the mass magnitude at the peak in Figure 1(a)). It is important to point out that for a given data set there is often some ambiguity about what constitutes amplitude variability and what constitutes phase variability (this will be discussed in more depth in Section 2). The problem of decomposing functional variability into amplitude and phase variability has been addressed by many authors (Kneip and Engel, 1995; Ramsay and Li, 1998; Wang and Gasser, 1999; Kneip et al., 2000; Gervini and Gasser, 2004, 2005; Kneip and Ramsay, 2008; Tang and M¨uller, 2008; Telesca and Inoue, 2008; Bigotand Gadat, 2010; Claeskens et al., 2010). All of these papers, however, have focused on independent and identically distributed samples of curves, but for the type of applications we have in mind the curves are not independent. For example, the growth curves in Figure 1(a) are correlated for indi- viduals with the same father. This type of design is common in evolutionary biology and quantitative genetics for the following reason. The variability observed in physical traits has two sources: genetic and environmental. Because environmental factors generally are not passed from one generation to the next, the evolution of phenotypes is driven largely by genetic variability (but see Skinner et al., 2010 and Manikkam et al., 2012 for a dis- cussion of epigenetic effects). Examining samples of genetically related individuals, like siblings or half-siblings, makes the genetic and environmental sources of variability math- ematically identifiable and therefore estimable, allowingbiologists to predict the evolution of traits in response to selection (Gomulkiewicz and Beder, 1996; Kingsolver at al., 2002). Therefore, it is important to possess statistical tools for the study of amplitude and phase variability of non-independent functional data. Some existing functional-data meth- ods handle non-independent or non-identically distributed curves, such as mixed-effects ANOVA models (Guo, 2002; Morris and Carroll, 2006; Di et al., 2009; Chen and Wang, 2011), but they do not specifically address phase variabilit y. To date, the problem of am- plitude/phase variability of functional traits has been addressed mostly in an ad-hoc way, by first aligning the curves with respect to some trait, and th en studying amplitude vari- 2