Heteroscedastic/Correlated Errors Option

Heteroscedastic/Correlated Errors Option

The ordinary least squares (OLS) fitting of a Joinpoint regression model is performed assuming that the errors are uncorrelated with constant variance.  When the errors are heteroscedastic and/or correlated, Joinpoint performs weighted least squares (WLS) analysis.

The error random variable in a model is homoscedastic if the variance of the error is constant; otherwise, the error is heteroscedastic. Often the homoscedastic assumption is violated, particularly when the variance of the error varies with time. The weights in WLS for uncorrelated but heteroscedastic errors are the reciprocal of the variance and can be specified in several ways. When the errors are correlated, Joinpoint conducts the weighted least squares analysis with the variance-covariance matrix provided by the user or for a special case where the errors are autocorrelated with lag one, the user specifies the autocorrelation coefficient or allows the program to estimate it from the data.

The “Heteroscedastic/Correlated Errors Option” allows the user to select an option for the WLS analysis, and provides the following options:

• Constant Variance (Homoscedasticity)
  • The "Constant Variance (Homoscedasticity)" option handles a situation where there are homogeneous errors. This indicates a model where errors are assumed to have constant variance.  No standard errors from the input file are allowed for this option. Under this option, you can specify a special correlation structure: uncorrelated or autocorrelated with lag one. Joinpoint allows the user to provide the autocorrelation parameter or estimates it from the data. For more information regarding the autocorrelated errors model, please reference the Joinpoint Regression for Correlated Data (below).

• Standard Error (Provided)
  • The “Standard Error (Provided)” option handles a situation where there are heteroscedastic errors according to the variance structure provided. This allows the user full flexibility to specify the standard error at each time period. These standard errors can come from the output of SEER*Stat. Under this option, you can specify a special correlation structure as under the “Constant Variance” option.

• Poisson Variance
  • The “Poisson Variance” option handles uncorrelated but heteroscedastic errors based on Poisson errors. This estimates the non-constant variance by assuming the dependent variable counts follow a Poisson distribution. 

• Variance-Covariance Matrix (Provided)
  
  • The “Variance-Covariance Matrix (Provided)” option handles a situation where the errors are correlated following a non-specific covariance structure. If this option is chosen, a variance-covariance matrix file must be specified by the user.  For more information regarding this option, please reference Variance-Covariance Matrix.

Restrictions for the variance-covariance matrix:

1.  Joinpoint will not accept a matrix filled entirely of zeroes.

2.  The number of rows and columns of the matrix must match the number of observations in the input file.

3.  Joinpoint currently only allows the variance-covariance option to be executed when the input data file has one cohort.

4.  The diagonal entries are variances, so they should be positive values.

5.  The matrix should be symmetric. To illustrate, the value on the second row and first column should be identical to the value on the first row and second column. 

Notes:

1. The Constant Variance (Homoscedasticity) option is automatically selected if your input data file contains only 2 fields.

Joinpoint Regression for Correlated Data

Autocorrelated errors

When the Constant Variance (Homoscedasticity) or Standard Error (Provided) options are selected, you have the following choices for an error correlation structure:

1. Uncorrelated

If you select "Uncorreleted", the program assumes the random errors in the regression model are uncorrelated and estimates the regression coefficients by ordinary least squares (unless the errors are heteroscedastic; see ).

 If you suspect negative autocorrelation, the uncorrelated errors model will suffice (see Kim et al., 2000).

Note: only the uncorrelated errors model can be used with the Pairwise Comparison.

2. First Order Autocorrelated estimated from the data:

If you select "First Order Autocorrelated estimated from the data", the autocorrelation parameter will be estimated separately for each by-group using the method described in Section 2.3 of Kim et al. (2000). Under this option, the autocorrelation parameter is estimated for the model with the default maximum number of joinpoints or the maximum number of joinpoints set by a user.

3. First Order Autocorrelated with correlation =

If you select "First Order Autocorrelated with correlation =", you must input an autocorrelation parameter (between -1 and 1) which represents the correlation between adjacent points. The program then assumes the random errors are autocorrelated and estimates the regression coefficients by weighted least squares. The autocorrelation model is based on the assumption that corr(e_i, e_j) = φ^|i-j| where e_i and e_j are the errors corresponding to the i^th and j^th points and φ is the autocorrelation parameter chosen. This option makes sense only with equally spaced points.

Although the autocorrelation may be estimated from the data, correcting for autocorrelation with this estimate may seriously reduce the power to detect joinpoints (see Section 3 of Kim et al. (2000)). We found in our simulations in Table IV of that paper that adjusting for autocorrelation was helpful in maintaining proper size of the tests of joinpoints when there was large autocorrelation. We also found that if there was no autocorrelation then the adjustment seriously affected the power of the test to detect joinpoints. For example, we see in Table IV (b) with φ = 0, the power goes from 90% to 68%. This is because it is difficult to differentiate between autocorrelation and joinpoints in a model.

1. If you suspect that your data are positively autocorrelated, we suggest using the “First Order Autocorrelated with correlation =" option to see how sensitive your results are to changes in autocorrelation. The option should be used as follows:
2. Fit the model with the uncorrelated errors option.
3. If the user suspects that there is positive autocorrelation in the data, then repeat the analysis trying several values of the autocorrelation parameter, say for example 0.1, 0.2, and 0.3. If the results are very similar with different values of the autocorrelation parameter, then the user knows their results will still hold if there is autocorrelation present. If the results change as the autocorrelation parameter changes, then the user may end up presenting the series of results, to show how the results depend on different assumptions about the autocorrelation.

If you suspect negative autocorrelation, the uncorrelated errors model will suffice (see Kim et al., 2000).

Note: only the uncorrelated errors model can be used with the Pairwise Comparison.