In general, there might have no correlation between the two. If there is a relationship, we do not need to investigate the antioxidant activity by using several different methods.

Another aspect that should be concerned is the mechanism of this two methods. DPPH assay is based on hydrogen donation that causes decoloration of DPPH. But beta carotene assay is not based on this. 

As I understand, DPPH shows a chemical antioxidant activity but beta carotene shows a kind of antioxidant activity more close to the practical use such as food industry.

If you want to know more and more specific, you  can read a book, Oxidation in Foods and Beverages and Antioxidant Applications. Volume 1: Understanding Mechanisms of Oxidation and Antioxidant Activity. Woodhead Publishing Limited., Cambridge, UK. pp. 345-346 (2010). Or you can read my publications.

Hope it helps you.

As I know, there is no correlation between 2,2-diphenyl-1-picrylhydrazyl (DPPH) radical scavenging and β-carotene bleaching . Total phenolic is correlated with β-carotene bleaching.

Article Total Phenolic Contents and Antioxidant Activities of Differ...

There is no correlation between DPPH radical scavenging activity and b-carotene/linoleic bleaching inhibition assay, each assay worked on different principles. 

Please read the following manuscript in the attached file.