Repeated measures, longitudinal d ata , and times series are all terms with a long history in statistics referring to measurements taken over time within a sampling unit. Add t o these the term functional data, used by Ramsay and Dalzell (1991) t o describe da t a consisting of samples of curves. Each curve or functional datum is defined by a set of discrete observations of a smooth underlying function, where smooth usually means being differentiable a specified number of times. A comparison of Analysis of Longitudinal Data by Diggle et al. (1994) with the more recent Functional Data Analysis by Ramsay and Silverman (1997) highlights the strong emphasis t hat the latter places on using derivative information in various ways. The more classic repeated measures da t a are typically very short time series as compared with longitudinal or functional data, and time series data are usually much longer. Most time series analysis is directed at the analysis of a single curve and is based on the further assumptions t h at the times at which the process is observed are equally spaced and th at the covariance structure for the series has the property of stationarity, or time invariance.
The Missing Data or Attrition Problem
Longitudinal or functional da t a are expensive and difficult t o collect if the time scale is months or years and the sampling units are people. Attrition due t o moving away, noncompliance, illness, death, and other events can quickly reduce a large initial sample down t o comparatively modest pro- portions. Much more information is often available about early portions of the process, and the amount of d at a defining correlations between early and late times can be much reduced.
Although there are techniques for plugging holes in d a t a by estimating missing values, a few of which are mentioned by CH and R , these methods
implicitly assume that the causes of missing data are not related t o the size or location of the missing data. This is clearly not applicable t o measure- ments missing by attrition in longitudinal data. Indeed, the probability of a datum being missing can be often related t o the features of the curve;
one can imagine that a subject with a rapidly increasing antisocial score in CH’s data over early measurements is more likely t o have missing later measurements. I am proposing, therefore, t o use the term missing data t o refer t o data missing for unrelated reasons and the term attrition t o indicate data missing because of time- or curve-related factors.
The data analyst is therefore faced with two choices, both problem- atical t o some extent. If only complete data records are used, then the sample size may be too limited t o support sophisticated analyses such as multilevel analysis. Moreover, subjects with complete data are, like people who respond t o questionnaires, often not typical of the larger population being sampled. Thus, an analysis of complete data, or a balanced design, must reconcile itself, like studies of university undergraduates, with being only suggestive by a sort of extrapolation of what holds in a more diverse sampling context.
Alternatively, one can use methods that use all of the available data.
The computational overhead can be much greater for unbalanced designs, and in fact the easily accessible software packages tend t o work only with balanced data. In any case, as already noted, results for later times will tend to both have larger sampling variance, due t o lack of data, and be biased if attrition processes are related t o curve characteristics. Thus, although using
”missing-at-random” hole-filling algorithms, such as developed by Jennrich and Schluchter (1986) and Little and Rubin (1987), may make the design balanced in terms of the requirements of software tools in packages such as SAS and BMDP, the results of these analyses may have substantial biases, especially regarding later portions of the curves, and this practice is risky.
I must say that my own tendency would be, lacking appropriate full data analysis facilities, t o opt to live with the bias problems of the complete data subsample rather than those that attrition and data substitution are all too likely t o bring.
Regist rat ion Problem
Figure 4.1 displays curves showing the acceleration in growth for ten boys, and we see there that there are two types of variation happening. Phase variation occurs when a curve feature, such as the pubertal growth spurt, occurs a t different times for different individuals, whereas the more familiar amplitude variation occurs when two units show differences in the charac- teristics of a feature even though their timings may be similar. We can see that averaging these curves results in a mean curve that does not resemble any of the actual curves; this is because the presence of substantial phase variation causes the average t o be smoother than actual curves.
In effect, phase variation arises because there are two systems of time
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Figure 4.1: Acceleration curves for the growth of 10 boys showing a mixture of phase and amplitude variation. The heavy dashed line is the mean curve, and it is a poor summary of curve shapes.
involved here. Aside from clock time, there is, for each child or subject, a biological, maturational or system time. In clock time, different children hit puberty at different times, but, in system time, puberty is an event that defines a distinct point in every child's growth process, and two children entering puberty can reasonably be considered at equivalent or identical system times, even though their clock ages may be rather different.
From this perspective, we can imagine a possibly nonlinear transforma- tion of clock time t for subject i, which we can denote as h i ( t ) , such t h a t , if two children arrive at puberty at ages tl and t2, then hl(t1) = hz(t2).
These transformations must, of course, be strictly increasing. They can be referred t o as time warping functions. If we can estimate for each growth curve a suitable warping function h such t ha t salient curve features or event are coincident with respect t o warped time, then we say that the curves have been registered. In effect, phase variation has been removed from the registered curves, and what remains is purely amplitude variation. The first panel of Figure 4.2 displays the warping functions hi(t) that register the 10 acceleration curves in Figure 4.1. The second panel of Figure 4.2 shows the registered acceleration curves in Figure 4.1.
The multileveling modeling methods considered by these three authors, and in most applications, concern themselves solely with amplitude varia- tion. However, it seems clear that children evolve in almost any measurable respect at different rates and at rates that vary from time t o time within individuals. The potential for registration methods t o clarify and enhance multilevel analyses seems important. Recent references on curve registra- tion are Ramsay and Li (1998) and Ramsay and Dalzell (1995).
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Figure 4.2: The left panel contains the warping functions for registering the growth acceleration curves in Figure 5.1, and the right panel contains the registered curves.
Serial Correlation Problem
Successive longitudinal observations usually display some amount of cor- relation, even after model effects are removed. Typically, this correlation between successive residuals decays fairly rapidly as the time values become more widely separated. This serial correlation is obvious in CH’s Table 3.1, for example, and we know from studies of human growth that heights measured 6 months apart have correlations between residuals of about mi- nus 0.4. This may be due t o ”catch-up” processes that ensure t h a t slow growth is followed by more intense growth episodes. Negative serial cor- relation causes actual observations t o tend to oscillate rapidly around a smooth curve that captures the longer-scale effects, whereas positive serial correlation results in slow smooth oscillations. Modeling these serial effects requires introducing some structure in the covariances among successive errors eij. Diggle et al. (1994) have an excellent discussion of these effects, as well as modeling strategies, and all three papers mention the problem.
We return t o this issue in the modeling sections.
Resolution of Longitudinal Data
T h e amount of information about a curve that is available in a set of ni
measurements is not well captured by ni itself. Let us consider other ways of thinking about this.
An event in a curve is a feature or characteristic that is defined by one or more measurable characteristics.
1. Levels are heights of curves and are defined by a minimum of one observation. The antisocial score of a child at time 1 in CH’s data is a level.
Longitudinal and Functional Data 93 2. Slopes are rates of change and require at least two observations t o
define. Both CH and R are concerned with slope estimation.
3. B u m p s are features having (a) the amplitude of a maximum or mini- mum, (b) the width of the bump, and (c) its location; they therefore are three-parameter features.
We may define the resolution of a curve as the smallest size on the time scale of the features t h at we wish t o consider.
Discrete observations are usually subject t o a certain amount of noise or error variation. Although three error-free observations are sufficient, if spaced appropriately, t o define a bump, even the small error level present in height measurements, which have a signal-to-noise ratio of about 150, means t h a t five observations are required t o accurately assess a bump, corresponding t o twice- yearly measurements of height. More noise than this would require seven measurements, and the error levels typical in many social science variables will imply the need for even more. This is why R says th at slope inferences cannot be made with only two observations. We may, therefore, define the resolution of the data t o be the resolution of the curve t h at is well-identified by the data. The five observations per individual considered by CH are barely sufficient t o define quadratic trend, tha t is, a bump, and confining inferences to slopes for these d a t a is probably safer, especially given their high attrition level.
However, the d a t a resolution is not always so low, and many d a t a col- lections methods under development in psychology and other disciplines permit the measurement of subjective states over time scales of hours and days rather than weeks and months. Brown and Moskowitz (1998) offer an interesting example of higher-resolution emotional state data. Equipment for monitoring physio!ogical or physical d at a are now readily available with time scales of seconds or milliseconds and with impressive signal-to-noise ratios. The optical motion tracking equipment t h at is used in studies such as Ramsay, Heckman, and Silverman (1997) has sampling rates of up t o 1,200 Hz and records positions accurate t o within a half a millimeter.