How do you allow for the baseline value?

Start here

Why does the baseline value matter in a randomised trial?

Randomisation makes the groups comparable on average, not identical. In any one trial the arms can start at different mean scores by chance, and a patient's score at 8 weeks depends strongly on where they started.

Regression to the mean

A single measurement is the patient's usual level plus variation on that day. Patients chosen because their first reading was high tend to have a lower second reading, even with no treatment, because the chance part of the first reading does not repeat. This is regression to the mean. Barnett, van der Pols and Dobson (2005) describe it as a statistical phenomenon that can make natural variation in repeated data look like real change, and note that its effect is larger with more measurement error and when follow-up is examined only in a subgroup selected on its baseline value.

Regression to the mean also explains why change scores do not fix baseline imbalance. Vickers and Altman (BMJ, 2001) wrote that analysing change does not control for baseline imbalance because of regression to the mean: patients with high baseline scores tend to fall more.

Interactive

Raise the entry cut-off and watch untreated patients appear to improve

No one in this population receives any treatment. The apparent fall comes only from selecting people whose first reading was high.

Three analyses

Final score, change score or ANCOVA: what does each one do with the baseline?

All three compare the arms at the end of the trial. They differ in how much of each patient's baseline value they subtract.

Why ANCOVA is usually preferred in randomised trials

1. No bias from chance imbalance. Vickers and Altman: ANCOVA adjusts each patient's follow-up score for their baseline score and is unaffected by baseline differences.
2. Smaller standard error. Frison and Pocock (1992) quantified the superiority of analysis of covariance over analysis of post-treatment means or mean changes, in both reduced variance and avoidance of bias.
3. Fewer patients. Vickers and Altman give an example with a correlation of 0.6 between baseline and follow-up: 85 patients for the final score, 68 for the change score and 54 for ANCOVA.
4. Regulatory advice. The EMA guideline says the baseline value of a continuous primary outcome should usually be included as a covariate, whether the outcome is the raw score or the change from baseline.

Interactive

Change the baseline imbalance and the correlation, and watch the three estimates and their standard errors

Each run is a new trial of 60 patients with a true effect of −4. The final score and change score move away from −4 in opposite directions when the arms start unequal. ANCOVA stays centred on −4 with the smallest standard error.

The method

How does ANCOVA adjust for the baseline?

ANCOVA fits one regression line per arm with a common slope. The treatment effect is the vertical gap between the two parallel lines, which is the difference expected between two patients with the same baseline score.

final score = a + b × baseline + c × group + error

group

0 for control, 1 for treated

b

common within-arm slope: how much the final score rises per point of baseline

c

treatment effect adjusted for baseline, the number to report with its 95% CI

adjusted mean = arm mean final − b × (arm mean baseline − overall mean baseline)

Control: 15.6 − 0.674 × (21.6 − 23.15) = 16.64. Treated: 13.3 − 0.674 × (24.7 − 23.15) = 12.26. The difference, −4.39, is c. Vickers and Altman note that ANCOVA, despite its name, is a regression method. With the change score as the outcome and baseline as the covariate, the estimate of c is identical, as the EMA guideline states: once baseline is in the model, choosing the raw score or the change is only a matter of interpretation.

ANCOVA as a test of significance

The F test for group asks whether the adjusted means differ. In the teaching data, after baseline is in the model, the group sum of squares is 81.54 on 1 df and the residual mean square is 7.78 on 17 df: F = 10.48, p = 0.005. With one grouping factor of two levels, F = t² (−3.24² = 10.48). One-way ANOVA on the final scores, which ignores baseline, gives F = 1.89, p = 0.19.

Assumptions

What must be true for ANCOVA to be valid?

The assumptions of linear regression, plus one that is specific to ANCOVA: the slope for baseline is the same in both arms.

1. Linearity: the final score rises in a straight line with the baseline score in each arm. Check a scatter plot. The EMA guideline advises a simple functional form, usually linear, in the absence of prior knowledge.
2. Homogeneity of regression slopes: no group × baseline interaction. Teaching data: slopes 0.62 (control) and 0.72 (treated), interaction p = 0.78.
3. Normal residuals with similar spread in both arms: teaching data residual SD 2.79, Shapiro-Wilk p = 0.86, Levene p = 0.19.
4. Independent patients, and a covariate measured without being affected by the treatment (measured before randomisation).

If the slopes differ, the treatment effect depends on the baseline value, and a single adjusted difference describes the effect only at the baseline where it is calculated. The EMA guideline says the primary model should not include treatment by covariate interactions, and that a substantial interaction expected in advance should be handled in the trial design, with separate estimates in the relevant subgroups.

Interactive

Make the slopes unequal and watch the treatment effect change with baseline severity

With parallel lines the gap is the same at every baseline. With unequal slopes the gap widens or narrows, and the interaction test tells you whether the data support that.

Choosing covariates

Which covariates, decided when, and which ones never?

Covariates for the primary analysis are chosen in the protocol, before any outcome data are seen, from variables measured before randomisation that are expected to influence the outcome.

• Prespecify: the EMA guideline requires covariates in the primary analysis to be pre-specified in the protocol, and advises only a few.
• Stratification factors: factors used to stratify the randomisation should normally be included (EMA). ICH E9 says the same: if factors stratify the design, account for them in the analysis.
• Not because the groups look unbalanced: baseline imbalance observed post hoc is not an appropriate reason to add a covariate (EMA). Senn (1994) called significance tests of baseline homogeneity philosophically unsound, of no practical value and potentially misleading, and recommended fitting prognostic variables named in the trial plan regardless of their baseline distribution.
• Never after randomisation: ICH E9 advises against adjusting the main analysis for covariates measured after randomisation, because they may be affected by the treatments. The EMA guideline gives duration of treatment, compliance and rescue medication as examples, and says such adjustment may hide or exaggerate the treatment effect.

Non-randomised groups

In 1967 Frederic Lord called attention to a basic source of confusion in interpreting comparisons of pre-existing groups, now known as Lord's paradox: change scores and ANCOVA can give different answers. Senn (2006) related the debate to Lord's paradox, and showed that equality of groups at baseline is not a necessary condition for ANCOVA to give unbiased estimates. He added that although many situations can be imagined in which ANCOVA is biased, it is very difficult to imagine one in which the change-score analysis would then be unbiased. In observational studies neither model removes confounding on its own.

Decide

Which adjustment fits your study?

Start from what you want to adjust for.

Comparison

Two pairs residents confuse

Change score or ANCOVA?

ANOVA or ANCOVA?

The formula

Worked example: ANCOVA by hand

Made-up teaching data: HAM-D in 10 control and 10 treated patients. Computed with Python (statsmodels 0.14.5).

1. Within each arm, work out deviations from the arm's own means. Add over both arms: Sxx (baseline) = 264.5, Sxy (cross-products) = 178.3, Syy (8-week) = 252.5.
2. Common slope b = Sxy ÷ Sxx = 178.3 ÷ 264.5 = 0.674.
3. Adjusted difference c = (13.3 − 15.6) − 0.674 × (24.7 − 21.6) = −2.3 − 2.09 = −4.39.
4. Residual SS = Syy − Sxy² ÷ Sxx = 252.5 − 120.19 = 132.31 on 20 − 3 = 17 df. Residual mean square = 7.78.
5. SE of c = √[7.78 × (1/10 + 1/10 + 3.1² ÷ 264.5)] = 1.356.
6. t = −4.39 ÷ 1.356 = −3.24, 17 df, p = 0.005. 95% CI = −4.39 ± 2.110 × 1.356 = −7.25 to −1.53.

Sxx, Sxy, Syy

within-arm sums of squares and cross-products

3.1

difference in mean baseline, treated − control

2.110

critical t for 17 df at 5%, two-sided

adjusted difference = −4.39 (95% CI −7.25 to −1.53)

Critical F(1, 17) at 5% = 4.45. The model explains 53% of the variance in 8-week scores (R² = 0.526). The baseline slope is 0.674 (95% CI 0.31 to 1.04).

In a real paper

Baseline adjustment in two published trials

Both articles: no retraction, correction or expression of concern found in PubMed, Europe PMC or scite (checked 15 September 2026).

Software

How to run it

One row per patient, with columns for group, baseline score and final score.

SPSS

• ANCOVA: Analyze > General Linear Model > Univariate. Final score in Dependent Variable, group in Fixed Factor(s), baseline score in Covariate(s).
• Adjusted means: the Estimated Marginal Means table gives the model-estimated mean and standard error for each group.
• Homogeneity of slopes: in Model, choose Build terms and add group, baseline and the group × baseline interaction. The full factorial model does not contain covariate interactions. Remove the interaction again for the final ANCOVA.

R

# d: one row per patient with group, baseline, final
fit <- lm(final ~ group + baseline, data = d)   # ANCOVA
summary(fit); confint(fit)

library(emmeans)
emmeans(fit, "group")                           # adjusted (estimated marginal) means

anova(lm(final ~ group * baseline, data = d))   # homogeneity of slopes

jamovi

• ANCOVA (jmv function ancova): final score as the dependent variable, group as a fixed factor, baseline as a covariate. The estimated marginal means option (emMeans) gives the adjusted means.
• Homogeneity of slopes: add the group × baseline term to the model terms (modelTerms) to test it, then remove it.

Reporting

How to report it

Say which covariates were prespecified, report unadjusted means by arm and the adjusted difference with its 95% CI, and state that assumptions were checked.

"The primary outcome, HAM-D at 8 weeks, was analysed by analysis of covariance with baseline HAM-D as a prespecified covariate. Mean baseline scores were 21.6 (SD 3.57) in the control arm and 24.7 (SD 4.08) in the treated arm. Unadjusted 8-week means were 15.6 and 13.3. The adjusted means were 16.6 and 12.3, an adjusted difference of −4.4 points (95% CI −7.3 to −1.5; F(1, 17) = 10.48, p = 0.005). There was no evidence that the effect of baseline differed between arms (interaction p = 0.78)."

By specialty

Baseline adjustment problems by specialty

Common myths

Common mistaken beliefs

• ✗Myth: Analysing change from baseline removes any baseline imbalance.

  ✓Vickers and Altman (2001): analysing change does not control for baseline imbalance because of regression to the mean. In the teaching trial it gave −5.4 against −4.39 from ANCOVA.

• ✗Myth: Test the baseline table with p-values and adjust only for what is significant.

  ✓Senn (1994) called such tests philosophically unsound and of no practical value. The EMA guideline says statistical testing for baseline imbalance has no role in a properly randomised trial.

• ✗Myth: ANCOVA is needed only when the arms are unbalanced.

  ✓Adjusting for a covariate strongly or moderately associated with the outcome generally improves the efficiency of the analysis (EMA), even with perfect balance.

• ✗Myth: The more covariates, the better the adjustment.

  ✓The EMA guideline advises only a few covariates, each justified and prespecified. Many covariates make the analysis less convincing.

• ✗Myth: Adjust for adherence or weight change to see the drug effect more clearly.

  ✓Variables measured after randomisation may be affected by the treatment. Adjusting for them may hide or exaggerate the effect (ICH E9, EMA). In the illustration, −0.80 became −0.24.

• ✗Myth: Percentage change from baseline is a fairer outcome.

  ✓Vickers (2001) found that percentage change from baseline had the lowest statistical power of four methods and was highly sensitive to changes in variance.

Why this matters to you

Five consequences of getting baseline adjustment wrong

1. Two p-values, one trial

   Without a prespecified analysis you can choose p = 0.19 or p < 0.001, and any choice made after seeing both is suspect.

2. A biased effect size

   With a chance imbalance, the final score understated and the change score overstated the adjusted effect of −4.39 points.

3. A larger, longer study than needed

   At a baseline-final correlation of 0.6, a final-score analysis needed 85 patients where ANCOVA needed 54 in Vickers and Altman’s example.

4. Questions at the viva

   "Why did you adjust for compliance?" and "Were your covariates in the protocol?" have only one acceptable answer each.

5. Misreading papers

   Within-arm improvement in patients selected for high scores is often regression to the mean. Only the adjusted difference between arms estimates the treatment effect.

History

History of baseline adjustment

1. 1886

   Francis Galton

   Published "Regression towards mediocrity in hereditary stature" in the Journal of the Anthropological Institute of Great Britain and Ireland.

2. 1967

   Frederic M. Lord

   In "A paradox in the interpretation of group comparisons" (Psychological Bulletin), called attention to a basic source of confusion in interpreting certain group comparison data: Lord’s paradox.

3. 1992

   L. Frison and S. J. Pocock

   Showed that analysis of covariance is the method of choice for pre- and post-treatment means, and quantified its advantage over post-treatment means and mean changes in variance and bias.

4. 1994

   Stephen Senn

   Argued that tests of baseline homogeneity are philosophically unsound, of no practical value and potentially misleading, and that prognostic variables should be named in the trial plan and fitted in an analysis of covariance.

5. 2001

   A. J. Vickers and D. G. Altman

   A BMJ Statistics Note recommended analysis of covariance as the preferred general approach for trials with baseline and follow-up measurements. The same year Vickers showed in a simulation study that percentage change from baseline is statistically inefficient.

6. 2002

   S. J. Pocock, S. E. Assmann, L. E. Enos and L. E. Kasten

   Surveyed 50 trial reports and found inconsistent covariate adjustment, no clear guidelines on covariate selection, and misuse of significance tests for baseline comparability.

7. 2005

   Adrian G. Barnett, Jolieke C. van der Pols and Annette J. Dobson

   Described regression to the mean and how to deal with it at the design and analysis stages (International Journal of Epidemiology).

8. 2006

   Stephen Senn

   In "Change from baseline and analysis of covariance revisited", showed that equal groups at baseline are not a necessary condition for ANCOVA to give unbiased estimates.

9. 2019

   Lei Clifton and David A. Clifton

   Showed that the change score is always correlated with the baseline score, that change scores do not address regression to the mean or baseline imbalance, and that one should adjust for baseline with ANCOVA.

Guidelines

Baseline covariates in guidelines

Reading a paper

Five checks on baseline adjustment in any trial

• Was the primary analysis (final score, change score or ANCOVA) and its covariates stated in the protocol or registry before the results?
• Is the baseline value of a continuous outcome included as a covariate?
• Are any covariates measured after randomisation (adherence, weight change, rescue medication, length of stay)?
• Are unadjusted means by arm reported alongside the adjusted difference and its 95% CI?
• Do the authors test the baseline table with p-values, or interpret within-arm improvement as a treatment effect?

Why is it called that?

Word surgery

Exam answers

Model answer outlines

Past papers

TNMGRMU questions on ancova and baseline adjustment

Questions from MD, MS, DM, MCh and Ph.D. papers that this chapter helps you answer. Tap a sitting to open the university's own question paper.

1. ×1ANCOVA test of significanceMD Community Medicine · Oct 2024