective type provides the role of important variables for a specified outcome in statistical analyses and the overall approach of the model building and model reporting steps in a study. In the statistical framework, the problems are classified into descriptive and inferential/analytical/confirmatory objectives. In the epidemiological framework, the analytical and prognostic problems are broadly classified into association, explanatory, and predictive objectives.19 These study objectives (figure 1) may be classified into six categories: (1) exploratory, (2) association, (3) causal, (4) intervention, (5) prediction and (6) clinical decision models in medical research.20

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Figure 1

Comparative assessments of developing and reporting of study objective types and models. Association measures include odds ratio, risk ratio, or hazard ratio. AUC, area under the curve; C, confounder; CI, confidence interval; E, exposure; HbA1C: hemoglobin A1c; M, mediator; MFT, model fit test; MST, model specification test; PI, predictive interval; R², coefficient of determinant; X, independent variable; Y, outcome.

The exploratory objective type is a specific type of determinant study and is commonly known as risk factors or correlates study in medical research. In an exploratory study, all covariates are considered equally important for the outcome of interest in the study. The goal of the exploratory study is to present the results of a model which gives higher accuracy after satisfying all model-related assumptions. In the association study, the investigator identifies predefined exposures of interest for the outcome, and variables other than exposures are also important for the interpretation and considered as covariates. The goal of an association study is to present the adjusted association of exposure with outcome.20 In the causal objective study, the investigator is interested in determining the impact of exposure(s) on outcome using the conceptual framework. In this study objective, all variables should have a predefined role (exposures, confounders, mediators, covariates, and predictors) in a conceptual framework. A study with a causal objective is known as an explanatory or a confirmatory study in medical research. The goal is to present the direct or indirect effects of exposure(s) on an outcome after assessing the model’s fitness in the conceptual framework.19 21 The objective of an interventional study is to determine the effect of an intervention on outcomes and is often known as randomized or non-randomized clinical trials in medical research. In the intervention objective model, all variables other than the intervention are treated as nuisance variables for primary analyses. The goal is to present the direct effect of the intervention on the outcomes by eliminating biases.22–24 In the predictive study, the goal is to determine an optimum set of variables that can predict the outcome, particularly in external settings. The clinical decision models are a special case of prognostic models in which high dimensional data at various levels are used for risk stratification, classification, and prediction. In this model, all variables are considered input features. The goal is to present a decision tool that has high accuracy in training, testing, and validation data sets.20 25 Biostatisticians or applied researchers should properly discuss the intention of the study objective type before proceeding with statistical analyses. In addition, it would be a good idea to prepare a conceptual model framework regardless of study objective type to understand study concepts.

Step 2: specify effect size measure according to study design (interpretation and practical value)

The study design provides information on the selection of study participants and the process of data collection conditioned on either exposure or outcome (figure 2). The appropriate use of effect size measure, tabular presentation of results, and the level of evidence are mostly determined by the study design.31 32 In cohort or clinical trial study designs, the participants are selected based on exposure status and are followed up for the development of the outcome. These study designs can provide multiple outcomes, produce incidence or incidence density, and are preferred to be analyzed with risk ratio (RR) or hazards models. In a case–control study, the selection of participants is conditioned on outcome status. This type of study can have only one outcome and is preferred to be analyzed with an odds ratio (OR) model. In a cross-sectional study design, there is no selection restriction on outcomes or exposures. All data are collected simultaneously and can be analyzed with a prevalence ratio model, which is mathematically equivalent to the RR model.33 The reporting of effect size measure also depends on the study objective type. For example, predictive models typically require reporting of regression coefficients or weight of variables in the model instead of association measures, which are required in other objective types. There are agreements and disagreements between OR and RR measures. Due to the constancy and symmetricity properties of OR, some researchers prefer to use OR in studies with common events. Similarly, the collapsibility and interpretability properties of RR make it more appealing to use in studies with common events.34 To avoid variable practice and interpretation issues with OR, it is recommended to use RR models in all studies except for case–control and nested case–control studies, where OR approximates RR and thus OR models should be used. Otherwise, investigators may report sufficient data to compute any ratio measure. Biostatisticians should educate investigators on the proper interpretation of ratio measures in the light of study design and their reporting.34 35

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Figure 2

Effect size according to study design.

Step 3: specify study hypothesis, reporting of p values, and interval estimates (interpretation and decision)

The clinical hypothesis provides information for evaluating formal claims specified in the study objectives, while the statistical hypothesis provides information about the population parameters/statistics being used to test the formal claims. The inference about the study hypothesis is typically measured by p value and confidence interval (CI). A smaller p value indicates that the data support against the null hypothesis. Since the p value is a conditional probability, it can never tell about the acceptance or rejection of the null hypothesis. Therefore, multiple alternative strategies of p values have been proposed to strengthen the credibility of conclusions.41 42 Adaption of these alternative strategies is only needed in the explanatory objective studies. Although exact p values are recommended to be reported in research studies, p values do not provide any information about the effect size. Compared with p values, the CI provides a confidence range of the effect size that contains the true effect size if the study were repeated and can be used to determine whether the results are statistically significant or not.43 Both p value and 95% CI provide complementary information and thus need to be specified in the statistical analysis section.24 44

Researchers often test one or more comparisons or hypotheses. Accordingly, the side and the level of significance for considering results to be statistically significant may change. Furthermore, studies may include more than one primary outcome that requires an adjustment in the level of significance for multiplicity. All studies should provide the interval estimate of the effect size/regression coefficient in the primary analyses. Since the interpretation of data analysis depends on the study hypothesis, researchers are required to specify the level of significance along with the side (one-sided or two-sided) of the p value in the test for considering statistically significant results, adjustment of the level of significance due to multiple comparisons or multiplicity, and reporting of interval estimates of the effect size in the statistical analysis section.45

Step 4: account for DGP in the statistical analysis (accuracy)

The study design also requires the specification of the selection of participants and outcome measurement processes in different design settings. We referred to this specific design feature as DGP. Understanding DGP helps in determining appropriate modeling of outcome distribution in statistical analyses and setting up model premises and units of analysis.4 DGP (figure 3) involves information on data generation and data measures, including the number of measurements after random selection, complex selection, consecutive selection, pragmatic selection, or systematic selection. Specifically, DGP depends on a sampling setting (participants are selected using survey sampling methods and one subject may represent multiple participants in the population), clustered setting (participants are clustered through a recruitment setting or hierarchical setting or multiple hospitals), pragmatic setting (participants are selected through mixed approaches), or systematic review setting (participants are selected from published studies). DGP also depends on the measurements of outcomes in an unpaired setting (measured on one occasion only in independent groups), paired setting (measured on more than one occasion or participants are matched on certain subject characteristics), or mixed setting (measured on more than one occasion but interested in comparing independent groups). It also involves information regarding outcomes or exposure generation processes using quantitative or categorical variables, quantitative values using labs or validated instruments, and self-reported or administered tests yielding a variety of data distributions, including individual distribution, mixed-type distribution, mixed distributions, and latent distributions. Due to different DGPs, study data may include messy or missing data, incomplete/partial measurements, time-varying measurements, surrogate measures, latent measures, imbalances, unknown confounders, instrument variables, correlated responses, various levels of clustering, qualitative data, or mixed data outcomes, competing events, individual and higher-level variables, etc. The performance of statistical analysis, appropriate estimation of standard errors of estimates and subsequently computation of p values, the generalizability of findings, and the graphical display of data rely on DGP. Accounting for DGP in the analyses requires proper communication between investigators and biostatisticians about each aspect of participant selection and data collection, including measurements, occasions of measurements, and instruments used in the research study.

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Figure 3

Common features of the data generation process.

Step 5: apply EBB methods specific to study design features and DGP (efficiency and robustness)

The continuous growth in the development of robust statistical methods for dealing with a specific problem produced various methods to analyze specific data types. Since multiple methods are available for handling a specific problem yet with varying performances, heterogeneous practices among applied researchers have been noticed. Variable practices could also be due to a lack of consensus on statistical methods in literature, unawareness, and the unavailability of standardized statistical guidelines.2 5 59 However, it becomes sometimes difficult to differentiate whether a specific method was used due to its robustness, lack of awareness, lack of accessibility of statistical software to apply an alternative appropriate method, intention to produce expected results, or ignorance of model diagnostics. To avoid heterogeneous practices, the selection of statistical methodology and their reporting at each stage of data analysis should be conducted using methods according to EBB practice.5 Since it is hard for applied researchers to optimally select statistical methodology at each step, we encourage investigators to involve biostatisticians at the very early stage in basic, clinical, population, translational, and database research. We also appeal to biostatisticians to develop guidelines, checklists, and educational tools to promote the concept of EBB. As an effort, we developed the statistical analysis and methods in biomedical research (SAMBR) guidelines for applied researchers to use EBB methods for data analysis.5 The EBB practice is essential for applying recent cutting-edge robust methodologies to yield accurate and unbiased results. The efficiency of statistical methodologies depends on the assumptions and DGP. Therefore, investigators may attempt to specify the choice of specific models in the primary analysis as per the EBB.

Step 6: report variable selection method in the multivariable analysis according to study objective type (unbiased)

Multivariable analysis can be used for association, prediction or classification or risk stratification, adjustment, propensity score development, and effect size estimation.61 Some biological, clinical, behavioral, and environmental factors may directly associate or influence the relationship between exposure and outcome. Therefore, almost all health studies require multivariable analyses for accurate and unbiased interpretations of findings (figure 1). Analysts should develop an adjusted model if the sample size permits. It is a misconception that the analysis of RCT does not require adjusted analysis. Analysis of RCT may require adjustment for prognostic variables.23 The foremost step in model building is the entry of variables after finalizing the appropriate parametric or non-parametric regression model. In the exploratory model building process due to no preference of exposures, a backward automated approach after including any variables that are significant at 25% in the unadjusted analysis can be used for variable selection.62 63 In the association model, a manual selection of covariates based on the relevance of the variables should be included in a fully adjusted model.63 In a causal model, clinically guided methods should be used for variable selection and their adjustments.20 In a non-randomized interventional model, efforts should be made to eliminate confounding effects through propensity score methods and the final propensity score-adjusted multivariable model may adjust any prognostic variables, while a randomized study simply should adjust any prognostic variables.27 Maintaining the event per variable (EVR) is important to avoid overfitting in any type of modeling; therefore, screening of variables may be required in some association and explanatory studies, which may be accomplished using a backward stepwise method that needs to be clarified in the statistical analyses.10 In a predictive study, a model with an optimum set of variables producing the highest accuracy should be used. The optimum set of variables may be screened with the random forest method or bootstrap or machine learning methods.64 65 Different methods of variable selection and adjustments may lead to different results. The screening process of variables and their adjustments in the final multivariable model should be clearly mentioned in the statistical analysis section.

Step 7: provide evidence for exploring effect modifiers (applicability)

Any variable that modifies the effect of exposure on the outcome is called an effect modifier or modifier or an interacting variable. Exploring the effect modifiers in multivariable analyses helps in (1) determining the applicability/generalizability of findings in the overall or specific subpopulation, (2) generating ideas for new hypotheses, (3) explaining uninterpretable findings between unadjusted and adjusted analyses, (4) guiding to present combined or separate models for each specific subpopulation, and (5) explaining heterogeneity in treatment effect. Often, investigators present adjusted stratified results according to the presence or absence of an effect modifier. If the exposure interacts with multiple variables statistically or conceptually in the model, then the stratified findings (subgroup) according to each effect modifier may be presented. Otherwise, stratified analysis substantially reduces the power of the study due to the lower sample size in each stratum and may produce significant results by inflating type I error.69 Therefore, a multivariable analysis involving an interaction term as opposed to a stratified analysis may be presented in the presence of an effect modifier.70 Sometimes, a quantitative variable may emerge as a potential effect modifier for exposure and an outcome relationship. In such a situation, the quantitative variable should not be categorized unless a clinically meaningful threshold is not available in the study. In fact, the practice of categorizing quantitative variables should be avoided in the analysis unless a clinically meaningful cut-off is available or a hypothesis requires for it.71 In an exploratory objective type, any possible interaction may be obtained in a study; however, the interpretation should be guided based on clinical implications. Similarly, some objective models may have more than one exposure or intervention and the association of each exposure according to the level of other exposure should be presented through adjusted analyses as suggested in the presence of interaction effects.70

Step 8: assessment of assumptions, specifically the distribution of outcome, linearity, multicollinearity, sparsity, and overfitting (reliability)

The assessment and reporting of model diagnostics are important in assessing the efficiency, validity, and usefulness of the model. Model diagnostics include satisfying model-specific assumptions and the assessment of sparsity, linearity, distribution of outcome, multicollinearity, and overfitting.61 72 Model-specific assumptions such as normal residuals, heteroscedasticity and independence of errors in linear regression, proportionality in Cox regression, proportionality odds assumption in ordinal logistic regression, and distribution fit in other types of continuous and count models are required. In addition, sparsity should also be examined prior to selecting an appropriate model. Sparsity indicates many zero observations in the data set.73 In the presence of sparsity, the effect size is difficult to interpret. Except for machine learning models, most of the parametric and semiparametric models require a linear relationship between independent variables and a functional form of an outcome. Linearity should be assessed using a multivariable polynomial in all model objectives.62 Similarly, the appropriate choice of the distribution of outcome is required for model building in all study objective models. Multicollinearity assessment is also useful in all objective models. Assessment of EVR in multivariable analysis can be used to avoid the overfitting issue of a multivariable model.18

Step 9: report type of primary and sensitivity analyses (consistency)

Numerous considerations and assumptions are made throughout the research processes that require assessment, evaluation, and validation. Some assumptions, executions, and errors made at the beginning of the study data collection may not be fixable13; however, additional information collected during the study and data processing, including data distribution obtained at the end of the study, may facilitate additional considerations that need to be verified in the statistical analyses. Consistencies in the research findings via modifications in the outcome or exposure definition, study population, accounting for missing data, model-related assumptions, variables and their forms, and accounting for adherence to protocol in the models can be evaluated and reported in research studies using sensitivity analyses.77 The purpose and type of supporting analyses need to be specified clearly in the statistical analyses to differentiate the main findings from the supporting findings. Sensitivity analyses are different from secondary or interim or subgroup analyses.78 Data analyses for secondary outcomes are often referred to as secondary analyses, while data analyses of an ongoing study are called interim analyses and data analyses according to groups based on patient characteristics are known as subgroup analyses.

Step 10: provide methods for summarizing, displaying, and interpreting data (transparency and usability)

Data presentation includes data summary, data display, and data from statistical model analyses. The primary purpose of the data summary is to understand the distribution of outcome status and other characteristics in the total sample and by primary exposure status or outcome status. Column-wise data presentation should be preferred according to exposure status in all study designs, while row-wise data presentation for the outcome should be preferred in all study designs except for a case–control study.24 32 Summary statistics should be used to provide maximum information on data distribution aligned with DGP and variable type. The purpose of results presentation primarily from regression analyses or statistical models is to convey results interpretation and implications of findings. The results should be presented according to the study objective type. Accordingly, the reporting of unadjusted and adjusted associations of each factor with the outcome may be preferred in the determinant objective model, while unadjusted and adjusted effects of primary exposure on the outcome may be preferred in the explanatory objective model. In prognostic models, the final predictive models may be presented in such a way that users can use models to predict an outcome. In the exploratory objective model, a final multivariable model should be reported with R² or area under the curve (AUC). In the association and interventional models, the assessment of internal validation is critically important through various sensitivity and validation analyses. A model with better fit indices (in terms of R² or AUC, Akaike information criterion, Bayesian information criterion, fit index, root mean square error) should be finalized and reported in the causal model objective study. In the predictive objective type, the model performance in terms of R² or AUC in training and validation data sets needs to be reported (figure 1).20 21 There are multiple purposes of data display, including data distribution using bar diagram or histogram or frequency polygons or box plots, comparisons using cluster bar diagram or scatter dot plot or stacked bar diagram or Kaplan-Meier plot, correlation or model assessment using scatter plot or scatter matrix, clustering or pattern using heatmap or line plots, the effect of predictors with fitted models using marginsplot, and comparative evaluation of effect sizes from regression models using forest plot. Although the key purpose of data display is to highlight critical issues or findings in the study, data display should essentially follow DGP and variable types and should be user-friendly.54 79 Data interpretation heavily relies on the effect size measure along with study design and specified hypotheses. Sometimes, variables require standardization for descriptive comparison of effect sizes among exposures or interpreting small effect size, or centralization for interpreting intercept or avoiding collinearity due to interaction terms, or transformation for achieving model-related assumptions.80 Appropriate methods of data reporting and interpretation aligned with study design, study hypothesis, and effect size measure should be specified in the statistical analysis section of research studies.