From 2nd June to today is about the same length of time as from 20th May to 1st June. (In fact, McCatty has had an extra game in that time.) By 20th May, Daniel Cabrera had been removed from the rotation, and Scott Olsen was on the disabled list. The youngsters Stammen and Detwiler entered the rotation and have remained since. The bullpen hasn't change dramatically, with Kip Wells going on the disabled list. So 20th May to 13th June involves much the same personnel.
First, let's use the old Big Bad Baseball Annual Quality Matrix (QMAX) to analyse the starters. (I like QMAX, because it actually uses individual starts, as opposed to lumping everything together, and offers a more nuanced picture than the Quality Start.)
Coach Success (Elite
Square Square) Hit Hard ---
St Claire 3 0 5 2
McCatty 6 2 1 2
The Elite Square is a subset of the Success Square, so don't add those together. In this case, it's a solid win for McCatty. This makes sense, after a fashion, because St Claire built his reputation working with veteran pitchers, and suddenly there weren't any in the Nationals' rotation. McCatty's more 'instinctual' approach might work better than St Claire's 'technical' methods with younger pitchers still learning the broad principles of pitching.
For relief, I took two elements. The Win Expectancy (WE) situation when the relief pitcher entered the game, and the net plus/minus after comparing the ideal result (no change to the leader's advantage in runs if the change occurs at the start of an inning; or, if the change is made mid-inning, all batters faced make outs) with the actual result.
Coach Average WE WE WE+
at entry effect
St Claire .254 -1.378 -1.79
McCatty .330 -3.230 -2.49
The WE+ is an attempt to translate the win expectancy effect to a common standard. The problem with the raw numbers is that the better performance of the starters under McCatty mean there is more win expectancy to lose than there was under St Claire. (Think of it as a reliever having an 8 per cent better chance of entering a close game under McCatty; there's just more win expectancy to lose to begin with.)
Basically, the bullpen under McCatty is having an even more difficult time than it was under St Claire. Some of that may be down to the fielding. If you take away the 12th June game, lost in part by Nick Johnson misplaying a foul pop-up, the WE+ number comes down to -1.88. Maybe the difference isn't so big.
On balance, from this evidence, the improvement in the rotation probably justifies the switch from St Claire to McCatty. I'd go so far as to say that a pitching coach really must stand or fall on how effective the rotation is.