Model-robust Inference for Continuous Threshold Regression Models

Summary

We study threshold regression models that allow the relationship between the outcome and a covariate of interest to change across a threshold value in the covariate. In particular we focus on continuous threshold models, which experience no jump at the threshold. Continuous threshold regression functions can provide a useful summary of the association between outcome and the covariate of interest, because they offer a balance between flexibility and simplicity. Motivated by collaborative works in studying immune response biomarkers of transmission of infectious diseases, we study estimation of continuous threshold models in this paper with particular attention to inference under model misspecification. We derive the limiting distribution of the maximum likelihood estimator, and propose both Wald and test-inversion confidence intervals. We evaluate finite sample performance of our methods, compare them with bootstrap confidence intervals, and provide guidelines for practitioners to choose the most appropriate method in real data analysis. We illustrate the application of our methods with examples from the HIV-1 immune correlates studies.

1. Introduction

Change point problems have arisen in many substantive areas, e.g. Alzheimer’s disease (Hall et al., 2000), heart diseases (e.g. Goetghebeur and Pocock, 1995; Pastor and Guallar, 1998; Carmody et al., 2014) and diabetes (e.g. Pastor-Barriuso et al., 2003). Our interests in change point analysis stem from collaborative works in infectious disease vaccine development. An important aspect of modern vaccine development is to identify human immune response biomarkers that are associated with the risk of infection/transmission or vaccine efficacy (Gilbert et al., 2008). The relationship between the risk of infection/transmission and immune biomarkers is often thought to be nonlinear (Tomaras and Haynes, 2014). In particular, it has been hypothesized that only an immune response above a certain quality and quantity threshold can result in protection from HIV-1 infection or transmission.

Change point problems cover a large number of interesting situations. A useful distinction (Hansen, 2000) can be made between two subclasses of change point models: change-point models and threshold models. In a change-point model (e.g. Muller, 1992; Loader et al., 1996; Csörgö and Horváth, 1997; Antoch et al., 2007; Gijbels, 2008; Zeileis et al., 2002) the samples have a natural order based on, for example, time or position on a chromosome. Change-point models are often used to detect structural changes along the natural axis. In a threshold model (e.g. Chan and Tsay, 1998; Hansen, 2000; Banerjee and McKeague, 2007; Fong et al., 2016) there is typically not a natural ordering of samples or the ordering is not essential to the problem, e.g. when there is only a single covariate. The goal for using threshold models is often to approximate a nonlinear relationship between the outcome and a covariate of interest, which we will refer to as change point covariate. Compared to nonparametric smoothing methods of modeling nonlinearity, threshold models have the advantage of being simple and easy to interpret. Our focus in this paper is on the estimation of threshold models; and because we view threshold models as a useful summary of the true relationship between outcome and change point covariate, particular emphasis is given to making valid inference when the true data-generating model differs from the assumed threshold models.

Threshold regression models come in many different flavors (Fong et al., 2016). We focus on two continuous threshold logistic regression models:

$$\text{segmented}:\text{ logit }\{\text{Pr}(Y=1|x,\mathit{z})\}={\beta }_1+{\mathit{\beta }}_{\mathit{z}}^T\mathit{z}+{\beta }_{x}x+{\beta }_{{(x-e)}_{+}}{(x-e)}_{+}$$ (1)

$$\text{hinge}:\text{ logit }\{\text{Pr}(Y=1|x,\mathit{z})\}={\beta }_1+{\mathit{\beta }}_{\mathit{z}}^T\mathit{z}+{\beta }_{{(x-e)}_{+}}{(x-e)}_{+}.$$ (2)

Here x is a single covariate, e is the threshold parameter, (x − e)₊ denotes (x − e) I (x > e), where I (·) is an indicator function, and z is a vector of additional covariates. Both models are continuous at the change point, and the hinge model is a special case of the segmented model by assuming zero slope before the threshold.

Much work has been done regarding hypothesis testing in threshold models (e.g. Davies, 1987; Ulm, 1991; Koziol and Wu, 1996; Xu and Adak, 2002; Mazumdar et al., 2003; Pastor-Barriuso et al., 2003; Antoch et al., 2004; Zheng and Chen, 2005; Vexler and Gurevich, 2006, 2009; Lee et al., 2011). Our goal in this paper is to develop model-robust methods for constructing confidence intervals for coefficients in continuous threshold models, which should provide proper coverage even when the true underlying model differs from the assumed model. In our view, this topic has not received enough attention in the literature with two notable exceptions. Banerjee and McKeague (2007) studied model-robust inference for discontinuous threshold models with cube root n convergence rate; and, more recently, Hansen (2015) studied model-robust inference for discontinuous threshold models. Our work builds upon the work of Hansen and makes the following unique contributions: first, we improve the coverage of Wald-type confidence intervals by deriving a correct asymptotic covariance matrix for the maximum likelihood estimator; and second, we propose a test-inversion confidence interval for the threshold parameter based on a profile likelihood ratio test and derive the asymptotic distribution of the test statistic. Finally, we evaluate the performance of the Wald, test-inversion and bootstrap confidence intervals in Monte Carlo studies both under true threshold models and misspecified models.

The balance of the paper is organized as follows. Sections 2 and 3 develop the asymptotic theories for estimation. Section 4 conducts simulation studies, and Section 5 uses real data examples from HIV-1 immune correlates studies to illustrate the application of threshold models. In Section 6, we discuss the choice of methods for constructing confidence intervals in practice.

2. Limiting distributions of maximum likelihood estimators

We first treat the more general segmented model. Let $\mathit{\beta }={({\beta }_1,{\mathit{\beta }}_{\mathit{z}}^T,{\beta }_x,{\beta }_{{(x-e)}_{+}})}^T$ and θ = (β^T, e)^T denote all parameters in the model. Let ℓ_θ denote the log likelihood under the working model (1). For a single observation ℓ_θ (y, x, z) = y log p_θ (x, z)+(1 − y) log {1 − p_θ (x, z)}, where p_θ (x, z) = expit {η_θ (x, z)} with expit (x) = 1/ (1 + exp (−x)) and ${\eta }_{\mathit{\theta }}(x,\mathit{z})={\beta }_1+{\mathit{\beta }}_{\mathit{z}}^T\mathit{z}+{\beta }_{x}x+{\beta }_{{(x-e)}_{+}}{(x-e)}_{+}$. Let θ̂_n ≡ argmax_θ∈Θ ℙ_nℓ_θ be the maximum likelihood estimator (MLE) of θ over domain Θ ⊂ R^D(θ), where ${\mathbb{P}}_n(·)=n^{-1}{\sum }_{i=1}^n(·)$ is the empirical average operator, and D (θ) is the dimension of θ. An effective algorithm (Fong et al., 2016) for finding the MLE is to alternate between updating {e, β_{(x − e)+}, β_x} and $\{{\beta }_1,{\mathit{\beta }}_{\mathit{z}}^T\}$. When updating {e, β_{(x − e)+}, β_x}, we use a smooth function, logistic function, to approximate the step function I (x − e) (Pastor-Barriuso et al., 2003), which allows the use of gradient-based methods to explore the parameter space.

We first show that under mild conditions, θ̂_n converges in probability to θ₀ ≡ argmax_θ∈Θ E_XYZ ℓ_θ, where the expectation is taken with regard to the joint distribution function of (Y, X, Z), F (y, x, z). If the model is correctly specified, i.e. the data-generating model and the working model match, θ₀ is the true parameter value of the data-generating model; otherwise, we say that the model is misspecified and ℓ_θ0 can be viewed as the projection of the data-generating model on the working model space using the Kullback-Leibler divergence metric. Whether or not the model is correctly specified, we always assume that θ₀ is unique, meaning that if the distributions of data generated from two models with parameters θ₁ and θ₂ are indistinguishable, it is necessarily true that θ₁ = θ₂.

Theorem 1

Assume we have i.i.d. observations (Y_i, X_i, Z_i) for i = 1, ⋯, n. Let E X² < ∞, E ‖Z‖² < ∞, and the distribution of X be absolutely continuous. Assume Θ is compact and θ₀ is unique. Under these assumptions, θ̂_n →_p θ₀.

Theorem 1 follows from Theorem 5.14 of van der Vaart (2000), which does not require a high degree of smoothness in the likelihood function. The key regularity condition to check is E sup_θ∈U ℓ_θ < ∞ for every sufficiently small ball U, which holds trivially in our model because ℓ_θ is continuous in θ and nonpositive. Adding the fact that θ̂_n maximizes the criterion function ℓ_θ, we have satisfied all the conditions of Theorem 5.14 of van der Vaart (2000).

We next study the asymptotic distribution of θ̂_n. Define x̃_θ ≡ {1, z, x, (x − e)₊, −β_{(x − e)+} I (x > e)}^T. Let ℓ̇_θ denote the first derivative of ℓ_θ with respect to θ. For any x ≠ e, ℓ̇_θ = (y − p_θ) x̃_θ.

Theorem 2

In addition to the assumptions of Theorem 1, assume the distribution of X has density function f_X (·). Denote the true mean of Y conditional on x and z by m₀ (x, z), i.e. m₀ (x, z) = E (Y | x, z). Then the second derivative matrix of E_XYZ ℓ_θ is

$$V_{\mathit{\theta }}=-{\mathrm{E}}_{X\mathit{Z}}{p}_{\mathit{\theta }}(1-p_{\mathit{\theta }}){\tilde{x}}_{\mathit{\theta }}{\tilde{x}}_{\mathit{\theta }}^T+\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill {\underset{¯}{0}}^T\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \underset{¯}{0}\hfill & \hfill [0]\hfill & \hfill \underset{¯}{0}\hfill & \hfill \underset{¯}{0}\hfill & \hfill \underset{¯}{0}\hfill \\ \hfill 0\hfill & \hfill {\underset{¯}{0}}^T\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\underset{¯}{0}}^T\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathrm{E}}_{X\mathit{Z}}(m_0-p_{\mathit{\theta }})I(x>e)\hfill \\ \hfill 0\hfill & \hfill {\underset{¯}{0}}^T\hfill & \hfill 0\hfill & \hfill -{\mathrm{E}}_{X\mathit{Z}}(m_0-p_{\mathit{\theta }})I(x>e)\hfill & \hfill {\beta }_{{(x-e)}_{+}}{f}_X(e){\mathrm{E}}_{\mathit{Z}}\{m_0(e,\mathit{z})-p_{\mathit{\theta }}(e,\mathit{z})\}\hfill \end{array}\right],$$ (3)

where 0̱ is a vector of 0’s of length D (β_z), D (β_z) being the dimension of β_z, and [0] is a D (β_z) × D (β_z) square matrix of 0’s. Further assume V_θ0 is non-singular. Denote $M_{\mathit{\theta }}={\mathrm{E}}_{XY\mathit{Z}}{\dot{\ell }}_{\mathit{\theta }}{\dot{\ell }}_{\mathit{\theta }}^T$. Then the limit distribution of θ̂_n is

$$\sqrt{n}({\hat{\mathit{\theta }}}_n-{\mathit{\theta }}_0){\to }_{d}N(0,V_{{\mathit{\theta }}_0}^{-1}{M}_{{\mathit{\theta }}_0}{V}_{{\mathit{\theta }}_0}^{-1}).$$

The derivation of (3) is detailed in Section A of the Supplementary Materials. The fact that V_θ is continuous in θ implies that E_XYZ ℓ_θ admits a second order Taylor expansion in a neighborhood of θ₀:

$${\mathrm{E}}_{XY\mathit{Z}}{\ell }_{\mathit{\theta }}={\mathrm{E}}_{XY\mathit{Z}}{\ell }_{{\mathit{\theta }}_0}+\frac{1}{2}{(\mathit{\theta }-{\mathit{\theta }}_0)}^T{V}_{{\mathit{\theta }}_0}(\mathit{\theta }-{\mathit{\theta }}_0)+o({\Vert \mathit{\theta }-{\mathit{\theta }}_0\Vert }^2).$$ (4)

Adding the fact that ℓ_θ is Lipschitz continuous in θ, the asymptotic normality follows from Theorem 5.23 of van der Vaart (2000).

Comparing our Theorem 2 with Hansen’s (2015) Theorem 2, we see that the last diagonal term of (3) is non-zero. This element vanishes along with the rest of the second term of V_θ when the model is correctly specified, but is often positive when the model is misspecified. Its presence suggests that the likelihood surface may have a smaller curvature along the change point parameter at θ₀ when the model is misspecified.

When the model is correctly specified, V_θ0 = −M_θ0 and the asymptotic variance is simply the inverse of the Fisher information. The asymptotic variance can be consistently estimated by ${\{{\mathbb{P}}_n{p}_{\hat{\mathit{\theta }}}(1-p_{\hat{\mathit{\theta }}}){\tilde{x}}_{\hat{\mathit{\theta }}}{\tilde{x}}_{\hat{\mathit{\theta }}}^T\}}^{-1}$ , which is always positive definite; Wald–type confidence intervals for β̂ can then be constructed.

When the model is misspecified, we can estimate M_θ0 by ${\mathbb{P}}_n{(y-p_{\hat{\mathit{\theta }}})}^2{\tilde{x}}_{\hat{\mathit{\theta }}}{\tilde{x}}_{\hat{\mathit{\theta }}}^T$ and the first term of V_θ0 by $-{\mathbb{P}}_n{p}_{\hat{\mathit{\theta }}}(1-p_{\hat{\mathit{\theta }}}){\tilde{x}}_{\hat{\mathit{\theta }}}{\tilde{x}}_{\hat{\mathit{\theta }}}^T$. The off-diagonal element in the second term of V_θ0 can be estimated by −ℙ_n (y − p_θ̂) I (x > ê). The last diagonal element in the second term of V_θ0 depends on four terms: β_{(x − e)+} can be estimated by β̂_{(x − e)+} ; f_X (e₀) can be estimated by a kernel density estimate at ê; both E_Z m₀ (e₀, z) and E_Z p_θ0 (e₀, z) are harder to estimate because it is (x, z, y) that we observe and not (e₀, z, y). Assuming the true mean function m₀ under model misspecification is a smooth function of x and z, we may approximate E_Z m₀ (e₀, z) by fitting a sufficiently flexible model to obtain an estimate of m₀ and use the plug-in estimate ℙ_n m̂₀ (ê, z).

The last term we need to estimate the variance is E_Z p_θ0 (e₀, z). A natural estimator is ℙ_np_θ̂ (ê, z), and it presents an interesting twist. Hansen (2015) discovered that in a segmented threshold linear regression model, the asymptotic distribution of η_θ̂ (x, z) does not center at η_θ0 (x, z) for x = e₀. The same phenomenon exists here, as the following theorem shows.

Theorem 3

Under the assumptions of Theorem 2, at every x and z,

$$\sqrt{n}\{p_{\hat{\mathit{\theta }}}(x,\mathit{z})-p_{{\mathit{\theta }}_0}(x,\mathit{z})\}{\to }_d{g}_{{\mathit{\theta }}_0}(N(0,V_{{\mathit{\theta }}_0}^{-1}{M}_{{\mathit{\theta }}_0}{V}_{{\mathit{\theta }}_0}^{-1})),$$

where, for $h={(h_{\mathit{\beta }}^T,h_e)}^T$

$$g_{\mathit{\theta }}(h)=p_{\mathit{\theta }}(1-p_{\mathit{\theta }})\left[h_{\mathit{\beta }}^T\right(\begin{array}{c}\hfill 1\hfill \\ \hfill \mathit{z}\hfill \\ \hfill x\hfill \\ \hfill {(x-e)}_{+}\hfill \end{array})+\{\begin{array}{cc}\hfill 0\hfill & \hfill \mathit{\text{if }}x<e\hfill \\ \hfill {(-h_e)}_{+}{\beta }_{{(x-e)}_{+}}\hfill & \hfill \mathit{\text{if }}x=e\hfill \\ \hfill -h_e{\beta }_{{(x-e)}_{+}}\hfill & \hfill \mathit{\text{if }}x>e\hfill \end{array}].$$

The proof of the theorem can be found in Section B of the Supplementary Materials. At x = e₀, the limit distribution is the sum of a mean 0 normal random variable and $p_{{\mathit{\theta }}_0}(1-p_{{\mathit{\theta }}_0}){\beta }_{{(x-e)}_{+},0}{\{-\sqrt{n}(ê-e_0)\}}_{+}$. At first look, this makes estimating E_Z p_θ0 (e₀, z) difficult; however, upon closer inspection we realize that $p_{{\mathit{\theta }}_0}(e_0,\mathit{z})=\text{expit}\{{\beta }_{1,0}+{\mathit{\beta }}_{\mathit{z},0}^T\mathit{z}+{\beta }_{x,0}{e}_0+{\beta }_{{(x-e)}_{+},0}{(e_0-e_0)}_{+}\}=\text{expit }({\beta }_{1,0}+{\mathit{\beta }}_{\mathit{z},0}^T\mathit{z}+{\beta }_{x,0}{e}_0)$, which can be well approximated by the plug-in estimator $\text{expit}({\hat{\beta }}_1+{\hat{\mathit{\beta }}}_{\mathit{z}}^T\mathit{z}+{\hat{\beta }}_xê)$.

3. Test-inversion confidence sets

The Wald-type confidence intervals of threshold model parameters can be constructed from the asymptotic distribution of the maximum likelihood estimator derived in the last section. Under correct model specification, it has long been recognized that the finite-sample performance of Wald-type confidence intervals may be misleading for some models due to parameter-effects curvature (Seber and Wild, 1989; Pastor-Barriuso et al., 2003). An alternative approach to constructing confidence sets is by inverting hypothesis tests. A likelihood ratio test-inversion confidence set is a collection of θ that is close to the MLE θ̂. More precisely, it is of the form

$$\{\mathit{\theta }:2{\ell }_n(\hat{\mathit{\theta }})-2{\ell }_n(\mathit{\theta })\le C_{D(\mathit{\theta })}(\alpha )\},$$ (5)

where ℓ_n is the sum of the log likelihood, θ̂ is the maximum likelihood estimator, and C_{D (θ)} (1 − α) is the (1 − α)^th quantile of the limit distribution of 2ℓ_n (θ̂) − 2ℓ_n (θ₀). Thus, the set (5) comprises parameter values that are consistent with the data as determined by the likelihood ratio test. The following theorem gives the asymptotic distribution of the likelihood ratio statistic in segmented threshold models.

Theorem 4

Under the assumptions of Theorem 2,

$$2{\ell }_n\left(\hat{\mathit{\theta }}\right)-2{\ell }_n({\mathit{\theta }}_0){\to }_d\sum _{j=1}^{D(\mathit{\theta })}{\lambda }_j{\chi }_1^2,$$

where ${\{{\lambda }_j\}}_{j=1}^{D(\mathit{\theta })}$ are the eigenvalues of $M_{{\mathit{\theta }}_0}^{1/2}{V}_{{\mathit{\theta }}_0}^{-1}{M}_{{\mathit{\theta }}_0}^{1/2}$.

Theorem 4 follows from the second order expansion (4) and the fact that the random quadratic form n (θ̂ − θ₀)^T V_θ0 (θ̂ − θ₀) can be written as a sum of weighted chi-squared distributions.

When the model is correctly specified, the limit distribution of this likelihood ratio statistic is simply a chi-squared distribution with D (θ) degrees of freedom, which is free of any parameters. When the model is misspecified, the limit distribution is a weighted sum of chi-square distributions with parameters that depend on the asymptotic distribution of the MLE.

Constructing a likelihood ratio test-inversion confidence set is typically not quite feasible unless D (θ) is rather small as the computational cost grows exponentially with D (θ). Fortunately, test-inversion confidence sets can also be based on profile likelihood (e.g. Davison, 2003, Section 4.5). Suppose θ= (β^T, γ)^T. Let pℓ_n (γ) = max_β ℓ (β, γ) denote the profile likelihood, also known as the concentrated likelihood of γ. A confidence set for γ can be constructed if we know the distribution of the profile likelihood ratio statistic. In threshold models, it is natural to let γ be e because the model can be easily fit for any given threshold value, while the same cannot be said for the slope parameters. We now state the limit distribution of the profile likelihood ratio statistic for the change point parameter, the proof of which is given in Section C of the Supplementary Materials.

Theorem 5

Under the assumptions of Theorem 2,

$$2p{\ell }_n(ê)-2p{\ell }_n(e_0){\to }_d\lambda {\chi }_1^2,$$

where $\lambda =-{[V_{{\mathit{\theta }}_0}^{-1}{M}_{{\mathit{\theta }}_0}{V}_{{\mathit{\theta }}_0}^{-1}]}_{ee}/{[V_{{\mathit{\theta }}_0}^{-1}]}_{ee}$. Here [·]_ee denote the last diagonal element of a square matrix.

When the model is correctly specified, λ is 1. When the model is misspecified, the limit distribution of the likelihood ratio statistic is not any simpler than the limit distribution of the parameter estimate. We note that Hansen (2015) had proposed the use of a test-inversion confidence set for the threshold parameter under model misspecification, but assumed λ = 1.

To construct test-inversion confidence intervals for e after finding the MLE by approximating the step function in the likelihood with a smooth function, we perform a grid search locally. Specifically, to find the upper bound, we test increasingly larger e in small increments starting from ê until the difference between 2pℓ_n (ê) and 2pℓ_n (e) drops below the critical value, and we find the lower bound similarly. If the MLE is found not through smooth approximation but by comparing profile likelihoods pℓ_n (e) on a global grid of candidate change points, the test-inversion confidence interval would come as a by-product and incur no additional computational burden. Overall it is still faster to compute the MLE by smooth approximation and find the test-inversion confidence interval by local grid search.

4. Monte Carlo studies

In this section we present two simulation scenarios. In scenario I the data are generated from threshold models; in scenario II the data are generated from a quadratic model. Results from an additional model misspecification scenario are shown in the Supplementary Materials.

4.1 Scenario I

The first batch of Monte Carlo experiments is conducted under correct model specification. We simulate data from the segmented and hinge logistic regression models (1) and (2) with β_z = log (1.4) = 0.34, β_x = −log (0.7) = 0.36 and β_{(x − e)+} = log (0.4) = −0.92 and e = 2.2. The intercepts are chosen so that the prevalence of Y is around 1/4 under each model. The covariate z is simulated from a standard normal distribution. We simulate x from a scaled and shifted gamma distribution with median 2.06 and interquartile range 1.76, as the asymmetry and heavy-tailedness of a gamma random variable better models the distribution of immune biomarker measurements. We experiment with sample sizes: 25, 50, 100, 250, 500, 2000 and 10,000.

Numerical summaries of the simulation results from 10,000 replicates are shown in Table 1, and the histograms of the sampling distributions are shown in the Supplementary Materials Figure D.1 and D.2. To compute relative bias of the slope estimates, we divide bias by the true value; to compute relative bias of the threshold estimate, we divide bias by the interquartile range of the x covariate. In addition to point estimates and relative bias, we show the width of 95% confidence intervals and actual coverage from three methods: ‘approx’, ‘model’ and ‘test inv’. The first two are symmetric Wald-type intervals, and the last is the asymmetric profile likelihood ratio test-inversion confidence interval for the threshold parameter only. The ‘approx’ method implements an approach from Pastor-Barriuso et al. (2003) that approximates the indicator function I (x > e) in the threshold models with a logistic function, and uses the observed information of the smoothed model to estimate the covariance matrix. The ‘model’ method is based on the asymptotic variance formula in Theorem 2 under correct model specification. The ‘test inv’ method is based on Theorem 5 under correct model specification.

Table 1.

Simulation results under correct model specification. β_z,0 = 0.34, β_x,0 = 0.36, β_(x−e)+,0 = −0.92 and e₀ = 2.2. Est: MC mean of parameter estimate; %bias: relative bias with respect to β₀ for the β′s and interquartile range of x for e. Three confidence interval methods are compared based on MC median width of 95% confidence interval and actual coverage (in parentheses). Approx: Wald CI based on asymptotic variance derived by approximating the indicator function I(x > e) in the threshold models with a smooth function; model: Wald CI based on asymptotic variance derived exactly; test inv: test inversion CI based on asymptotic distribution derived exactly.

n			Segmented				Hinge
		Est(%bias)	Approx	Model	Test Inv	Est(%bias)	Approx	Model	Test Inv
βz
25		0.47(41)	2.29(97)	2.33(98)		0.48(42)	2.18(98)	2.23(98)
50		0.39(15)	1.42(95)	1.43(95)		0.37(9)	1.41(96)	1.42(96)
100		0.36(8)	0.95(95)	0.95(95)		0.35(5)	0.96(95)	0.96(96)
250		0.35(4)	0.58(95)	0.58(95)		0.34(3)	0.59(95)	0.59(95)
500		0.34(1)	0.41(95)	0.41(95)		0.34(1)	0.41(95)	0.42(95)
2000		0.34(0)	0.20(95)	0.20(95)		0.33(−1)	0.21(95)	0.21(95)
10000		0.34(0)	0.09(96)	0.09(96)		0.34(0)	0.09(96)	0.09(96)
βx
25		0.64(61)	6.30(95)	7.17(96)
50		1.10(174)	3.93(93)	4.57(95)
100		1.11(178)	2.31(93)	2.74(95)
250		0.79(98)	1.29(87)	1.52(91)
500		0.59(47)	0.89(84)	1.04(88)
2000		0.42(6)	0.47(87)	0.52(90)
10000		0.40(1)	0.23(93)	0.23(93)
β(x−e)+
25		−1.17(28)	10.58(97)	11.91(98)		−1.27(39)	3.23(94)	4.33(96)
50		−1.97(115)	6.61(93)	7.35(96)		−1.35(47)	2.39(97)	3.11(99)
100		−2.04(123)	4.03(93)	4.42(95)		−1.42(55)	1.87(95)	2.41(97)
250		−1.58(73)	2.09(94)	2.21(96)		−1.22(33)	1.20(91)	1.46(94)
500		−1.26(37)	1.33(94)	1.38(95)		−1.02(12)	0.85(89)	1.01(93)
2000		−0.96(5)	0.62(94)	0.63(95)		−0.93(2)	0.45(93)	0.49(95)
10000		−0.92(1)	0.28(94)	0.28(94)		−0.92(0)	0.21(95)	0.22(96)
e
25		2.23(2)	1.09(47)	2.80(84)	3.28(96)	1.78(24)	1.36(50)	5.32(98)	2.85(89)
50		2.16(2)	0.91(41)	2.16(73)	3.32(94)	1.93(15)	1.18(55)	3.75(98)	3.04(96)
100		2.22(1)	0.84(39)	1.92(69)	3.60(91)	2.12(4)	1.04(55)	2.61(92)	2.91(96)
250		2.29(5)	0.77(45)	1.59(70)	2.58(90)	2.18(1)	0.85(60)	1.70(87)	2.11(93)
500		2.29(5)	0.69(53)	1.28(73)	1.72(89)	2.18(1)	0.72(68)	1.23(88)	1.43(93)
2000		2.23(2)	0.50(72)	0.72(84)	0.80(92)	2.19(1)	0.49(85)	0.62(92)	0.65(95)
10000		2.20(0)	0.29(89)	0.33(91)	0.34(94)	2.19(0)	0.26(93)	0.28(94)	0.28(95)

Under the segmented model, we see that bias decreases as the sample size increases for all parameters. The degree of bias differs between parameters. For example, at n = 500 the relative bias is 47%, 37%, 5% and 1% for β̂_x, β̂_{(x − e)+}, ê and β̂_z, respectively. This fits our intuition that non-smoothness of threshold models makes estimation more difficult for parameters associated with the change point covariate, especially in small to moderate samples. Among the methods for confidence intervals construction, ‘approx’ works reasonably well for β_z and β_{(x − e)+} but undercovers β_x and e. ‘Model’ has improved coverage for β_x and e, yet the coverage for e is still unsatisfactory. Finally, the coverage for e is substantially improved in ‘test inv’. In summary, ‘model’+‘test inv’ provide reasonable finite sample coverage for both the slope and threshold parameters.

Under the hinge model, we see similar results with some notable differences: β̂_{(x − e)+} is much less biased in the hinge model than in the segmented model under this simulation setting. For example, at n = 500, the relative bias is 37% and 12% under the segmented and hinge model, respectively. However, the relative bias of ê under the segmented model remains small at 2% even when n drops down to 50 and 25, but the relative bias of ê in a hinge model increases to 15% and 24%, respectively, at those sample sizes. In addition, while the precision of β̂_z as measured by confidence interval width is unchanged between the two models, the precision of β̂_{(x − e)+} and ê is much higher in the hinge model than in the segmented model. For example, the confidence interval for β_{(x − e)+} at n = 500 has a median width of 1.38 and 1.01 under the segmented and hinge model, respectively, which corresponds to a relative efficiency of 1.87.

To study the impact of threshold location on model estimation, we compared three hinge models with e ∈ {2.2, 1.5, 1} and three segmented models with the same threshold values. The results, summarized and discussed in more detail in Section D.4 of the Supplementary Materials, show that for the hinge model the threshold estimate is least biased when the true threshold is at 1.5, which is ~ 30 percentile of the distribution of x; while for the segmented model the threshold estimate is least biased when the true threshold is 2.2, which is near the median of X.

4.2 Scenario II

The second batch of experiments is conducted under the scenario that the model is misspecified. We simulate data from a quadratic model:

$$\text{logit }\{\text{Pr}(Y=1|x,z)\}=\alpha +\text{log }(1.4)z-x+0.3x^2.$$

The covariates are simulated as in scenario I, and the intercept is also chosen such that there are about a quarter of cases. Figure 1 shows a simulated dataset with n = 500 along with the fitted models. To determine θ₀ under the change point working models, we simulate datasets of sample size 100, 000 and take the median of 2,000 replicates. In the segmented model, β_z,0 = 0.34, β_x,0 = −0.040, β_{(x − e)+,0} = 1.49, and e₀ = 2.82; in the hinge model, β_z,0 = 0.34, β_{(x − e)+,0} = 1.46, and e₀ = 2.87. These values approximate the asymptotic limit of the threshold model parameter estimates and are used to help define finite sample biases of the MLE with respect to the population limit.

Figure 1.

A sample dataset under scenario II, n = 1000. The lines show the estimated linear combinations η_θ̂(x, 0) (left) and estimated probabilities p_θ̂(x, 0) (right) based on the fitted models in the left and right panels, respectively.

Numerical summaries of the simulation results for sample sizes ranging from 25 to 10,000 from 10,000 replicates are shown in Table 2 and 3, and the histograms of the sampling distributions are shown in the Supplementary Materials Figure D.3 and D.4. In addition to the estimate and relative bias, we compare five methods for constructing confidence intervals: ‘model’, ‘robust’, ‘test inv’, ‘boot basic’, and ‘boot bca*’. The first three are analytical confidence intervals, and the last two are bootstrap confidence intervals. ‘Model’ and ‘robust’ are based on the asymptotic variance formula in Theorem 2 under correct model specification and model misspecification, respectively. Under the ‘robust’ method, to estimate E_Z m₀ (e₀, z), we model the effect of x through a natural cubic spline with two degrees of freedom. The ‘test inv’ method is based on Theorem 5 under model misspecification. ‘Boot basic’ and ‘boot bca*’ are implemented with the help of the R package boot. Results from 2000 bootstrap replicates are reported. The basic bootstrap method is a member of the pivotal family of bootstrap methods (Carpenter and Bithell, 2000). Related to the basic bootstrap method is the studentized bootstrap method, whose performance is unstable in our models and not included in the table. Among the non-pivotal family of bootstrap methods, we find that the bias corrected (bc) percentile method (Efron, 1981) and the bias corrected and accelerated method (bca) (Efron, 1987) have similar performance when the sample size is 250 or 500. In ‘bca*’, we report results from bca when the sample size is less than or equal to 500; and results from bc when the sample size is larger because bca requires more bootstrap replicates when the sample size is larger. Another member of the non-pivotal family is the percentile method, whose overall performance is not as good as bc and bca, hence not included in the table.

Table 2.

Simulation results when model is misspecified, segmented model fit, β_z,0 = 0.34, β_x,0 = −0.040, β_(x−e)+,0 = 1.49 and e₀ = 2.82. Est: MC mean of parameter estimate; %bias: relative bias with respect to β₀ for the β′s and interquartile range of x for e. Five confidence interval methods are compared based on MC median width of 95% confidence interval and actual coverage (in parentheses). Model: Wald CI based on asymptotic variance dervied assuming correct model specification; robust and test inv: Wald and test inversion CI based on asymptotic variance dervied without assuming correct model specification; boot basic: basic bootstrap CI; boot bca*: bias corrected (and accelerated when n < 2000) bootstrap CI.

n		Est(%bias)	Model	Robust	Test Inv	Boot basic	Boot bca*
βz
25		0.45(35)	2.58(99)	2.48(95)		3.66(97)	3.71(92)
50		0.41(21)	1.71(96)	1.76(95)		2.29(99)	2.21(96)
100		0.36(8)	1.12(95)	1.18(96)		1.32(99)	1.30(97)
250		0.35(3)	0.68(95)	0.71(96)		0.73(97)	0.72(96)
500		0.34(2)	0.48(94)	0.48(95)		0.49(96)	0.49(95)
2000		0.34(0)	0.23(95)	0.24(95)		0.24(96)	0.24(95)
10000		0.34(0)	0.10(95)	0.10(95)		0.11(95)	0.10(95)
βx
25		−1.51(3677)	6.76(99)	8.99(93)		8.25(79)	7.91(81)
50		−1.23(2979)	4.06(97)	9.21(94)		7.42(86)	7.16(86)
100		−0.85(2044)	2.23(94)	7.09(96)		6.34(93)	5.93(92)
250		−0.44(1017)	1.28(84)	4.05(96)		3.53(88)	3.24(94)
500		−0.24(491)	0.89(78)	2.43(96)		2.05(82)	1.78(94)
2000		−0.09(117)	0.45(75)	0.97(99)		0.90(86)	0.85(96)
10000		−0.05(37)	0.20(70)	0.38(99)		0.40(90)	0.39(96)
β(x−e)+
25		3.60(142)	12.89(98)	11.96(91)		13.01(72)	13.27(79)
50		3.41(129)	8.26(96)	11.03(91)		10.77(79)	11.50(82)
100		3.10(108)	5.28(96)	10.23(96)		7.80(88)	8.35(81)
250		2.41(62)	2.60(95)	6.49(98)		5.86(96)	4.85(85)
500		1.97(33)	1.61(94)	2.89(98)		3.51(96)	1.79(88)
2000		1.59(7)	0.73(94)	0.82(98)		0.99(95)	0.75(94)
10000		1.51(1)	0.31(95)	0.32(97)		0.34(95)	0.32(95)
e
25		2.43(22)	3.31(79)	3.35(64)	2.72(70)	2.80(42)	1.82(57)
50		2.51(18)	2.53(72)	6.02(77)	3.28(82)	2.93(44)	2.21(65)
100		2.69(7)	1.93(63)	6.82(80)	3.79(88)	3.02(46)	2.54(77)
250		2.80(1)	1.38(57)	6.05(87)	4.24(94)	2.91(56)	2.76(90)
500		2.83(1)	1.06(57)	4.50(92)	3.93(97)	2.52(64)	2.51(94)
2000		2.81(0)	0.58(54)	1.94(99)	1.80(99)	1.59(77)	1.57(96)
10000		2.80(1)	0.27(51)	0.77(98)	0.75(98)	0.78(83)	0.76(96)

Table 3.

Simulation results when model is misspecified, hinge model fit, β_z,0 = 0.34, β_(x−e)+,0 = 1.46 and e₀ = 2.87. Est: MC mean of parameter estimate; %bias: relative bias with respect to β₀ for the β′s and interquartile range of x for e. Five confidence interval methods are compared based on MC median width of 95% confidence interval and actual coverage (in parentheses). Model: Wald CI based on asymptotic variance derived assuming correct model specification; robust and test inv: Wald and test inversion CI based on asymptotic variance derived without assuming correct model specification; boot basic: basic bootstrap CI; boot bca*: bias corrected (and accelerated when n < 2000) bootstrap CI.

n		Est(%bias)	Model	Robust	Test Inv	Boot basic	Boot bca*
βz
25		0.43(29)	2.64(99)	2.31(94)		3.98(98)	3.84(94)
50		0.38(12)	1.66(96)	1.58(94)		2.19(100)	2.11(97)
100		0.36(7)	1.11(95)	1.09(95)		1.26(99)	1.24(97)
250		0.34(2)	0.68(95)	0.68(95)		0.71(97)	0.71(96)
500		0.34(0)	0.47(95)	0.47(95)		0.49(96)	0.49(95)
2000		0.34(0)	0.23(95)	0.23(95)		0.24(96)	0.24(95)
10000		0.34(0)	0.10(96)	0.10(95)		0.11(95)	0.10(95)
β(x−e)+
25		1.95(33)	4.73(98)	3.07(92)		7.64(85)	7.89(84)
50		2.11(44)	3.65(96)	3.19(93)		6.72(81)	5.58(84)
100		2.19(50)	2.88(94)	3.15(94)		6.12(78)	3.74(85)
250		1.82(24)	1.66(93)	2.07(95)		4.39(84)	2.52(93)
500		1.61(10)	1.13(90)	1.47(94)		2.37(86)	1.73(96)
2000		1.48(1)	0.55(88)	0.73(94)		0.83(89)	0.78(96)
10000		1.46(0)	0.25(86)	0.33(94)		0.35(91)	0.34(96)
e
25		2.25(35)	6.29(98)	2.90(65)	2.18(57)	3.20(43)	2.78(68)
50		2.63(14)	3.41(94)	3.44(83)	3.17(81)	3.20(59)	2.98(85)
100		2.87(0)	2.19(85)	3.15(90)	3.23(91)	2.84(63)	2.76(89)
250		2.89(1)	1.38(78)	2.32(96)	2.35(97)	2.18(76)	2.30(94)
500		2.87(0)	0.98(78)	1.68(98)	1.65(98)	1.67(82)	1.71(96)
2000		2.86(1)	0.50(79)	0.83(97)	0.80(97)	0.83(87)	0.83(96)
10000		2.86(1)	0.22(77)	0.37(95)	0.36(96)	0.38(90)	0.38(95)

Under the segmented model, bias decreases with sample size. The relative bias of β̂_x is artificially high because β_x,0 = −0.040 is very close to 0. As in scenario I, the biases of β̂_x and β̂_{(x − e)+} are higher than those of ê and β̂_z. Among the analytical methods for confidence intervals construction, ‘model’ works reasonably well for β_z and β_{(x − e)+} but undercovers β_x and e, substantially so for e. The undercoverage is not helped by increasing sample size. ‘Robust’ improves the coverage for both β_x and e, bringing it close to the nominal level, but it shows some degree of overcoverage, particularly when the sample size is 2000 or above. ‘Test inv’ provides further improvement in the coverage for e by improving coverage and reducing the confidence interval width.

Among the bootstrap methods for constructing confidence intervals, ‘basic’ provides good coverage for β_z and β_{(x − e)+} but undercovers β_x and e. ‘Bca*’ improves the coverage for β_x and e but sees some pullback in the coverage for β_{(x − e)+}, e.g. 88% at n = 500. Overall, ‘bca*’ is preferable over ‘basic’. Comparing the ‘bca*’ bootstrap interval with the ‘robust’+‘test inv’ analytical interval, we see that the bootstrap intervals are typically shorter and have lower coverage for β_x, β_{(x − e)+}, and e when sample sizes are small. This difference decreases when the sample size reaches 2000 and disappears at n = 10, 000.

Under the hinge model, we see similar results with some interesting differences, some of which are also seen in scenario 1. First, as in scenario 1, β̂_{(x − e)+} is much less biased in the hinge model fit than in the segmented model fit. For example, at n = 500, the relative bias is 33% and 10% under the segmented and hinge model, respectively. Second, there is less of an overcoverage problem in the hinge model fit than in the segmented model fit. Third, the width of the ‘robust’, ‘test inv’ and ‘bca*’ intervals are similar to each other. In some cases, the ‘robust’ method has the shorter intervals and better coverage. For example, for β_{(x − e)+} the ratios between the width of the ‘robust’ and ‘bca*’ intervals are 0.39, 0.82 and 0.94 for n = 25, 250 and 2000, respectively; and the coverage probabilities are 92%, 95% and 94% for the ‘robust’ intervals and 84%, 85% and 96% for the ‘bca*’ intervals.

5. Application in HIV-1 immune correlates of risk studies

Developing an effective HIV-1 vaccine is currently one of the greatest public health challenges. One strategy to accelerate development is to study so-called immune correlates of risk (Gilbert et al., 2008; Plotkin and Gilbert, 2012), which are biomarkers measuring immune responses to vaccination or natural infection that are associated with the risk of HIV-1 infection. As part of this strategy, Permar et al. (2015) measured antibody-based immune responses in blood samples from a cohort of HIV-1 infected pregnant women in the era before the advent of antiretroviral therapy (Rich et al., 2000) to study their association with the risk of mother-to-child transmission (MTCT) of HIV-1. An array of immunological assays was conducted to measure binding antibodies, neutralizing antibodies, antibody-dependent cell-mediated cytotoxicity, and antibody avidity, the last of which refers to the combined effect of multiple antibody binding affinities. Permar et al. showed that the risk of transmission of HIV-1 from mothers to children was associated with binding antibodies against the V3 region of the HIV-1 envelope protein as well as neutralization activities against the so-called tier 1 HIV-1 strains, which are relatively easy to neutralize compared to other circulating strains of HIV-1. The latter finding came as a surprise because the weakly neutralizing antibodies had not been associated with protection against HIV-1 infection in vaccine studies. A potential explanation for the association of weakly neutralizing antibodies with MTCT risk was the finding that maternal antibodies could indeed neutralize concomitant autologous virus strains (Moody et al., 2015).

Permar et al. used logistic regression models with linear terms to select immune responses that are associated with MTCT risk. Our goal here is to further refine our understanding of the association between MTCT risk and the covariate NAb_SF162LS (x), which is a measure of the neutralizing activities against a tier 1 HIV-1 strain SF162LS. The dataset comprises 236 samples, each corresponding to one infected pregnant woman. There are 79 transmitters and 157 non-transmitters. In addition to NAb_SF162LS, all models adjust for birth (z), which indicates whether the birth was C-section or vaginal.

We first seek to detect nonlinearity by testing the null hypothesis β_{(NAb_SF162LS−e)+} = 0 against the alternative β_{(NAb_SF162LS−e)+} ≠ 0 in a segmented model using a maximum of scores test statistic (Fong et al., 2015). The null is rejected with p-value 0.003. We then fit four logistic regression models to the data. Two of them model NAb_SF162LS through natural cubic splines with two and three degrees of freedom, respectively; and the other two models NAb_SF162LS through the hinge and segmented threshold effects, respectively. The fitted risks are shown as functions of NAb_SF162LS in Figure 2. The nonlinearity of the relationship between MTCT risk and NAb_SF162LS is apparent in the two cubic spline fits. Table 4 reports the estimated parameters and 95% confidence intervals for the two threshold models. The two models have similar estimates for the slope of the birth covariate. In the segmented model fit, both β̂_{NAb_SF162LS} and β̂_{(NAb_SF162LS−e)+} are not significant according to either model-robust Wald or BCa bootstrap confidence intervals. We thus focus on interpreting the hinge model fit.

Figure 2.

Estimated log odds ratio as functions of NAb_SF162LS in the MTCT example.

Table 4.

Threshold regression model fits in the MTCT example. OR: odds ratio; robust/BCa: model-robust and bias corrected or accelerated bootstrap 95% CI.

	segmented model		hinge model
	OR (robust)	(BCa)	OR (robust)	(BCa)
birthVaginal	1.23 (0.63, 2.39)	(0.60, 2.39)	1.24 (0.64, 2.40)	(0.61, 2.45)
NAb_SF162LS	1.71 (0.38, 7.62)	(0.88, 4.80)
(NAb_SF162LS-chngpt)+	0.40 (0.10, 1.56)	(0.14, 1.38)	0.67 (0.51, 0.88)	(0.27, 0.87)
chngpt	6.09 (2.75, 9.42)	(4.81, 7.33)	7.37 (5.14, 9.61)	(6.02, 10.46)

In the hinge model fit, β̂_{(NAb_SF162LS−e)+} is significant by both model-robust Wald and BCa confidence intervals. The model also suggests that NAb_SF162LS is not associated with MTCT risk when it is less than 7.37, although the estimated confidence intervals for the threshold are fairly wide by either the model-robust test-inversion method or BCa. Above the threshold, one unit increase in NAb_SF162LS is associated with an odds ratio of 0.67 (95% robust confidence interval: 0.51, 0.88). In contrast, in a logistic regression model fit with linear NAb_SF162LS, the estimated odds ratio for one unit increase in NAb_SF162LS is attenuated to be 0.82 (95% robust confidence interval: 0.71, 0.95).

As a second example we apply threshold models to the immune correlates dataset from the RV144 HIV-1 vaccine trial (Haynes et al., 2012). The results are shown in the Supplementary Materials Section F. Results from both of these examples suggest that, when appropriate, hinge threshold models are particularly attractive compared with generalized linear models and segmented threshold models. Our applications deal with covariates that are measured by sensitive immunoassays. Assay readouts in the lower range are likely to be dominated by experimental noises rather than biological signals. By using hinge threshold models we can get a more accurate and less attenuated picture of the relationship between immune responses and the outcome of interest than generalized linear models.

6. Conclusions

In this paper we investigate the asymptotic behavior of continuous threshold model parameter estimates and use the knowledge to construct two types of model-robust analytical confidence intervals. To provide guidance for choosing the most appropriate inference procedures in real data analyses, we conduct simulation studies to evaluate these methods as well as bootstrap confidence intervals. Three main conclusions can be drawn: (1) When using model-robust intervals, profile likelihood ratio test-inversion intervals should replace Wald intervals for the threshold parameter. (2) Among the commonly used bootstrap confidence interval methods, bias corrected (accelerated) intervals have the best performance. (3) Analytical confidence intervals assuming correct model specification have low coverage when the model is in fact misspecified, hence overly optimistic. Both model-robust and bootstrap confidence intervals provide a remedy to the problem, but neither is perfect at moderate sample sizes in the segmented model: model-robust intervals have overcoverage and bootstrap intervals have undercoverage of some parameters. Thus, the choice of method will depend on which error is more tolerable in a given application. In the hinge model model-robust intervals work better overall.

When the sampling distributions are skewed and heavy-tailed, the bootstrap distributions intervals are asymmetric around the point estimates by construction. The model-robust analytical intervals, on the other hand, are always symmetric. In the hinge model, the sampling distributions are close to being normal, and the length of the model-robust and bootstrap intervals are close to each other as well. In the segmented model, the sampling distributions are more skewed and heavy-tailed. As the model-robust intervals are constructed based on the second moment only, they tend to be longer than the bootstrap intervals. In addition, when the skewness of the sampling distributions and the asymmetry of the bootstrap intervals go in a coordinated direction, we get good coverage; but if they go in the wrong direction, we get undercoverage, and that is what happens with, e.g. the bca intervals of β̂_{(x − e)+}.

Our numerical studies suggest that the hinge model may be estimated with higher precision than the segmented model. To investigate this further, we generate data from a hinge model as specified in scenario I, and fit the segmented model. Results are summarized in Table D.1 in the Supplementary Materials, where, for the ease of comparison, we include a replicate of the hinge model fit results. Based on the results at n = 10, 000, it takes twice as many samples to fit a segmented model to achieve the same level of precision in β̂_{(x − e)+} as it does to fit a hinge model. For ê, the ratio is (0.36/0.28)² = 1.65. Under model misspecification, in scenario II, β_x,0 is −0.04, small enough that a hinge model fit seems a sufficient summary of the relationship between the outcome and x. Indeed, β_{(x − e)+,0} under the segmented model is close to that under the hinge model, so is e₀. Based on results from n = 10, 000 in Table 2 and 3, the relative efficiency of β̂_{(x − e)+} is close to 1 and the relative efficiency of ê is (0.75/0.36)² = 4.34. In scenario II, the hinge model fit and the segmented model fit are very different; thus, it is not as meaningful to compare the precision of estimation.

Our theoretical results depend on the assumption that θ₀ is unique. When θ₀ is not unique, we have a nonregularity problem, which is connected with several other such problems as listed by McKeague and Qian (2015). They include post-model-selection inference, the Davies problem (of which threshold models are an example), and when a parameter is on the boundary of the parameter space. A unifying theme of these problems is that the asymptotic distributions of parameter estimates are discontinuous near the boundary between identifiability and non-identifiability in the parameter space, while their finite sample distributions are continuous. We illustrate this with a simulation study on the segmented model in Section D.5 of the Supplementary Materials. The approach of McKeague and Qian, which builds upon local asymptotics results, is also promising for threshold models.

Threshold models are useful in practice because they are more flexible than simple linear models and more interpretable than spline or polynomial models. To make judicious use of threshold models in practice, we should compare the threshold model fit with fits from more flexible smooth models to ensure that the trends they represent match each other. It is also important to note that the meanings of the threshold model parameter estimates depend on the model, and we should be careful not to over-interpret them. To illustrate, we present an example based on a hybrid hinge-quadratic model in Section G of the Supplementary Materials.

A useful alternative to the threshold models is the transition models proposed by Pastor-Barriuso et al. (2003), which use either logistic function [exp {(x − e)/ε} − 1]/[exp {(x − e)/ε+1] or hyperbolic function $\sqrt{{(x-e)}^2+{\epsilon }^2}/(x-e)$ in the place of the indicator function 2I (x > e) − 1. As ε → 0, the two functions approach the indicator function. The parameter ε may be specified beforehand or estimated from the data. When it is estimated from the data, the resulting model fit may look very different from their threshold model counterparts, especially around the threshold. Formal methods for guiding the choice between threshold models and transition models are worth further investigation.