# Particle movement

## Functional ANOVA

Factors:

- heavy/light
- initial location: 2
- speed: 30/50/60

5 replicates under each treatment

Records:

- distance of particle to center of the machine over time
- speed of particle over time

Some missing data is observed.

Question: Is there a difference in the patterns of distance or speed that depend on the factors?

## Consulting response

Generally, the best strategy is to summarize the functional response (distance or time) with some summary statistic, e.g. area under the curve, average distance from center, etc. These summaries should generally be motivated by the scientific questions. Then use these summaries in an ANOVA.

The functional response may be able to be decomposed using an (seasonal, trend, and irregular with loess) STL approach. Obtaining summary statistics from these decompositions, e.g. trend slope, seasonal frequency/amplitude, or error variance, would allow an ANOVA approach as above. But obtaining this decomposition may be difficult due to the missing data.

More sophisticated functional analysis approaches will likely be too difficult.

The missing data is a real concern. It seems unlikely the missingness is ignorable and therefore a reasonable analysis should account for the mechanism that caused the missingness. It is not as simple as imputing some values.

For more information on dealing with missingness in an STL decomposition, see here.

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