Analysis of Continuous and Binary Trial Outcomes

Overview

Trial outcomes should be analysed using methods aligned with:

• outcome type;
• estimand;
• randomisation design;
• baseline measurement;
• follow-up completeness;
• effect scale.

For continuous outcomes, baseline-adjusted ANCOVA is often efficient.

For binary outcomes, report absolute and relative effects.

A p-value alone is not a trial result.

Why this matters

Common errors include:

• comparing follow-up means without baseline adjustment;
• testing within-group change;
• reporting odds ratios as risk ratios;
• reporting only percentages without denominators;
• presenting statistical significance without clinical magnitude.

The effect estimate and confidence interval should answer the treatment question directly.

When to use these methods

Use ANCOVA when:

• a numerical follow-up outcome is measured;
• baseline value is available;
• treatment was randomised;
• linear-model assumptions are acceptable.

Use binomial models or direct two-group estimators for binary outcomes.

For common outcomes, risk difference and risk ratio are usually more interpretable than odds ratio.

Research question and variables

Continuous estimand:

Adjusted mean difference in follow-up outcome between assigned groups.

Binary estimands:

• risk difference;
• risk ratio;
• odds ratio.

The binary threshold should be clinically justified and prespecified.

Data structure

Trial outcome variables

simulated data

Variable	Role
group	Assigned treatment
baseline	Baseline continuous outcome
outcome_cont	Follow-up continuous outcome
binary_event	Follow-up success indicator
site	Randomisation stratum

Statistical assumptions

Continuous outcome analysis

The ANCOVA model is:

$\beta_1$ is the adjusted treatment mean difference.

Baseline adjustment improves precision when baseline predicts follow-up.

Binary effect measures

The odds ratio is:

$$OR= \frac{p_1/(1-p_1)} {p_0/(1-p_0)}$$

OR can be much farther from one when outcomes are common.

Worked example

Worked analysis in R

trial <- read.csv(
  "data/part-13-trial-data.csv"
)

continuous_data <- trial[
  !is.na(trial$outcome_cont),
]

continuous_fit <- lm(
  outcome_cont ~ group +
    baseline +
    factor(site),
  data = continuous_data
)

coef(
  summary(continuous_fit)
)

confint(
  continuous_fit
)

binary_table <- with(
  continuous_data,
  table(
    group,
    binary_event
  )
)

risk_control <- binary_table["0", "1"] /
  sum(binary_table["0",])

risk_intervention <- binary_table["1", "1"] /
  sum(binary_table["1",])

risk_difference <-
  risk_intervention -
  risk_control

risk_ratio <-
  risk_intervention /
  risk_control

odds_ratio <- (
  binary_table["1", "1"] *
  binary_table["0", "0"]
) / (
  binary_table["1", "0"] *
  binary_table["0", "1"]
)

c(
  risk_control = risk_control,
  risk_intervention = risk_intervention,
  risk_difference = risk_difference,
  risk_ratio = risk_ratio,
  odds_ratio = odds_ratio
)

Selected analysis output

Adjusted mean difference = -5.478
95% CI = -6.324 to -4.631
p < 0.001

Control risk = 0.207
Intervention risk = 0.584
Risk difference = 0.377
Risk ratio = 2.819
Odds ratio = 5.373

Visualising trial effects

Missing outcomes

Complete-record analysis is shown for teaching.

A primary trial analysis should use the prespecified missing-data approach.

Sensitivity analyses can include:

• multiple imputation;
• mixed models;
• delta adjustment;
• tipping-point analysis.

The analysis should remain aligned with the estimand’s intercurrent-event strategy.

Interpreting the output

Presenting the results

Suggested reporting

Report group counts, observed outcome summaries, adjusted effect, confidence interval, analysis population, covariates and missing-data sensitivity.

ICH E9 emphasises effect estimation aligned with the estimand [1].

Baseline-adjusted analysis, the limitations of change scores and sensitivity analysis for missing trial outcomes are discussed in methodological guidance [2], [3], [4].

Common mistakes

Knowledge check

Why use ANCOVA?

Answer and reasoning

It estimates the between-group follow-up difference while improving precision through baseline prognosis.

What does risk difference measure?

Answer and reasoning

The absolute difference in outcome probabilities between groups.

Why can OR exaggerate RR?

Answer and reasoning

Odds diverge from probabilities when outcomes are common.

Does within-group improvement prove treatment benefit?

Answer and reasoning

No. The direct between-group contrast is required.

What should accompany every effect estimate?

Answer and reasoning

A compatible confidence interval and clear analysis population.

Key take-home messages

References

1. International Council for Harmonisation. ICH E9: statistical principles for clinical trials. Geneva: ICH; 1998.
2. Vickers AJ, Altman DG. Analysing controlled trials with baseline and follow-up measurements. BMJ. 2001;323:1123–1124. https://doi.org/10.1136/bmj.323.7321.1123
3. Senn S. Change from baseline and analysis of covariance revisited. Stat Med. 2006;25(24):4334–4344. https://doi.org/10.1002/sim.2682
4. Morris TP, Kahan BC, White IR. Choosing sensitivity analyses for randomised trials with missing data. BMC Med Res Methodol. 2014;14:8. https://doi.org/10.1186/1471-2288-14-8