In terms of claims paid out, it would be good to know companies' premiums to claims paid out ratio is. For health insurance, there is a huge range. Southern Cross pays out around 86c for every dollar collected, compared to Nib's 60c per dollar.
I have tried and failed to find data for other kids of insurance. I suspect because Southern Cross explicitly advertises their payout rate, other companies feel they need to provide that data as well. But for other types of insurance it's just not publicly available.