I installed nginx with --prefix=/home/jefftk/nginx so I'm not seeing any permission problems.  The issue is that relative paths default to where you installed nginx, and you installed in a directory you don't have access to except as root.

I think what would fix this would be switching the `error_log logs/error.log` in `pagespeed_test.conf.template` to an absolute path populated by `nginx_system_test.sh`.  Something like:

    pagespeed_test.conf.template:
    -error_log  logs/error.log  debug;
    +error_log  "@@ERROR_LOG@@ debug;
    
    nginx_system_test.sh:
    + ERROR_LOG="$this_dir/error.log"
    ...
    +  | sed 's#@@ERROR_LOG@@#'"$ERROR_LOG"'#' \

Does that fix it for you?