Change from baseline cannot use ANCOVA

Question

I have the following model:

final count ~ baseline count + factor

The factor has four levels and there is an interaction between baseline count and factor. This rules out an ANCOVA analysis.

I have thought about doing a percentage change analysis: (final count - baseline count)/baseline count ~ factor

Firstly, is this the right thing to do? If not, is there a better way to approach this problem?

Answer 1

Other questions have addressed modelling the change score, final count - baseline count, as the outcome. In that answer I show why using the change score is non-sensical if you want to include the baseline on the right hand side. Assessing interactions with treatments (or equivalently your factors here) is one situation in which you want to include the baseline on the right hand side. The same arguments apply to not using percent change in this situation.

Even if you did not want to assess interactions with your factors, I would not recommend modelling percent change. Percent change has some undesirable properties. Here are a few off-hand:

• it is asymetric, e.g. a change from 4 to 5 is 25%, but a change from 5 to 4 is -20%
• it is undefined if the baseline is zero
• the variance is not easily defined. You can rewrite the change score as $1 - \text{Post}/\text{Pre}$, so the variance is only defined by the ratio of the Pre and Post values.

For Poisson distributed count data, a simpler change metric is $Z = 2 \cdot (\sqrt{\text{Post}} - \sqrt{\text{Pre}})$. If Pre and Post are from the same Poisson distribution with a mean not too small $Z$ has a normal distribution. (Above 5 is the typical recommendation, but IMO it does not behave too badly with means as low as 2~3. The tails are fatter, but it is still close to symmetric.)