Pairwise Comparison

The main goal of the comparability test is to compare two sets of trend data whose mean functions are represented by joinpoint regression. Specific interests are testing:

1. whether two Joinpoint regression functions are identical (test of coincidence) or
2. whether the two regression mean functions are parallel (test of parallelism)

and the details can be found in Kim et al. (2004):

H-J. Kim, M. P. Fay, B. Yu, M. J. Barrett and E. J. Feuer (2004), Comparability of Segmented Line Regression Models, Biometrics, 1005-1014.

Consider the Joinpoint regression mean function at the j^th year of the i^th group, x = x_ij,

\(\mu_i \left( x \right) = \beta_{i,0} +\ \beta_{i,1}x\ +\ \delta_{i,1} ( x - \tau_{i,1} )^+ +\ \cdots\ +\ \delta_{i,K_i} ( x - \tau_{i,K_i} )^+,\)

where k_i is the unknown number of joinpoints, the T's are the unknown joinpoints, the β's and the δs are the regression coefficients, and a⁺ = a for a > 0 and 0 otherwise. In general, the two groups may have different numbers of joinpoints, say k_i for the i^th group, but the program fits both groups with a larger model with k_max joinpoints.

Computation Notes:

• These tests are selected using the Pairwise Comparison option under the Advanced Analysis Tools tab.

• Using these tests could substantially increase the computation time.

• Autocorrelated errors models are not available with these tests.

• All 2-way combinations of the innermost by-variable are tested.

What are the hypotheses?

The null hypotheses for the test of coincidence and the test of parallelism are

\(H_0: \left( \tau_{1,1}, \ldots, \tau_{1, k_{max}}, \beta_{1,0}, \beta_{1,1}, \delta_{1,1}, \ldots, \delta_{1, k_{max}} \right) = \left( \tau_{2,1}, \ldots, \tau_{2, k_{max}}, \beta_{2,0}, \beta_{2,1}, \delta_{2,1}, \ldots, \delta_{2, k_{max}} \right)\)

and

\(H_0: \left( \tau_{1,1}, \ldots, \tau_{1, k_{max}}, \beta_{1,1}, \delta_{1,1}, \ldots, \delta_{1, k_{max}} \right) = \left( \tau_{2,1}, \ldots, \tau_{2, k_{max}}, \beta_{2,1}, \delta_{2,1}, \ldots, \delta_{2, k_{max}} \right),\)

respectively.

What is the test static?

The test statistic with k = k_max is

\(F_k = \frac{\left( RSS_{H_{0,k}} - RSS_{H_{1,k}} \right) / d_{1,k}}{RSS_{H_{1,k}} / d_{2,k}},\)

where RSS denotes the residual sum of squares obtained from the least squares fitting, and d_1,k and d_2,k are appropriate degrees of freedom.

How is the k_max chosen?

The recommended data driven choice of k_max is the largest number of joinpoints estimated under the null and alternative models:

\(\hat{K}^+ = \max \left( \hat{K}_0, \hat{K}_1, \hat{K}_2 \right),\)

where \(\hat{K}_0\) is the number of joinpoints estimated for the two groups combined under the null model, and \(\hat{K}_1\) and \(\hat{K}_2\) are the numbers of Joinpoints estimated separately.

How is the p-value estimated?

The p-value of the test is estimated using the permutation distribution of the test statistic. The residuals, obtained under the null model, are permuted to generate the permutation distribution of the test statistic, and the p-value is estimated as the proportion of the permutation data sets whose test statistic values are greater than or equal to the original test statistic value.

Significance Level

Set the significance level for the test of coincidence or the test of parallelism.

Max Number Of Randomly Permuted Data Sets

The program performs a permutation procedure for the test of coincidence or the test of parallelism. Since fitting all N! possible permutations of the data would take too long, the program takes a Monte Carlo sample of these N! data sets, using a random number generator. Specify the size of the Monte Carlo sample of permuted data sets.

The minimum allowed number of permutations is 1000.  The maximum number is 10,000.  For greater consistency in the p-values obtained if one were to change the seed for each run, we strongly recommend running the program for at least 4499 permutations.  You may use a smaller number of permutations to speed up the calculations, but the permutation test may produce less consistent results.  Unless you have a good understanding of the implications of changing the number of permutations, we recommend against it.