Consider an RCT with individuals "i" in 2 arms ("group", with 0 = control and 1 = treatment) in which one metric outcome ("score") is collected at baseline ("pre") and after some treatment ("post").
In an ANCOVA model, the two explanatory variables "pre" and "group" are mapped to the dependent variable "post" as follows:
post_i ≈ a + b * pre_i + c * group_i
Now, Van Breukelen (reference below) claims that ANCOVA is equivalent to testing the group by time interaction c' in the model
score_i,t ≈ a' + b' * time_i,t + c' * group_i * time_i,t
with time = 0 denoting the pretest and time = 1 the posttest time point.
I tried to verify Van Breukelen's statement on a toy dataset in R:
```R
# toy dataset
df_wide |t|)
#(Intercept) 2.7090 1.5362 1.763 0.176
#pre 0.1700 0.6354 0.268 0.806
#group 0.8030 0.8282 0.970 0.404
# reshaping data, ensuring correct coding of the time variable
df_long
For randomized experiments with pretest (baseline) and posttest (follow-up) measurements, usually ANCOVA should be preferred with pretest score as covariate. The dependent variable can be either the posttest score or the difference between the pretest and the posttest score (as long as the pretest covariate is in the model, significance testing of the group factor will lead to the same results in both cases). The reason for preferring ANCOVA over repeated-measures ANOVA is that the latter has lower power and some assumptions (most prominently homogeneity of variances) that are unlikely to be met in several cases. I am unsure about what you mean by repeated-measures ANOVA with pretest score as covariate, I do not think that it is feasible.
Some references you may find helpful:
Vickers AJ, Altman DG. Statistics notes: Analysing controlled trials with baseline and follow up measurements. BMJ 2001; 323 (7321): 1123-1124.
http://www.ncbi.nlm.nih.gov/pubmed/11701584
European Agency for the Evaluation of Medicinal Products (EMEA). Points to Consider on Adjustment for Baseline Covariates. London: EMEA, 2003.
http://www.emea.europa.eu/docs/en_GB/document_library/Scientific_guideline/2009/09/WC500003639.pdf
Cristian, i appreciate your effort to contribute but you did not address my question. I'm aware of the power issue, and I was not in doubt whether to prefer "ANCOVA over repeated-measures ANOVA". (Btw. I never mentioned any "repeated-measures ANOVA with pretest score as covariate" and I'm not sure why you came up with that...) My question was why the two cited models don't yield the same coefficient estimates as they should if Van Breukelen were right...
Maybe the title of my question was misleading; i changed it...
Hello Thomas Grischott. I cannot access the Van Breukelen article from home, but I cannot imagine he stated that those two models are equivalent. In the context of the two-group pre-post design, the usual methods people use are:
The Group*Time interaction from the Between-Within ANOVA is equivalent to the t-test (or ANOVA) on the change scores. See the tables in this conference paper, for example:
- http://support.sas.com/resources/papers/proceedings14/1798-2014.pdf
Also, ANCOVA as above but with Y = Post-Pre (rather than Post) gives the same result for the Group effect. (That is also shown in the conference paper.) Perhaps Van Breukelen was talking about one of those equivalencies?
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