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: (1) Here, g(x) is some “smooth” regression function that needs to be estimated from the data and ϵ_i 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_λ(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_λ(x) and its derivative, . A common estimate of g_λ(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_λ-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_λ(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_λ 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_λ(x). Here, the smoothing spline function g_λ(x) is a natural cubic spline with knots at data locations x₁ … x_n. 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:

1. Adding a random intercept component to summarize subject-specific correlation
2. Applying a P-spline [13] as the underlying smoothing function
3. Constructing a simultaneous 95% confidence limit addressing multiple-comparison issues

Model specification

Let y_ij denote measurements on subject i = 1, 2, … n at time x_ij, j = 1,2…n_i. We model the responses as, (2) where g_λ(x_ij) is a smooth function indexed by a smoothing parameter λ and ε_ij 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_ij 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_λ. 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_λ. Let B_m(x_ij;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_λ: where a_m is a vector of coefficients, and B_m(x_ij;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_d is a matrix such that Δ_d constructs the vector of d^th difference of the coefficients a i.e., Δa_m = a_m – a_{m - 1}; Δ²a_m - 2a_{m - 1} + a_{m - 2} 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 (3) 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_d 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, (4) where and .

The model in (4) above has three components. Xβ represents the fixed overall effect while Z_Bu corresponds to the smoothing function and b_i, 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_λ(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_REML) ± 2, where λ_REML 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 = [X_x Z_Bx].

Estimate and variability bands of the derivatives

The derivatives of the smooth function g_λ(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_min, λ_max 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_REML 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_λ(·) on a grid of x-values, and define A 100(1 – α)% simultaneous confidence band for g_λ(x_grid) is where is the corresponding empirical best linear unbiased predictor (EBLUP) obtained from the mixed model framework. Here, q_λ(1 – α) is the (1 – α) quantile of the random variable at a smoothing level λ, i.e., (5) which is the supremum on the set . The quantile q_λ(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_λ(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: (6)

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:

1. 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.
2. 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.
3. 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_i chosen to be equally spaced in the interval [0, 1] with where is random noise, is the subject-specific random intercept and x_ij 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_i = 10 for i = 1, …N. For each simulated trial, three different SiZer maps were generated at 100 levels of EDF.

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Table 1. Finding features: Simulation study-1 with varying variability.

https://doi.org/10.1371/journal.pone.0220165.t001

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).

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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.

https://doi.org/10.1371/journal.pone.0220165.g002

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_i equally spaced in [1:20) with where x_ij 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_i = 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.

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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.

https://doi.org/10.1371/journal.pone.0220165.g003

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.

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Fig 4. Boxplot-summary of three SiZer maps: Time to detect a true plateau in the data.

https://doi.org/10.1371/journal.pone.0220165.g004

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.

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Table 2. Summary of baseline characteristics-IeDEA study by d4T and non-d4T based regimen.

https://doi.org/10.1371/journal.pone.0220165.t002

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.

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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).

https://doi.org/10.1371/journal.pone.0220165.g005

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.

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Table 3. Estimated weeks at which HIV-patients experienced non-increasing weight.

https://doi.org/10.1371/journal.pone.0220165.t003

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.