Churn rate is the fraction of subscribers in any period who fail to remain engaged with the company. If the rate of growth ((total likes – new likes)/new likes) exceeds the churn rate (unlikes/total likes), the fan page will grow. Analyzing the causation of these events isn’t always easy with the metrics that Facebook provides. You do know what you post and how often, so using that data along with the growth rate and churn rate you can see which posts are leading to growth or attrition. In this chart the green bar represents a post, the higher the bar the more posts in that one day.

There is definitely a relationship between posts and attrition as you can see the blue line jumps when ever people see a post show up in their feed. And you can get a sense of which posts lead to the most growth. In this case the three posts on 8/9 were promotional based which drove so much growth. And churn rose on 8/28 – 8/29 where two posts a day on two consecutive days had editorial content that must not have resonated or may have been too frequent.
You can download this data from your own Facebook page in the Insights section. All you need is the Lifetime Total Likes, Daily New Likes and Daily Unlikes.