Longitudinal parameters
Comparison of samples of curves
In biomedical research, a prevalent challenge is effectively comparing and describing sample curves Given N subjects, with nj measurements taken for the j-th subject, we can model this scenario to analyze the data accurately.
In the study of random processes represented by functions \( g_j \), estimating a "longitudinal average" curve is crucial, as it differs from a cross-sectional ordinary average curve that may distort typical time courses, particularly by broadening peaks (Muller and Ihm, 1985) Shape-invariant modeling offers an alternative approach, where minimal assumptions about "invariant shapes" allow for different scaling of curves This method iteratively refines nonparametric shapes and scaling parameters by pooling residuals at corresponding times across the sample of curves, using spline functions for model improvement (Lawton, Sylvestre, and Maggio, 1972; Stutzle et al., 1980; Kneip and Gasser, 1986) Additionally, principal components techniques have been explored for stochastic processes, further enhancing the analysis of curve samples (Castro, Lawton, and Sylvestre, 1986).
Parametric regression models applied to the data yield a vector of regression coefficients for each subject, allowing for further analysis of their distribution This enables the estimation of means to create a characteristic curve for the sample, and facilitates cluster analysis to identify subgroups or discriminant analysis to distinguish features between two groups In contrast, nonparametric models do not provide direct parameters, but longitudinal parameters can often be defined, serving as key characteristics in various applications.
In analyzing curves and their derivatives, it is essential for individual curves to exhibit consistent characteristics, often defined by the location and size of peaks and zeros These longitudinal parameters hold significant meaning in various fields; for instance, in studying the human growth curve, the timing and magnitude of peak velocity during puberty are biologically relevant Chapter 9 will explore longitudinal parameters and how they differentiate boys and girls within growth curves Additionally, in the context of heart pacemakers, it is crucial to estimate a-points, which are the times when the pacemaker's relative frequency output drops below a specific threshold, such as the 0.95 limit indicating a 5% decline in frequency This estimation is akin to determining the location of zeros in the data.
The following analysis will be carried out conditional on gj' 1 e we assume that just one curve is given and that it is non-random setting of (8.1), for specific models, e.g
In the context of smooth zero expectation stochastic processes, more effective estimators for identifying the location and size of peaks in G can be derived compared to simply averaging the proposed estimates of the corresponding coordinates However, our focus here is on estimating the coordinates of each individual peak independently of sample information Further exploration of models of this type is crucial for accurately estimating longitudinal parameters in curve samples.
Estimating the location and size of a peak or an a-point, such as the dose where the probability of an effect reaches 0.5 in dose-response curves, is of significant interest in statistical analysis This estimation corresponds to determining the ED50 or EDa Nonparametric sequential procedures, including stochastic versions of Newton methods, are employed for this purpose Notable examples are the Robbins-Monro procedure for locating an a-point and the Kiefer-Wolfowitz procedure for identifying a peak, both of which contribute to effective statistical modeling and analysis.
In practical applications, traditional measurement procedures can be overly time-consuming, as each result must be obtained before proceeding to the next measurement To address this issue, Staniswalis and Cooper explore kernel methods for analyzing dose-response curves and estimating effective doses (EDa).
Sequential methods are not appropriate for longitudinal data as they rely on the previous outcomes of independent variables for new measurements Instead, Muller (1985a) proposed an alternative approach that involves estimating the coordinates of peaks and a-points directly from the kernel-estimated curve This raises the question of whether it is beneficial to determine the location of a peak by finding the zero of its derivative Additionally, the estimation of peaks in the derivatives of longitudinal data has been explored by Jørgensen et al (1985) and Silverman (1985).
Definition of longitudinal parameters and consistency 124
In the context of a fixed design regression model characterized by triangular Ll.d errors, we consider the function g(v), which, for a specific value of v approaching zero, exhibits a unique maximum at a point 9 within the interval [0, 1] We establish the estimator for identifying the location of this maximum as max g(v)(x), where x is constrained to the interval [0, 1].
9 - inf[te[O,l]: g(v)(t) - max g(v)(xằ) xe[O,l] where g(v)(.) is the kernel estimate (4.4) for g(v)(.) The estimate for the size of the maximum is g(v)(9) Analogous definitions hold for minima
Assuming that g(v) has a unique zero in r, we define the estimator for the location of the zero by r - inf[te[O,l]:g(V)(t) - 0)
Measurability of these estimators can be established longitudinal parameters
Consistency follows from the uniform consistency of the curve estimates g(v) Assume that g e ~k([O,l]), k ~ v+2, and
(S.3) for a nonnegative null sequence Pn If g(v+2)(9) < 0, this implies
This can be seen as follows: For any interval [91,92 ] s.t ge[91,92 ] there exists £: > ° s.t g(v)(91 ) + £: < g(v)(9) - £:, g(v)(92 ) + £: < g(v)(9) - £: because of the strict concavity of g(v) in some interval around 9 Choosing n so large that suplg(V)_g(V)I < £: a.s then implies 9 e [9 1,92 ] a.s
The convergence rate discussed in (S.3) allows for a more precise understanding of the convergence outlined in (S.4), particularly regarding peaks and zeros We present this finding along with a similar result for zeros, as referenced in Muller (1955a), without providing a detailed proof.
Lemma S.l Under (S.3), assume there are a,b,c and P s.t 0 0 and (S.14):
To derive the distribution of the size of a peak, we assume that we have an estimator 8 of the location satisfying A
In this study, we consider the scenario where 8 is easily derived from (S.3) We assume that conditions (S.10), (S.11), and (S.14) are applicable, and we define b as tn - 1/(2k + 1) for g while estimating the peak size using g(8) This analysis is further refined through a Taylor expansion.
A A A 1 A(2) A g(O) - g(O) - 2 g (0*)(0-0)2 for some mean value 0* between 0 and A D Since under the assumptions sup Ig(2)(x)_g(2)(x)1 a 0, xe[O,l] it follows that and we conclude by (8.16) that where
Therefore nk/(2k+1) (g(0)-g(9ằ - op(l), and it follows from (4.16):
Theorem 8.2 Under (8.10), (8.11) and (8.14), choosing b_sn- 1/(2k+3) for estimating 9, and choosing b - tn- 1/(2k+1) for estim~ting g(.),
The scaling of bandwidth for estimating the location of a peak differs from that used for estimating its size This distinction arises because locating a peak is asymptotically equivalent to estimating a zero in the derivative.
When utilizing the kernel K e Mo,k n ,,1 (:R) for estimating the function g, the optimal bandwidths take the form of cn-1/(2k+1) Conversely, if the kernel K(l) is employed for estimating g(l), where K(l) is part of M1,(k+1)', the optimal bandwidths are adjusted to the form of cn-1/(2k+3).
The joint asymptotic distribution of the location and size of a peak, characterized by a normal distribution with zero covariances, can be derived and utilized to create asymptotic confidence ellipsoids for peaks Assuming the kernel K is symmetric, the Cramer-Wold device can be applied to the expression AXn + p.Yn, where Xu is defined as nk/(2k+3)(9_0), allowing for the analysis of peak behavior in statistical modeling.
129 bandwidth b - sn- 1/(2k+3), and Yn - nk/(2k+1) (g(8) -g(fJằ, employing any consistent estimator 8 for the location and bandwidth b - tn- 1/(2k+1) for g, one finds with constants cx,cy : n fSi ( 1 (1) ( fJ - u
~Xn + pYn - i~l si-1 cx~nk/(2k+3) sn- 1/(2k+3) K sn- 1/(2k+3) ) k/(2k+1) 1 (fJ-u ))
The first term exhibits a standard limiting distribution, while the covariance terms are calculated using the expression f K (1) (sn-1/(2k+3) fJ-u) K (tn-1/(2k+1) fJ-u) du - 0 This is due to the symmetry of K and the anti-symmetry of K(l).
Theorem 8.3 Under the assumptions of Theorem 8.2, using a symmetric kernel function, i 1 ](820)
From these results it is obvious that increasing the order of the kernel
K reduces the asymptotic mean squared errors for longitudinal parameters,
'" for fJ as derived from (8.16), and
(8.22) for g(8) as derived from (8.19) A generalization of these results to local bandwidth choice in the spirit of 7.6, 7.7 is possible, applying weak convergence of multiparameter stochastic processes (Bickel and Wichura,
Minimum variance kernels for equations (8.21) and (8.22) can be developed, resulting in polynomials (5.22) for ~ - 0 and ~ -1, respectively The bias observed in (8.19) indicates that peaks are generally underestimated, while (8.16) reveals that the bias in peak location tends to favor the less steep side; for symmetric peaks, the asymptotic bias disappears According to (8.17), the bias in the location of a zero typically leans towards the less steep slope, with the asymptotic bias vanishing when the slope is constant near the zero A simulation comparison of various kernels regarding peak size and location estimation, conducted by Muller (1985a), highlights the significant advantages of higher-order kernels, showing potential improvements of approximately 50% in mean squared error (MSE) for peak size estimates Additionally, peak size estimates improve considerably when using kernels of order k - 4 or 6 compared to k - 2, and higher-order kernels also enhance location estimates, yielding MSE gains of 30-50%, though with minimal improvements in mean deviation.
For constructing confidence ellipsoids, it is essential to either estimate or neglect the bias, with neglecting being the more practical approach The covariance matrix must be estimated using the appropriate estimators, such as (7.1) or (7.2) for u, alongside a kernel estimator g(2)(8) for g(2)(9) Following the assumptions of Theorem 8.3, this method is asymptotically valid when A P approaches 0 Additionally, the bandwidth for g(2) should be selected using the factor method outlined in (7.17).
The choice between directly estimating the location of a peak or using a zero in the first derivative hinges on the available bandwidth selection methods When optimal local bandwidths are applied, both approaches yield identical limit distributions A simulation study reveals that utilizing IMSE-optimal bandwidths, as outlined in section 4.4, results in improved mean squared error (MSE) and average deviation for peak location estimates when employing the derivative zero method.
Nonparametric estimation of the human height growth curve 131
Introduction
This article explores the application of nonparametric regression methods to analyze the human height growth curve, utilizing data from the Zurich Longitudinal Growth Study conducted between 1955 and 1978 The findings of this nonparametric analysis are detailed in the work of Largo et al.
This chapter summarizes and discusses the findings from key studies by Gasser et al (1984, 1985), focusing on the estimation of derivatives in growth curves It also compares parametric and nonparametric models, examines smoothing splines versus kernel estimators, defines longitudinal parameters, and addresses the phenomenon of growth spurts.
In 1978, Falkner and Tanner edited a comprehensive three-volume monograph that provided an overview of human growth knowledge, highlighting that the midgrowth spurt (MS) was not yet recognized as a distinct phenomenon in contrast to the well-documented pubertal growth spurt (PS) Tanner (1981) offered a historical perspective on human growth modeling, while Goldstein (1986) discussed the differences between classical MANOVA growth curve models and parametric/nonparametric regression modeling, referencing the Zurich longitudinal growth study Nonparametric methods, such as cubic smoothing splines and kernel estimates, were utilized to address issues like the quantification of MS and PS, particularly in comparing boys and girls Key endocrinological questions remain, including the triggers of PS onset and the basis of MS, with Sizonenko (1978) suggesting that a better phenomenological understanding of these spurts could illuminate human growth regulation The PS is characterized by significant growth velocity and acceleration, peaking around 12 years for girls and 14 years for boys, driven by hormonal changes that enhance growth velocity while initiating ossification in the epiphyses, ultimately halting growth Numerous studies have investigated various aspects of the PS, including its onset, peak intensity, and duration, and there is ongoing interest in understanding its relationship with adult height.
The menarcheal syndrome (MS), first noted around the age of seven in the early 20th century, saw diminished discussion with the rise of parametric models in research due to advancements in computing technology Recent studies have revived interest in the MS, although its cause remains largely unknown, with some hypotheses suggesting an endocrinological basis, such as the DHEA hypothesis A quantification study conducted by Gasser et al in 1985 found no significant sex differences in the timing or intensity of the MS.
The Zurich longitudinal growth study, conducted at the University Children's Hospital in Zurich alongside auxology units in Brussels, London, Paris, and Stockholm, provides a comprehensive analysis of growth patterns For an in-depth understanding, refer to the works of Falkner (1960) and Largo et al (1978) This study employs nonparametric regression methods, which are particularly effective in addressing the uncertainty surrounding growth patterns in different subgroups, such as boys and girls.
Choice of kernels and bandwidths
The kernel estimator was utilized to estimate growth curves and their derivatives, highlighting the significance of selecting appropriate bandwidths for accurate curve estimation In growth curve applications, it is crucial to identify genuine peaks in the derivatives while avoiding oversmoothing that may obscure fluctuations in the curve To prevent the emergence of spurious peaks, it is advisable to carefully manage the impact of random noise suppression.
As a method of bandwidth choice for
The pilot method outlined in section 7.2 utilized a parametric pilot estimator, specifically the individually fitted Preece-Baines model II, as detailed in section 2.3 This approach was implemented to determine the optimal global bandwidth choice for the 133 growth curves.
The study examined whether to estimate a separate optimal bandwidth for each individual curve or to apply a uniform bandwidth across a sample of 90 children, comprising 45 females and 45 males While estimating individual bandwidths results in a lower overall Integrated Mean Squared Error (IMSE), it can also exaggerate differences between curves, potentially leading to misleading variations, especially between sexes, due to differing error variances To mitigate these artificial inter-individual variations in curve estimates, the researchers opted for a consistent bandwidth solution for the entire sample.
In this study, a consistent bandwidth of j-1 was selected for all 90 curves, with an error variance estimated by setting u² at 0.3, which was considered an upper limit of the true error variance This choice resulted in a slightly over-smoothed bandwidth To achieve variance stabilization, local bandwidths were adjusted so that n approximated Wi²(t, bt) minus a constant for all t, where Wi(t, bt) represents the i-th kernel weight with bandwidth bt at estimation point t For more comprehensive insights into this methodology, refer to Gasser et al (1984b) It is important to note that these bandwidths were not tailored to local variance or curvature as outlined in section 7.6.
A finite evaluation was conducted to compare the choice of different kernels and smoothing splines using the method outlined in section 4.4, employing the average Preece-Baines model II as the true curve, based on the average parameters from 90 individual fits (Gasser et al., 1984b) The results for an error variance of u² = 0.3 are presented in Table 9.1 Bias, variance, and mean squared error (MSE) were calculated by averaging the estimated MSE over 201 equidistant points within the interval [4, 18].
In the Zurich longitudinal growth study involving 34 participants, a finite evaluation was conducted on various kernels and smoothing splines, assuming the average Preece-Baines model II for boys represents the true growth curve With a parameter value of u² set to 0.3, the optimal smoothing parameters were determined, and the Integrated Mean Squared Error (IMSE) was assessed over the interval [4, 15] The analysis utilized kernels defined by equation (5.22) with a bandwidth of ~ - 1, as referenced in Gasser et al (1984b).
IVAR IMSE IBIAS2 IVAR IMSE 11.6 lS.9 29.7 20.9 50.6
The study utilized optimal kernels (5.22) with a degree of approximately -1, revealing that the smoothing spline outperforms kernels of order (v+2) but falls short compared to those of order (v+4) when variance stabilization is applied, as detailed in Table 4.2 for jittered designs Consequently, it is recommended to employ kernels of order (k - v+4) in conjunction with variance stabilization The findings indicated that the resulting bandwidths were maximal during the prepubertal stage and minimal during the pubertal stage.
Maximal bandwidth is observed in the prepubertal range with annual measurements, while minimal bandwidth occurs during puberty with semi-annual measurements The shift from maximal to minimal bandwidth is influenced by variance stabilization Notably, the kernel estimator acts as a weighted average across intervals [t-b, t+b], revealing unexpectedly large bandwidths.
135 spurious peaks is not likely On the other side, the height of peaks will be underestimated
The separate determination of bandwidths for girls and boys revealed smaller values for girls, likely due to less pronounced performance scores (PS) Utilizing these distinct bandwidths would have exacerbated the existing disparities between the groups Optimal bandwidths were established by calculating 10 values for Integrated Mean Squared Error (IMSE) near the anticipated minimum and then refining the minimum through spline interpolation (IMSL routines) However, caution is advised as spline overshooting can lead to misleading minima.
The heart pacemaker study highlighted in section 3.6 revealed a significant inhomogeneity between the two sets of curves, as illustrated in Figures 3.2 and 3.3 This discrepancy indicated the need for a distinct approach, leading to the determination of the average IMSE optimal bandwidth for each curve sample.
The analysis utilized the Rice criteria to estimate the performance of each sample, employing the positive optimal Epanechnikov kernel for calculations A significant difference was observed in the decline of pacemaker frequency, measured at 95% confidence levels, after implantation across two groups To compare the 100(1-a)% confidence intervals, a two-sample comparison with censored data was necessary, as not all pacemakers experienced the same degree of frequency loss during the observation period Censoring is crucial in estimating longitudinal parameters, particularly in studies where observation periods differ among individuals, necessitating careful consideration of these variations in the analysis.
Comparison of parametric and nonparametric regression 135
In the comparison of parametric and nonparametric approaches, the parametric method often exhibits global or local lack of fit, which can be significant depending on the application While this lack of fit is usually crucial for descriptive analysis, it may be acceptable when comparing groups or identifying subgroups through estimated parameter vectors However, the parametric approach is not ideal for exploratory data analysis, as it fails to reveal features not included in the model Additionally, assessing bias and variance through residual analysis is challenging with this method.
In the nonparametric approach, the estimator exhibits a slower rate of convergence, with bias and variance that can be predicted locally Specifically, there is a significant downward bias at peaks and a notable upward bias at troughs Additionally, the local variance can be estimated with ease.
02 l: W' (t) i-1 L where 0 is the variance estimator (7.1) or (7.2) and Wi(t) are the kernel weights by which the observations Yi are to be multiplied
Figures 9.1 to 9.3 present a detailed comparison between the kernel estimator (solid line) and the Preece-Baines model II (dashed line) regarding the first and second derivatives of the human growth curve Each figure also includes a smaller graph that depicts the estimated growth curve along with cross-sectional percentiles.
Fig 9.1 As Fig 2.1, for another boy al (1984b) Fig 9.1-9.3 from Gasser et
AGE IN YEARS Fig 9.2 As Fig 2.1, for a girl
RGE IN YERRS Fig 9.3 As Fig 2.1, for a boy
The parametric Preece-Baines (PB) model fails to account for the menarcheal stage (MS) and struggles to accurately model the initial four years, a period intentionally excluded by its authors Analysis reveals that the omission of the MS results in a poor fit during the rising phase of the pubertal stage (PS), with an estimated onset for girls occurring, on average, 0.76 years earlier than the kernel method However, the kernel method is also known to produce an early onset bias, indicating that the PB model's earlier estimate is primarily influenced by this bias Hauspie et al (1980) previously noted the PB model's tendency for premature onset estimations Additionally, the kernel method underestimates peak sizes, and since both PB and kernel methods align closely on peak sizes for the PS, this bias is largely reflected in the PB model as well Table 9.2 presents the average values for onset timing (T6) and height at T6 (HT6) for both boys and girls, comparing the PB and kernel methods.
Table 9.2 presents estimates of the onset time T6 for peak strength (PS), along with the corresponding heights (HT6) and velocities (VT6) for both boys and girls, based on a comparison between the kernel estimate and the Preece-Baines model II (PB) The analysis includes data from 45 boys and 45 girls, with Spearman correlation (r) denoted for the findings These results are derived from the study by Gasser et al (1984b).
To validate the hypothesis that bias in the PB estimate of onset arises from inadequate modeling of the MS, a stepwise linear regression was conducted using T6(Kerne1)-T6(PB) as the dependent variable and six independent longitudinal parameters estimated through the kernel method The analysis revealed that the variable with the highest R² value (R² = 0.35) was the timing of the end of the MS, supporting the initial hypothesis.
MS seems to be related to this lack of fit To find fuxther influences on
The poor performance of the PB model prompted a secondary stepwise linear regression analysis, using the estimated residual variance of the PB model as the dependent variable This analysis was conducted separately for boys and girls, revealing that for boys, only the intensity of the variable was significant.
MS (as characterized by the amplitude of the second derivative) remains in the regression equation; for girls, three variables, two characterizing the
MS and one the intensity of the PS, remain in the equation, so that the MS influences the overall fit of the PB model
Table 9.3 illustrates the correlation between the kernel and PB methods for T8, focusing on the timing of maximal velocity during the PS, marked by a zero in the second derivative For boys, the average velocity VT8 is 0.38 cm/yr lower in the kernel method compared to the PB method Stepwise regression analysis indicates that the intensity of the PS, measured by the difference between VT8(Kernel) and VT6(Kernel), is the most significant variable positively correlating with the difference in velocities (VT8(PB) - VT8(Kernel)) Overall, the PB model shows only a slight improvement over the kernel estimate concerning peak size bias, particularly in cases of high peaks, as detailed in Gasser et al (1984b).
Table 9.3 As Table 9.2, but for the time T8 where maximal pubertal growth velocity is attained From Gasser et al (1984b)
Sex Kernel PB Kernel PB Kernel PB x m 13.91 14.00 161.4 162.6 8.313 8.697 f 12.22 12.09 150.4 150.2 6.996 7.090 s m 9505 8861 6.629 6.337 8218 1.034 f 8066 7771 5.968 5.856 9519 1.039 r m 971 942 911 f 873 888 974
To assess bias in growth velocity during puberty for boys and girls, individual curves are aligned by shifting them horizontally to match the sample average timing of maximal velocity (T8) This alignment allows for a consistent comparison of pubertal peak velocities Heights at T8 (HT8) are then adjusted by adding a constant to each aligned curve Subsequently, the available data is analyzed using kernel estimation with a small bandwidth, separately for each gender The results indicate differences in peak velocity estimates when compared to traditional methods, with all values expressed in cm/yr, as reported by Gasser et al (1984b).
This indicates that both methods suffer from a large bias, which is expected and predictable for the kernel method, but a negative surprise for the PB method
The alignment of curves involves using a longitudinal parameter, denoted as 9(1), with its value for the j-th curve represented by 9 j (i) and the true sample average as "8(1) By substituting the unknown true longitudinal parameters with their nonparametric estimates, as outlined in section 8.2, the aligned curves can be averaged cross-sectionally to generate realistic "average" curves around the typical point 9(1) to which they are aligned.
In the context of estimating longitudinal parameters, such as zeros or extrema of curves and their derivatives, a coherent approach involves defining these parameters consistently across individual curves By establishing a sufficient number of these parameters, a plausible heuristic definition of a longitudinal average curve can be created This is achieved by forming convex combinations of the aligned curves, ensuring that the resulting average accurately reflects the characteristics of the data.
Estimation of growth velocity and acceleration
The growth velocity and acceleration of a boy without metabolic syndrome (MS) are illustrated in Fig 9.4, showing that he reaches an adult height of 183.4 cm and achieves peak velocity of the pubertal growth spurt (PS) at 15.2 years, which is 1.5 years later than the average for boys This case is rare, with only two out of 90 children not exhibiting signs of MS, supporting earlier theories of a consistent negative acceleration prior to the PS (Prader, 1978, 1982) Overall, the metabolic syndrome is more clearly observed and quantified in the second derivative of growth data.
It is not feasible to directly compare derivative estimates with raw data; however, the closest approach involves using first and second-order difference quotients as defined in section 7.4 A comparison of kernel and PB estimates with these difference quotients is illustrated in Figure 9.5, following the work of Gasser et al (1984a).
Fig 9.4 As Fig 2 1, for a boy without MS From Gasser et al (1984a)
In Figure 9.5, similar to Figure 2.1, the triangles illustrate first-order difference quotients above and second-order difference quotients below The triangles at the upper and lower boundaries indicate values that exceed the boundary coordinates Notably, the variance of second-order difference quotients significantly increases during the half-yearly measurement range, as reported by Gasser et al (1984a).
Evaluating the accuracy of various methods in relation to difference quotients is challenging due to significant variability, particularly in half-yearly measurements for individuals aged 9 to 18 years For instance, the variance observed in second-order difference quotients highlights this complexity.
A more accurate comparison can be made by examining the cross-sectional averages of PB, kernel fits, and difference quotients for 45 boys and girls, as indicated by Gasser et al (1984a) Refer to Fig 9.6 for boys and Fig 9.7 for girls for visual representation of these findings.
Figure 9.6 illustrates the cross-sectional averages of velocity and acceleration for n-45 girls, showcasing the averaged difference quotients (first order above and second order below) as solid lines Additionally, dashed lines represent the averaged kernel estimates, while dotted lines indicate the averaged fits of the Preece-Baines model These findings are derived from Gasser et al (1984a).
AGE IN TEARS Fig 9 7 As Fig 9.6, but for n-45 boys
The comparison of curve estimates to raw velocities and accelerations reveals that the kernel method consistently aligns more closely with difference quotients than the parametric PB model, particularly in the prepubertal stage and for ages 0 to 4 years This highlights the inadequacy of the parametric model and underscores the advantages of the nonparametric approach in this context Additionally, the sawtooth pattern observed in the difference quotients is attributed to their correlation structure, where neighboring first and second order difference quotients exhibit negative correlations, while second nearest neighbors show positive correlations.
Longitudinal parameters for growth curves
Largo et al (1975) systematically examined longitudinal parameters for growth curves for the first time, focusing on key points derived from the first or second derivative of individual growth curves with biological significance Gasser et al (1985b) identified specific longitudinal parameters at the zeros or extrema of the kernel estimate of the second derivative, denoting the corresponding timings as T1 to T9.
Tl age of four years (fixed, therefore not a longitudinal parameter, included for comparison purposes);
T2 age of maximal acceleration during MS;
The T3 age is defined as the point at which the average acceleration, calculated as (AT2 + AT4)/2, is reached during the maximum speed (MS) Meanwhile, T4 marks the age at which maximal deceleration occurs at the conclusion of the maximum speed phase.
TS age of last minimum in the acceleration curve before T6 (often coinciding with T4);
T6 age of onset of PS, zero acceleration at the beginning of the PS; T7 age of maximal acceleration during PS;
TS age of maximal velocity during PS, estimated as a zero of the acceleration;
T9 age of maximal deceleration at the end of the PS
T2 and T4 serve as the natural parameters marking the beginning and conclusion of the MS, while T6 and T9 indicate the start and end of the PS These longitudinal parameters are illustrated through a real example curve, as shown in Figure 9.S.
For each time point from Tl to T9, we estimate the corresponding heights (HT1 to HT9) using kernel estimation for the curve Additionally, we calculate the velocities (VT1 to VT9) and accelerations (AT1 to AT9) through kernel estimates of the respective derivatives, noting that AT6 and AT5 are defined as zero In addition to HT1 to HT9, we also determine relative heights (H%T1 to H%T9), expressed as percentages of the actual height compared to adult height, resulting in a total of 45 longitudinal parameters for each individual curve.
Figure 9.8 illustrates the longitudinal parameters derived from kernel estimates of velocity (above) and acceleration (below) for a girl The times T1-T9, defined in the text, are identified as zeros or extrema of the estimated acceleration curve, indicated by dashed vertical lines Notably, T4 coincides with T5, a common occurrence for girls but infrequent for boys The timings T6*, T8*, and T9* correspond to T6, T8, and T9, respectively, and are determined from the estimated velocity curve The data is sourced from Gasser et al (1985b), with measurements expressed in years for age, centimeters for height, centimeters per year for velocity, and centimeters per year squared for acceleration.
In addition to these 45 parameters, 15 derived parameters were computed which are functions of the original longitudinal parameters of special biological interest These derived parameters were:
Measure of the skewness of the pubertal growth spurt The skewer the peak is (AT9 > AT7) , the smaller is this measure which is always positive
Measure for the duration of the phase of declining acceleration during PS
Measure for the duration of the PS
AT7-9 - AT7-AT9 Acceleration amplitude of PS, measure for the intensity of the PS HT9-6 - HT9-HT6 height gain during PS
H%T9-6 - H%T9-H%T6 relative height gain during PS
HT7-6 - HT7-HT6 height gain during the first phase of PS
(increasing acceleration) H%T7-6 H%T7-H%T6 relative height gain during the first phase of PS HT9-7 - HT9-HT7 height gain during the phase of declining acceleration of the PS H%T9-7 - H%T9-H%T7 relative height gain during T9-T7
VTS-6 - VTS-VT6 height of pubertal velocity peak over prepubertal velocity VTS-6 x T9-6
A crude measure of the additional height gain during the PS
Amplitude of acceleration during the MS, a measure of intensity of the MS
Duration of the MS Increase in acceleration from the end of the MS to peak acceleration during the PS
The PS can be divided into two distinct subphases: the initial phase from T6 to T7, which focuses on maximal acceleration, and the second phase from T7 to T9, characterized by declining acceleration and conclusion We can analyze 60 parameters—many of which are highly correlated—by calculating means, standard deviations, and correlations separately for boys and girls Additionally, hypothesis tests can be employed to compare the equality of means between the two groups.
More information about the distribution of interesting parameters ever sub samples can be obtained by applying kernel density estimates in order to estimate the density f of the distribution
(Rosenblatt, 1956; Parzen, 1962) are given by
In the study, kernel estimators were utilized to compare the density parameters TS, AT7 (PS) and T4, AT2 (MS) between boys and girls, as referenced by Gasser et al (1985b) The observations, denoted as Xl to Xn, are characterized by a density function f, with a bandwidth b and a kernel function K typically selected from K ~ 0, K ∈ Mo,2, in accordance with equation (7.19) The bandwidths were manually selected by examining various smoothed curves.
14? selecting the seemingly most appropriate one As an example, the densities for T8 (time of pubertal velocity peak) for boys and girls are displayed in Fig 9.9
Fig 9.9 Estimated probability densities for T8 (peak velocity of the PS) by the kernel method Solid line: n-45 girls, dotted line: n-45 boys From Gasser et al (1985b)
The density distributions for boys and girls exhibit remarkable similarities, although the density for boys is shifted towards higher values, indicating that boys experience a later and more intense peak shift (PS) compared to girls.
Growth spurts
Table 9.4 contains sample means, standard deviations and ranges of selected derived variables, separately for boys and girls, and p values for the two sample comparison by the Wilcoxon test
Only a few interesting findings are shortly discussed here (for more details, compare Gasser et al, 1985a,b): The PS peaks are clearly skew (AT?
The growth patterns in boys and girls differ significantly, particularly in terms of acceleration and peak strength (PS) Boys exhibit larger acceleration values (AT?) and (AT9) compared to girls, but girls experience earlier puberty, leading to a higher proportion of VT6 within VT8 This results in a more pronounced deceleration of VT8-6 and VT6 for girls, causing their PS peak to be more skewed than that of boys Consequently, the PS peak for girls is not merely a smaller version of that of boys; it possesses a distinct structure that reflects these underlying growth differences.
Table 9.4 Mean values and standard deviations for selected longitudinal parameters p values for Wilcoxon test of pairwise comparisons between boys and girls are indicated (** : p < 10-', * : p < 10-2 ) Based on 45 m, 45 f From Gasser et a1 (1985a, 1985b)
Research indicates that the timing, duration, and intensity of the PS are largely uncorrelated, aligning with findings from Largo et al (1978) and Zacharias and Rand (1983) This lack of correlation may account for the wide variety of observed PS Additionally, AT7 and AT9, along with their skewness (-AT7/9), consistently show a high correlation, suggesting that elevated hormone levels are likely to result in a significant impact.
The acceleration of 149 is accompanied by a time lag leading to rapid ossification, resulting in significant deceleration Consequently, the peak observed is less skewed, as the proportion of VT8-6 within VT8 is relatively high The endocrinology of the PS is well-documented, as evidenced by studies conducted by Sklar, Kaplan, and Grumbach (1980) and Sizonenko, Paunier, and Carmignac (1976).
Research indicates that the timing and intensity of menarche (MS) are not influenced by sex, contradicting earlier findings by Bock and Thissen (1980) and Berkey, Reed, and Valadian (1983) These studies utilized a triple-logistic model and variable knot cubic splines, respectively, both of which encountered significant fitting issues The triple-logistic model fails to account for the latency period between menarche and puberty (PS), which averages 1.1 years for boys and only 0.3 years for girls Meanwhile, the variable knot cubic spline method is heavily reliant on the selection of knots; insufficient knots, as evidenced in Berkey et al (1983), hinder the model's ability to accurately represent both MS and PS, resulting in poor overall fit Consequently, the critical nature of knot selection complicates the application of this method in research.
The endocrinological basis of multiple sclerosis (MS) remains unclear; however, the DHEA hypothesis proposed by Molinari, Largo, and Prader in 1980 is compelling This hypothesis suggests that levels of the adrenal hormone dehydroepiandrosterone (DHEA) rise in individuals with MS.
Between the ages of 6 to 8 years, children experience growth spurts that appear to be independent of sex, as noted in various studies (de Peretti and Forest, 1976; Reiter, Fu1dauer and Root, 1977; Sizonenko, Paunier and Carmignac, 1976; Sizonenko, 1978) Notably, the parameters of these growth spurts are uncorrelated with those of puberty, suggesting that they occur independently in terms of timing and intensity This independence may be linked to the separate release of adrenal hormones, such as DHEA, and gonadal hormones that initiate puberty (Sklar, Kaplan and Grumbach, 1980) However, a challenge to this hypothesis arises from the observation that despite rising DHEA levels beyond age 8, growth deceleration occurs at the end of the growth spurt.
Another question of interest is how adult height is influenced by various growth phenomena Correlations of adult height with selected longitudinal parameters and derived parameters are given in Table 9.5
Table 9.5 Rank correlations between adult height and various longitudinal parameters (45 m, 45 f) Longitudinal parameters as explained in text HTO is height at the age of four weeks (**: p < 10-", * : p < 10- 2 ) From Gasser et a1 (1985a)
Sex HTl HT2 HT3 HT4 HT5 HT6 HT7 HT8 m 78** 74** 70** 70** 67** 80** 85** 92** f 67** 45** 51** 60** 64** 72** 81** 91**
VTl VT2 VT3 VT4 VT5 VT6 VT7 VT8 m 54** 43** 48** 55** 33** 45** 24 10 f 40** 21 30* 17 17 05 13 24
Adult height is influenced primarily by prepubertal growth velocities and heights achieved at various stages, rather than by the timing, duration, or intensity of growth spurts (PS) In fact, a PS is not essential for attaining normal adult height, and an early PS can result in premature ossification, potentially leading to shorter adult stature Interestingly, there is a strong correlation between height at four years and eventual adult height, indicating that early growth in infancy and early childhood plays a significant role in determining final height.
Further applications
Monitoring and prognosis based on longitudinal
In the medical field, common challenges arise that are also relevant in other domains Longitudinal medical data are collected not only to describe and assess time-dependent physiological or pathological processes but also for patient monitoring and prognostic classification Prognostic data typically include a vector of covariates such as age, sex, and age at diagnosis, alongside longitudinal observations for each patient The primary objective is to extract key longitudinal parameters from the time course data and incorporate them into the covariate vector These combined vectors are then analyzed using discriminant analysis techniques to identify variables that effectively differentiate between groups with favorable and unfavorable prognoses, with methods like CART (Breiman et al., 1982) being a viable option.
In medical research, the classification of new cases can be effectively achieved using classification trees, as highlighted by the work from 1985 Beyond traditional longitudinal parameters, the variability of observations is crucial for classification and prognosis, as indicated by equations (7.1) and (7.2) To enhance classification accuracy, it is essential to extract and select parameters that minimize the misclassification rate, typically assessed through cross-validation methods (Breiman et al., 1982) The selection of optimal longitudinal parameters necessitates collaboration with medical researchers and a combination of subject-matter insights and trial-and-error approaches Importantly, longitudinal parameters must be applicable to all subjects within the study sample.
When analyzing longitudinal medical data, it's crucial to consider whether data transformation is necessary before conducting statistical analysis Transforming data can enhance interpretation from a subject-matter perspective; for instance, body weight can be adjusted relative to an individual's ideal weight In cases where small value differences are more significant than those of larger values, the inverse transformation Yi ~ l/Yi may be suitable Additionally, data transformation can improve graphical representations, making it easier to identify when a patient is at risk.
Patient monitoring encompasses a wide range of scenarios, including transplant oversight and intensive care unit management It can be challenging to identify the time courses that may lead to critical situations To address this, one can utilize discriminant analysis to replicate the decision-making process of physicians based on longitudinal data.
Postoperative monitoring of kidney transplant patients is crucial for detecting potential rejection reactions Clinicians assess whether a patient is developing a rejection response, which may necessitate immediate immunosuppressive therapy, although this should be avoided unless clearly indicated Key clinical signs such as fever, tenderness, graft enlargement, and elevated blood pressure, along with serum measurements of creatinine and urea, as well as urine volume, are routinely evaluated Daily monitoring of serum creatinine and other indicators is essential for assessing kidney function A Kalman filter approach, developed by A.F.M Smith and colleagues, utilizes time series data of serum creatinine to identify abrupt changes associated with rejection reactions.
Nonparametric regression offers alternative procedures that do not require prior knowledge of which features in observed time courses are linked to rejection reactions By employing one-sided kernel estimators with boundary kernels, past measurements can be utilized to predict current values These predicted values can then be compared to the actual observed values for the current day The resulting differences from various measurements can be analyzed using discriminant analysis techniques, such as stepwise logistic regression or GART, allowing for variable selection and comparison of the algorithm's classification accuracy.
153 retrospective clinical assessment of whether there was a rejection reaction at the current day, and minimizing the misclassification rate.
Estimation of heteroscedasticity and prediction
Heteroscedasticity is a common issue in longitudinal data analysis, as illustrated in Figure 4.1 To address this problem and enhance both parametric and nonparametric regression estimators, it is essential to estimate the variance function This estimation can be achieved using kernel estimators, as outlined by Carroll (1982) and Muller and Stadtmuller (1987b) In the basic model, assuming that the variance function is a smooth function of time, consistent kernel estimators can be derived for the variance function.
(10.1) i.e by applying the usual kernel smoother (4.4) for v - 0 to ui2 , where ui 2 is the "raw variance" near ti, based on (7.1) or (7.2), i.e
The estimate (10.1) for the muscular activity data of Fig 4.1 is displayed in Fig 10.1
The estimator ~2(t) is effective for selecting local bandwidth in the presence of heteroscedasticity, as outlined in equation (7.18) Additionally, in linear regression models, the estimator (10.1) facilitates efficient adaptation to heteroscedasticity When constructing confidence intervals for kernel estimates, as discussed in section 7.5, it is essential to replace ~2 with ~2(t) to ensure that local confidence bands expand in areas with higher error variance.
The study aimed to compare muscular activity measurements of a patient before and after a specific treatment To determine if the new measurements significantly differed from the previous ones, prediction intervals for measurements at a defined force were constructed By denoting the kernel estimate of the regression curve as g(t) and the kernel weights as Wi(t), the estimators of variance were derived Given that the variance of any measurement at time t is denoted as σ²(t), the 100(1-α)% local prediction interval was established, neglecting bias and assuming normal errors.
95% prediction intervals (10.4), 95% confidence interva~s (not depending on the normal error assumption) based on
" and kernel estimate g(t) for the muscular activity data are displayed in Fig 10.2 aD
48 &II au lDD 120 14D t&O taD :zaG 220 248 280 :zea 3DB 3aa MD
Fig 10.1 Estimator (10.1) of variance functi~ for musc~lar activity data using Epanechnikov kernel K • 4 (1-x2)1[_l,l] and b 90
Fig 10 2 Kernel estimate for muscular activity data (Epanechnikov kernel , b - 40), solid line, 95% confidence intervals (10.5)
(Epanechnikov kernel, b 90 for ~2( ằ, dashed line; 95% prediction intervals (10.4), dotted line.
Further developments
The following remarks concern problems which are of current (and maybe future) research interest
(1) Estimation of conditional distributions and conditional functiona1s
In the context of random design, researchers such as Beran (1981), Stute (1986a,b), and Dabrowska (1987) have explored the estimation of the entire conditional distribution function, rather than merely estimating the regression function g(x) = E(Y | X = x) This approach is equally applicable to fixed design scenarios Stute (1986a) introduces estimators for the conditional distribution function G(y | x) as P(Y ≤ y | X = x), utilizing a kernel function K, a bandwidth b, and the marginal empirical distribution function Fn of the variables (Xi) Moreover, conditional functionals Y(G(â | x)) can be estimated by Y(Gn(â | x)), with Stute demonstrating that conditional quantiles are asymptotically normally distributed.
These ideas can be transported to the fixed design case where one might consider estimating the error distribution function at a fixed point t, Et(y)
One could then define estimators for quanti1es u E (0,1),
Et,n(Y) ~ u} and in such a way obtains local prediction intervals which do not depend on the normality assumption made in (10.4)
Traditionally, it has been assumed that measurement errors are uncorrelated or independent; however, this assumption is often questionable in practical applications Real-world dependence structures can be effectively modeled using m-dependence, autoregressive, or moving average schemes Research into dependent errors has been conducted by scholars such as Collomb (1985b) and Hart and Wehrly (1986) A key inquiry in this field is how to adjust nonparametric curve estimators to account for the dependence structure of the errors.
The Alternating Conditional Expectations (ACE) algorithm, introduced by Breiman and Friedman in 1983, aims to iteratively and nonparametrically transform the x- and y-axes in a bivariate relationship to achieve linearity This process relies significantly on effective smoothing techniques to derive the necessary transformations A key consideration in this context is the applicability of residual analysis methods, typically used for parametric models, to nonparametric regression, particularly in identifying dependencies among errors.
This is another major field of current and future research Some ideas are discussed in 3.6 and 8.1.
Consistency properties of moving weighted averages
Local weak consistency
We consider here the usual fixed design regression model
Yi,n - g(ti,n) + £i,n (11.1) with triangular array errors £i,n 1.1.(1 for each n, E£i,n - 0 and g:
A ~ R, A c Rm, corresponding to the multivariate case (6.1) The notation and assumptions are the same as in 6.l average estimator
In the study of DVg(t), significant findings regarding weak and universal consistency are presented in the works of Muller (1983, 1987c), with some results summarized here without proofs Additionally, Stadtmuller (1982, 1986a) and Muller (1983) provide insights into local almost sure convergence Stadtmuller further explores the limiting distribution of the appropriately normalized maximal deviation, expressed as suplg(x) - g(x)1 for m - 1 and v - 0.
Decomposing we first consider the deterministic part which is handled by a Taylor expansion as usual Assume that tEA
Lemma 11.1 Let g E Clvl(A) and (Wi,v(tằ satisfy
To get conditions for local weak consistency, we need results for convergence in probability of weighted averages A first result is Theorem 1 of Pruitt (1966)
Let EI £ I < "', ~ IWi v(t)1 ~ L < '" and max IWi v(t)1 0, i=l ' l~i~n '
By combining equations (11.3) and (11.5), we find that the limit inferior as n approaches a of J i-1 is approximately equal to |Wi,v(t)|, leading to the conclusion that this holds true as a approaches infinity when |v| > 0 Consequently, Lemma 11.1 and Lemma 11.2 are applicable only when v equals 0 For cases where |v| > 0, a different result is necessary, with the proof resembling that of Lemma 11.2.
Let EIÊl r < "', i-1 ~ IWã ~,v (t)l r 0, n'" '" for an r satisfying 1~r~2
Combining Lemmas 11.1-11.3 we arrive at
Theorem 11.1 Let the requirements of Lemma 11.1 be satisfied If I v I = 0, assume that the requirements of Lemma 11.2 are satisfied, if Ivl > 0, let the requirements of Lemma 11.3 be satisfied Then
Further results along these lines can be given for MSE consistency To apply the results to kernel estimates requires one further step weights are according to (6.2)
In this context, we define \( b \) as a function of \( n \), specifically \( b = b(n) = b_1(n) = \ldots = b_m(n) \), which represents a vector comprising \( m \) instances of \( b(n) \) Additionally, we assume that the conditions outlined in equations (6.3) through (6.7) are met Furthermore, for \( B = 2md(T)b \), the design is characterized by a high degree of homogeneity.
(1l.8) which implies n i~lIWi,v(t) Il(llti -tll>B} o (11.9) and
Lemma 11.4 Let an integer p ~ o and a multiindex v ~ 0 be given Then f 0 O~lctl~p, ct v f Kv(x) xctdx
T ct = v implies for the kernel weights (11.7): n f O(Qn) O~lctl~p, ct v
1 + O(Qn) i-l ' v! ct = v where Qn : = [nl/mblvl]-l If K c A is compact, O-terms are uniform over t E K
Proof Let Mt y (t-x)!b, x EM} By an integral transformation we get
I i-l ~ b-Ivl-m f Ai K ( t-x ) (ti-t)ct(-l)lctldx-blctl-lvlf Kv(x) xctdx v b T I n t-tã
~ blctl-Ivl L f IKv(x)1 I( ~)ct - xctldx i=l Ait b
If a = 0, this is bounded by O(Qn)' If lal > 0, use T C n U Ai t (which holds uniformly for all tEKcA) to show that this expression is bounded by i=l
Corollary 11.1 Assume that tEA, g E "lvl(A) and n1/mbl v l co as n co, and that in addition to (6.3)-(6.7), (11.8) is in force
A If v - 0 and EI£I < co, then g(t) ~ g(t)
B If v > 0, EI£l r < co for some 1 1 is straightforward, since the stochastic part of the proof is not affected
This article utilizes an exponential inequality as presented in Lamperti's 1966 work, specifically in Lemma 1 The proofs are adapted from Muller's 1983 research, with additional insights from Muller and Stadtmüller's 1987 findings Furthermore, related results on uniform convergence are attributed to the studies conducted by Cheng and Ling in 1981 and Georgiev in 1984.
Lemma 11.5 Assume that the (£i n) satisfy I£i nl ~ M < co, and E(£i,n) ~ Ri,n,
, n ' l~i~n Then it holds for Sn - ~ £i n and all x E [O,2/M] i-I ' E( exp (xSnằ ~ exp(3ãx 2 • j-1 n ~ Rã J, 2 n ) (11.11)
For the stochastic part of the maximal deviation sup Ig(t)-g(t)1 one tE[O,l] obtains (boundary effects play no role for the stochastic part)
Theorem 11.2 Assume that EI£i,nl s ~ M < 00 for some s > 2, and that the weight functions Wi(t) (suppressing index v) satisfy for some 0 ° max IWi(t)1 ~ c n- l uniformly for t E [0,1) l~i~n
Finally suppose that there is a sequence an ~ 0, and constants '7 E (0,s-2) and K > 1/2 S.t for all t E [0,1):
Then sup Ig(V)(t)-E(g(V)(tằ1 - O(an ) a.s tE[O,l]
Proof Defining ~ - 3/6, r = S-'7 and I - [0,1), consider a sequence of n-~- neighborhoods Un covering I needs O(n~) sets Un and II II eo : - sup I ã I : tE[O,l)
Choosing proper middle-points Tn for Un one
The crucial part is the third term Define Bn(t) :- aii.2 m~x IWi(t) In2/ r l~l~n
(log n)2, '7n(t) - an.Bn(t) and apply Lemma 11.5 to the random variables Bn(t)Wi(t)(ci-E(ciằ, choosing x = (.Bn (t)n2/ r max IWi(t)I)-1/2 l~i~n
Observing P(Sn > a) ~ e-axE(eXSn), one obtains for any c.onstant T > 0:
163 n2/t max \Wã(t)\ l:Si:Sn ~ with suitable constants c1 ,cz > O
The result follows from the Borel-
For those focused solely on probability bounds or assuming a linear scheme for the variables, the proof indicates that the factor n²/(s-I) in equation (11.14) can be substituted with nl/(s-I) This modification reduces the moment requirements for the variables, simplifying the conditions necessary for the analysis.
Theorem 11.2 can be easily specialized for kernel estimates
Corollary 11.2 Assume that g E ~k([ 0,1]), K E Mil, k and that the error variables satisfy E\!:i,n\S :S M < 00 with some given s > 2 Assume that K is Lipschitz continuous on R
A If b satisfies for some 0 > 0 and some I) E (0,s-2): lim inf nb l +o+1I > 0 n->ô> lim inf nbk -II > 0 n->ô> lim inf (nb 211+l /log n)1/2 n- 2/(s-I) > 0 n->ô>
(11.18) then it holds on any compact interval I C (0,1) for the kernel estimator (4.4) that ifk>1I a.s ifk-II
B If k > II and s > 4+2/k and if we choose b - (log n/n)1/(2k+l), then we have
In case that one is interested in bounds in probability, condition (11.18) can be relaxed to lim inf(nb2v+l /log n)1/2 n-l/(s-~) > o n~ (11.18')
Corollary 11.2B then requires the weaker condition EI£i,nl s S M < m for s >
Using the modified kernel estimator with boundary kernels in the boundary regions allows for the extension of results to the interval [0,1) when condition (5.24) is met, as demonstrated with kernels in (5.23) This approach ensures that equation (11.12) is satisfied and maintains favorable bias behavior across the entire interval [0,1] Consequently, in Corollary 11.2, the interval can be selected as [0,1].
FORTRAN routines for kernel smoothing and differentiation 165
Structure of main routines KESMO and KERN
The programs mentioned are designed for kernel estimation and differentiation (v-O-3) using various estimators (4.4) Users can select from different orders of kernels and choose between two bandwidth options: FAC-CV, which integrates the factor method for bandwidth selection with cross-validation for v = 0, and FAC-R, which combines the factor method with the Rice criterion for v = 0 According to the simulation study in section 7.4, FAC-R provides the optimal bandwidth choice for derivatives Additionally, the program accommodates nonequidistant data and offers two boundary modification options, allowing for bandwidth adjustments either similar to the interior or increased (stationary) bandwidth in boundary regions, as detailed in section 5.8.
The graphical charts illustrate the logical structure of the programs, with numbers indicating their sequence The primary subroutine, KESMO, takes various options as input, such as bandwidth selection and the computation of confidence intervals The secondary subroutine, KERN, is responsible for executing the actual kernel smoothing process.
A short description of the programs follows
Description Main subroutine: options, data and auxiliary arrays transferred; calls other subroutines
Estimation of error variance according to (7.2)
Determines optimal bandwidth FAC-CV by (7.17), (7.11)
Determines optimal bandwidth FAC-R by (7.17), (7.12)
Determines minimum of a given array of function values by quadratic interpolation
Determines factor (7.17) for factor method of bandwidth choice
Determines moments of kernel function for FAK
C Relations between KESMO and KERN
Determines integral over kernel function squared for FAK
9 KERN Main kernel smoothing subroutine
10 KEWEIR Computes kernel weights (adapted from a program by
11 KOEFF Computes coefficients for kernel function
12 RM Auxiliary function for KOEFF
13 KOEFFD Computes kernel coefficients for NUE-O, KORD-3,5
14 KEFFBO Computes coefficients for kernel function at boundary
(is called for each point within boundary region if boundary modification is requested)
Solves linear system of equations (adapted from a program by Rutishauser)
The usual notation used for the kernels and bandwidths in the text translates as follows into the parameters of the programs: text program v NUE k KORD
WOPT short description order of derivative to be estimated order of kernel used smoothness of kernel (NKE-1 "MIN VAR"
NKE-2 "OPT" etc.) bandwidth optimal bandwidth reference in text (4.4),(5.22)
The article presents a comprehensive list of 15 subroutines and functions developed in standard FORTRAN 77, adhering closely to the 66 standard While the programs have not been optimized, there is potential for faster algorithms, particularly when using a rectangular kernel and equidistant designs The cross-validation process can be time-consuming, especially with a large number of bandwidths (NWID) and boundary modifications for larger bandwidths The programs are designed to be self-explanatory and include error checks The original KERN and its subroutines were collaboratively developed with Th Gasser, while most other programs were created with Th Schmitt Additionally, the KERSOL program for solving linear systems is an adapted version of Rutishauser's LIGLEI routine.
A packaged subroutine from a program library can replace the routine MINS, which is used to determine the minimum value of a function from a specified array of function values.
Listing of programs
SUBROUTINE KESMO (NUE, KORD, NKE, NBO, NALPHA, BIIIIN, BIo'MAX NWID, OPTlO, XIN, YIN, XOU, YOU, CONFL, CONFU, N, Nl, M VARI, VAR, NAIN, NBIN, BWNUE, NERR1, NERR2, NERR3
WKAR1, WKAR2, WKAR3, WKAR4, WKAR5, XW1, XW2)
KERNEL SMOOTHED CURVE INCLUDING BOUNDARY MODIFICATION AND CORRESPONDING
100(1-ALPHA)X CONFIDENCE INTERVAL (BANDWIDTH CHOICE BY FACTOR-RICE-METHOD
OR FACTOR-CV-METHOD) VERSION 10/87 TS,HGM
ORDER OF DERIVATIVE TO BE ESTIMATED ORDER OF KERNEL USED
*** REQUIREMENT NUE KORD BOTH ODD OR BOTH EVEN, KORD GE NUE+2 ***
*** REQUIREMENT: KORD+2*(NKE-l)-1