Are the Trends Changing Joinpoint Regression Analysis

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Are the Trends Changing?Joinpoint Regression
Analysis

• Hyune-Ju Kim
• Syracuse University
• Joint Work with
• M. Fay, E. Feuer, B. Yu, M. Barrett and D.
  Midthune
• National Cancer Institute

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Outline

• Motivation
• Joinpoint Regression (segmented line regression)
• Model
• Fitting a joinpoint regression model
• Determining the number of joinpoints
• Software Joinpoint
• http//srab.cancer.gov/joinpoint/

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• I. Motivation

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Polynomial model vs. linear model
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Change-point model 1 Jump model
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Change-point model 2 Joinpoint model
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Joinpoint Regression Model

• Piecewise linear regression
• Segmented line regression
• Broken line regression
• Spline regression
• Joinpoint
• Change-points
• Changeover points
• Joins
• Knots

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Motivating Example
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Year-by-Year Fit
RSS19.41 for 1991-2000
RSS22.86 for 1969-1990
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Search for Min RSS
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Questions

• How can we impose the constraint that the two
  phases are continuous at the change-point?
• Where is/are the change-point(s)?
• How do we know if a two-phase model is preferred
  to a three-phase model?

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• II. Joinpoint Regression

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II-1. Mathematical Model
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k-joinpoint model
Model y ß0ß1xd1(x-t1)dk(x-tk) error
ß2ß1d1
ß3ß1d1d2
t2
ß1
t1
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Questions of Interests

• Fitting a joinpoint regression model
• Point estimates of ?, ? and ?
• Grid search (Lerman, 1980)
• Hudsons method (Hudson, 1966)
• Confidence interval for the joinpoints and
  regression coefficients
• Large sample theory (Feder, 1975 Hinkley 1971)
• Determining the number of joinpoints, k.
• Permutation procedure, BIC

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II-2. Model Fitting
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Lermans Grid Search (Lerman (1980), Applied
Statistics)

• If the joinpoints are fixed at (t1,,tk), then
  fit a least square (LS) regression model using
  covariates
• x, (x-t1),,(x-tk)
• and get a residual sum of squared error (SSE).
• Search all possible combinations of (t1,,tk) to
  find the point (t1,,tk) which minimizes the SSE.
  
• The LS estimates of the regression coefficients
  ß and ? are the estimates corresponding to the
  estimated joinpoints.

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Fit of 1-joinpoint model
Model E(yx) ß0ß1xd1(x-t1)
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Fit of 1-joinpoint model
Model E(yx) ß0ß1xd1(x-t1)
?
t11991
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Confidence intervals
95 Confidence interval for the joinpoints x
SSE(x) ? MinSSE ? (1 F(0.95, k, p)) , where
F(0.95, k, p) is the 95th percentile of an
F-distribution with k and p degrees of freedom,
and pn-2(k1)
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Parameter estimates (Grid search)
Model 1 1 Joinpoint(s) Number of
Observations 32 Number of
Parameters 4 Degrees of Freedom
28 Sum of Squared Errors
0.00084419 Mean Squared Error
0.00003015 Estimated Joinpoint(s) Join Pt
Estimate (95 Confidence Interval)
1 91.0000000000 ( 90.0000000000 ,
93.0000000000 ) Estimated Regression
Coefficients (Beta) Parameter
Standard Parameter Estimate Error
Z Prob t Intercept1
4.989298 0.014415 346.125422 0.000000
Intercept2 6.204406 0.066682
93.044993 0.000000 Slope1
0.004244 0.000181 23.479840 0.000000
Slope2 -0.009109 0.000694
-13.118777 0.000000 Slope2-Slope1
-0.013353 0.000717 -18.610506
0.000000 Annual Percent Change (APC) Segment
Range APC (95 Confidence
Interval) 1 69 - 91 0.425282 (
0.388046 , 0.462532 ) 2 91 - 100
-0.906767 ( -1.047844 , -0.765489 )
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Hudsons method (Hudson (1966), JASA))

• Partition the data points into k1 Segment.
• Fit the unconstrained LS line for each segment.
• Calculate the intersection of the regression
  lines for the neighboring segments.
• If the intersections are not in the right
  locations, then adjust them to either end.
• The estimated joinpoints are the intersections
  for the partition with the minimum SSE.

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Parameter estimates (Hudsons algorithm)

• Model 1 1 Joinpoint(s)
• Number of Observations 32
• Number of Parameters 4
• Degrees of Freedom 28
• Sum of Squared Errors 0.00079300
• Mean Squared Error 0.00002832
• Estimated Joinpoint(s)
• Join Pt Estimate (95 Confidence
  Interval)
• 1 91.4254830000 ( 90.0000000000 ,
  93.0000000000 )
• Estimated Regression Coefficients (Beta)
• Parameter Standard
  Z Prob t
• Parameter Estimate Error
• Intercept1 4.996409 0.013429
  372.060337 0.000000
• Intercept2 6.274229 0.065980
  95.093460 0.000000
• Slope1 0.004150 0.000167
  24.808813 0.000000

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II-3. Determining the number of joinpoints

• Hypothesis testing
• H There are k joinpoints
• H There are k joinpoints.
• Information based model selection
• BIC (Bayesian Information Criteria)

0
0
1
1
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Permutation Test

• Consider the null and alternative hypothesis
• H There are k joinpoints
• H There are k joinpoints.
• Step 1 Choose the test statistic.
• Step 2 Compute the value of the test statistic
  by fitting the null and alternative models and by
  computing the residual sum of squares.
• Step 3 Assess if the observed amount of
  reduction in error is significant enough to
  choose a model with a larger number of joins.

0
0
1
1
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Test Statistic

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Example Data 1 JP vs. 2 JP

• Null fit (1 JP) Joinpoint at 1991, Mean
  Squared Error1.2761
• Alternative fit (2 JP) Joinpoints at 1973,
  1991, MSE0.9296
• T6.2185

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• Question Is T6.2185 significant enough to
  reject the null hypothesis of 1 joinpoint in
  favor of the alternative hypothesis of 2
  joinpoints?
• Answer We need to find the null distribution of
  T(y) and the p-valueP(T(y)T), where T is the
  observed value of the test statistic.
• Problem The null distribution of T(y) is not
  known, even asymptotically, when the location of
  the joinpoints are not known.
• Our solution Permutation test (Kim et al. (2000)
  Statistics in Medicine)

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Idea of Permutation
(3)
(1)
(2)
(3)
(2)
(1)
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Fits for the permuted data sets
Joinpoint est1991 RSS34.9579 T0.9212
Joinpoint est1991 RSS35.1528 T4.4592
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Empirical distribution of the Test Statistic

• T 6.219
• Number of permutations 30
• ( of T-values ? T) 2
• 2/30 0.0667 ? P-value

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Estimation of the p-value
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Determining the number of joinpoints

• Method 1 Sequential application of the
  permutation tests
• Pre-specify kmin,,and kmax.
• Start from testing H0 kk0 vs. H1 kk1, where
  k0 kmin and k1 kmax.
• Use permutation test to make a decision
• If H0 is rejected, then test for H0 kk0 kmin
  1 vs. H1 kk1.
• If H0 is not rejected, then test for H0 kk0
  vs. H1 kk1 kmax -1.
• Stop when k is determined.
• Note The a-level for each test
  a/(kmax-k0)
• Method 2 Bayesian information criterion
• k argmin B(k) argmin log(SSE(k)/n) (2k2)
  (log n)/n ,
• where the minimum is taken over k such that 0?
  k ? kmax.

?
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Example Data (Method 1)

• Number of permutations 4499
• kmin 0 and kmax 3
• Final selected model two-joinpoint model

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Example Data (Method 1)
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Example Data (Method 2)

• kmax 3
• The model with the smallest BIC is selected.
• Final selected model three-joinpoint model

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Fit for the 3 JP model
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References

• Kim HJ, Fay MP, Feuer EJ, and Midthune D (2000),
  Permutation Tests for Joinpoint Regression with
  Applications to Cancer Rates,, Statistics in
  Medicine, 19, 335-351.
• Weighted LS to handle heteroscadastic and
  correlated errors
• Power study via simulations
• Kim HJ, Fay MP, Yu B, Barret MJ and Feuer EJ.
  (2004), Comparability of segmented line
  regression models, Biometrics 60, 1005-1014.
• Yu B., Barrett MJ, Kim HJ, and Feuer EJ. (2006),
  Estimating joinpoints in continuous time scale
  for multiple change-point models, To appear in
  Journal of Computational Statistics and Data
  Analysis

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• III. Software Joinpoint

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Data Input
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Data used
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Input tab Default Settings
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Joinpoints tab Default Settings
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Execute Session
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Results - Graph
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Results - Data
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Results Model Estimates
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Results Permutation Tests
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Current Developments

• Early stopping rule can reduce the number of
  permutations while controlling the resampling
  risks
• Comparability test enables us to determine if two
  or more groups share the common joinpoint model
• Clustering 18 contiguous age groups so that the
  age groups within the same cluster share the
  common joinpoint model.