I don’t think your KPIs should be any different on LinkedIn vs another channel.
The KPIs you should use depend on your goal.
Is your goal lead generation? Then Cost Per Lead (CPL) is your KPI.
Is traffic your goal? Then the number of new visitors to your site is your goal.
The same applies across any channel: social, email, events, etc. The appropriate KPIs depend more on your goals than the medium involved.

Our KPIs on LinkedIn
At Net-Results, where I’m founder & CEO, we advertise on LinkedIn. Our goal is lead generation. Our core KPI is CPL (again, Cost Per Lead). Secondary KPIs include number of leads and conversion rate.
Here’s a screenshot of one of our currently running LinkedIn “sponsored content” ads that’s converting pretty well with acceptable CPL…

Some Perspective
LinkedIn, in my opinion, is best suited for B2B lead generation efforts. Your cost per action may be too high to be sustainable for other purposes, ymmv.

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Michael Ward

I'm founder & CEO @NetResults, the 1st choice of people buying marketing automation for the 2nd time.