Analysis of Change Scores

Overview

A change score is the difference between follow-up and baseline for one participant:

$$\text{change}=Y_{\text{follow-up}}-Y_{\text{baseline}}$$

Change scores can be clinically intuitive. They are useful for:

• describing individual change;
• analysing one-group mean change;
• comparing mean change between independent groups;
• constructing responder definitions when justified.

However, in a randomised trial with a continuous follow-up outcome, baseline-adjusted ANCOVA is commonly more precise and less sensitive to chance baseline imbalance than a simple change-score comparison [1], [2].

The correct analysis depends on the estimand, design, baseline–follow-up relationship and measurement reliability.

Why this matters

A common incorrect strategy is:

1. test whether Group A changes significantly;
2. test whether Group B changes significantly;
3. conclude the groups differ because only one within-group test is significant.

The treatment effect is not the difference between two p-values; separate within-group baseline comparisons can be highly misleading [3]. It is the direct between-group contrast in follow-up or change.

Change scores also contain baseline measurement error with a negative sign. Participants selected for extreme baseline values tend to move towards the population mean on repeat measurement, even without an intervention. This is regression to the mean, a well-described source of misleading change interpretations [4].

When to use change scores

A change-score comparison can be reasonable when:

• change itself is the prespecified estimand;
• baseline and follow-up are measured on the same scale;
• two measurements are available per participant;
• change-score variability is acceptable;
• and the design supports a direct between-group comparison.

Change may be particularly natural for:

• cumulative dose;
• growth;
• change from a stable calibration baseline;
• or a clinically defined change threshold.

Do not assume change is always more informative than follow-up.

Research question and variables

The change-score estimand is intervention minus usual-care mean change.

The ANCOVA estimand is intervention minus usual-care mean follow-up at a common baseline value under the model.

These estimands can differ when:

• baseline means differ;
• the baseline coefficient is not exactly 1;
• measurement error is present;
• or model assumptions differ.

Data structure

Baseline, follow-up and change

simulated data

group	Baseline mean (SD)	Follow-up mean (SD)	Change mean (SD)
Usual care	145.5 (7.3)	134.7 (9.2)	−10.8 (6.7)
Intervention	155.7 (6.4)	135.3 (6.6)	−20.4 (8.5)

Note: Change = follow-up minus baseline. Values are simulated.

The intervention group begins approximately 10.2 units higher. This imbalance strongly affects the raw change comparison.

Statistical assumptions

Regression to the mean

When participants are selected because of an extreme baseline value, some extremeness reflects random measurement variation.

On repeat measurement, values tend to move towards the population mean even without treatment.

This phenomenon can produce apparent improvement and is not corrected by a paired test.

Mathematical coupling

Change contains baseline:

$$D=Y_1-Y_0$$

Correlating $D$ with $Y_0$ creates mathematical coupling because $Y_0$ appears in both variables. A negative association can arise mechanically.

Do not interpret a simple baseline-versus-change correlation as evidence that participants with high baseline values benefit more.

Worked example

Reproducible R analysis

Base R provides t.test() for the change-score comparison and lm() for baseline-adjusted ANCOVA [5], [6].

trial <- data.frame(
  group = factor(
    rep(c("Usual care", "Intervention"), each = 20),
    levels = c("Usual care", "Intervention")
  ),
  baseline = c(
    161,151,147,148,146,151,138,140,138,138,141,144,131,145,159,149,140,152,145,145,
    169,162,154,158,144,157,153,155,149,165,162,146,158,149,152,156,151,159,161,153
  ),
  followup = c(
    147,141,136,144,135,141,119,129,136,116,121,146,130,136,147,143,135,135,126,130,
    134,138,125,136,138,132,127,142,134,136,154,134,142,131,143,135,132,126,131,135
  )
)

trial$change <- trial$followup - trial$baseline

change_fit <- t.test(
  trial$change[trial$group == "Intervention"],
  trial$change[trial$group == "Usual care"],
  var.equal = FALSE
)

followup_fit <- t.test(
  trial$followup[trial$group == "Intervention"],
  trial$followup[trial$group == "Usual care"],
  var.equal = FALSE
)

# Base R provides `t.test()` and `lm()` for these analyses [5], [6]
ancova_fit <- lm(
  followup ~ baseline + group,
  data = trial
)

change_fit
followup_fit
summary(ancova_fit)
confint(ancova_fit)

R change-score output

Welch Two Sample t-test
t = -3.9754, df = 36.148, p-value = 0.0003223
95 percent confidence interval:
 -14.49688  -4.70312
sample estimates:
mean in group Intervention = -20.4
mean in group Usual care   = -10.8

R unadjusted follow-up output

Welch Two Sample t-test
t = 0.2365, df = 34.600, p-value = 0.8145
95 percent confidence interval:
 -4.55302   5.75302
sample estimates:
mean in group Intervention = 135.25
mean in group Usual care   = 134.65

R ANCOVA coefficient output

                         Estimate Std. Error t value Pr(>|t|)
(Intercept)               52.5025    24.6387   2.131 0.0398
baseline                   0.5648     0.1690   3.341 0.0019
groupIntervention         -5.1608     2.8376  -1.819 0.0771

groupIntervention  95% CI: -10.9104 to 0.5888

Reproducibility checks

Interpreting the output

Presenting the results

These analyses answer related but different model-based questions.

Suggested reporting

Lord's paradox MMed extension

Different analyses of baseline and follow-up data can produce different conclusions because they condition on different quantities and imply different models.

This is often called Lord’s paradox.

The solution is not to choose the most favourable analysis. The protocol should define:

• estimand;
• randomisation or causal structure;
• baseline role;
• model;
• and primary analysis before outcome comparison.

Common mistakes

Knowledge check

What is the correct treatment-effect contrast for change scores?

Answer and reasoning

It is the difference between group mean changes, not whether each group’s within-group change is separately significant.

Why can extreme baseline selection create apparent improvement?

Answer and reasoning

Part of the extreme baseline value may be random error. Repeat measurements tend to be less extreme even without treatment, producing regression to the mean.

Why do change-score and ANCOVA estimates differ?

Answer and reasoning

Change-score analysis effectively constrains the baseline coefficient to 1, while ANCOVA estimates the baseline–follow-up relationship. Baseline imbalance and measurement error therefore affect them differently.

Which method is commonly preferred for a randomised trial with a continuous baseline and follow-up outcome?

Answer and reasoning

Baseline-adjusted ANCOVA is commonly preferred because it often improves precision and handles chance baseline imbalance appropriately under the model.

Does p = 0.077 prove no intervention effect?

Answer and reasoning

No. The confidence interval includes moderate benefit and a small effect favouring usual care. The estimate remains imprecise.

Key take-home messages

References

1. Vickers AJ, Altman DG. Analysing controlled trials with baseline and follow up measurements. BMJ. 2001;323(7321):1123–1124. https://doi.org/10.1136/bmj.323.7321.1123
2. Senn S. Change from baseline and analysis of covariance revisited. Stat Med. 2006;25(24):4334–4344. https://doi.org/10.1002/sim.2682
3. Bland JM, Altman DG. Comparisons against baseline within randomised groups are often used and can be highly misleading. Trials. 2011;12:264. https://doi.org/10.1186/1745-6215-12-264
4. Barnett AG, van der Pols JC, Dobson AJ. Regression to the mean: what it is and how to deal with it. Int J Epidemiol. 2005;34(1):215–220. https://doi.org/10.1093/ije/dyh299
5. R Core Team. Student’s t-test. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/t.test.html
6. R Core Team. Fitting linear models. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/lm.html