Presenting Multivariable Analysis

Overview

A multivariable table is not a list of variables that “remained significant”.

It should communicate:

• the model’s purpose;
• outcome and analysis population;
• predictor coding;
• reference categories;
• functional forms;
• adjusted effect estimates;
• confidence intervals;
• model uncertainty;
• diagnostics or validation;
• missing-data method.

The presentation differs according to whether the model is intended for:

• causal or treatment-effect estimation;
• explanatory association;
• clinical prediction.

Why this matters

A regression table can look complete while leaving readers unable to interpret it.

Examples of missing information include:

• whether age is per one year or ten years;
• which sex category is the reference;
• whether BMI is linear or modelled with splines;
• which variables are held fixed;
• whether odds ratios are adjusted;
• how many participants and events were analysed;
• whether the model was selected by stepwise procedures;
• whether a prediction model was internally validated.

Good presentation begins with a prespecified model, not table formatting.

When to use this framework

Use it for:

• linear regression;
• logistic regression;
• ordinal or multinomial models;
• Poisson or negative-binomial regression;
• Cox regression;
• prediction models.

The exact metrics depend on the model family and objective.

What the table must identify

Outcome and time point

State the outcome in the table title.

Example:

Adjusted associations with 12-week systolic blood pressure.

Model type

State:

• linear regression;
• binary logistic regression;
• Cox proportional-hazards model;
• negative-binomial regression.

Analysis sample

Report:

• total analysed;
• events for binary or survival outcomes;
• clusters if applicable;
• records excluded;
• imputation or complete-record approach.

Predictor coding

For every predictor, show:

• unit;
• contrast;
• reference;
• transformation;
• non-linear term;
• interaction.

Effect measure

Use the correct model scale:

• linear coefficient or mean difference;
• odds ratio;
• rate ratio;
• hazard ratio;
• cumulative odds ratio.

Confidence intervals

Report 95% CIs unless another level is justified.

Model fit or performance

Choose metrics aligned with the purpose.

For explanatory linear regression:

• adjusted $R^2$;
• residual standard error;
• diagnostics.

For prediction:

• calibration;
• discrimination;
• Brier score;
• validation.

Data example

Simulated multivariable results table

simulated data

Table 1. Adjusted associations with 12-week systolic blood pressure

Predictor	Adjusted coefficient	95% CI	p-value
Baseline systolic blood pressure, per 1 mmHg	0.41 mmHg	0.26 to 0.56	$p<0.001$
Telemonitoring versus usual care	−4.52 mmHg	−5.78 to −3.26	$p<0.001$
Age, per 10 years	1.10 mmHg	0.22 to 1.99	0.015
BMI, per 5 kg/m²	1.57 mmHg	0.61 to 2.52	0.002
Adherence, per score unit	−1.98 mmHg	−2.40 to −1.57	$p<0.001$
Male versus female	2.08 mmHg	0.81 to 3.35	0.002

Note: The calculations underlying these simulated coefficients, confidence intervals and p-values are not shown here.

Notes: Outcome is clinic systolic blood pressure at 12 weeks. Coefficients are mutually adjusted for all predictors shown. Analysis includes 180 participants. BMI = body mass index; CI = confidence interval.

Scaling predictors

The analysis may estimate age per one year, but the table can present a ten-year contrast:

$$10\beta_{\text{age}}$$

The confidence interval endpoints are multiplied by ten on the linear outcome scale.

For logistic regression, the ten-year odds ratio is:

$$e^{10\beta_{\text{age}}}$$

Do not multiply a one-year odds ratio by ten.

Categorical predictors

A three-category variable requires two coefficient rows when one category is the reference.

Example:

The table should not omit the reference row if readers could misunderstand the comparison.

Non-linearity

A spline or polynomial cannot be represented adequately by one ordinary coefficient.

Options include:

• overall p-value for the predictor;
• test of nonlinear contribution;
• figure of adjusted predictions;
• clinically meaningful contrasts.

Do not list spline-basis coefficients as though they are interpretable effects.

Interactions

If an interaction is present, main effects are conditional.

Report:

• interaction coefficient;
• effect measure scale;
• simple effects at clinically relevant modifier values;
• confidence intervals;
• interaction plot.

Do not report separate subgroup p-values as evidence of heterogeneity.

Worked example

Crude and adjusted estimates

Whether to show crude estimates depends on the purpose.

For observational causal analysis, showing crude and adjusted effects can help readers understand confounding, but a percentage change between them is not a sufficient confounding criterion.

For randomised primary analyses, the adjusted effect may be primary, with group summaries presented separately.

Avoid a sequence of many models designed to show when a coefficient becomes significant.

Variable selection

The Methods and table footnotes should state whether predictors were:

• prespecified;
• selected by stepwise methods;
• penalised;
• retained as a full model;
• selected through cross-validation.

Univariable screening can bias coefficients and uncertainty and is not recommended as a general model-building strategy [1].

Model fit and diagnostics

A table may include:

• adjusted $R^2$;
• AIC;
• residual standard error;
• event count;
• dispersion;
• calibration slope;
• c-statistic.

These metrics should not be mixed without explanation.

Diagnostics can be reported in text or supplement:

Residual plots showed no material non-linearity or heteroscedasticity. One influential observation was retained after source verification; exclusion did not materially alter the treatment estimate.

Prediction-model presentation

For a prediction model, a coefficient table is insufficient.

Report:

• full equation or executable code;
• intercept;
• transformations;
• predictor definitions;
• apparent and validated performance;
• calibration plot;
• clinical utility;
• external validation.

TRIPOD+AI applies to regression and machine-learning prediction models [2].

Interpreting the model

Presenting the results

Suggested reporting

A clear multivariable table should be reconstructable: outcome, sample, coding, effect scale, reference categories, adjustment set and uncertainty must all be explicit [3]. Detailed regression reporting should also preserve the full model specification and clinically interpretable contrasts [4].

Common mistakes

Knowledge check

Why report age per ten years?

Answer and reasoning

It can provide a more clinically interpretable contrast while representing the same fitted linear coefficient.

What must be shown for a categorical predictor?

Answer and reasoning

All non-reference contrasts and the reference category.

Can spline-basis coefficients be interpreted directly?

Answer and reasoning

Usually not. Report the curve, overall contribution and meaningful predicted contrasts.

Does an adjusted coefficient prove causality?

Answer and reasoning

No. Causal interpretation requires suitable design, variable selection and assumptions.

What additional information is needed for prediction models?

Answer and reasoning

The complete equation, validation, calibration, discrimination, prediction error and clinical utility.

Key take-home messages

References

1. Heinze G, Wallisch C, Dunkler D. Variable selection—a review and recommendations for the practicing statistician. Biom J. 2018;60(3):431–449. https://doi.org/10.1002/bimj.201700067
2. Collins GS, Moons KGM, Dhiman P, Riley RD, Beam AL, Van Calster B, et al. TRIPOD+AI statement: updated guidance for reporting clinical prediction models that use regression or machine learning methods. BMJ. 2024;385:e078378. https://doi.org/10.1136/bmj-2023-078378
3. Lang TA, Altman DG. Basic statistical reporting for articles published in biomedical journals: the SAMPL guidelines. Int J Nurs Stud. 2015;52(1):5–9. https://doi.org/10.1016/j.ijnurstu.2014.09.006
4. Harrell FE Jr. Regression modeling strategies. 2nd ed. Cham: Springer; 2015. https://doi.org/10.1007/978-3-319-19425-7