Estimated Regression Coefficients (Beta)

The output is a combination of the two parameterizations (see Table 1). The estimates of (β₀,β₁,δ₁,...,δ_k) come from the grid-search method. The estimates of (β₀₁,β₁₁,...,β_0,k+1,β_1,k+1) are calculated based on Table 1.

However, the standard errors of the regression coefficients are estimated under the GP model (Equation (2)) without continuity constraints. Following Lerman's implementation, (Lerman; 3rd paragraph, page 79, 1980; Feder, Section 4, page 69, 1975), the data points that are on the joinpoints are deleted. Then conditioned on the partition implied by the estimated joinpoints (τ₁,...,τ_k) , the standard errors of (β₀₁,β₁₁,...,β_0,k+1,β_1,k+1) are calculated using unconstrained least square for each segment. If there are segments with zero or one observation (not including the joinpoints at both ends), then a generalized inverse is used to calculated the covariance matrix. The standard error of the difference in slopes, δ_j, is the square root of the sum of the squared standard errors (variance) for the two consecutive slopes β_{1j and} β_1,j+1

The test statistic (U) is the parameter estimate divided by the standard error. The test statistic has a t distribution with d degrees of freedom where d is defined as follows. Let n_J be the number of data points that are on joinpoints. The effective number of data points is n^*=n-n_J. Let k₀ and k₁ be the number of segments with zero or one observation. The effective number of parameters (rank of the design matrix) is p^*=2(k+1)-2k₀-k₁ and the degrees of freedom d=n^*-p^*. For the default option where the minimum number of data points between two joinpoints (excluding any joinpoint that falls on an observation) is two, k₀=k₁=0 and d=n^*-2(k+1). For testing H₀: β = 0 the p-value is calculated as 2{1-t_d( |U| )}, where t_d is a t distribution with d degrees of freedom.

If a standard error cannot be calculated, then the associated statistics will not be displayed.