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We present a new statistical modelling approach where the response is afunction of high frequency count data. Our application is about investigatingthe relationship between the health outcome fat mass and physical activity (PA)measured by accelerometer. The accelerometer quantifies the intensity ofphysical activity as counts per epoch over a given period of time. We use datafrom the Avon longitudinal study of parents and children (ALSPAC) whereaccelerometer data is available as a time series of accelerometer counts perminute over seven days for a subset of children. In order to compareaccelerometer profiles between individuals and to reduce the high dimension afunctional summary of the profiles is used. We use the histogram as afunctional summary due to its simplicity, suitability and ease ofinterpretation. Our model is an extension of generalised regression of scalarson functions or signal regression. It allows also multi-dimensional functionalpredictors and additive non-linear predictors for metric covariates. Theadditive multidimensional functional predictors allow investigating specificquestions about whether the effect of PA varies over its intensity, by gender,by time of day or by day of the week. The key feature of the model is that itutilises the full profile of measured PA without requiring cut-points definingintensity levels for light, moderate and vigorous activity. We show that the(not necessarily causal) effect of PA is not linear and not constant over theactivity intensity. Also, there is little evidence to suggest that the effectof PA intensity varies by gender or whether it happens on weekdays or onweekends.

Modelling a response as a function of high frequency count data: the association between physical activity and fat mass