Normalising data for a learning experiment on alpine skiing

Question

We are currently performing a large learning experiment on alpine skiing. In this experiment, we test skiers on three different slalom courses. We compute performance by calculating the average of the two best runs for each of the three courses and divide it by the time they use to ski the hill straight down ('straight gliding'). I have added this figure for illustration:

Doing so, we see that good skier have a negative ratio score, meaning that they are faster when they ski a slalom course with turns. That’s cool! After this, we randomly stratify the skiers into two groups. One group of skiers will ski all the three courses on a single day, in random order. The other group will only ski one course per day. Three days after this training block, we test the skiers again in the same three courses to compare their learning. To this end, we are going to fit an ANCOVA model. My concern, however, is that we can (in theory) observe negative learning from pre- to post-test because of different snow conditions. I don’t think we ever will but I just started to think if it would be better to normalize the data to better able to compare performance across different days. Does this sound reasonable? If so, what can I do?

Answer 1

Let me try, by first writing down a linear model for the experiment. That should also help in clearing up misunderstandings! so let $$ y_{ickd}= \mu + \beta_{cd} + \epsilon_{ickd} $$ (as a preliminary model) where $i$ is individual, $c$ is course, $k$ is replicate run and $d$ is day. We really want to compare the ratio $$ \theta_{cd}=\frac{\mu + \beta_{cd}}{\mu + \beta_{c_0d}} $$ where $c_0$ is the straight glide course. Ideally, this should not depend on $d$, but for a preliminary analysis keep it. You propose as a normalization to replace this with $$ \DeclareMathOperator{\E}{\mathbb{E}} \frac{y_{ickd}}{y_{ic_0kd}}=\frac{\mu + \beta_{cd} + \epsilon_{ickd}}{\mu + \beta_{c_0d} + \epsilon_{ic_0kd}} $$ Some problems with this is that it is noisy, as we divide by the random error term, and we are doing it at the level of the data itself, which obviously is more noisy that doing it at the level of estimated parameters!

So let us try to rewrite the model using the interest, or focus, parameter (see https://www.mn.uio.no/math/forskning/prosjekter/focustat/) $\theta_{cd}$ defined above. Then $$ y_{ic_0kd}= \mu + \beta_{c_0d} + \epsilon_{ic_0kd} $$ for the straigh gliding course, and $$ y_{ickd}= \theta_{cd} (\mu + \beta_{c_0d}) + \epsilon_{ickd} $$ for the other. Problem is that this is no longer a linear model, we could solve that by estimating it with nonlinear least squares, but there might be a better option.

My intuition (which would have to be checked with some exploratory data analysis) is that for time data, an additive error model is not the best, maybe a multiplicative error is more natural. That could lead to analyze the logarithm of the response time, or maybe better, using a generalized linear model with log link function. The family function is less important, it could be gamma, or even normal ... But now concentrating on the expectation model, $$ \E y_{ic_0kd} = e^{\mu+\beta_{c_0 d} } $$ and $$ \E y_{ickd} = e^{\theta^*_{cd} \mu+\beta_{c_0 d}} $$ (note that definitions of parameters have changed). Now the comparison is concentrated to only one parameter, making for easier interpretation and more effctive inference. This model must now be extended to incorporate the different training methods ... and other variations can be contemplated, too.