Estimating joinpoints in continuous time scale for multiple change-point models
Introduction
It is of great importance to describe the trend of cancer incidence and mortality data. The joinpoint regression model, which is composed of a few continuous linear phases, is often useful to describe changes in trend data. Suppose that for the observations , the responses , with and for random errors . The joinpoint regression models assume that, in each segment, the follows a linear modelwhere , and is continuous throughout , such thatAs the response is continuous at the change points, we call model (1) the joinpoint model and the 's joinpoints (JPs). This model is also called segmented-line regression model or piecewise linear model (Kim et al., 2004). An alternative parameterization of the JP model (1) iswhere and if and 0 otherwise. This parameterization implicitly satisfies the continuity of at .
The current estimation method is the grid search (LGS) method proposed by Lerman (1980), which is implemented by Joinpoint software developed by U.S. National Cancer Institute (http://www.srab.cancer.gov/joinpoint). Although the LGS method can be refined such that the JPs could occur at the middle point or quarterly point between two data points, the computation time for finer grid increases dramatically. Hence, the LGS method is practical only when the JPs occur at the observed data points. Hudson (1966) described the continuous algorithm in detail for a one-JP model and discussed its extension to a model with more than two JPs, which is not straightforward. Our aims in this paper are to describe the details of the extension to a multiple JP model and to compare computational efficiencies of these two fitting methods.
Several alternative methods have been proposed to estimate the locations of the change points for single series in different contexts. For example, Quandt (1958) and Quandt and Ramsey (1978) proposed the procedure of estimating a single change point without continuity constraint at response in economics settings, Hinkley, 1969, Hinkley, 1971 discussed the estimation and inference for the joinpoints in one-joinpoint models, Smith (1975), Carlin et al. (1992), Slate and Turnbull (2000) and Tiwari et al. (2005) use Bayesian approaches to estimate the change points under different scenarios. Most of the available methods estimate the single change/join point. The proposed method in the paper estimates the multiple joinpoints in continuous scale, hence it provides a better fit.
The rest of the paper is organized as follows: The model formulation and notation are described in Section 2 and Hudson's method for a one-JP model is reviewed in Section 3. In Section 4, Hudson's method is extended to a multiple JP model and the issues arising in the implementation are discussed. Then the multiple JP model is applied to colorectal cancer incidence data for men under age 65 from the SEER nine registries. The relative merits of different approaches are discussed in the final section.
Section snippets
Model formulation and notation
Let the th segment denoted by for and . For each segment , we define that where , and the weight matrix . Let Notice that and . Then, the JP model (1) can be expressed as with constraints (2), where , . Let . To fit this model, we find the estimates
Review of Hudson's method: one JP
In this section, we first summarize Hudson's algorithm for the 1-JP model. The procedure to estimate is described as follows:
- (a)
For the partition , fit the least square (LS) regression for each segment. Let The unconstrained weighted LS estimates are
- (b)
Let be the solution to the equation . If , then is called in the “right” place. That
Estimation of multiple JP model in continuous scale
For a K-JP model, there are segments, and K JPs. The kth JP divides segments and . Recall that the unconstrained LS estimates that minimize in (4) areWhen the , are independent, then is block diagonal andand .
The th JP is obtained by solving equation . Let denote the location of
Application
In the “Annual Report to the Nation on the Status of Cancer”, jointly released by the National Cancer Institute (NCI), the American Cancer Society (ACS), the North American Association of Central Cancer Registries (NAACCR), and the Centers for Disease Control and Prevention (CDC), including the National Center for Health Statistics (NCHS), the rate of new cancer cases and deaths for all cancers combined as well as for most of the top 10 cancer sites were reported. The joinpoint regression
Discussion
In this paper, we discuss the computational details of estimating multiple joinpoints in continuous time scale, and compare the computational efficiencies of the two fitting methods, the Hudson's method and the Lerman's grid search method.
In summary, the Hudson's method takes longer time than the basic grid search where only the data points serve as the grid points, but it is more efficient than a grid search with more than four points inserted between the consecutive data points. To illustrate
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