Motivation

  • Growth curve analysis = Change score analysis = Difference analysis
  • Analyze change over time
  • Baseline measurement to delineate group differences and treatment effects

Notation

Group Pre Post
trt \(X_{trt}\) \(Y_{trt}\)
ctrl \(X_{ctrl}\) \(Y_{ctrl}\)

\(\bar{Y_{trt}} = mean(Y_{trt})\)

Possible Procedures

Post Measurement Difference

\(Y_{diff} = Y_{trt} - Y_{ctrl}\)

Group Pre Post
trt \(X_{trt}\) \(Y_{trt}\)
ctrl \(X_{ctrl}\) \(Y_{ctrl}\)
  • Advantages
    • Only one vector
    • Possible if baselines balanced (big But!)
    • Report difference in original units
  • Disadvantages
    • Bias: ignores baseline measurements
    • Reduced power

Pre-Post Difference

\(T = Y_{trt} - X_{trt}\)

and

\(C = Y_{ctrl} - X_{ctrl}\)

Group Pre Post
trt \(X_{trt}\) \(Y_{trt}\)
ctrl \(X_{ctrl}\) \(Y_{ctrl}\)
  • Advantages
    • One vector per group
    • Possible if baselines are balanced (but!)
    • (Welch’s) t-test
    • Report difference in original units
  • Disadvantages
    • Bias: ignores correlation between \(X\) and \(Y\)
    • No adjustment for imbalance due to regression to the mean
    • Loss of statistical power

Percentage Change

\(T = (Y_{trt} - X_{trt}) / X_{trt}\)

and

\(C = (Y_{ctrl} - X_{ctrl}) / X_{ctrl}\)

Group Pre Post
trt \(X_{trt}\) \(Y_{trt}\)
ctrl \(X_{ctrl}\) \(Y_{ctrl}\)
  • Advantages
    • One vector per group
    • Possible if baselines are balanced (but!)
    • (Welch’s) t-test
    • Report difference in % change
  • Disadvantages
    • Bias: ignores correlation between \(X\) and \(Y\)
    • Power loss

Difference in Means for Change

\[\begin{align} DiM &= (\bar{Y_{trt}} - \bar{X_{trt}}) \\ &- (\bar{Y}_{ctrl} - \bar{X}_{ctrl}) \end{align}\]

Group Pre Post
trt \(X_{trt}\) \(Y_{trt}\)
ctrl \(X_{ctrl}\) \(Y_{ctrl}\)
  • Advantages
    • One vector
    • Report difference in original units
    • Adjusts for initial imbalance
  • Disadvantages
    • Bias: ignores correlation between \(X\) and \(Y\)
    • Power loss

ANCOVA

Analysis of covariance = regression:

\(Y = \beta_0 + \beta_1 \times X + \beta_2 \times Group\),
where \(Group\) is a vector of length \(N\), with a 0 for each member of the ctrl group, 1 otherwise1

  • Advantages
    • A model
    • Predictions are possible
    • Report difference in original units or standard deviations
    • Best stat. power

Addendum

Why Are Repeat Measures ‘Better’ than Two Groups?

  • Repeat measures within a group are, on average, higher correlated than measures between groups
  • Higher correlation between \(X\) and \(Y\) means less variation between \(X\) and \(Y\)
  • Smaller differences between \(X\) and \(Y\) can more easily be detected if variance is low
  • More power = fewer participants required

Why Is ANCOVA Better Suited than Any Other Analysis?

  • ANCOVA not only adjusts for initial ‘static’ imbalance (as does Difference in Means for Change)
  • ANCOVA also adjusts for initial imbalance due to regression towards the mean

Regression to the Mean

In repeat measures designs, extreme values (e.g., very low/high blood pressure) tend to become less extreme in the follow-up -> they regress to the mean

Take-away Message

Learn regression

Literature

Vickers and Altman (2001) and the literature therein

Vickers, A. J., and Altman, D. G. (2001). Analysing controlled trials with baseline and follow up measurements. BMJ 323, 1123–1124. doi: 10.1136/bmj.323.7321.1123