We're having the same problem also, awstats stopped working this past weekend on Saturday the 21st.
After cpanel tech-support checked also found at; /usr/local/apache/domlogs/
owner permissions had been changed by Litespeed to nobody.nobody
When we switch to Apache, they wrote a script to restore permissions to root.username
But when we switch back to Litespeed, in less than 16 secs, it changes permissions again back to nobody.nobody and that makes the logs files not writable.
For some strange reason, we have Litespeed on other servers and this problem is not there, plus on this server all has been fine with Litepseed for last 2 years and no such issue, therefore puzzle now is what happened to Litespeed this past weekend to start changing permissions to nobody.nobody for logs files at; /usr/local/apache/domlogs
Any ideas? Any help?