I would advise to use the ANCOVA. Anyway, it should lead to the same conclusions. Statisticians have criticized the use of change scores. See:

Senn, S. 2006. Change from baseline and analysis of covariance revisited. Statistics in Medicine 25: 4334-4344.

The benefits of an ANCOVA are usually said to be the following:

- If one has an observational study in which the groups have differences in baseline values, ANCOVA can remove a potential bias from the results.

- if one has randomised trials where, by virtue of randomization, any real group differences in baseline values are unlikely, then ANCOVA reduces the amount of unexplained variation in the data. This reduces the error variance and makes the F-test of group difference on the final value of the outcome variable mores sensitive; the power of the F-test is increased.

But, a.o., Brown has criticized its use:

Brown, K.S. 2005. Analysis of covariance. In Encyclopaedia of Biostatistics, 2n Ed., P. Armitage and T. Colton. Chichester, England: Wiley.

Hi Jos Feys,

Thanks for sharing your opinion.

I am familiar with the article from Senn. But I also read Dimitrov et al 2003 'Pretest-posttest designs and measurement of change'. They don't want to discard change scores right away. I quote:

"The gain scores, D = Y2 − Y1, represent the dependent

variable in ANOVA comparisons of two or

more groups. The use of gain scores in measurement

of change has been criticized because of the (generally

false) assertion that the difference between scores is

much less reliable than the scores themselves [5,14,15].

This assertion is true only if the pretest scores and the

posttest scores have equal (or proportional) variances

and equal reliability. When this is not the case, which

may happen in many testing situations, the reliability of

the gain scores is high [18,19,23]. The unreliability of

the gain score does not preclude valid testing of the null

hypothesis of zero mean gain score in a population of

examinees. If the gain score is unreliable, however, it

is not appropriate to correlate the gain score with other

variables in a population of examinees [17]. An important

practical implication is that, without ignoring the

caution urged by previous authors, researchers should

not always discard gain scores and should be aware of

situations when gain scores are useful."

Hi,

I would go for a multi-level linear model, like the following (in R):

lmer(strength ~ 1 + age * time + (1 + time|Part))

In words, this is a multilevel model with

• age and time as population-level predictors, including an interaction effect
• Intercept and time as participant-level predictors. (You cannot make age a participant-level predictor, because it does not vary on that level)

These are the advantages:

1. You are working with age as a metric variable and not loosing information due to the artificial grouping.

2. You get participant-level trajectories, which you can plot to verify your assumptions

3. You will get correlations between participant-level Intercepts (which is each participants starting strength) and time. That is important, because participants who are already quite trained will gain less strength.

You can even simplify things further by estimating a multiplicative model, where your (exponentiated) estimates become rates of improvement. You can doe so easily by setting a logarithmic link function (and perhaps a better suited distribution, such as teh Gamma distribution)

lmer(strength ~ 1 + age * time + (1 + time|Part), family = Gamma(link = log))

For an introduction to multilevel and generalized linear models see my online book: https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher

Martin Schmettow

Thank you for your suggestion. But how can you compare the two age groups with your suggested model? By looking at the interaction effect time*age? But still, what will this say...?

You might like to have a look at the longitudinal chapter in this, there is a discussion of mixed models versus GEE; they are not just different estimation procedures but different models. and it explains the conditions under-which you will get different results.

Book Developing multilevel models for analysing contextuality, he...

I also gave quite a long answer to this question a while back

https://www.researchgate.net/post/Where_can_I_find_good_material_on_the_difference_between_mixed_models_and_gee_models

What I suggest is that you don't use age as a group, but as a metric variable (years). Then you have a model with time as a factor and age as metric. The following chapters explain what you need: 5.4.3, 5.5, 6.1, 6.8.

Hi Martin Schmettow

I tried your model and it is indeed a solution to handle the small group of 5. Now, I have one big group of 35 and since age differentiates a bit in the older group as well (58 y until 77 y) it is more informative about the true effect of age. But I guess you can only do this when you assume that there is a linear response between age and strength gain (which is not the case: I assume strength increases only go down from a certain age and not gradually from 25y).

Best,

Sara