PS-SiZer map to investigate significant features of body-weight profile changes in HIV infected patients in the IeDEA Collaboration.

OBJECTIVES: We extend the method of Significant Zero Crossings of Derivatives (SiZer) to address within-subject correlations of repeatedly collected longitudinal biomarker data and the computational aspects of the methodology when analyzing massive biomarker databases. SiZer is a powerful visualization tool for exploring structures in curves by mapping areas where the first derivative is increasing, decreasing or does not change (plateau) thus exploring changes and normalization of biomarkers in the presence of therapy.
METHODS: We propose a penalized spline SiZer (PS-SiZer) which can be expressed as a linear mixed model of the longitudinal biomarker process to account for irregularly collected data and within-subject correlations. Through simulations we show how sensitive PS-SiZer is in detecting existing features in longitudinal data versus existing versions of SiZer. In a real-world data analysis PS-SiZer maps are used to map areas where the first derivative of weight change after antiretroviral therapy (ART) start is significantly increasing, decreasing or does not change, thus exploring the durability of weight increase after the start of therapy. We use weight data repeatedly collected from persons living with HIV initiating ART in five regions in the International Epidemiologic Databases to Evaluate AIDS (IeDEA) worldwide collaboration and compare the durability of weight gain between ART regimens containing and not containing the drug stavudine (d4T), which has been associated with shorter durability of weight gain.
RESULTS: Through simulations we show that the PS-SiZer is more accurate in detecting relevant features in longitudinal data than existing SiZer variants such as the local linear smoother (LL) SiZer and the SiZer with smoothing splines (SS-SiZer). In the illustration we include data from 185,010 persons living with HIV who started ART with a d4T (53.1%) versus non-d4T (46.9%) containing regimen. The largest difference in durability of weight gain identified by the SiZer maps was observed in Southern Africa where weight gain in patients treated with d4T-containing regimens lasted 59.9 weeks compared to 133.8 weeks for those with non-d4T-containing regimens. In the other regions, persons receiving d4T-containing regimens experienced weight gains lasting 38-62 weeks versus 55-93 weeks in those receiving non-d4T-based regimens. DISCUSSION: PS-SiZer, a SiZer variant, can handle irregularly collected longitudinal data and within-subject correlations and is sensitive in detecting even subtle features in biomarker curves.

Introduction

In the study of changes in longitudinal biomarkers in response to therapy or disease progression, it is useful to be able to identify the periods in time where changes occur. A key challenge arising from this effort is to isolate the underlying features of interest (say marker decreases or increases) in the presence of potentially large data variation. For example, in a data set of weight measurements in HIV-infected individuals initiating antiretroviral therapy (ART), which forms the core illustration of our methods in this paper, a scatterplot involving a mere 1% of the data (Fig 1 top left panel) is largely indecipherable. The situation does not improve when a “spaghetti” plot is generated (Fig 1, top right panel). However, a plot of the median weight at binned time points (Fig 1, bottom left panel) starts picking up the rapid early weight gains following ART initiation, but is less informative about the time when these gains reach a plateau and the possibility of long-term weight decreases possibly resulting from treatment toxicity or treatment failure.

Fig 1

Four different visualizations of weight changes t (kg) after antiretroviral therapy initiation in involving data from 1% of HIV-infected patients from the IeDEA database (2,000 patients, 46,207 observations).

The upper-left panel (a) represents a scatter plot; the upper-right panel (b) shows a spaghetti plot; the lower-left panel includes a plot of the mean weight over time; Lower-right panel represents smooth curves estimated at 3 different smoothing parameter values.

The bottom right panel in Fig 1 includes three smooth weight trajectories at different values of a smoothing parameter estimated via a penalized spline regression method (see Ruppert et al., [1]), which appear to capture the well-known features in such data involving rapid weight increase and subsequent plateau [2]. However, it is unclear what the durability of weight gain is or whether there are decreases in weight after long-term exposure to therapy. In addition, each smoothing level produces a slightly different fit, particularly with respect to the timing of reaching the plateau in weight gain. As noted in Marron and Zhang [3], a hurdle in the application of smoothing methods is the selection of the smoothing parameter, because interesting features that are present in the data may be visible after applying some smoothing techniques or at some levels of smoothing but disappear in others, so choosing among the various smoothing techniques or the level of smoothing can be critical in extracting relevant features from the data; and of course, there is a tremendous computational burden associated with such data analyses, as the above conclusions were drawn from only about 1% of the underlying database.

The Significant Zero Crossings of Derivatives (SiZer) [4] was proposed to address many of the aforementioned issues [5]. It is a useful Exploratory Data Analysis tool for understanding the significant features resulting from smoothed curves. SiZer simultaneously studies a family of smooth curves under a wide range of smoothing parameters (bandwidths) and produces inference on a smoothed version of the underlying curve viewed at varying levels of smoothing. The standard implementation of SiZer [5] is based on the local linear smoother with a kernel-type smoothing method for a single predictor and a single outcome data [6]. The SiZer map graphically explores structures in curves under study by mapping areas where the curve is significantly increasing, decreasing or does not change by studying its first derivative. Statistical inference is based on the derivatives of the smoothed curve by constructing confidence intervals at each location and also at each level of the smoothing parameters. The technique assembles these analyses at a wide range of smoothing parameters and synthesizes them in a single “map” where increase, decrease and plateau regions are identified by different colors, presenting an attractive global visualization of the data under many smoothing scenarios.

A number of extensions of the SiZer methodology have been proposed to increase inference precision [7, 8]. Of greater relevance to this paper is the extension proposed by Park and colleagues [9], which relaxes the assumption of independent errors, thus ignoring spurious features which are caused by the existence of dependence in the data. These and other authors further extended the SiZer method into the area of time series [10, 11]. Another relevant extension is the SiZer for Smoothing Splines (SS-SiZer) [3], which uses splines to enhance detection of true features in the data.

Despite its attractiveness as a data visualization tool, the SiZer map has not been used widely in biological applications. This is unfortunate, since biological processes frequently involve changes in various measures (most notably biomarkers), which evolve over time, in response to disease progression or initiation and/or modification of clinical therapies. One important reason for this is the fact that the SiZer and its extensions do not account for within-subject correlation. This frequently arises in longitudinal settings from measurements obtained repeatedly on the same individuals. A further technical complication is that the timing of these measurements becomes increasingly less regular with the passing of time (as subjects miss or reschedule clinical visits). It should be clear that this is a much different problem from time-series analysis, since longitudinal data are obtained from the same sample of study subjects repeatedly over time. Thus, neither the originally proposed method of SiZer nor its extensions in the area of time series fully address the challenges posed by longitudinal data.

It is a core aim of this paper to extend the method of SiZer maps to account for within-subject correlation in the setting of irregularly collected longitudinal biomarker data, since SiZer offers an appealing global data visualization technique which can be tremendously useful in answering many important biological questions about the evolution of these data over time. We accomplish this by proposing a semiparametric extension of the SiZer methodology, named Penalized Spline SiZer (PS-SiZer), which combines a penalized spline regression model [10] with an embedded linear mixed-model representation of the marker evolution, coupled with methods which increase the computational efficiency of the standard SiZer. These computational advances are particularly attractive when analyzing massive databases with hundreds of thousands of patients and millions of observations.

The paper is organized as follows: In the Methods we give a brief overview of the core ideas of the SiZer methodology as originally proposed [1, 5] along with the SS-SiZer methodology of Marron and Zhang [3] and present the proposed penalized spline PS-SiZer procedure for longitudinal data. Two simulation studies are presented in the Results where the proposed methodology is compared with the local linear smoother LL-SiZEr [1, 5] and the SS-SiZer [3] in, respectively, detecting changes and plateaus in longitudinal biomarker data. These are followed by the analysis of a large database obtained from hundreds of thousands of HIV-infected patients enrolled in care and treatment programs around the world participating in the International Epidemiology Databases to Evaluate AIDS (IeDEA) Collaboration, where body weight measurements were collected repeatedly at each clinic visit. The clinical interest of this analysis is to describe the pattern of body weight changes after initiation of antiretroviral therapy as a surrogate of treatment effectiveness and to determine the durability of weight gain by detecting a plateau in weight increases and the presence of possible decreases after long-term exposure to various therapeutic modalities. We conclude the paper with a brief discussion of our findings.

Methods

SiZer

More formally, for a given set of observed data and a smoothing function g(x), we can consider a non-parametric regression model as follows: Here, g(x) is some “smooth” regression function that needs to be estimated from the data and ϵ is the random error component with variance . The smooth function g(x) may be a non-parametric regression function indexed by a smoothing parameter λ (bandwidth) as g(x) [1].

Local linear smoother SiZer: LL-SiZer

The LL-SiZer model specification [5] considers a family of smooth functions indexed by the smoothing parameter as in Ruppert, Wand and Carroll [1]. A significant feature in the data is detected from the confidence limits of the first derivatives of the fitted model at each level of λ.

The LL-SiZer applies the local linear regression method of Fan and Gijbels [12] to estimate g(x) and its derivative, . A common estimate of g(x) at each location of x is given by the equation where argmin is the minimum of the sum jointly over the regression coefficients, a0 and a1. A line is fitted to the data for each x using K-weighted least-squares, where K(·) is a Gaussian kernel. A SiZer map is then constructed by changing the value of the smoothing parameterλ. The estimated regression function of g(x), and are obtained to construct a family of smooth functions at various levels of the smoothing parameter. Confidence limits for are obtained as where q is a suitably defined Gaussian quantile [7]. In the SiZer map, a pixel at x and a specific smoothing level λ is colored blue if the confidence interval suggests that (implying that the curve at x is increasing, red if the confidence interval suggests that (implying that the underlying curve is decreasing) and purple if the confidence interval contains zero (implying that no significant change in the curve can be detected).

SiZer for smoothing splines (SS-SiZer)

The SiZer for Smoothing Splines (SS-SiZer) [3] is an extension of a kernel-type estimation to the smoothing spline estimation. SS-SiZer incorporates the smoothing spline model and estimates the regression function by minimizing where λ is the smoothing spline parameter that determines the smoothness of the regression estimate and represents the roughness of the underlying function g(x). Here, the smoothing spline function g(x) is a natural cubic spline with knots at data locations x … x. The smoothing parameter, λ acts similarly as the bandwidth in the LL-SiZer presented in the previous section.

SS-SiZer constructs point-wise confidence limits to produce the map. In our research, we apply the simultaneous confidence limit to the SS-SiZer model to address the multiplicity comparison issue (see next section). Otherwise, the interpretation remains the same as in the SS-SiZer maps [3]. For other implementation details, such as the expression of first derivative estimate and its standard error, the reader is referred to the paper by Marron and Zhang [3].

The penalized SiZer (PS-SiZer)

In this section, we present our extension of the SiZer map to handle data that arise in the longitudinal setting. In the proposed model, we consider subject-specific correlation arising from data obtained repeatedly on the same individuals. We utilize an approach similar to the standard SiZer in which a family of smooth functions is used at various levels of smoothing parameters λ. We enhance the underlying model through the use of a computationally efficient smoothing model (presented below). In the PS-SiZer map, we also apply simultaneous confidence limits to resolve the issues related to multiple comparisons. Our proposed methodology extends SiZer in the following areas:

Adding a random intercept component to summarize subject-specific correlation

Applying a P-spline [13] as the underlying smoothing function

Constructing a simultaneous 95% confidence limit addressing multiple-comparison issues

Model specification

Let y denote measurements on subject i = 1, 2, … n at time x, j = 1,2…n. We model the responses as, where g(x) is a smooth function indexed by a smoothing parameter λ and ε is a vector of random normal error terms with mean 0 and variance . The model in (2) extends the basic model in (1) by adding the random subject-specific component , a normally distributed random intercept with mean 0 and variance which accounts for the within-subject correlation in the repeatedly collected measurements y in subject i. As this model is a member of the family of linear mixed models, a major advantage from its use is the ability to handle longitudinal data at irregularly spaced time points [14]. In this paper we use the P-spline model of Eilers & Marx [13], as the underlying smoothing method to estimate the function g. The P-spline model specification includes B-splines as the bases functions with evenly spaced knots with the difference penalty applied directly to the B-spline regression parameters to control the smoothness of the function g. Let B(x;p) denote B-spline basis of degree p with k′ + 1 internal knots. The number of B-splines is M = k′ + 1+p in the regression, resulting in the following approximation of the smooth function g: where a is a vector of coefficients, and B(x;p) is the B-spline basis function of degree p. For the penalty term, the P-spline model of Eilers and Marx uses a base penalty on higher-order finite differences, [13]. Consequently, the difference penalty matrix with order d can be written as, . Here, Δ is a matrix such that Δ constructs the vector of d difference of the coefficients a i.e., Δa = a – a; Δ2a - 2a + a and so on.

The second component of the model is the addition of a subject-specific random effect . This results in the penalized least square objective function minimizing where and

Mixed model representation

The minimization problem discussed in the previous section can be handled using the mixed-model framework [15]. Eq (3) can be turned into a regular mixed model by making use of the mixed-effect model framework discussed in detail in [1, 16–18] among others. Let us first consider the difference matrix, Δ that has dimension The penalty matrix is singular and has rank . A singular value decomposition of leads to with U are the eigenvectors and is the diagonal matrix of eigenvalues in non-increasing order. Thus, eigenvalues are strictly positive and the remaining d are zeros. Hence, U and Λ can be represented as U = [U+, U0] and respectively. The dimension of U+ is now with corresponding non-zero elements of vector . Consequently, we can rewrite Ba as and The mixed-model representation of the smooth function is , where . Our final model, including the random intercept, is of the form, where and .

The model in (4) above has three components. Xβ represents the fixed overall effect while Zu corresponds to the smoothing function and b, subject-specific random intercept, measures the random departure of subject i from the overall effect. The estimates of the parameters and the random coefficients are obtained as the best linear unbiased predictors (BLUP) from the mixed model using the restricted maximum likelihood (REML) criterion for the variance components. Eq (4) can thus be solved using any standard mixed-model software. We utilized the R-package mgcv::gam [19], which provides a computationally feasible approach to the parameter estimation in Eq (4). We obtained the estimate of g(x), the mean population curve at x, and the quantities of interest to generate the PS-SiZer map as the most crucial component in the PS-SiZer map is to estimate the first derivatives of the fitted functions (i.e. and the variance of and associated confidence bands. The gam() function from the R library mgcv was used to estimate the function g(x) at varying levels of the smoothing parameter λ. The PS-SiZer map includes a number of levels of the smoothing parameters. In the present application, we used a range between log10(λ) ± 2, where λ is the estimated smoothing parameter obtained via the REML approach using mixed model representation of the P-spline model. At each smoothing level, the resultant smoothing function component is obtained and extracted from the subsequent model fitting.

Inference

In the previous sections the point estimate for the model parameters were discussed, yet we are also interested in finding the confidence intervals for the quantities derived from them, such as an estimate of smooth function, and the first derivatives of the smooth function, . We describe the estimate of the covariance matrix for the smoothing parameters specified by [18]. Let contain all the fixed and random effects from the smooth term only and let be the corresponding model matrix. Let Z be the random effect model matrix excluding the columns corresponding to the smooth terms and be the corresponding random effect covariance. The covariance matrix is . Therefore, the estimated covariance matrix (Σ) for the parameters is where is the positive semi-definite matrix of the coefficients for the smooth terms. The standard error of the smooth function estimate, with C = [X Z].

Estimate and variability bands of the derivatives

The derivatives of the smooth function g(x) are obtained by defining Here, and . The first derivative estimate of is: The estimated standard error of is .

Confidence bands

The construction of the PS-SiZer map involves a family of smooth functions based on the confidence bands of the derivatives . We used 100 values of the smoothing parameter λ on the logarithmic grid to construct the PS-SiZer map. The range of the smoothing parameters λ, λ was chosen as, The number 2 above is arbitrary. However, the range of the smoothing parameters shown above spans 4 orders of magnitude giving us a picture of the estimated function at a sufficient smoothing span. We obtained the λ from the REML estimate of the variance components using the same P-spline methodology.

The PS-SiZer can be viewed as a collective summary of a large number of hypothesis tests so multiple-testing issues must be addressed. We follow Ruppert et al., [1] who showed that the penalized spline has fairly straightforward simulation-based simultaneous confidence bands which can be used in situations when multiplicity testing is carried out. Suppose we want a simultaneous confidence band for g(·) on a grid of x-values, and define A 100(1 – α)% simultaneous confidence band for g(x) is where is the corresponding empirical best linear unbiased predictor (EBLUP) obtained from the mixed model framework. Here, q(1 – α) is the (1 – α) quantile of the random variable at a smoothing level λ, i.e., which is the supremum on the set . The quantile q(1 – α) was approximated using N = 10,000 simulations. The N simulated values were sorted from smallest to largest, and the one with rank (1 – α)N was used as q(1 – α). For a PS-SiZer map, we obtained the 95% quantile of Eq (4) based on a simulation of size N at each level of λ. The confidence limits for were obtained as follows:

Results

Simulation studies

In practice, the fundamental function of a SiZer map is to detect the underlying features in the data. For this reason, it is natural to compare the SiZer maps according to which ones detect the correct number of underlying features.

We have conducted Monte Carlo simulation studies to evaluate the performance of PS-SiZer map under various scenarios. The key objective of this simulation study was to compare the PS-SiZer with the LL-SiZer and SS-SiZer. Our simulation studies were designed to mimic the HIV data analyzed later as part of the illustration of the methodology. In doing so, we use the concept of effective degrees of freedom (EDF) to encapsulate the complexity of the model as the actual degrees of freedom are not defined for semiparametric models. We use the established method for estimation of the EDF, i.e., the trace of the smoother matrix (see Hastie and Tibshirani [20] as cited in Chauduri and Maron[5]).

Using these simulated data, the relative performance of the SiZer maps was evaluated by the following approaches:

By making the SiZer maps comparable at a similar level of Effective Degrees of Freedom. For this reason all three SiZer maps (PS-SiZer with LL-SiZer and SS-SiZer) were generated with the same range of EDF.

By comparing the performance of the three SiZer maps at various levels of EDF according to which flags more features of a curve when a curve changes its status (increasing, decreasing or stable) from one to another.

By comparing the performance of the three SiZer maps which are most sensitive to detect plateaus that are really present in the data.

In this research, performance of the PS-SiZer maps are presented in two different simulation studies: “Simulation Study 1” which addresses item 2 above and “Simulation Study 2” which addresses item 3.

Simulation study 1. Longitudinal data were simulated as x chosen to be equally spaced in the interval [0, 1] with where is random noise, is the subject-specific random intercept and x denote the time of measurement. The function sin (5 π(x + 5)) is a periodic function which has five features. By this term, we mean changes in the curve from increasing to decreasing or vice versa. The quantity, is a function to control the spread of the periodic sine function, which has the effect of diminishing the size of the features at higher time intervals.

We compared three SiZer maps through the various levels of combination of error variance and subject-specific variance, respectively (Table 1). For each scenario of different variance combination, 50 trials were generated consisting of N = 100 subjects each and the number of observations per subject was n = 10 for i = 1, …N. For each simulated trial, three different SiZer maps were generated at 100 levels of EDF.

Table 1

Finding features: Simulation study-1 with varying variability.

		SiZer Maps
Variability (σε2:σb2)	Number of features detected	LL-SiZer	SS-SiZer	PS-SiZer
5.0: 2.0	Five	14%	2%	51%
	Four	34%	47%	88%
2.0: 5.0	Five	4%	10%	30%
	Four	32%	62%	85%
5.0: 15.0	Five	0%	5%	8%
	Four	24%	40%	68%

Proportions are from 50 simulation data sets.

Table 1 represents the mean proportion of features detected by the 50 simulated data sets at various levels of and . All three maps detected the first three features in the data most of the time. We were mainly interested to find how sensitive PS-SiZer is to detect the fourth and the fifth features of the true curve compared to LL-SiZer and SS-SiZer, as these were significantly diminished by the addition of the phasing-out component in the data as described above. When subject-specific variation is small (i.e., ), PS-SiZer detected all five features 51% of the time, whereas the SS-SiZer and LL-SiZer were able to detect all features 2% and 14% of the time respectively. At the same variability level, four features were detected by PS-SiZer 88% of the time, compared to 47% and 34% by SS-SiZer and LL-SiZer respectively. When the subject-specific variation is high (i.e., ), PS-SiZer was still able to detect four features in the data almost 68% of the time compared to 40% and 24% by SS-SiZer and LL-SiZer respectively. Interestingly, the fifth feature was not detected by LL-SiZer at all in this variability level, compared to 8% by PS-SiZer and 5% by SS-SiZer (Table 1).

Results from the above table are illustrated in Fig 2. Three maps were generated for each of the three methods under comparison from a randomly chosen trial from out of the 50 trials generated in the simulation study . In the Figure, the x-axis is represents time and the y-axis the EDF or the scale of smoothing of the three maps. As it is clear from the Figure, all three maps were able to clearly flag the dominant first and second features (blue and then red regions in the maps). However, in the majority of smoothing levels, LL-SiZer could not flag the third or fourth features as being statistically significant and did not detect the fifth feature at any level of EDF as mentioned above in the description of Table 1. SS-SiZer detected all four features for a large proportion of smoothing parameters and was able to detect the fifth feature only at higher levels of EDF, i.e. undersmoothing. By contrast, PS-SiZer detected all five features at the majority of smoothing parameter levels (Fig 2).

Fig 2

Simulation study 1.

Upper-left panel: True function; Upper-right panel: LL-SiZer map. Lower-left panel: SS-SiZer map. Lower-right panel: PS-SiZer map. For the SiZer maps, vertical axis represents 100 levels of EDF and the horizontal axis represents time.

Simulation study 2

In this simulation study, our aim was to illustrate how sensitive PS-SiZer map is compared to LL-SiZer and SS-SiZer in detecting the plateau of an increasing function. The true curve and the first derivative of the curve are presented in the top left panel of Fig 3. The data were generated as x equally spaced in [1:20) with where x are time measurements as before, is independent random noise and is the subject-specific random intercept. In a manner similar to Simulation Study 1, we generated 50 simulated trials, each with N = 100 subjects and involving n = 10 equally spaced time points i = 1, …, N. We consider an error variance and a subject-specific random variation . For each simulated trial, three SiZer maps were generated at 100 levels of EDFs.

Fig 3

Simulation study 2.

Upper-left panel: True function and first derivative; Upper-right panel: LL-SiZer map; Lower-left panel: SS-SiZer map; Lower-right panel: PS-SiZer map. The vertical axis represents the 100 levels of EDF and the horizontal axis represents the time.

The function used in this example had a true plateau at time . The sensitivity of the SiZer maps was calculated at each level of EDF by following exploring at which point in all three SiZer maps, a blue region changed to a purple region at each level of EDF. The process was repeated for 50 simulation trials. The summary of the first time point where the plateau was detected by the three SiZer maps is presented as a box plot (Fig 4). The box plot summary shows that the PS-SiZer map detects the earliest time point of the plateau at x ~ 5.89 the closest estimate of the true value. By contrast, LL-SiZer and SS-SiZer detected the plateau of the curve at x > 6. The true data curve and the resulting three SiZer maps from a randomly selected simulated trial are presented in Fig 3. Three maps were able to plot the pattern of the curve by moving from the blue region to the purple region at all levels of EDF.

Fig 4

Boxplot-summary of three SiZer maps: Time to detect a true plateau in the data.

Combined, Simulation Studies 1 and 2 demonstrate that the PS-SiZer map not only detects the significant changes of the true curve, but is also sensitive enough to detect the true time point where the curve reaches its plateau. Even though all three SiZer maps were able to detect the dominant features of the underlying curves, (that is, the trajectory of the curve from significantly increasing–blue area–to decreasing–red area–to non-significant–purple area), they were not able to detect less pronounced features at almost all levels of the EDF and were less sensitive than PS-SiZer in locating the true plateau of the curve.

Illustration

As an illustration of the proposed methodology of the PS-SiZer, we analyze data on weight changes in people living with HIV who initiate ART. In addition to detecting features in the data corresponding to body weight increases after the start of therapy, an important clinical question pertains to the durability of weight gains under different treatment regimens. More specifically, we explore possible differences in the durability of weight gain between stavudine (d4T) containing ART regimens versus non-d4T-containing regimens. Previous literature suggests that d4T is associated with lipodystrohy, a problem with the way the body produces and stores fat [21] and long-term weight loss compared to other regimens such as, for example, those containing Tenofovir [22] a regimen which is increasingly used as a first-line antiretroviral drug, particularly in the Southern Africa region.

The present study includes data on 185,010 adults living with HIV from five regions within the IeDEA collaboration [23]: Southern Africa (65.6% of the cohort), East Africa (21.9%), West Africa (8.3%), Central Africa (3.2%) and Asia Pacific (0.9%). Baseline demographic data of IeDEA patients identified by region and by d4t-containing versus non-d4T-containing regimen are shown in Table 2. In the sequel, we present in detail results from the Southern Africa IeDEA region. Results from the remaining four regions are presented in less detail and are left for the supplementary material.

Table 2

Summary of baseline characteristics-IeDEA study by d4T and non-d4T based regimen.

	d4T Regimen				Non-d4T regimen
	N	Age (years)	Female (%)	Baseline Body weight (kg)	N	Age (years)	Female (%)	Baseline Body weight (kg)
Overall	98160	36(30–42)	64152 (65)	55.0 (48–62)	86850	36 (30–42)	50682 (58)	55.0(49–62)
Asia Pacific	963	35(29–40)	410 (43)	51.0 (45–58)	751	34 (29–42)	181 (24)	57.7(50–56)
Central Africa	2839	37 (31–44)	2008 (70)	56.0 (49–65)	3045	37 (31–44)	2118 (51)	56.0 (50–65)
East Africa	30990	37 (31–43)	20017 (78)	54.0 (48–61)	9571	37 (31–43)	5758 (22)	55.0 (49–62)
Southern Africa	55192	35 (30–42)	36227 (49)	55.0 (48–62)	66295	35 (30–42)	38137 (51)	55.0 (49–62)
West Africa	8176	39 (32–42)	5490 (55)	55.0 (48–64)	7188	41 (37–42)	4488 (45)	57.0 (50–65)

Summaries are median (IQR) or n (%)

PS-SiZer Maps for the Southern IeDEA region were generated for each ART group, i.e., one map each for the groups of patients initiating ART with a regimen containing or not containing d4T. To address the issue of durability of weight gain, we need to determine the first time point (in weeks from start of ART) at which weight gain stops, i.e., the time when either weight stops increasing or starts to decrease. The PS-SiZer map provides an overall visual representation of the longitudinal weight change after the start of ART. However, to reach a conclusion on the durability of weight increases after ART start, we need to decide on a single optimum level of smoothing. Our algorithm does not depend on a specific smoothing technique. Here we have used a P-spline (13) PSR model for its computational efficiency and flexibility for correlated data. In addition, we took advantage of re-expressing the PSR model as a linear mixed effect model (1). The REML estimate of the mixed model is used to obtain the optimum smoothing parameter.

Statistical analyses were performed using SAS Software 9.3 and R software (2.13.2). SAS was used to create the analysis datasets for each of the five IeDEA regions. The user defined R-functions and the R package SiZer [24] was used to generate LL-SiZer maps. To generate SS-SiZer and PS-SiZer maps, user defined R-functions and the R package mgcv::gam [19] was used.

The PS-SiZer map for the Southern-Africa region is presented in Fig 5. The corresponding maps generated for each of the remaining four IeDEA regions are presented in S1–S4 Figs. Each Figure is divided into four panels. The smoothed trajectories of weight after ART initiation at the optimum level of smoothing for the two regimens and the smoothed first derivative of the weight change over time is the two types of regimens are shown in the top row. The PS-SiZer maps for d4T-containing and non-d4T-containing ART regimens are shown in the top row. In each PS-SiZer map, the vertical axis represents the level of smoothing and the horizontal axis the time in weeks since the start of ART as described in the Methods. For example, for d4T-containing regimens, at a medium level of smoothing (0.5–1.0), body weight increases for about 60 weeks, as reflected by the blue color on the left of the PS-SiZer map. The area to the right of the blue region is colored purple, indicating that no more significant increases in body weight are evident after about 60 weeks from the start of ART. There are red regions in the map at most of the smoothing levels indicating possible weight decreases. Similarly, at very high smoothing levels, (i.e., for values of the smoothing parameter λ > 1.0), the entire map is blue, following purple and red indicating weight increases, then stable or decreasing for the entire follow-up period. The PS-SiZer map of body-weight changes among individuals initiating ART with a non-d4T-containing regimen shows that, at lower smoothing levels, there are some blue and purple areas suggesting an intermittent weight increase. Otherwise, the map consists of mostly blue areas (indicating weight increases) for medium and higher levels of smoothing for up to about 150 weeks after ART initiation. This indicates that patients starting ART with a non-d4T-containing regimen experience sustained body-weight increases for a period possibly double that of patients treated with d4T-containing regimens.

Fig 5

Southern Africa: Plots of the weight change and its first derivative (top row) and PS-SiZer Maps (bottom row), for d4T-containing and non-d4T-contiainging ART regimens (left and right column respectively).

To reach a conclusion about the comparison of the durability of weight changes in the d4T-containing versus not-containing ART regimens, we choose the PS-SiZer analysis (top-row of the PS-SiZer map) at the optimum level of smoothing for the Southern Africa IeDEA region (Fig 5). This analysis shows that weight in patients treated with d4T-containing ART regimens increased rapidly after ART initiation and plateaued afterwards. Consulting the first derivative (Fig 5 top-row right panel), we observed that the 95% CI of the curve includes zero after 59.9 weeks in the group of patients who received a d4T-containing regimen compared to 133.8 weeks for patients treated with non-d4T-containing regimens. A numerical summary of these results is shown in the first row of Table 3.

Table 3

Estimated weeks at which HIV-patients experienced non-increasing weight.

	Durability of weight gain
	Weeks after ART start (95% confidence interval)
IeDEA Region	d4T-based regimen	Non-d4T regimen
Southern Africa	59.92 (57.56, 62.27)	133.82 (131.08, 136.56)
East Africa	52.92(50.76, 55.08)	84.88 (80.57, 89.19)
West Africa	43.94 (39.43, 48.45)	92.87 (86.59, 99.14)
Central Africa	61.92 (54.86, 68.98)	60.92 (53.23, 68.37)
Asia-Pacific	38.94 (34.45, 43.43)	54.92 (46.69, 63.15)

Similar analyses are presented in S1–S4 Figs for the East-Africa, West Africa, Central-Africa, and Asia-Pacific IeDEA regions respectively. The SiZer maps corresponding to the East and West Africa regions are very similar. For d4T-containing regimens (panel a1 in S1 and S2 Figs), blue areas are followed by purple areas after about 50 weeks for most levels of smoothing, indicating significantly increasing weight during this period. After this point, weight gain diminishes. By contrast, the blue areas in the SiZer maps corresponding to the non-d4T-containing regimens (S1 and S2 Figs) extend past week 80, indicating that weight continues to increase past 80 weeks after initiation of ART. Analyses at the optimum smoothing level produced the estimated curves of weight measurements are shown in S1 and S2 Figs and rows 2 and 3 in Table 3. For East Africa, results at the optimal smoothing levels showed that the weight in patients treated with d4T-containing regimens did not significantly increase after 52.9 weeks compared to 84.9 weeks for patients treated with non-d4T-containing regimens.

For West Africa, the results are similar, with d4T-containing regimens estimated to weight gain for 43.9 weeks versus 92.9 weeks for the non-d4T-containing regimens.

Results were similar in analyses from data in the Central-Africa and Asia-Pacific IeDEA regions (S3 and S4 Figs respectively) but the differences between the two regimens were less pronounced. Analyses of data from the Central Africa IeDEA region are shown in S3 Fig and in row 4 of Table 3. The estimated duration of weight increases in the Central Africa region was 61.9 weeks for d4T-containing ART regimens versus 60.9 weeks for non-d4T-containing regimens.

Results from the analyses of data in the Asia Pacific IeDEA region are presented in Fig 3 and row 5 of Table 3 The estimated duration of weight gain in d4T-containing regimens was 38.9 weeks versus 54.9 weeks in non-d4T-containign regimens.

Discussion

This paper presents a significant extension of the SiZer methodology, the penalized SiZer or PS-SiZer. Current SiZer methods, such as the standard LL-SiZer [5] and SS-SiZer [3] do not account for the correlation induced by repeated measurements obtained on the same patient, which invariably arise in longitudinal settings with particular frequency in biomarker data. In addition, to developing a SiZer variant which can take into account within-subject correlation, our efforts were also centered on developing computationally efficient methods to address analyses involving massive databases from tens of thousands of subjects and millions of individual measurements.

The fundamental motivation of the originally proposed SiZer map is to detect the underlying features in the data and present a global visualization of changes in quantitative data for a spectrum of smoothing levels. The key goal of this research is to show propose a SiZer variant which can detect more real features in data in the context of data collected repeatedly from the same subjects at irregular time points longitudinally. From the simulation results, it was evident that both the standard LL-SiZer formulation and the SS-SiZer method, while able to detect large dominant features in the data, missed more subtle features, because neither method appropriately addresses within-subject variability. This results in wider confidence intervals and a diminished sensitivity when features in the data become attenuated (i.e., smaller changes from increases to decreases or vice versa).

Marron & Zhang [3] have also attempted to compare these two maps by carrying out a number of simulations studies. The authors concluded that the original local linear version (here, LL-SiZer) of the SiZer and the smoothing spline SiZer (here, SS-SiZer) often performed similarly, without one method dominating the other in all cases. Similar findings were observed in our own simulation studies. By contrast, the PS-SiZer maps identified more underlying features in the simulation data than the other two SiZer map methods at a wide range of smoothing levels. In addition, both LL-SiZer and SS-SiZer detected a plateau in the data later compared to the PS-SiZer map, which detected the plateau almost exactly at the true time that it occurred in the simulated data. The simulation studies thus clearly demonstrate that, at a wide range of smoothing levels, PS-SiZer was more sensitive to small features in the data than the other two methods, presumably due to its improved ability to account for the presence of correlation in the data. More recently, Chen & Wang[25] proposed a new method for using the penalized spline approach for functional mixed effects models with varying coefficients. Their focus is different from our approach, which is used for the discovery of features in the underlying population regression function, by expanding the applicability of the SiZer approach to longitudinal designs where the P-spline model of Eilers & Marx [13] is used to estimate the population regression curve.

The main idea of SiZer maps is to detect significant changes in the data by mapping areas where the 95% confidence intervals of the first derivative is significantly different from zero. The combination of the penalized spline regression model with random intercepts in the PS-SiZer map results in narrower confidence intervals, which, in turn, lead to more sensitive detection of even less prominent features present in the data compared to standard SiZer maps. In summary, PS-SiZer is a reasonably accurate addition to the family of SiZer map methods particularly when analyzing data from longitudinal settings.

In the application of the PS-SiZer methodology, we analyzed a database involving more than 185,000 adult HIV-infected patients and well over two million longitudinal weight measurements. Our ability to handle such a large data set, underlines the computational advantages of the proposed methodology. In addition to a global visualization of the data, the PS-SiZer analysis produced meaningful clinical results by showing that the durability of weight gain experienced by after starting ART with regimens containing d4T is likely significantly shorter than among persons who start ART with regimens which do not contain d4T.

Specifically, within the Southern Africa region, weight increases in the former regimens were observed to end after about 60 weeks from initiating of ART compared to almost 133 weeks (2.5 years) among patients who started ART with regimens not containing d4T. While the clinical importance of this finding is less pronounced, given the almost universal phasing out of stavudine as a first-line regimen, weight gain among people living with HIV is a relevant topic, particularly with the wide adoption of integrase inhibitors and dolutegravir in particular, as main line antiretroviral therapies, all of which are known to result in significant weight gain in these patients [26-28].

These analyses underscore the power of the methodology to detect meaningful features in the data and can address similar questions with other biomarkers, particularly in situations where normalization of the marker is of significant clinical importance. For example, the durability of increases in CD4-positive T-lymphocytes after ART initiation [29] or normalization of inflammatory factors [30] is of major clinical significance in the setting of antiviral treatment of people living with HIV as are numerous other cases of biomarkers, where the timing of normalization of the marker following initiation of therapy can be estimated by the PS-SiZer based on repeatedly obtained measures obtained on the same subjects over time.

East Africa: Plots of the weight change and its first derivative (top row) and PS-SiZer maps (bottom row), for d4T-containing and non-d4T-contiainging ART regimens (left and right column respectively).

(TIF)

Click here for additional data file.

West Africa: Plots of the weight change and its first derivative (top row) and PS-SiZer maps (bottom row), for d4T-containing and non-d4T-contiainging ART regimens (left and right column respectively).

(TIF)

Click here for additional data file.

Central Africa: SiZer maps (top row) and plots of the weight change and its first derivative (bottom row), for d4T containing and non-d4T-contiainging ART regimens (left and right column respectively).

(TIF)

Click here for additional data file.

Asia Pacific: Plots of the weight change and its first derivative (top row) and PS-SiZer maps (bottom row), for d4T-containing and non-d4T-contiainging ART regimens (left and right column respectively).

(TIF)

Click here for additional data file.

10 Sep 2019

PONE-D-19-18970

SiZer Map to investigate significant features of body-weight profile changes in HIV infected patients in the IeDEA Collaboration

PLOS ONE

Dear Dr. Yiannoutsos,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

We would appreciate receiving your revised manuscript by Oct 25 2019 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter.

To enhance the reproducibility of your results, we recommend that if applicable you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). This letter should be uploaded as separate file and labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. This file should be uploaded as separate file and labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. This file should be uploaded as separate file and labeled 'Manuscript'.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

We look forward to receiving your revised manuscript.

Kind regards,

Ram Chandra Bajpai, Ph.D.

Academic Editor

PLOS ONE

Journal Requirements:

1. When submitting your revision, we need you to address these additional requirements.

Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

http://www.journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and http://www.journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2. In your Methods section, please provide additional information about the data set used fro illustration (for example, citing the appropriate references, and describing the data sets in more detail).

3. Our internal editors have looked over your manuscript and determined that it may be within the scope of our Mathematical Modelling of Infectious Disease Dynamics Call for Papers. The Collection will encompass a diverse range of research articles on using mathematical models to better understand infectious diseases. Additional information can be found on our announcement page: https://collections.plos.org/s/mathematical-disease-dynamics. If you would like your manuscript to be considered for this collection, please let us know in your cover letter and we will ensure that your paper is treated as if you were responding to this call. If you would prefer to remove your manuscript from collection consideration, please specify this in the cover letter. " 2) please request the following from the authors and do not ping for follow up: "Please note that PLOS ONE has specific guidelines on software sharing (http://journals.plos.org/plosone/s/materials-and-software-sharing#loc-sharing-software) for manuscripts whose main purpose is the description of a new software or software package. In this case, new software must conform to the Open Source Definition (https://opensource.org/docs/osd) and be deposited in an open software archive. Please see http://journals.plos.org/plosone/s/materials-and-software-sharing#loc-depositing-software for more information on depositing your software.

4. We note that you have indicated that data from this study are available upon request. PLOS only allows data to be available upon request if there are legal or ethical restrictions on sharing data publicly. For information on unacceptable data access restrictions, please see http://journals.plos.org/plosone/s/data-availability#loc-unacceptable-data-access-restrictions.

In your revised cover letter, please address the following prompts:

a) If there are ethical or legal restrictions on sharing a de-identified data set, please explain them in detail (e.g., data contain potentially identifying or sensitive patient information) and who has imposed them (e.g., an ethics committee). Please also provide contact information for a data access committee, ethics committee, or other institutional body to which data requests may be sent.

b) If there are no restrictions, please upload the minimal anonymized data set necessary to replicate your study findings as either Supporting Information files or to a stable, public repository and provide us with the relevant URLs, DOIs, or accession numbers. Please see http://www.bmj.com/content/340/bmj.c181.long for guidelines on how to de-identify and prepare clinical data for publication. For a list of acceptable repositories, please see http://journals.plos.org/plosone/s/data-availability#loc-recommended-repositories.

We will update your Data Availability statement on your behalf to reflect the information you provide.

5. Thank you for stating the following in the Acknowledgments Section of your manuscript:

[Data collection was funded by the NIH National Institute of Allergies and Infectious

Diseases (NIAID). Samiha Sarwat was supported in part by Grant Number TL1 TR000162 (A.

Shekhar, PI) from the National Institutes of Health, National Center for Advancing Translational

Sciences, Clinical and Translational Sciences Award]

We note that you have provided funding information that is not currently declared in your Funding Statement. However, funding information should not appear in the Acknowledgments section or other areas of your manuscript. We will only publish funding information present in the Funding Statement section of the online submission form.

* Please remove any funding-related text from the manuscript and let us know how you would like to update your Funding Statement. Currently, your Funding Statement reads as follows:

[All authors' institutions received direct or indirect research funding by the National Institutes of Health supporting this study.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript and no funds were directly provided to any of the authors.]

6. Thank you for stating the following in the Competing Interests section:

[The authors have declared that no competing interests exist.].

* We note that one or more of the authors are employed by a commercial company: 'Bayer U.S., LLC'.

Please provide an amended Funding Statement declaring this commercial affiliation, as well as a statement regarding the Role of Funders in your study. If the funding organization did not play a role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript and only provided financial support in the form of authors' salaries and/or research materials, please review your statements relating to the author contributions, and ensure you have specifically and accurately indicated the role(s) that these authors had in your study. You can update author roles in the Author Contributions section of the online submission form.

Please also include the following statement within your amended Funding Statement.

“The funder provided support in the form of salaries for authors [insert relevant initials], but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The specific roles of these authors are articulated in the ‘author contributions’ section.”

If your commercial affiliation did play a role in your study, please state and explain this role within your updated Funding Statement.

2. Please also provide an updated Competing Interests Statement declaring this commercial affiliation along with any other relevant declarations relating to employment, consultancy, patents, products in development, or marketed products, etc.

Within your Competing Interests Statement, please confirm that this commercial affiliation does not alter your adherence to all PLOS ONE policies on sharing data and materials by including the following statement: "This does not alter our adherence to  PLOS ONE policies on sharing data and materials.” (as detailed online in our guide for authors http://journals.plos.org/plosone/s/competing-interests) . If this adherence statement is not accurate and  there are restrictions on sharing of data and/or materials, please state these. Please note that we cannot proceed with consideration of your article until this information has been declared.

* Please include both an updated Funding Statement and Competing Interests Statement in your cover letter. We will change the online submission form on your behalf.

Please know it is PLOS ONE policy for corresponding authors to declare, on behalf of all authors, all potential competing interests for the purposes of transparency. PLOS defines a competing interest as anything that interferes with, or could reasonably be perceived as interfering with, the full and objective presentation, peer review, editorial decision-making, or publication of research or non-research articles submitted to one of the journals. Competing interests can be financial or non-financial, professional, or personal. Competing interests can arise in relationship to an organization or another person. Please follow this link to our website for more details on competing interests: http://journals.plos.org/plosone/s/competing-interests

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: In this manuscript, the authors presented an improved method for SiZer, PS-SiZer, which aims at increasing sensitivity and accuracy in detecting relevant features. On simulated data, PS-SiZer has better sensitivity than existing methods. On real data, PS-SiZer can find useful features and biological inferences can be made through these features. The manuscript will be good for publication if the authors solve the following issues:

1. In Table 1, the authors show that PS-SiZer detect a much higher fraction of features than the other two methods. In Figure 2, the authors show that PS-SiZer can detect all features in the simulated study. In Figure 3, the estimated time to reach the plateau is much smaller in PS-SiZer than in other two methods. Taking these results together, is PS-SiZer overly sensitive? If not, can the authors perform another simulation to prove that?

2. The authors need to increase the resolution of figure 1a and 1b, and make figure 1c axis ticks bigger. There are also some typos, e.g. in page 10 "a family of smoonth".

Reviewer #2: Manuscript ID: PONE-D-19-18970

Title: SiZer Map to investigate significant features of body-weight profile changes in HIV infected patients in the IeDEA Collaboration

Summary:

The paper extends the method of SiZer maps to detect the time reaching a plateau when analyzing irregular longitudinal data by using penalized spline regression model (PS-SiZer).

Strengths:

1. It is nice to realize how the penalized spline regression model can be converted into the mixed-model framework to increase the computational efficiency.

2. Methodology development is well described from model specification, inference, estimate of first derivate, to confidence band.

3. Two simulation studies indicate better performance of PS-SiZer in detecting peaks and the time at peaks.

Comments:

1. Please clarify what Effective Degrees of Freedom (EDF) is and the purpose.

2. It is unclear how the model addresses the irregular longitudinal data.

3. The method section indicates the use of R package, mgcv::gam, as mixed model to estimate g(x) function, but data analysis used R package SiZer. Please clarify how g(x) function is estimated.

4. Is any difference of the proposed method vs “A Penalized Spline Approach to Functional Mixed Effects Model Analysis” by Chen and Wang (Biometrics, 2010)?

5. Please illustrate more how the optimum smoothing parameter is estimated using the ‘Rule of thumb’ approach.

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org. Please note that Supporting Information files do not need this step.

13 Dec 2019

Response to reviewers in attached file.

Regarding any conflict of interest of Dr. Sarwat, the work was completed during her doctoral research at Indiana University and prior to the commencement of her employment with Bayer. She has no conflicts and no commercial interests related to this work and this work is unrelated to her present employment.

As stated in the cover letter, we would like this manuscript to be considered for the special issue of Modeling in Infectious Diseases Call for Papers.

Submitted filename: PS-SiZer_ResponsesToTheReviewers.docx

Click here for additional data file.

31 Jan 2020

PONE-D-19-18970R1

PS-SiZer Map to investigate significant features of body-weight profile changes in HIV infected patients in the IeDEA Collaboration

PLOS ONE

Dear Dr. Yiannoutsos,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Reviewer have highlighted that authors did not incorporate previous comments. Therefore, authors should clearly mention that how they have  incorporated those comments in the manuscript.

We would appreciate receiving your revised manuscript by Mar 16 2020 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter.

To enhance the reproducibility of your results, we recommend that if applicable you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). This letter should be uploaded as separate file and labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. This file should be uploaded as separate file and labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. This file should be uploaded as separate file and labeled 'Manuscript'.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

We look forward to receiving your revised manuscript.

Kind regards,

Ram Chandra Bajpai, Ph.D.

Academic Editor

PLOS ONE

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: The responses were not incorporated in the revised manuscript. Here is one example for illustration.

1. Please clarify what Effective Degrees of Freedom (EDF) is and the purpose.

Response: We have clarified that effective degrees of freedom encapsulate the complexity of the model as the actual degrees of freedom are not defined for the semiparametric models. We use the established method for their estimation, i.e. the trace of the smoother matrix (see Hastie & Tibshirani (1990) as cited in Chauduri & Marron (1999, pp. 812)).

The revised manuscript did not have any change to better explain EDF.

By making the SiZer maps comparable at a similar level of “Effective Degrees of Freedom” (EDF). For this reason all three SiZer maps (PS-SiZer with LL-SiZer and SS-SiZer) were generated with the same range of EDFs.

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org. Please note that Supporting Information files do not need this step.

3 Feb 2020

The point about the EDFs is well taken and we apologize for having complicated the review. We have added the text into the body of the revised paper. To facilitate with identifying the changes made in the document, we have added comments in the margins where the precise response and the responded-to comment were identified.

Submitted filename: PS-SiZer_ResponsesToTheReviewers-with-clarification.docx

Click here for additional data file.

26 Feb 2020

PS-SiZer Map to investigate significant features of body-weight profile changes in HIV infected patients in the IeDEA Collaboration

PONE-D-19-18970R2

Dear Dr. Yiannoutsos,

We are pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it complies with all outstanding technical requirements.

Within one week, you will receive an e-mail containing information on the amendments required prior to publication. When all required modifications have been addressed, you will receive a formal acceptance letter and your manuscript will proceed to our production department and be scheduled for publication.

Shortly after the formal acceptance letter is sent, an invoice for payment will follow. To ensure an efficient production and billing process, please log into Editorial Manager at https://www.editorialmanager.com/pone/, click the "Update My Information" link at the top of the page, and update your user information. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, you must inform our press team as soon as possible and no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

With kind regards,

Ram Chandra Bajpai, Ph.D.

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: (No Response)

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #2: No

25 Mar 2020

PONE-D-19-18970R2

PS-SiZer Map to investigate significant features of body-weight profile changes in HIV infected patients in the IeDEA Collaboration

Dear Dr. Yiannoutsos:

I am pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please notify them about your upcoming paper at this point, to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

For any other questions or concerns, please email plosone@plos.org.

Thank you for submitting your work to PLOS ONE.

With kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Ram Chandra Bajpai

Academic Editor

PLOS ONE