Semiparametric regression modeling of the global percentile outcome

Abstract

When no single outcome is sufficient to capture the multidimensional impairments of a disease, investigators often rely on multiple outcomes for comprehensive assessment of global disease status. Methods for assessing covariate effects on global disease status include the composite outcome and global test procedures. One global test procedure is the O’Brien’s rank-sum test, which combines information from multiple outcomes using a global rank-sum score. However, existing methods for the global rank-sum do not lend themselves to regression modeling. We consider sensible regression strategies for the global percentile outcome (GPO), under the transformed linear model and the monotonic index model. Posing minimal assumptions, we develop estimation and inference procedures that account for the special features of the GPO. Asymptotics are established using U-statistic and U-process techniques. We illustrate the practical utilities of the proposed methods via extensive simulations and application to a Parkinson’s disease study.

1. Introduction

For many diseases with multifaceted symptoms and deteriorations, there does not exist a single outcome that can comprehensively quantify the global disease severity. One example is the NINDS Exploratory Trials in Parkinson’s Disease (NET-PD) Long-term Study 1 (LS-1), a multicenter, placebo-controlled clinical trial. The primary goal was to examine whether the nutritional supplement creatine could slow the progression of PD (Kieburtz et al., 2015). However, it is well known that PD is complex and multidimensional, and no single outcome can provide a thorough summary of the disease severity or progression. The NET-PD LS-1 steering committee defined the primary outcome as the rank-sum of five individual measures, namely change from baseline to five years in Schwab England Activities of Daily Living, 39-Item Parkinson’s Disease Questionnaire, Ambulatory Capacity, Symbol Digit Modalities, and the year-5 measurement of modified Rankin Scale (Elm and Investigators, 2012; Kieburtz et al., 2015). Situations when a single primary outcome is insufficient are also common for other diseases with complex symptoms, such as stroke (Tilley et al., 1996), Alzheimer’s disease (Lim et al., 2016), and heart failure (Felker and Maisel, 2010).

When scientific interest centers on an overall assessment of the global disease status, one may consider a combined analysis of the outcomes, instead of separate analyses of each outcome (Tilley et al., 2014; Pocock et al., 1987). One approach to achieve a combined analysis is by first integrating the multiple outcomes into a univariate composite outcome. When all the individual outcomes are numeric, composite scores are often formed as a linear combination of the outcomes through summation, averaging, or the composite z or t score. However, when some of the individual outcomes are skewed or involve outliers, the meaningfulness of such composite scores may be undermined (Strauss et al., 2006; Vanderploeg, 2014).

Another approach to achieve a combined analysis is to adopt a global test procedure. For continuous outcomes, the Hotelling’s T² is analogous to the two-sample t-test and evaluates the equivalence in the mean vector of the multiple outcomes. O’Brien (1984) proposed a global statistical test (GST) by first transforming the multiple outcomes to a univariate rank-sum score. Subjects are first ranked with respect to each individual outcome; the rank-sum score is then calculated by summing up the outcome-specific ranks for each subject and then compared between different groups. The GST was shown to be robust to skewed distributions and was often more powerful than the Hotelling’s T² (O’Brien, 1984). Huang et al. (2005) examined the theoretical properties of the GST and extended the methods to Behrens-Fisher problems. Ramchandani et al. (2016) studied a global rank test based on different composite scoring functions for multiple, possibly censored, outcomes.

The rank-sum GST has been successfully applied in many clinical studies (Fereshtehnejad et al., 2015; Simuni et al., 2015; Hinson et al., 2017; Luo et al., 2016), including the LS-1 study (Kieburtz et al., 2015). However, existing statistical methods for the global rank-sum are mainly designed for two-sample or K-sample hypothesis testings. They do not accommodate the identification of essential prognostic factors of the overall disease status from a collection of patient characteristics, the evaluation of global treatment efficacy while adjusting for confounders, or the examination of any potential moderators of treatment efficacy. Such analytical tasks are often of interest in clinical settings (Pocock et al., 1987) and would be greatly facilitated by solid statistical tools for regression modeling.

In this paper, we seek to fill this gap by developing sensible regression strategies. While the global rank-sum score is not directly interpretable, we focus on a global percentile outcome (GPO), defined as the average of the subjects’ percentile rank for each outcome. As such, the GPO ranges between 0 and 1 and facilitates easy interpretation. Considering the formulation of this global percentile outcome, it is important to adopt modeling frameworks that require weak restrictions on the distribution of the GPO and on the form of the relationship between the GPO and covariates. We consider herein the transformed linear model and the monotonic index model (Han 1987, Sherman 1993, Cavanagh and Sherman 1998, Fan et al. 2017), both of which are semiparametric and pose relaxed model assumptions. Research efforts have been devoted to estimation methods under these modeling frameworks, but existing works are only designed for a single outcome. The GPO, however, is a composite outcome formed by multiple individual outcomes. As we shall show later, the GPO is not directly observable and needs to be estimated, and the estimated GPO is no longer independent across subjects. As such, existing regression strategies do not account for the special feature of the GPO. Moreover, even with a single outcome, there remain some challenges in variance estimation and inference under the monotonic index model. Existing variance estimation methods often require smoothing, which can cause uncertainty due to the need to specify a bandwidth parameter.

In this work, our goal is to develop inferential strategies for the GPO under the flexible transformed linear model and the monotonic index model. In Section 2, we propose estimation and inference procedures that will account for the special features of the GPO. We first consider the situation when all individual outcomes are continuous, then accommodate scenarios when some of the individual outcomes involve ties or have a discrete distribution. The proposed methods are rigorously justified by theoretical results on consistency and asymptotic normality, and we develop consistent variance estimation procedures. The numerical performance of the proposed methods is evaluated via extensive simulations in Section 3. In Section 4, we apply the proposed methods to the NET-PD LS-1 study to examine the risk factors on global disease progression.

2. Methods

2.1. Formulation of the global percentile outcome

To illustrate the motivation, we first consider a commonly used transformed linear model for a single outcome Y_i,

$$f\left(Y_i\right)={\tilde{X}}_i^{\top }{\beta }_{L0}+{\epsilon }_i,$$

where ${\tilde{X}}_i={\left(1,X_i^{\top }\right)}^{\top },X_i={\left(X_{i1},X_{i2},\dots ,X_{ip}\right)}^{\top }$ is a vector of prognostic factors for subject i (i = 1, 2,…, n), f(·) is a monotonic increasing link function, and errors ε_i’s are of mean 0 and assumed to be independent, and identically distributed (i.i.d.). Note that this model can be equivalently written as

$$h\left({\mathcal{P}}_i\right)={\tilde{X}}_i^{\top }{\beta }_{L0}+{\epsilon }_i,$$ (1)

where ${\mathcal{P}}_i=F\left(Y_i\right)$ is a random variable representing the percentile level of Y_i in the study population, F(·) is the marginal cumulative distribution function (CDF) of Y_i, and h(·) is the marginal quantile function of f(Y_i).

Our focus lies in the situation when there are M outcomes Y₁, …, Y_M, each with marginal CDF F_m(y) = P(Y_m ≤ y), m = 1, 2, …, M, where M is an integer that does not grow with the sample size. Without loss of generality, suppose that higher values represent worse disease severity for all outcomes. To integrate information from these M outcomes, we propose a global percentile outcome (GPO) as

$${\mathcal{P}}_i=M^{-1}\left\{F_1\left(Y_{i1}\right)+F_2\left(Y_{i2}\right)+\cdots +F_M\left(Y_{iM}\right)\right\},$$ (2)

which represents the composite percentile across M outcomes for subject i.

If some of the individual outcomes are deemed more clinically important than the other outcomes, one could pre-specify weights based on prior knowledge and define a weighted version of the GPO as ${\mathcal{P}}_i={\sum }_{m=1}^M{w}_m{F}_m\left(Y_{im}\right)$, where ${\sum }_{m=1}^M{w}_m=1\text{ and }{w}_m>0\text{ for }m=1,2,\dots ,M$. Without loss of generality, we focus on the unweighted definition of the GPO below, which corresponds to the situation with w_m ≡ 1/M, but all estimation and inference procedures also apply to the weighted version.

GPO ranges between [0, 1], and bears straightforward interpretation as the global level of subject i in the study population, when all M outcomes are considered simultaneously. When each of the M outcomes captures valuable information about certain aspects of the disease, but none of them is conclusive, the GPO effectively combines the information from them to achieve comprehensive assessments. Further, the GPO is robust to outliers and/or skewed distributions and invariant to monotone transformations. For now, we consider situations when all Y_m’s are continuous outcomes and do not involve ties. The situation when some of these outcomes involve ties or have a discrete distribution will be discussed in Section 2.5.

As the average of M uniformly distributed random variables, the distribution of ${\mathcal{P}}_i$ is generally non-uniform and depends on M as well as the association between Y₁, …, Y_M. See the left column of Fig. 1 for the histogram of ${\mathcal{P}}_i$ formed by three correlated outcomes, Y₁, Y₂ and Y₃, that follow the Frank copula (top row) or the Clayton copula (bottom row). As it is challenging to correctly specify the distribution of the GPO, it is desirable to adopt modeling strategies that rely on weak distributional assumptions.

Fig. 1.

Histograms of ${\mathcal{P}}_i$ and $h\left({\mathcal{P}}_i\right)$, where ${\mathcal{P}}_i$ is formed by three correlated continuous outcomes that follow the Frank copula (top panel) or the Clayton copula (bottom panel).

In practice, the true marginal CDFs of the Y_m’s in the population are often unknown, preventing us from observing the true ${\mathcal{P}}_i$. We can derive an empirically estimated $\widehat{{\mathcal{P}}_i}$ as

$$\begin{gathered} {\widehat{\mathcal{P}}}_i=M^{-1}\left\{{\widehat{F}}_{1n}\left(Y_{i1}\right)+{\widehat{F}}_{2n}\left(Y_{i2}\right)+\cdots +{\widehat{F}}_{Mn}\left(Y_{iM}\right)\right\} \\ =\frac{1}{n\times M}\sum _{j=1}^n\left\{I\left(Y_{j1}\le Y_{i1}\right)+I\left(Y_{j2}\le Y_{i2}\right)+\cdots +I\left(Y_{jm}\le Y_{iM}\right)\right\}, \end{gathered}$$ (3)

where ${\widehat{F}}_{mn}(\cdot )$ denotes the empirical CDF of Y_m, m = 1, 2, …, M. Also note that ${\widehat{\mathcal{P}}}_i$ is proportional to the global rank-sum in O’Brien (1984).

2.2. Transformed linear model

We first consider a transformed linear model (1), where ε_i has an unspecified distribution with zero mean and finite variance. The monotonic increasing link function h(·) can be pre-specified as the identity link or other commonly used link functions. The choice of h(·) is relevant to the model mis-specification and estimation efficiency. In preliminary simulations, we find that the quantile function of a truncated normal distribution with zero mean and variance 1, truncation points ±3, is a sensible choice of h(·). Other link functions, such as the logit link, may also be considered. Fig. 1 compares the histogram of ${\mathcal{P}}_i$ and $h\left({\mathcal{P}}_i\right)$ for correlated outcomes when h(·) is the truncated normal link.

Under model (1) and with a pre-specified h(·), we adopt the following estimating equation to obtain the estimator of regression coefficient, denoted as ${\widehat{\mathit{\beta }}}_L$,

$$\frac{1}{n}\sum _{i=1}^n{\tilde{X}}_i\left\{h\left({\widehat{\mathcal{P}}}_i\right)-{\tilde{X}}_i^{\top }{\widehat{\beta }}_L\right\}=0.$$ (4)

While the equation can be easily solved using standard software, it is not straightforward to derive the asymptotic distribution and variance of ${\widehat{\beta }}_L$. This is because ${\widehat{\mathcal{P}}}_i’s$ are constructed based on the empirical CDFs, which use all the observations. As such, ${\widehat{\mathcal{P}}}_i’s$ are not independent across subjects. In Supplemental Material A, we use the technique of Hájek projection for U-statistics to sort out the asymptotic distribution of ${\widehat{\beta }}_L$, which leads to the theorem below.

Theorem 1. Under model (1) and regularity conditions A1–A3 in Supplemental Material A, we have ${\widehat{\beta }}_L\stackrel{p}{\to }{\beta }_{L0}$ and

$$\sqrt{n}\left({\widehat{\beta }}_L-{\beta }_{L0}\right)\stackrel{d}{\to }\mathit{\text{Normal}}\left(0,A^{-1}B{A}^{-\top }\right)$$

where $A=-E\left({\tilde{X}}_i{\tilde{X}}_i^{\top }\right),B=E\left\{\left(s_{1i}+s_{2i}\right){\left(s_{1i}+s_{2i}\right)}^{\top }\right\}$, $s_{1i}=E\left\{{\tilde{X}}_j{h}^{\prime }\left({\mathcal{P}}_j\right){\xi }_{ji}\mid {\mathcal{Z}}_i\right\}$, and $s_{2i}={\tilde{X}}_i\left\{h\left({\mathcal{P}}_i\right)-{\tilde{X}}_i^{\top }{\beta }_L\right\}$.

The detailed proof, coupled with the definition of ξ_ji and ${\mathcal{Z}}_i$, can be found in Supplemental Material A. For matrix B, s_2i represents the variability in the estimating function when ${\mathcal{P}}_i$ is known, and s_1i corresponds to the additional variability due to the estimated $\widehat{{\mathcal{P}}_i}$. The sandwich-type variance covariance matrix can be estimated explicitly, by replacing A and B with their respective empirical counterparts.

2.3. Monotonic index model

The transformed linear model features easy implementation and convenient variance estimation. However, one disadvantage is that it requires a prespecified link function. In practice, there is often little prior knowledge to inform the selection of h(·), and mis-specification of this link function may lead to misleading inference. To avoid the specification of h(·), we consider a monotonic index model (Han, 1987; Cavanagh and Sherman, 1998) for GPO as follows,

$${\mathcal{P}}_i=\zeta \left(X_i^{\top }{\beta }_0,{\epsilon }_i\right).$$ (5)

Here, ε_i is an i.i.d. error term with an unspecified distribution, and ζ(a₁, a₂) is an unspecified bivariate function that is monotonically increasing in both components. The regression coefficient β₀ links covariates explicitly to the GPO under minimal assumptions. Model (5) includes many commonly used models as special cases, such as model (1) which corresponds to ζ(a₁, a₂) = h⁻¹(a₁ + a₂). Because of the unspecified ζ(·), we require the that ∥β∥² = 1 to ensure model identifiability. Therefore, number of unknown parameters in β₀ equals p − 1, where p is the dimension of X. Without loss of generality, we shall treat θ ≡ (β₂, …, β_p)^⊤ as free parameters below and let θ₀ ≡ (β₂₀, …, β_p0)^⊤.

Considering the monotonicity of the link function, we extend the rank estimation strategies (Han, 1987; Cavanagh and Sherman, 1998) for parameter estimation and set the objective function as

$$G_n(\beta )=\frac{1}{n(n-1)}\sum _{k\ne i}g\left({\widehat{\mathcal{P}}}_i\right)I\left(X_i^{\top }\beta \ge X_k^{\top }\beta \right)$$ (6)

where $\sum _{k\ne i}I\left(X_i^{\top }\beta \ge X_k^{\top }\beta \right)+1$ gives the sample rank of $X_i^{\top }\beta$ among the n subjects. g(·) is a monotonic increasing function, and its choice does not affect the consistency of $\widehat{\beta }$, but affects the efficiency. We set g(·) also as the quantile function of a truncated normal distribution with zero mean, σ² = 1 and truncation points ±3, which rendered favorable numerical performance in our exploratory simulation studies.

We can obtain $\widehat{\theta }$ by maximizing (6) through optimization algorithms that accommodate non-smooth objective functions, such as the Nelder–Mead algorithm in R package optim, and then obtain the corresponding $\widehat{\beta }$. The non-intercept part of ${\widehat{\beta }}_L$ can serve as a nice initial value after being standardized to have the unit norm. To ensure locating the global maxima, it is beneficial to implement the optimization algorithm with multiple initial values, which can be obtained by adding random errors to the estimator from the transformed linear model. Moreover, since the sign of β₁ is uncertain given θ = (β₂, …, β_p)^T, we may use a covariate with known direction as X₁, after verifying this direction in the transformed linear estimator. When the direction is uncertain for all elements of X, one should implement the optimization twice, once with a positive sign and once with a negative sign for β₁, and retain the maximizer with larger fitted objective function.

Deriving the asymptotic distribution of $\widehat{\theta }$ and $\widehat{\beta }$ was complicated by the non-smoothness of G_n(·) and the dependence between ${\widehat{\mathcal{P}}}_1\dots ,{\widehat{\mathcal{P}}}_n$. To tackle these, we show that Γ_n(θ) = G_n(θ) − G_n(θ₀) is asymptotically equivalent to ${\Gamma }_n(\theta )={\overline{\Gamma }}_{1n}(\theta )+{\Gamma }_{2n}(\theta )$ in the vicinity of θ₀. The first part ${\overline{\Gamma }}_{1n}\left(\theta \right)$ is a zero-mean order-3 U process accounting for the difference between $\widehat{{\mathcal{P}}_i}$ and ${\mathcal{P}}_i$, and the second part Γ_2n(θ) is an order-2 U process involving the true ${\mathcal{P}}_i$ only. Define $\mathcal{G}(\theta )=E\left[g\left({\mathcal{P}}_1\right)I\left\{X_1^{\top }\beta (\theta )\ge X_3^{\top }\beta (\theta )\right\}\right]$, $\Gamma \left(\theta \right)=\mathcal{G}\left(\theta \right)-\mathcal{G}\left({\theta }_0\right)$, and let V be the second derivative of Γ(θ) evaluated at θ₀. We apply the U-process decomposition technique (Sherman, 1994) to show that ${\overline{\Gamma }}_n(\theta )=\frac{1}{2}{\theta }^{\top }V\theta +\frac{1}{\sqrt{n}}{\theta }^{\top }{W}_n+o_p\left(\Vert \theta {\Vert }^2\right)+o_p\left(n^{-1}\right)$ in the vicinity of θ₀, where W_n constitutes two parts that correspond to ${\overline{\Gamma }}_{1n}(\theta )$ and Γ_2n(θ) respectively and asymptotically follows a zero-mean normal distribution with variance covariance matrix Δ. Along these arguments, we obtained the following Theorem; the detailed proof is detailed in Supplemental Materials B.

Theorem 2. Under model (5) and conditions B1–B4 in Supplemental Material B, we have $\widehat{\theta }\stackrel{p}{\to }{\theta }_0$, and $\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)$ converges to a normal distribution with mean 0 and variance covariance matrix V⁻¹ΔV⁻¹.

In addition to the exact objective function in (6), we considered a smoothed objective function, which may be easier to solve numerically. Specifically, the indicator functions $I\left(X_i^{\top }\beta \ge X_k^{\top }\beta \right)$ are approximated by Φ{(X_i − X_k)β/C_n}, where C_n is a bandwidth parameter, and Φ(·) is the CDF of a standard normal distribution. One may let C_n = C × n^−1/3 for a constant C, and the resulting objective function is

$$K_n(\beta )=\frac{1}{n(n-1)}\sum _{k\ne i}g\left({\widehat{\mathcal{P}}}_i\right)\Phi \left\{\left(X_i-X_k\right)\beta /C_n\right\},$$ (7)

which can be optimized via standard optimization algorithms. We evaluate the performance of this smoothed objective function in our numerical studies.

2.4. Variance estimation for the rank estimator

Although the variance covariance matrix of $\widehat{\beta }$ is derived theoretically in Supplemental Material B, the formula involves unknown components that are difficult to estimate explicitly. This issue often arises with non-smooth objective function or estimating equations. Moreover, the standard bootstrap technique does not apply here, as it creates many ties and complicates the pairwise comparisons in the objective function. To resolve these issues, following the resampling procedures in Jin et al. (2001) and Peng and Huang (2008), we propose a perturbation algorithm to achieve a consistent variance estimation for $\widehat{\beta }$.

Let ω_i be an i.i.d. positive random variable with mean 1 and variance 1 for i = 1, 2, …, n, such as ω_i ~ Exponential(1). We propose to first perturb the estimated GPO as

$${\widehat{\mathcal{P}}}_i^{*}=\frac{1}{n}\sum _{j=1}^n{\omega }_j{M}^{-1}\left\{I\left(Y_{j1}\le Y_{i1}\right)+I\left(Y_{j2}\le Y_{i2}\right)+\cdots +I\left(Y_{jM}\le Y_{iM}\right)\right\}.$$

Next, using the same set of ${\left\{{\omega }_i\right\}}_{i=1}^n$, we perturb the objective function G_n(β) as

$$G_n^{*}(\beta )=\frac{1}{n(n-1)}\sum _{k\ne i}{\omega }_i{\omega }_{k}g\left({\widehat{\mathcal{P}}}_i^{*}\right)I\left(X_i^{\top }\beta \ge X_k^{\top }\beta \right),$$ (8)

the maxima of which is denoted by ${\widehat{\beta }}^{*}$. In practice, one can repeat the perturbation for B times to obtain ${\widehat{\beta }}^{*b}$, b = 1, 2, …, B, where B can be set as a reasonably large number such as 200 or 400. The sample variance covariance matrix of ${\left\{{\widehat{\beta }}^{*b}\right\}}_{b=1}^B$ can serve as a desirable estimator for the variance covariance matrix of $\widehat{\beta }$. The rationale of this perturbation scheme lies in the fact that the conditional distribution of ${\widehat{\beta }}^{*}$ given the observed data is asymptotically equivalent to the unconditional distribution of $\widehat{\beta }$. Formal justifications are provided in Supplemental Material C.

2.5. Handling outcomes with ties and ordinal outcomes

Up to this point, we have focused on situations where all Y_m, m = 1, 2, …, M are continuous without ties. Our estimation and inference procedures can be extended to situations where some of these outcomes involve ties or have a discrete distribution. First, consider the situation where all individual outcomes are continuous, but some of them involve ties. We can replace the empirical CDF by

$${\widehat{F}}_{mn}(y)=\frac{1}{2n}\sum _{j=1}^n\left\{I\left(Y_{jm}\le y\right)+I\left(Y_{jm}<y\right)+1\right\},$$ (9)

where ${\widehat{F}}_{mn}\left(Y_{im}\right)$ gives the rank of Y_im in {Y_1m, …, Y_nm} using the fractional ranking strategy for ties. Next, consider the situation when an individual outcome, say Y_m, is binary or ordinal. We adopt F(y) = {P(Y_m ≤ y) + P(Y_m < y)}/2 instead of the standard CDF, P(Y_m ≤ y), in the definition of GPO in (2). Similarly, in the calculation of ${\widehat{\mathcal{P}}}_i$, we use (9) in place of the empirical CDF.

It is worth noting that one may have more than one individual outcome that involves ties or has discrete distributions, as long as the resulting GPO combining all individual outcomes remains continuous. With these modifications, the previous estimation and inference procedures continue to work.

3. Simulation study

We conducted extensive simulations to examine the finite-sample performance of the proposed transformed linear estimator ${\widehat{\beta }}_L$ and rank estimator $\widehat{\beta }$. We considered four covariates, where X₁ ~ Unif (−1, 1), X₂ ~ Bernoulli(0.5), X₃ ~ Normal(0, 1²), and X₄ ~ Gamma(shape = 2, rate 2). Four correlated outcomes, Y₁, Y₂, Y₃ and Y₄, were generated from the Clayton copula with parameter 2, and marginal distributions of these outcomes are Y₁ ~ Normal(0, 1²), Y₂ ~ Exponential(1), Y₃ ~ t(df = 3), and Y₄ ~ Unif(−1, 1), respectively.

The following four different setups regarding the relationship between the GPO and covariates were considered:

• Setup 1: ${\mathcal{P}}_i=H_1\left\{X_i^{\top }{\beta }_0+{\epsilon }_i\right\}$, ε_i ~ Normal(0, 0.8²),

• Setup 2: ${\mathcal{P}}_i=H_2\left\{X_i^{\top }{\beta }_0+{\epsilon }_i\right\}$, ε_i ~ Exponential(1),

• Setup 3: ${\mathcal{P}}_i=H_3\left\{\text{exp}\left(X_i^{\top }{\beta }_0\right)+{\epsilon }_i\right\}$, ε_i ~ Unif (0.5, 1.5),

• Setup 4: ${\mathcal{P}}_i=H_4\left[\text{exp}\left(X_i^{\top }{\beta }_0\right)/\left\{1+\text{exp}\left(X_i^{\top }{\beta }_0\right)\right\}\times {\epsilon }_i\right]$, ε_i ~ Unif (0.5, 1.5),

where H_s(·) are all monotone functions for s = 1, 2, 3, 4. As such, the monotonic index model (5) is correctly specified under all four setups. The transformed linear model (1) is subject to mis-specification under all setups, and the degree of mis-specification is larger under Setups 3–4 than that under Setups 1–2. The first two setups still follow the transformed linear model, but the link function is subject to mis-specification. The true links $H_j^{-1}(x)$ under Setups 1–2 were plotted in Figure S1 of Supplemental Material D, along with the assumed link, to show the degree of mis-specification. Setups 3–4 represent more complex effect patterns than specified by the model. We let the true values β₀ = (0.5, −0.707, 0, 0.5) with ∥β₀∥² = 1.

Under each setup, we ran 1000 simulations with sample sizes of n = 200 or n = 400. For each simulated dataset, we implemented the perturbation resampling method with B = 200 to obtain the standard error estimates. R codes for implementing the proposed method can be found at https://github.com/rli1010/GPOreg. Table 1 summarizes simulation results for the proposed estimators, including empirical biases (BIAS), empirical standard deviations (ESD), average of perturbation resampling-based standard errors (ASE), and the empirical coverage probabilities (ECP) of 95% Wald-type confidence intervals and empirical rejection rates (ERR) for testing the null hypotheses $H_0^j$ : ${\beta }_0^{\left(j\right)}=0$ for j = 1, 2, 3, 4. Without loss of generality, we adopted the quantile function of the truncated normal distribution with truncation points ±3 for the h(·) and g(·) functions. To facilitate the comparisons between ${\widehat{\beta }}_L$ and $\widehat{\beta }$, we standardized the transformed linear estimator ${\widehat{\beta }}_L$ by its L2-norm; and obtained the variance estimation for the standardized estimator using the Delta method.

Table 1.

Simulation results for the transformed linear estimator and the rank estimator, where β₀ = (0.5, −0.707, 0, 0.5).

n		Transformed linear estimator					Rank estimator
		BIAS	ESD	ASE	ECP	ERR	BIAS	ESD	ASE	ECP	ERR
		Setup 1
200	${\beta }_0^{(1)}$	0.000	0.089	0.086	0.941	1.000	−0.011	0.102	0.102	0.942	0.986
	${\beta }_0^{(2)}$	−0.001	0.073	0.072	0.931	1.000	0.013	0.086	0.088	0.937	1.000
	${\beta }_0^{(3)}$	−0.002	0.059	0.057	0.942	0.058	−0.002	0.066	0.068	0.956	0.044
	${\beta }_0^{(4)}$	−0.024	0.080	0.073	0.915	1.000	−0.002	0.098	0.095	0.930	0.996
400	${\beta }_0^{(1)}$	0.003	0.063	0.062	0.929	1.000	−0.006	0.071	0.073	0.947	1.000
	${\beta }_0^{(2)}$	−0.005	0.053	0.051	0.927	1.000	0.005	0.061	0.062	0.947	1.000
	${\beta }_0^{(3)}$	0.000	0.043	0.041	0.934	0.066	0.000	0.047	0.048	0.948	0.052
	${\beta }_0^{(4)}$	−0.022	0.053	0.052	0.916	1.000	−0.003	0.066	0.068	0.950	1.000
		Setup 2
200	${\beta }_0^{(1)}$	0.007	0.078	0.077	0.944	1.000	−0.011	0.088	0.092	0.953	0.994
	${\beta }_0^{(2)}$	−0.017	0.063	0.061	0.901	1.000	0.003	0.072	0.078	0.958	1.000
	${\beta }_0^{(3)}$	0.001	0.055	0.053	0.936	0.064	0.000	0.058	0.062	0.960	0.040
	${\beta }_0^{(4)}$	−0.051	0.064	0.063	0.845	1.000	−0.007	0.082	0.087	0.953	1.000
400	${\beta }_0^{(1)}$	0.012	0.056	0.054	0.923	1.000	−0.003	0.060	0.064	0.962	1.000
	${\beta }_0^{(2)}$	−0.015	0.045	0.043	0.904	1.000	0.005	0.051	0.053	0.952	1.000
	${\beta }_0^{(3)}$	0.000	0.038	0.038	0.955	0.045	0.000	0.039	0.043	0.969	0.031
	${\beta }_0^{(4)}$	−0.044	0.044	0.044	0.806	1.000	−0.001	0.057	0.060	0.953	1.000
		Setup 3
200	${\beta }_0^{(1)}$	0.002	0.034	0.033	0.938	1.000	−0.001	0.037	0.040	0.956	1.000
	${\beta }_0^{(2)}$	−0.013	0.028	0.026	0.893	1.000	0.003	0.030	0.031	0.952	1.000
	${\beta }_0^{(3)}$	0.000	0.025	0.023	0.936	0.064	0.000	0.027	0.028	0.952	0.048
	${\beta }_0^{(4)}$	−0.025	0.028	0.027	0.828	1.000	0.001	0.035	0.036	0.959	1.000
400	${\beta }_0^{(1)}$	0.001	0.024	0.024	0.938	1.000	−0.002	0.026	0.027	0.964	1.000
	${\beta }_0^{(2)}$	−0.016	0.019	0.019	0.834	1.000	0.001	0.021	0.022	0.949	1.000
	${\beta }_0^{(3)}$	0.000	0.017	0.016	0.943	0.057	−0.001	0.018	0.019	0.963	0.037
	${\beta }_0^{(4)}$	−0.026	0.020	0.019	0.720	1.000	0.000	0.023	0.025	0.957	1.000
		Setup 4
200	${\beta }_0^{(1)}$	0.016	0.074	0.072	0.929	1.000	−0.004	0.084	0.086	0.957	0.993
	${\beta }_0^{(2)}$	−0.009	0.063	0.060	0.912	1.000	0.009	0.073	0.074	0.949	1.000
	${\beta }_0^{(3)}$	0.001	0.051	0.049	0.935	0.065	0.002	0.054	0.056	0.946	0.054
	${\beta }_0^{(4)}$	−0.049	0.068	0.065	0.865	1.000	−0.006	0.085	0.085	0.939	0.996
400	${\beta }_0^{(1)}$	0.017	0.052	0.051	0.912	1.000	0.000	0.057	0.060	0.952	1.000
	${\beta }_0^{(2)}$	−0.015	0.042	0.042	0.913	1.000	0.005	0.048	0.051	0.955	1.000
	${\beta }_0^{(3)}$	0.000	0.034	0.035	0.951	0.049	0.000	0.037	0.039	0.964	0.036
	${\beta }_0^{(4)}$	−0.049	0.049	0.046	0.801	1.000	−0.003	0.056	0.059	0.942	1.000

We observe in Table 1 that the rank estimator $\widehat{\beta }$ is virtually unbiased under all settings, and the empirical bias tends to shrink when the sample size increases. The perturbation resampling-based standard errors agree with the empirical standard deviations quite well and, as expected, decrease with the sample size at the $\sqrt{n}$ rate. The ECPs are close to the nominal level of 95% under both sample sizes. The ERR’s for testing ${\beta }_0^{(j)}=0$ are close to 0.05 when ${\beta }_0^{(j)}=0$. The ERR’s are quite high when ${\beta }_0^{(j)}$ equals 0.5 or −0.707, suggesting satisfactory power of the proposed Wald test.

The transformed linear estimator ${\widehat{\beta }}_L$ has reasonable performance, but it tends to have larger bias than $\widehat{\beta }$. This is likely because the transformed linear model poses stronger model assumptions that are subject to mis-specification. The ESDs tend to be slightly smaller than their counterparts for the rank estimator, and the ASEs provide good approximation to the ESD. However, the ECPs are sometimes lower than the nominal level and do not improve when the sample size increases.

We conducted additional simulation studies for the transformed linear model under Setups 1 and 3, using various link functions h(·). The results were displayed in Table S1 of Supplemental Material D. Under Setup 1 where the transformed linear model holds, the true link used in the data generation led to the best performance in terms of efficiency and coverage. When the link function and/or the model form were mis-specified, the truncated normal link and the logit link showed better performance than the identity link. The performance of the transformed linear estimator was reasonable when the degree of mis-specification is small (Setup 1) but worsened when the degree of mis-specification increases (Setup 3).

Furthermore, we have conducted simulations for the rank estimator by adopting other choices of g(·), including the identity link and the rank of $\widehat{{\mathcal{P}}_i}$. The results are summarized in Table S2 of Supplemental Material D. The performance of the rank estimator was satisfactory and largely comparable using different choices of g(·). The truncated normal link rendered ESD and root mean squared error (RMSE) that were slightly smaller, though the difference was very small. The identity link was slightly inferior than the truncated normal link and the rank in terms of efficiency.

We also examined the performance of the kernel-smoothed objective function in (7). Results of the corresponding estimators are displayed in Table 2, where the top panel represents the results under the exact objective function, and the bottom panels correspond to the kernel-smoothed objective function with different C’s in C_n = C × n^−1/3. We observe a bias–variance trade-off when n = 200, with increasing bias and decreasing variance when C gets larger. When the bandwidth is too large (e.g., C = 3), the bias becomes non-ignorable, and the convergence rates (CR) decrease dramatically. However, the bias–variance trade-off becomes less noticeable when n = 400, where the results are not sensitive to the choice of C. These results suggest that the performance of the kernel-smoothed estimator is very similar to that of the original estimator when the bandwidth is not too large.

Table 2.

Simulation results of the rank estimator under Setup 4, with and without kernel smoothing (bandwidth C_n = C × n^−1/3), where β₀ = (0.5,−0.707, 0, 0.5). Column CR reports the convergence rate.

C		n = 200						n = 400
		BIAS	ESD	ASE	ECP	ERR	CR	BIAS	ESD	ASE	ECP	ERR	CR
Exact	${\beta }_0^{(1)}$	−0.004	0.084	0.086	0.957	0.993	0.997	0.000	0.057	0.060	0.952	1.000	0.995
	${\beta }_0^{(2)}$	0.009	0.073	0.074	0.949	1.000		0.005	0.048	0.051	0.955	1.000
	${\beta }_0^{(3)}$	0.002	0.054	0.056	0.946	0.054		0.000	0.037	0.039	0.964	0.036
	${\beta }_0^{(4)}$	−0.006	0.085	0.085	0.939	0.996		−0.003	0.056	0.059	0.942	1.000
0.1	${\beta }_0^{(1)}$	−0.009	0.079	0.087	0.960	0.998	1.000	−0.004	0.059	0.060	0.948	1.000	1.000
	${\beta }_0^{(2)}$	0.007	0.072	0.075	0.956	1.000		0.002	0.050	0.051	0.954	1.000
	${\beta }_0^{(3)}$	0.000	0.053	0.055	0.963	0.037		0.001	0.038	0.038	0.964	0.036
	${\beta }_0^{(4)}$	−0.002	0.082	0.084	0.955	0.999		−0.003	0.057	0.059	0.945	1.000
0.25	${\beta }_0^{(1)}$	−0.008	0.077	0.084	0.960	0.999	1.000	−0.004	0.058	0.058	0.941	1.000	1.000
	${\beta }_0^{(2)}$	0.006	0.069	0.073	0.960	1.000		0.003	0.049	0.050	0.949	1.000
	${\beta }_0^{(3)}$	0.000	0.052	0.054	0.965	0.035		0.001	0.037	0.037	0.961	0.039
	${\beta }_0^{(4)}$	−0.003	0.080	0.082	0.948	1.000		−0.002	0.056	0.057	0.940	1.000
0.5	${\beta }_0^{(1)}$	−0.006	0.074	0.079	0.957	1.000	1.000	−0.004	0.057	0.055	0.929	1.000	1.000
	${\beta }_0^{(2)}$	0.008	0.067	0.069	0.950	1.000		0.003	0.048	0.047	0.942	1.000
	${\beta }_0^{(3)}$	0.001	0.051	0.051	0.950	0.050		0.001	0.036	0.036	0.953	0.047
	${\beta }_0^{(4)}$	−0.002	0.077	0.078	0.942	1.000		−0.002	0.055	0.055	0.932	1.000
1	${\beta }_0^{(1)}$	−0.004	0.071	0.074	0.946	1.000	1.000	−0.002	0.056	0.052	0.925	1.000	1.000
	${\beta }_0^{(2)}$	0.010	0.065	0.065	0.945	1.000		0.004	0.047	0.045	0.939	1.000
	${\beta }_0^{(3)}$	0.001	0.051	0.050	0.943	0.057		0.001	0.036	0.035	0.947	0.053
	${\beta }_0^{(4)}$	0.000	0.075	0.073	0.935	1.000		−0.001	0.054	0.052	0.930	1.000
2	${\beta }_0^{(1)}$	0.005	0.067	0.069	0.948	1.000	0.978	0.001	0.055	0.051	0.921	1.000	1.000
	${\beta }_0^{(2)}$	0.020	0.065	0.064	0.943	1.000		0.009	0.046	0.045	0.936	1.000
	${\beta }_0^{(3)}$	0.002	0.055	0.053	0.941	0.059		0.001	0.038	0.036	0.948	0.052
	${\beta }_0^{(4)}$	0.005	0.073	0.070	0.938	1.000		0.002	0.052	0.050	0.930	1.000
3	${\beta }_0^{(1)}$	0.046	0.058	0.064	0.914	1.000	0.595	0.010	0.053	0.049	0.913	1.000	0.967
	${\beta }_0^{(2)}$	0.051	0.067	0.066	0.889	1.000		0.019	0.047	0.045	0.934	1.000
	${\beta }_0^{(3)}$	0.005	0.060	0.059	0.933	0.067		0.001	0.040	0.039	0.946	0.054
	${\beta }_0^{(4)}$	0.004	0.073	0.069	0.924	1.000		0.008	0.051	0.050	0.925	1.000

Finally, we examined the performance of proposed estimators in the situation where some of the outcomes are discrete. The data generation scheme for Y₁ and Y₂ remained the same. The marginal distribution of Y₃ follows Bernoulli(0.5), and Y₄ ∈ {1, 2, 3, 4} marginally follows Multinomial{4; (0.1, 0.5, 0.15, 0.25)}. The resulting ${\mathcal{P}}_i$ is still continuously distributed, and we generated data to follow the aforementioned Setup 4. As seen in Table 3, the performance of both estimators is comparable to their counterparts when all the individual outcomes are continuous.

Table 3.

Simulation results under Setup 4, where some of the outcomes are discrete and β₀ = (0.5, −0.707, 0, 0.5).

n		Linear transformation estimator					Rank regression estimator
		BIAS	ESD	ASE	ECP	ERR	BIAS	ESD	ASE	ECP	ERR
200	${\beta }_0^{(1)}$	0.007	0.080	0.076	0.925	1.000	−0.011	0.085	0.088	0.948	0.996
	${\beta }_0^{(2)}$	−0.020	0.065	0.061	0.889	1.000	0.007	0.074	0.074	0.941	1.000
	${\beta }_0^{(3)}$	−0.003	0.052	0.050	0.940	0.060	−0.002	0.056	0.057	0.957	0.043
	${\beta }_0^{(4)}$	−0.057	0.069	0.067	0.843	1.000	−0.002	0.085	0.087	0.935	0.998
400	${\beta }_0^{(1)}$	0.013	0.054	0.054	0.939	1.000	−0.003	0.058	0.062	0.963	1.000
	${\beta }_0^{(2)}$	−0.023	0.043	0.044	0.878	1.000	0.003	0.050	0.052	0.943	1.000
	${\beta }_0^{(3)}$	0.002	0.038	0.036	0.940	0.060	0.001	0.040	0.040	0.953	0.047
	${\beta }_0^{(4)}$	−0.058	0.049	0.048	0.749	1.000	−0.004	0.060	0.060	0.945	1.000

4. Real data example

We applied the proposed methods to the aforementioned NET-PD LS-1 study (Kieburtz et al., 2015), where the primary outcome of global disease progression was defined as the rank-sum of five individual measures at the 5-year visit (Elm and Investigators, 2012). Some of the individual outcomes were reversely coded, such that higher values were worse for all individual outcomes. Application of the O’Brien’s GST yielded a two-sided p-value of p = 0.45, suggesting no significant difference between the creatine arm and the placebo arm (Kieburtz et al., 2015). In the following analysis, we aimed to evaluate whether and how baseline covariates affect the disease progression for early stage PD patients. Our analytic data consisted of n = 704 subjects with complete data. Besides the treatment arm (creatine vs. placebo), we considered the following baseline covariates in the regression model: race (non-Hispanic White vs. others), gender (male vs. female), age of symptom onset (OnsetAge), Total Functional Capacity (TFC), levodopa equivalent dose (LED), Beck Depression Inventory score (BDI), and the Scales for Outcomes of Parkinson’s Disease Cognition (SCOPA-COG) score. These covariates were identified as potentially important risk factors in Bega et al. (2015) and/or our exploratory analyses. The continuous covariates were centered at the mean and standardized by their respective standard deviation.

Table 4 presents the estimated regression coefficients (COEF), standard errors (SE), and p-values under the transformed linear model (left panel) and the monotonic index model (right panel), respectively. Both estimators were restricted to have the unit norm, such that the COEFs represent the relative importance of individual risk factors. The results under the two models, as shown in Table 4, were largely consistent, though some differences were also noted. The SEs were comparable under the two models. Both methods identified younger age of symptom onset, lower baseline functional capacity, and worse baseline cognition status as prognostic factors of faster PD progression. The use of creatine (treatment) was insignificant under both models, and the COEF was closer to 0 under the monotonic index model. The monotonic index model also detected significant racial effect, where the non-Hispanic White race was associated with slower disease progression. This association was not captured by the transformed linear model.

Table 4.

Data analysis: estimated regression coefficients, standard error, and p-values.

	Transformed linear model			Monotonic index model
	COEF	SE	p	COEF	SE	p
OnsetAge	0.571	0.139	<.001	0.480	0.138	0.001
TFC	−0.318	0.125	0.011	−0.214	0.102	0.036
SCOPACOG	−0.555	0.150	<.001	−0.550	0.134	<.001
BDI	0.151	0.118	0.203	0.161	0.117	0.167
LED	0.133	0.107	0.214	0.150	0.143	0.293
Race	−0.458	0.311	0.142	−0.611	0.286	0.033
Treatment	−0.121	0.218	0.580	0.000	0.203	0.998

Fig. 2 displays the scatter plots with the estimated score $X_i^T\widehat{\beta }$ on the x-axis and Table S2 of ${\widehat{\mathcal{P}}}_i$ or $h\left({\widehat{\mathcal{P}}}_i\right)$ on the y-axis. The solid curve represents the smoothed Lowess estimate. In both columns, the slope showed some curvature and appeared flatter for scores below −1. This pattern suggests potential model mis-specification under a (transformed) linear model, and the results under the monotonic index model are thus preferred for this data application.

Fig. 2.

Data analysis: scatter plot of ${\widehat{\mathcal{P}}}_i$ vs. $X_i^T\widehat{\beta }$ (left column) and $h\left({\widehat{\mathcal{P}}}_i\right)$ vs. $X_i^T\widehat{\beta }$ (right column), overlaid with the lowess curve.

5. Discussion

In this paper, we proposed regression strategies for the global percentile outcome (GPO), defined as the composite percentile of multiple individual outcomes. Regression models were formulated to associate the covariates directly to the GPO under relaxed and realistic assumptions, accounting for the fact that the distribution of GPO is non-normal and non-uniform. While the transformed linear model is appealing due to its simple implementation and easy interpretation, the monotonic index model has advantages in flexibility and robustness. Both estimators have been shown to have satisfactory numerical performance and asymptotic properties. Although the proposed methods were motivated by Parkinson’s disease studies, they can be readily applied to many disease fields where it is of interest to examine covariate effects on the global disease burden.

When the extent of mis-specification is small in our simulation studies, the transformed linear model may render smaller standard errors, which is desirable for small datasets. It is also easy to fit with a unique zero-crossing and an explicit variance formula. However, due to its stronger model assumptions, it may show larger bias and lower coverage rates when the degree of mis-specification increases. In practice, there is usually limited prior information about the underlying link function. Therefore, it is desirable to conduct some diagnostic procedures regarding the form of the link function, similar to what we did in Fig. 2 for the motivating example, to evaluate the plausibility of the assumed link. When the sample size is moderate to large, we recommend to adopt the rank estimator for more robust conclusions.

In special cases where an additional condition is satisfied for the covariate vector X, the transformed linear estimator is also robust to the mis-specified link. Li and Duan (1989) showed that if the distribution of X is elliptical symmetric and satisfies condition (A.2) therein, the slope part of ${\widehat{\beta }}_L$ is consistent up to a multiplicative scalar. Specifically, the slope part would be consistent after standardized to the unit norm. In applications where the elliptical symmetric assumption can be verified for the regressors, the transformed linear model can be fitted without concerns about the link violation.

Our modeling procedure examines covariate effects on the global disease level, as captured by a composite outcome based on ranks and percentiles. The direction of covariate effects on the individual outcomes may not always agree with its counterpart on the global outcome, especially in the presence of additional covariates and/or small sample sizes. Therefore, the results only pertain to the global disease level and may not apply simultaneously to all the individual outcomes. If the effect of a covariate on one outcome does not agree with those on other outcomes, the proposed method can still be applied, with the understanding that the inclusion of this outcome in the GPO may lead to a smaller covariate effect estimate. Following the regression modeling of GPO, one could analyze the individual outcomes in secondary analyses to gain more specific insights.

We have focused on complete data, but the proposed method could be extended to handle missing data when combined with multiple imputation. Moreover, it is of interest to extend the proposed modeling strategies to correlated observations in clustered data and longitudinal data. These directions are beyond the scope of the current work but merit future research.