Functional linear regression analysis for longitudinal data

Fang Yao; Hans-Georg M\"uller; Jane-Ling Wang

REVIEW

Functional linear regression analysis for longitudinal data

Fang Yao, Hans-Georg Müller, Jane-Ling Wang

Statistics Theory · Submitted 6 Mar 2006 · 31 references

Pith's verdict: Solid, influential paper that correctly extends FPCA-based functional regression to sparse longitudinal data; the confidence-band theory is asserted rather than proved, but the consistency results hold up.

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CitationFang Yao, Hans-Georg M\"uller, Jane-Ling Wang. "Functional linear regression analysis for longitudinal data." arXiv:math/0603132 (2006). https://pith.science/paper/4QE4GTWH

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Editor's note

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Things you should know: this is the Yao–Müller–Wang paper that introduced the PACE-style conditional-expectation approach to functional linear regression with sparse longitudinal data. The core claim—that you can consistently estimate the regression function β(s,t) and predict a response trajectory from a few noisy predictor measurements, as n grows—is well supported. Theorems 1 and 2 have real proofs, and the machinery (smoothing, eigen-estimation, cross-covariance) is imported from the authors' companion JASA paper [31], which is appropriate.

What's actually new: prior functional regression required dense or complete trajectories; this paper explicitly targets designs with 3–5 measurements per subject. The conditional-expectation score predictor (10)–(12) is a genuine extension of the authors' own PACE method to the regression context. The functional R² is a reasonable addition, though not the heart.

The main soft spot is Theorem 3, the asymptotic normality supporting the pointwise confidence bands. The proof is one sentence: 'Analogous to the proof of Theorem 4 in [31] with slight modifications.' That's not enough. The variance (13) includes the conditional prediction variance Ω_M of the latent scores, but the statement needs to show that estimation error in μ_Y, φ_k, P, and ζ̂−ζ̃ is asymptotically negligible in the standardized difference. The stress-test note is right about this. Also, Gaussianity (A5) is doing real work: without it, ζ̃ is only a best linear predictor, and the distributional result (25) does not follow. The simulation tests mixture scores for prediction error but never checks interval coverage under non-Gaussianity. So the bands are the least supported part of the paper. That said, the bands are a secondary feature; the consistency results are the load-bearing claim and they hold.

Minor: the simulation is narrow (one covariance design, no comparison to James et al.'s sparse method), and no code/data are provided. For 2005 that's not disqualifying.

Bottom line: this paper deserves a serious referee. It's a well-constructed extension of a known framework, with honest asymptotics for the core estimators. The referee should push for a real proof of Theorem 3 or an explicit statement that the bands are heuristic. I'd cite it, and I'd probably bring it to a reading group focused on functional data or longitudinal methods.

Referee report

Significance

If fully established, the paper would extend functional linear regression to sparse longitudinal designs, a practically important setting. The manuscript supplies explicit rates in Theorem 1, a clear conditioning mechanism that borrows strength across subjects for score prediction, and a useful functional R2 measure. The proof structures for Theorems 1 and 2 are plausible, and the two applications are informative. However, the strongest new inferential claim, the pointwise confidence bands, rests on a proof that is deferred by analogy to the companion paper [31], and the Gaussianity assumption is only stress-tested through prediction errors rather than through the coverage of the proposed bands. These gaps affect the central advertised inference result.

Major objections (3)

[Section 6 / Appendix A.4, Theorem 3] The proof of the asymptotic normality result (25) is a single sentence deferring to Theorem 4 of [31]. The setting here is genuinely different: the standardized difference (\hat Y*_{KM}(t)-E[Y*(t)|X*])/\sqrt{\hat\omega_{KM}(t,t)} includes an estimated response mean, estimated response eigenfunctions \hat\varphi_k, estimated regression coefficients \hat P_{KM}, and plug-in score estimates \hat\zeta*_m-\tilde\zeta*_m. The manuscript does not show that these estimation errors are asymptotically negligible relative to the conditional variance \omega_{KM}. Since the confidence bands (13) are a central advertised contribution, this is load-bearing. Please provide a complete proof of Theorem 3 or state the additional assumptions under which the estimation errors are negligible and verify them.
[Section 6, assumption (A5); Section 4, simulation study] The Gaussian assumption (A5) is used twice: to identify \tilde\zeta* as the conditional expectation and to obtain the normal distribution of \tilde\zeta*-\zeta*. The robustness simulation only compares RMSPE under mixture scores; it does not evaluate the coverage of the bands (13). Because the manuscript claims asymptotic pointwise confidence bands for predicted trajectories, coverage under non-Gaussian sparse data should either be proved or reported in simulations before the bands are recommended for general use.
[Appendix A.3, Lemma A.1] The key rates (37)-(40), including uniformity over k and m, are largely imported from the companion paper [31] rather than proved here. These rates feed directly into the bound for Q1(n) in (44) and hence into Theorem 1. The sentence 'Part of the proof follows that of Theorem 2 in [31]' does not make clear which statements are established in this manuscript and which are borrowed. Please either prove the uniform rates or quote the precise results from [31] and verify that the assumptions there match those of the present sparse-data setting.

Minor objections (5)

[Section 6, paragraph before (B5)] The sentence defining M and K says 'the numbers M = M(n) and K = M(n)'; the second equality should read K = K(n).
[Appendix A.1, definition of I_i and delta_j^X] The definition of I_i uses {j : rho_j = rho_i}, while delta_j^X is defined using |rho_l - rho_j|; the index notation should be made consistent, for example by defining I_j and rho_l accordingly.
[Appendix A.3, Lemma A.3 proof] The display following (42) uses '\infty\sum_{M+1}\infty\sum_{K+1}' as summation bounds; this should be written as \sum_{m=M+1}^\infty \sum_{k=K+1}^\infty.
[Section 3.2] The matrix H = cov(\zeta*_M, \tilde U*|L*,S*) is described with a transpose, but the dimensions and notation should be made explicit so that the expression H\Sigma_{U*}^{-1}H^T is unambiguous as an M x M matrix.
[Section 5, applications] Figures 4 and 9 display 95% bands, but the manuscript does not report the bandwidth choices or the number of grid points used in the applications; reporting these tuning details would improve reproducibility.

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The paper

The paper claims

Sparse data still support consistent functional regression.

The reading

The paper claims that functional linear regression—where a response curve is modeled as a linear functional of a predictor curve—can be made to work when each subject contributes only a few noisy measurements at irregular times. It proposes estimating the regression surface and predicting new response trajectories through functional principal component curves, recovering the sparse per-subject scores by conditional expectation rather than by integrating over each observed curve. The central result is consistency: as the number of subjects grows, the estimated regression function converges to the true one in integrated squared error, and predicted trajectories converge to the best predictor available from the sparse observations. This matters because longitudinal studies—clinic visits, liver disease follow-up, blood pressure and body mass index—routinely produce sparse, irregular data for which earlier functional regression methods required dense or complete trajectory sampling.

What carries the argument

The load-bearing device is the conditional-expectation estimate of a new subject's functional principal component scores, $\hat{\zeta}_m^* = \hat{\rho}_m \hat{\psi}_m^{*T} \hat{\Sigma}_{U^*}^{-1}(\tilde{U}^* - \hat{\mu}_X^*)$ from (11), which extends the best linear prediction formula (10) to sparse data. This step replaces the usual integral approximation of scores with a prediction that borrows strength from the whole sample, which is what makes the method work when only three to five measurements per trajectory are available. Around this step the paper assembles the basis representation $\beta(s,t)=\sum_{k,m}(\sigma_{km}/\rho_m)\psi_m(s)\phi_k(t)$, the smoothed cross-covariance surface $\hat{C}$, and the Gaussian variance identity $\Omega_M = D - H\Sigma_{U^*}^{-1}H^T$ that drives the confidence bands.

Extended reading notes

Core claim

The paper's central claim is that estimation and prediction in the functional linear model remain consistent under sparse, irregular longitudinal sampling. For the regression surface, Theorem 1 states that with $\hat{\beta}(s,t)$ defined in (8) from estimated eigenfunctions, eigenvalues, and a smoothed cross-covariance surface, $\int_T \int_S [\hat{\beta}(s,t)-\beta(s,t)]^2\,ds\,dt \to 0$ in probability as the number of subjects $n\to\infty$ under eigenvalue-separation and bandwidth conditions, with a uniform version under a stronger summability assumption. Theorem 2 states that the predicted trajectory $\hat{Y}^*_{KM}(t)$ in (12) converges to the target $\tilde{Y}^*(t)$, the conditional-expectation predictor built from the sparse measurements of a new subject's predictor trajectory. Under joint Gaussianity of the predictor's principal component scores and measurement errors, Theorem 3 gives asymptotic normality of the prediction error and justifies the pointwise confidence bands in (13). The paper also defines a functional coefficient of determination $R^2 = \sum_{k,m}\sigma_{km}^2/\rho_m / \sum_k \lambda_k$ and proves that its plug-in estimator is consistent.

Load-bearing premise

The load-bearing premise is that the predictor's principal component scores and measurement errors are jointly Gaussian, because this is what makes the best linear score predictor equal to the conditional expectation and supplies the variance formula $\Omega_M = D - H\Sigma_{U^*}^{-1}H^T$ behind the confidence bands.

Editorial extensions

If this is right

Regression-function estimation remains consistent when per-subject measurement counts are bounded and irregular, provided the number of subjects grows and the eigenstructure is well separated.
Response trajectories for new subjects can be predicted from sparse predictor measurements alone, with predictions converging to the best available conditional predictor.
Asymptotic pointwise confidence bands for predicted trajectories are available under joint Gaussianity of scores and errors; the paper's simulation evidence indicates the predictions remain useful when Gaussianity fails.
The functional coefficient of determination $R^2$ gives a consistent, interpretable measure of the variance of the response trajectories explained by the predictor, and the paper demonstrates it on two longitudinal datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The conditional-expectation score predictor could be extended to non-Gaussian settings via an elliptical or rank-based version of the best linear predictor; consistency would likely carry over, but the exact normal bands would need a different derivation.
Because the method permits predictor and response measurements at different times, it is a natural template for regression between two longitudinally measured biomarkers when visit schedules and missingness differ between the two outcomes.
A practical danger point is the effective number of subjects with at least two measurements: the theory needs only a positive probability of multiple measurements, but users should verify that this subset is large enough to identify the covariance eigenfunctions.
A direct test of the Gaussian assumption's role would be to measure the coverage of the nominal bands (13) under skewed or heavy-tailed scores; the paper's robustness simulations address prediction error, not band coverage.

Original abstract

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.

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We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.

background contribution method result interpretation main claim

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s0	We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time.	contribution 1.00	0.91	0.14	asserted 0.99
s1	Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects.	background 0.93	0.21	0.03	asserted 1.00
s2	In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit.	background 1.00	0.10	0.03	hedged possibility 0.57
s3	We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations.	method 0.90	0.93	0.09	asserted 1.00
s4	This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory.	interpretation 0.36	0.77	0.14	asserted 0.97
s5	The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories.	contribution 0.67	0.72	0.05	asserted 0.99
s6	Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function.	result 0.85	0.94	0.26	asserted 1.00
s7	Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories.	contribution 0.72	0.96	0.13	asserted 1.00
s8	A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case.	contribution 0.97	0.94	0.07	asserted 1.00
s9	The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.	contribution 0.55	0.93	0.11	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.42.

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Figures

**Figure 1.** Left panel: Observed individual trajectories (solid ) and the smooth estimate of the mean function for albumin (thick solid ). Right panel: Corresponding observed individual trajectories (solid ) and the smooth estimate of the mean function for prothrombin time (thick solid ), for the primary biliary cirrhosis (PBC) data

**Figure 2.** Left panel: Smooth estimates of the first (solid ) and second (dashed ) eigenfunctions for albumin, accounting for 87% and 8% of total variation. Right panel: Smooth estimates of the first (solid ) and second (dashed ) eigenfunctions for prothrombin time, accounting for 54% and 33% of total variation, for the PBC data

**Figure 3.** Estimated regression function (8), where the predictor (albumin) time is s (in days), and the response ( prothrombin) time is t (in days), for the PBC data

**Figure 4.** Observed values (circles) for prothrombin times (not used for prediction), predicted curves (solid ) and 95% pointwise bands (dashed ), for four randomly selected patients, where bands and predicted curves are based on one-curve-leave-out analysis and sparse noisy albumin measurements (not shown) of the predictor function

**Figure 5.** Estimated pointwise coefficient of determination Rb 2 (t) (19) as a function of time t for the PBC data (left panel, time is duration of study in days) and for the BLSA data (right panel, time is age in years). Research Center bi-annually. Time corresponds to age of each subject and is measured in years. On each visit, systolic blood pressure (in mm Hg) and body mass index (BMI = weight in kg/height in m2 ) were assessed. Since both measurements are highly variable, the data are noisy, and as many participants missed scheduled visits, or were seen at other than the scheduled times,

**Figure 7.** Left panel: Smooth estimates of the first (solid ), second (dashed ) and third (dash-dot) eigenfunctions for BMI, accounting for 90%, 6% and 3% of total variation. Right panel: Smooth estimates of the first (solid ), second (dashed ), third (dash-dot) and fourth (dotted ) eigenfunctions for SPB, accounting for 76%, 15%, 4% and 2% of total variation, for the BLSA data

**Figure 8.** Estimated regression function (8), where the predictor (BMI) time is s (in years), and the response (SBP) time is t (in years), for the BLSA data. then, as n tends to infinity, remainder terms such as θn and ϑn in (41) will disappear. Appropriately modified versions of the following theoretical results still hold. Since in this case the true K and M are finite, most of the technical assumptions that are needed to handle the infinite case would not be needed, such as (A1)–(A3), (A6)–(A7) and (B5), and K(n),M(n) would be assumed to converge to the true K and M instead of infinity, as…

**Figure 9.** Observed values (circles) of systolic blood pressure (SBP) that are not used for the prediction, predicted curves (solid ) and 95% pointwise bands (dashed ), for four randomly selected patients, where bands and predicted curves are based solely on sparse and noisy measurements of the predictor function (BMI, not shown). (B5) in Appendix A.1. For simplicity, we suppress the dependency of M and K on n in the notation. The consistency of βˆ (8) is obtained as follows. Theorem 1. Under (A1) and the assumptions of Lemma A.1 and (B5) (see Appendix A.1), limn→∞ Z T Z S [βˆ(s,t) − β(s,t)]

Showing 8 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:math/0603132 by the authors.

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[ref] developed a functional regression procedure for sparse longitudinal predictor and response processes using conditional principal component scores.

The Cost of Discretization in Functional Linear Regression: Minimax Rates and Adaptation · Related work · 0.85

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [31]

Yao, F., M ¨uller, H.-G. and W ang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590. MR2160561 F. Yao Department of Statistics Colorado State University Fort Collins, Colorado 80523 USA H.-G. M ¨uller J.-L. W ang Department of Statistics University of California, Davis Davis, California 95616 USA e...

View on Pith 2005

[2] [1]

Besse, P., Cardot, H. and Ferraty, F. (1997). Simultaneous nonparametric re- gressions of unbalanced longitudinal data. Comput. Statist. Data Anal. 24 255–

View on Pith 1997

[3] [2]

Boularan, J., Ferr ´e, L. and Vieu, P. (1995). A nonparametric model for unbal- anced longitudinal data, with application to geophysical d ata. Comput. Statist. 10 285–298. MR1363099

View on Pith 1995

[4] [3]

Cardot, H., Ferraty, F., Mas, A. and Sarda, P. (2003). Testing hypotheses in the functional linear model. Scand. J. Statist. 30 241–255. MR1965105

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