Generally speaking, you can control the logging level using the TraceFlags.
TraceFlags have overriding authority over all other tracing methods. See Summer '15 Release notes - Receive Debug Logs Predictably.
As per Does the Summer ’15 release remove the ability to set test logging levels with the DebuggingHeader?, the current modifications seem a bit off to me, but they can be made to work.
So long story short, use the Developer Console Debug > Change Log Levels option to set your user level trace flags. You can also set specific trace flags for apex classes and triggers. These will be applied when capturing asynchronous test results.
If you want to go further, you can still exert control of the logging levels through other means. One option is to not have no active TraceFlags set when running the tests. This also requires having Developer Console closed, as it will set active TraceFlags when opened.
You can perform a synchronous test run using the Apex API and set a specific logging levels in the DebuggingHeader. If your interested in this approach I've made a tool that allows you to call this method for selected test classes or methods with defined logging levels.