galli et al. , 2010), while others use extremely flexible nonparametric f amilies (Telesca and Inoue, 2008; Ramsay and Li, 1998). We will follow an intermediate approach, using the semiparametric family of interpolating monotone Hermite splines (Fritsch and Carlson, 1980), although the proposed method can be implemented with any other warping family. Monotone interpolating Hermite splines are defined as follo ws (more details are given in Web Appendix C). For a subject i , let τ i ∈ R r be a vector of “landmarks” in I = [ a, b ] , with a < τ$_{i}$ 1 < · · · < τ$_{ir}$ < b ; the τ s can be, for example, the locations of the peaks and valleys of the observed curve. Let τ$_{0}$ ∈ R r be a knot vector, usually taken as the mean of the τ $_{i}$s. For given values s$_{i}$ 0 , . . . , s$_{i,r}$ $_{+1}$, there exists a unique piecewise cubic function w$_{i}$ ( t ) such that w$_{i}$ ( a ) = a , w$_{i}$ ( b ) = b , w$_{i}$ ( τ 0 $_{j}$) = τ$_{ij}$ for all j , w ′ $_{i}$( a ) = s$_{i}$ $_{0}$, w ′ $_{i}$( b ) = s$_{i,r}$ $_{+1}$, and w ′ $_{i}$( τ 0 $_{j}$) = s$_{ij}$ for all j . This function w$_{i}$ ( t ) aligns the individual features τ i with the average features τ$_{0}$ in a smooth way, so it is useful for “landmark registration” ( Bookstein, 1997). For w$_{i}$ ( t ) to be strictly monotone increasing the derivatives s$_{ij}$ s must satisfy certain conditions, given in Fritsch and Carlson (1980). But for curve-alignment purposes only the τ $_{i}$s are specified; in that case Fritsch and Carlson (1980) provi de an algorithm that produces a vector of derivatives s$_{i}$ that satisfy the sufficient conditions for w$_{i}$ ( t ) to be monotone increasing. Since the algorithm is deterministic, s$_{i}$ is a function of τ i and τ$_{0}$ , therefore w$_{i}$ ( t ) is entirely parameterized by τ i and τ$_{0}$ . In this paper, instead of specifying τ i for each curve and taking τ$_{0}$ = ¯ τ , we will specify τ$_{0}$ and treat the τ $_{i}$s as unobserved random effects. Our family of warping functions W τ $_{0}$, then, is an r -dimensional space ( r will usually be small). In general, it is not problematic to specify a reasonable τ$_{0}$ for a given data set; for example, for the curves in Figure 1(a) a single knot at τ 0 = 15 will provide reasonable warping flexibility, and the rest of the v ariation will be considered amplitude variability. For other warping families, such as monotone B-splines (Telesca and Inoue, 2008) or smooth monotone transformations (Ramsay and Li, 1998), the number and placement of the knots are harder to specify because they are not directly associated 6