An official website of the United States government

Weighted BIC (WBIC)

While the Data Dependent Selection (DDS) internally uses BIC or  BIC3 based on the empirically determined cut-off values for the selection statistics, the weighted BIC combines BIC and BIC3 using a weighted penalty term based on the data characteristic.  That is, it assigns a harsher penalty, making the selection rule close to BIC3, when change sizes are relatively large and a less penalty, making the selection rule close to BIC, otherwise. 

In addition to the notations used for the description of the DDS, let \(y_i = (y_{j_1 + 1}, ... , y_{j_2})^T\) and consider the partial R2 as

\(R^2_{i,i+1} = \frac{\left\{ y_i^T \left( I - H_0 \right)z_i \right\}^2}{\left\{ y_i^T \left( I - H_0 \right)y_i \right\} \left\{z_i^T \left( I - H_0 \right)z_i \right\}}\).

 

Then, define the weighted BIC as

 

\(WBIC(k) = ln\left(\frac{SSE(k)}{\# Obs}\right) + ((2 + R^2_{max})k + 2) \frac{ln(\# Obs)}{\# Obs}\),

 

where \(R^2_{max} = max_{i=1,...,k} R^2_{i,i+1}\).

 

There is a simplified version of the WBIC calculation which is displayed in a footnote on the Model Selection Tab when WBIC is selected:

  • WBIC is calculated as (BIC * (1 – Weight)) + (BIC3 * Weight).   Weight not applicable for the 0 Joinpoint model.

When WBIC is  selected, the Test For Number of Joinpoints table on the Model Selection tab of the Joinpoint output will contain the following values:

  • BIC
  • BIC3
  • Weight
  • WBIC