In terms of real numbers, it's very hard to analyze how much a manager is worth to their team. There are some that try to analyze how much a manager goes over their team's Pythagorean win expectancy, others who look at how proficient they are at making in-game moves, and still more that try to "quantify" the intangibles of managing a clubhouse. With the advent of the instant replay as an active force in the game today, we have at our disposal data that can easily be translated into real results.
With the wonderful data provided by Baseball-Reference, we can see what the result of each replay is in terms of Win Probability Added. The best way to analyze how Joe Girardi has done so far is not to look at calls that are blatantly wrong--one would expect the "replacement manager" to make the correct decision to challenge. I will also not look at calls that were upheld; one could not say that the WPA was Girardi's fault because the call was correct to begin with. One would have to look at the questionable calls to understand what he is worth to the team. So far, Joe Girardi has called for 11 challenges; nine have been overturned and two have been upheld. Of those, I'll only take a look at the overturned plays that were extremely close or questionable.
April 13th vs. Red Sox | LI: 1.76 | WPA: +.02
In the bottom of the fourth inning of a game where the Yankees led the Red Sox 2-1, Francisco Cervelli (in the very play that injured him) hit a ground ball with runners at the corners and one out into what would have been an inning-ending double play. With a very close play at first, Girardi challenged the out call and it was subsequently overturned. Although this run was the "deciding run" in a 3-2 game, the effect in terms of WPA was only 2%. It was a close play, and any challenge involving a run scoring is usually a productive one.
April 24th vs. Red Sox | LI: 1.13 | WPA: -.03
When I just said that any challenge that ends in a run scored is usually a productive run, this play would rule as an exception. In the top of the third inning, the Yankees led the Red Sox 3-0 with runners at the corners and one out. Felix Doubront threw a wild pitch which would have scored the runner from third, but the umpire ruled that the ball hit Carlos Beltran and the bases would be loaded with one out. Girardi challenged this ruling and instead got the wild pitch and the run. I didn't choose this play because it was a close one, but because Girardi should, arguably, never have challenged the call in the first place. With a three run lead in the second inning against the struggling Felix Doubront, it would have been more productive to have the bases loaded with one out instead of getting the run on the wild pitch. WPA states that this cost the Yankees 3% in win expectancy, though they did win the game. This will probably be the most deciding factor in value for managers when deciding to challenge: whether the situation is better with the overturned call or without it.
May 2nd vs. Rays | LI: 3.44 | +.18
This was easily the most valuable close call of the season. In the 13th inning of a 5-5 game against the Rays, a would-be double play was ruled a fielder's choice as the first base umpire thought that Mark Teixeira moved off the base. In a very close play, the challenge overturned the call and pretty much killed a possible rally for the Rays. The Tampa Bay did end up winning 10-5 due to a five-run 14th inning, but the WPA for this close play was astonishing: +.18. Challenges in extra innings will almost always have the highest WPA changes, so getting the call right here is crucial.
If you tally the three plays reviewed here, Joe Girardi would have a +.17 WPA over this first part of the season, putting him on pace for close to one win over "replacement level" throughout the course of the season. Some may argue that attributing value to Girardi himself is wrong; often times he just gets the thumbs-up from Tony Pena and he goes with it. In many cases, that is true. There were eight other plays that worked just in that manner; Girardi merely decided to challenge because he got the go-ahead and it was either completely wrong or completely right. That's why I don't attribute "value" to him for that; those are decisions that a replacement manager would make in his stead. What really determines his value is making the call in a very tight spot, as well as making the decision as to whether the overturned call will actually give him the advantage (like in the hit by pitch scenario). But generally speaking, Joe Girardi has been proficient at deciding to challenge at the right time and for the right reasons. Although this facet of the game is brand new, it appears that Girardi has not been fazed.