| dc.description.abstract | This thesis presents a contribution to the active research area of functional data analysis
(FDA) and is concerned with the analysis of data from complex experimental designs in
which the responses are curves. High resolution, closely correlated data sets are encountered
in many research fields, but current statistical methodologies often analyse simplistic
summary measures and therefore limit the completeness and accuracy of conclusions drawn.
Specifically the nature of the curves and experimental design are not taken into account.
Mathematically, such curves can be modelled either as sample paths of a stochastic process
or as random elements in a Hilbert space. Despite this more complex type of response, the
structure of experiments which yield functional data is often the same as in classical experimentation.
Thus, classical experimental design principles and results can be adapted to the
FDA setting.
More specifically, we are interested in the functional analysis of variance (ANOVA)
of experiments which use orthogonal designs. Most of the existing functional ANOVA approaches
consider only completely randomised designs. However, we are interested in more
complex experimental arrangements such as, for example, split-plot and row-column designs.
Similar to univariate responses, such complex designs imply that the response curves
for different observational units are correlated.
We use the design to derive a functional mixed-effects model and adapt the classical
projection approach in order to derive the functional ANOVA. As a main result, we derive
new functional F tests for hypotheses about treatment effects in the appropriate strata of the
design. The approximate null distribution of these tests is derived by applying the Karhunen-
Lo`eve expansion to the covariance functions in the relevant strata. These results extend
existing work on functional F tests for completely randomised designs.
The methodology developed in the thesis has wide applicability. In particular, we consider
novel applications of functional F tests to gait analysis. Results are presented for two
empirical studies. In the first study, gait data of patients with cerebral palsy were collected
during barefoot walking and walking with ankle-foot orthoses. The effects of ankle-foot
orthoses are assessed by functional F tests and compared with pointwise F tests and the
traditional univariate repeated-measurements ANOVA. The second study is a designed experiment
in which a split-plot design was used to collect gait data from healthy subjects. This
is commonly done in gait research in order to better understand, for example, the effects of
orthoses while avoiding confounded analysis from the high variability observed in abnormal
gait. Moreover, from a technical point of view the study may be regarded as a real-world
alternative to simulation studies. By using healthy individuals it is possible to collect data
which are in better agreement with the underlying model assumptions.
The penultimate chapter of the thesis presents a qualitative study with clinical experts
to investigate the utility of gait analysis for the management of cerebral palsy. We explore
potential pathways by which the statistical analyses in the thesis might influence patient
outcomes.
The thesis has six chapters. After describing motivation and introduction in Chapter
1, mathematical representations of functional data are presented in Chapter 2. Chapter 3
considers orthogonal designs in the context of functional data analysis. New functional F
tests for complex designs are derived in Chapter 4 and applied in two gait studies. Chapter
5 is devoted to a qualitative study. The thesis concludes with a discussion which details the
extent to which the research question has been addressed, the limitations of the work and the
degree to which it has been answered. | en_US |
