**Basic Method**

Kim, H-J, Fay, M.P., Feuer, E.J., and Midthune, D.N. (2000) "Permutation Tests for Joinpoint Regression with Applications to Cancer Rates", Statistics in Medicine 19, 335-351. (correction: 2001;20:655). Correction to Table 1(a) of Kim, et al. is provided as a PDF at http://surveillance.cancer.gov/documents/joinpoint/table1.pdf.

This is a paper where joinpoint regression is applied to describe cancer rates and the permutation test is proposed to determine the number of significant joinpoints. The grid search of Lerman (1980) was used to fit the segmented regression function and the p-value of each permutation test is estimated using Monte Carlo methods, and the overall asymptotic significance level is maintained through a Bonferroni adjustment.

**Background Papers for Basic Method**

Feder, P.I. (1975) "On Asymptotic Distribution Theory in Segmented Regression Problems: Identified Case", Annals of Statistics 3, 49-83.

Feder studied asymptotic properties of the least squares estimators in multi-segment regression and proved that under some technical conditions on the independent variable, the least squares estimators are consistent and asymptotically normal.

Lerman, P.M. (1980) "Fitting Segmented Regression Models by Grid Search", Applied Statistics 29: 77-84.

Lerman proposed the grid search method to fit segmented line regression where the joinpoint estimates occur at discrete grid points, and studied asymptotic inference using asymptotic normality proved by Feder (1975).

Hudson, D. (1966) "Fitting segmented curves whose join points have to be estimated", Journal of the American Statistical Association 61, 1097-1129.

Hudson proposed a procedure to fit a segmented regression curve whose joinpoints can be estimated anywhere in the data range. In the first stage, the model is fit to every feasible partition of the data without imposing continuity across the change-points. We then examine the locations of the intersection points of the least square lines. When the intersection point of the two adjacent least squares lines is not between the two data points which separate the segments, we make an adjustment in the fit by mathematically imposing the continuity constraints at one of the two boundary data points. The final estimates are obtained by searching for the global minimum of residual sum of squares.

**Fitting Joinpoints in Continuous Time**

Yu, B., Barrett, M., Kim, H-J, and Feuer, E.J. (2007) "Estimating Joinpoints in Continuous Time Scale for Multiple Change-Point Models", Computational Statistics and Data Analysis 51, 2420-2427.

In this paper, we extend the Hudson's continuous fitting method to a multiple joinpoint model and discuss some practical issues in the implementation. We also compare computational efficiencies of the Lerman's grid search and the Hudson's continuous fitting.

**Early Stopping Rules for the Permutation Test**

Fay, M.P., Kim, H-J, and Hachey, M. (2007) "On using Truncated Sequential Probability Ratio Test Boundaries for Monte Carlo Implementation of Hypothesis Tests", Journal of Computational and Graphical Statistics 16 (4), 946-967.

Joinpoint selects a final model conducting a series of permutation tests and we can save computation time by using sequential stopping boundaries. This paper proposes a truncated sequential probability ratio test boundary to stop resampling when the early replications indicate a large enough or small enough p-value, and studies its properties.

**Comparing Two Joinpoint Regression Lines**

Kim, H-J, Fay, M.P., Yu, Binbing, Barrett, M.J., and Feuer, E.J. (2004) "Comparability of segmented line regression models", Biometrics 60, 1005-1014.

We propose a procedure to compare two segmented line regression functions, especially to test (i) whether two segmented line regression functions are identical or (ii) whether the two mean functions are parallel allowing different intercepts. A general form of the test statistic is described and then the permutation procedure is proposed to estimate the p-value of the comparability test.

**AAPC**

Clegg, L.X., Hankey, B.F., Tiwari, R., Feuer, E.J., Edwards, B.K. (2009) "Estimating average annual percent change in trend analysis", Statistics in Medicine 28(29): 3670-8.

**AAPC CI**

Kim H-J, Luo J., Chen H.S., Green D., Buckman D., Byrne J., Feuer E.J. (2017) "Improved Confidence Interval for Average Annual Percent Change in Trend Analysis", Statistics in Medicine 36(19): 3059-74.

**Studies on the performance of Joinpoint**

Kim, H-J, Yu, B., and Feuer, E.J. (2008) "Inference in segmented line regression: A simulation study", Journal of Statistical Computation and Simulation 78:11, 1087-1103.

Via simulations, this paper empirically examines small sample behavior of asymptotic confidence intervals and tests, based on Feder (1975)'s asymptotic normality of least squares estimators in joinpoint regression, studies how the two fitting methods, the grid search and the Hudson's continuous fitting algorithm, affect these inferential procedures and also assesses the robustness of the asymptotic inferential procedures.

Kim, H-J, Yu, B., and Feuer, E.J. (2009) "Selecting the number of change-points in segmented line regression", Statistica Sinica 19(2):597-609.

In this paper, we show that under some conditions, the number of joinpoints selected by the permutation procedure of Joinpoint is consistent. Via simulations, the permutation procedure is compared with some information-based criteria such as Bayesian Information Criterion (BIC).

**Model Selection**

J. Kim and H.-J. Kim (2016), "Consistent Model Selection in Segmented Line Regression", *Journal of Statistical Planning and Inference* **170**, 106-116.

N.R. Zhang and D. O. Siegmund (2007), "A Modified Bayes Information Criterion with Applications to the Analysis of Comparative Genomic Hybridization Data", *Biometrics* **63**, 22–32.