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Kila Busts Out

To say that Kila Ka’aihue’s first 71 plate appearances in the Major Leagues this year were disappointing is a major understatement. After the 26-year-old tore up the league to the tune of a .322/.465/.601 line, good for a 174 wRC+ that qualifies as ridiculous even in the minor leagues, Ka’aihue had been utterly awful entering today’s game, showing no power and hardly walking at all. Kila’s overall MLB line sat at .164/.211/.224 entering today’s extra inning thriller against Detroit, a .200 wOBA that screams AAAA player.

Today, though, Ka’aihue was key in the Royals’ 4-3 victory over the Tigers, walking twice, homering, and doubling in six plate appearances, compiling .202 WPA. That marks only Ka’aihue’s second double and second homer, and, perhaps more remarkably, the two walks raises his total to only 6 in 77 MLB plate appearances. Games like these were Ka’aihue’s signature in the minor leagues, but this is only his second multi-hit game, first multi-walk game, and first multi-XBH hit game in the majors in 2010.

Regardless of how well Ka’aihue performs down the stretch, we’re not going to get enough plate appearances to truly evaluate his true talent in the majors. That’s really unfortunate, because Ka’aihue clearly had nothing left to learn in the minors by at least the All-Star Break. Instead, the Royals will go into next season with at best an unclear picture of Ka’aihue’s true talent and at worst a distorted one.

It will be important for evaluators both inside and outside the Royals organization to remember two things. First, and I think the Royals have clearly demonstrated this (a little too strongly), is that minor league success is not a simple translation to the minors. Obviously, the Royals weren’t convinced by Ka’aihue’s 2008, which almost matched his 2010, nor his decent 2009, and it took another 400 PAs of fantastic performance in AAA in 2010 to finally earn a spot on the MLB club. Expecting Kila to be a star is probably just as unreasonable an expectation, however, and the list of players that have torn up the minors to go on to fizzle in the majors is quite long, and the list of players that unexpectedly play well in the majors after unimpressive minor league careers isn’t exactly tiny.

It remains to be seen if the Royals will understand the second point, which is the ever-repeated argument of sample size, sample size, sample size. Given how poorly Kila started out the season, it would be a surprise if his final line is much more than average. The Royals aren’t going to know how he handles the Majors after only 150 or 200 PAs, and if Kila finishes with something like a .315 wOBA and gets sent back to the minors for the opening of the 2011 season, the Royals will have made the wrong decision.


Cust Succeeding Without Power

Before the season, it appeared that Jack Cust’s career with the Oakland Athletics was over. After a relatively unimpressive 2009 campaign (.342 wOBA, 0.9 WAR, 612 PAs), the A’s designated Cust for assignment at the end of Spring Training. Cust went unclaimed and seemed destined to rot in AAA Sacramento. But the Eric Chavez project ended as a complete failure (.268 wOBA in 123 PAs), resulting in Cust’s return to Oakland.

Cust’s return has been a rousing success. He’s still not making contact often (38.2% K/AB), but when he does, he’s making things happen. Entering play yesterday, Cust had a .267/.391/.436 slash line, and he added a 2/4 game with a HR against the Indians last night. With that home run, Cust has only 10 home runs in 278 plate appearances – not bad for the typical player, but Cust hit 84 HR in his first three MLB seasons. The low home run total marks a bit of a decline for Cust, and we can see that in his ISOs: .248, .245, .177, .169 from 2007 to 2010.

Despite this power drop, Cust’s 2010 wRC+ of 134 compares well with 144 wRC+ in 2007 and his 132 wRC+ in 2008 largely because Cust is having a remarkable amount of success on balls in play. His .386 BABIP is the highest mark of his career, besting the .355 mark which helped power his career year of ’07 and his solid career mark of .334. Naturally, this is a huge reason why Cust’s line looks as good as it does. My four factors method would project Cust’s line at a slightly above average .335 wOBA with his career .334 BABIP. If Cust’s average were 52 points below his career average instead of 52 points above, the method predicts a .303 BABIP, right around replacement level for a DH.

It shouldn’t be that surprising, however, that Cust is seeing relative large fluctuations in his BABIP, as he quite rarely hits the ball in play. In fact, the only player to put the ball in play less often this year is Mark Reynolds, whose 46.2% BIP rate is the only one besides Cust to be under 50% this year among players with at least 200 platte appearances. Reynolds is actually seeing the opposite kind of variance to Cust, as Reynolds’s .279 BABIP is 51 points below his career BABIP. Here is a spreadsheet with these BIP rate numbers for the whole league to date in 2010.

BABIP is a fickle beast, and the fact that players such as Cust see even fewer balls in play than the average player makes it even more prone to wild fluctuations. This season, Cust is the benefactor. Next year, though, the A’s might get more of a 2010 Mark Reynolds BABIP than what they’ve seen from Cust this year, and then Cust may come crashing down to Earth if his former power doesn’t return.


Defense and Slumps

A statistically minded baseball fan typically does not have to go far to find criticisms of the defensive metrics available. Certainly, some of that criticism is valid – defensive metrics are by no means perfect, and they particularly have flaws when looking at single seasons of data. This piece at Baseball Prospectus (subscription required) by Colin Wyers does a fantastic job of framing the problems and biases involved with defensive metrics, as well as a possible answer.

One issue presented with defensive metrics I believe is invalid, however, is the idea that inconsistencies in defensive ratings from year to year somehow render defensive systems useless, whether it be UZR, TotalZone, +/-, or any other stat. The idea seems to be based on a traditional baseball idea that while pitching and hitting can slump, speed and defense both tend to hold constant. According to this idea, seeing a player post UZRs of -3, +6, -9, and then +2 in four consecutive seasons would represent a problem with the metric as opposed to simply the ups and downs of single season. Such fluctuations in wRAA aren’t uncommon, but such things can be explained away by hot streaks and slumps over the course of multiple seasons.

Sky Andrecheck analyzed the idea of “defense doesn’t slump” last year. Based on the idea that the distribution of probabilities of outs on balls in play is bimodal – that is, most are either sure hits or sure outs – the standard error for fielders is smaller than that for hitters.

From an individual player’s standpoint, the average fielder has about 500 balls in play in his area over the course of the season (of course, this varies by position, and we can adjust accordingly) . Using the numbers above, we see that the average fielder has a standard error of about .23*SQRT(500) = 5.14 outs over the course of a season. This means that he is prone to make about 5 or so more or 5 or so less plays in a season than his true talent would usually call for. This corresponds to a difference of about 4 runs in a season. While this is fairly small, it does show that random variability can play a part in a fielder’s performance just as it can for hitters

Sky’s work, to me, effectively proves that we should expect some variability in defensive abilities. There’s also another element to the variation we see in defensive metrics that he doesn’t look at, and that’s the physical and mental aspects of the game. If a player tweaked his hamstring but doesn’t tell his manager, we could see a decrease in his range that would be unexplained by the information available. If a player is uncomfortable in a certain park, for whatever reason, he could slump if he receives an abnormal amount of chances at that park in that season. There are probably a multitude of other reasons as well.

The issue of chances is likely one of the reasons that we don’t typically recognize fielding slumps. I posed the question of why slumps can’t occur on defense on twitter, and the omnipresent Colin Wyers pointed out that it is “because a player’s ability to get to a batted ball informs our thinking about whether or not he should have made a play.” A player who is slumping on defense won’t get to balls that he may have gotten to normally, and that’s terribly difficult for fans to point out. Announcing crews aren’t going to have convenient stats like “Prince Fielder has one hit in his last 24 at bats” for defense.

Basically, I think that there’s plenty of reason to believe that defense slumps, and although we don’t really know the magnitude, I also don’t see why it wouldn’t be near that of the slumps we see for batters, or even greater due to the fact that all defensive chances aren’t created equal nor distributed at a regular schedule. Similarly, I would imagine that defense sees hot streaks as well – perhaps my focus on poor fielding is because I’m a pessimist. Regardless, as our ability to evaluate defense evolves, I predict that we won’t look at fluctuations in defensive metrics as a sign of incorrectness, and instead we will learn to accept that, for whatever reason, it’s not fair nor reasonable to accept that a player is the same quality defender against every ball in play in every game in every season.


Don’t Intentionally Walk Teixeira

In the fifth inning of yesterday’s game between the Seattle Mariners and the New York Yankees, the Mariners decided to intentionally walk Mark Teixeira with runners on second and third and one out in a 1-0 game. With Alex Rodriguez out of the lineup, that brought up Robinson Cano with the bases loaded. Cano hit a grand slam, putting the Yankees up 5-0. At that point, the Yankees win expectancy was over 95%, and a formerly close game was effectively over.

The Yankees scoring after an intentional walk of Teixeira isn’t exactly something new. Yesterday marked Teixeira’s fourth intentional walk. Three times, including yesterday, the result was a grand slam. The other resulted in a run-scoring wild pitch, a strikeout, and then a two-RBI single. Overall, Teixeira and both runners in front of him have scored every single time that he’s been intentionally walked.

Obviously, we wouldn’t expect this to continue to happen every single time, but that doesn’t make the intentional walk a good strategy in this case. Here’s what The Book has to say:

If all batters have equal ability, intentionally walking a batter to set up a double play, force, or other situation is at best a break-even move (or insignficantly better than a break-even move). Doing so early in the game is counterproductive, since it increases the odds of a big inning more than it increases the odds of a scoreless inning.

Now, all batters in the Yankees lineup aren’t of equal ability, but when we look at those after Teixeira, they are all very good. Alex Rodriguez and Robinson Cano are typically the two hitting after Teixeira, making them the important two to examine when it comes to analyzing the value of the intentional walk. The third batter also matters, but far less; this season it has been a mix of players such as Jorge Posada, Marcus Thames, Nick Swisher, and Randy Winn. Overall, though, what matters is that the hitters that come to bat after Mark Teixeira are well above average, particularly Rodriguez and Cano.

Using the Markov chain calculator at Tom Tango’s website, we can take a look at the run expectancy for the composite line of Alex Rodriguez and Robinson Cano, which is essentially what managers are looking at after walking Mark Teixeira. The results, unsurprisingly, suggest that the Yankees are probably going to score a lot of runs after walking Teixeira, even in the situation with one out and runners and second and third, the situation most conducive to a productive intentional walk. Here’s what the calculator spits out.

Runners on 2nd, 3rd, 1 out: 1.578 runs
Bases loaded, 1 out: 1.986 runs

So opposing managers are essentially forfeiting .4 runs for the opportunity to say “Well, I didn’t let Mark Teixeira beat me!” at the post game press conference. The odds of scoring at least 1 run in an inning don’t justify the managers’ decisions either.

Runners on 2nd, 3rd, 1 out: .707
Bases loaded, 1 out: .752

Intentionally walking Mark Teixeira is simply not a smart move. The Yankees lineup behind Teixeira, particularly Cano and Rodriguez, is simply too good, which just magnifies the fact that the intentional walk adds to the possibility of a big inning. Four times, opposing managers have attempted to get out from a jam by walking Teixeira. Four times, at least three runs have scored. If a manager wants to win, his best option is to pitch to Teixeira and hope for the best.


Lou Piniella Retires

Although it won’t impact the 2010 playoff race, the retirement of Cubs manager Lou Piniella certainly qualifies as high profile news. Piniella’s 23 years as manager makes him one of only 21 managers in MLB history to manage for at least 20 years, and ties him for the 13th most years as manager with a pair of Hall of Famers in Walter Alston and Harry Wright. The only eligible manager with more time than Piniella to miss the Hall of Fame is Gene Mauch, who is also the only one without a pennant or World Series title.

Piniella won his one and only World Series championship in 1990 with the Cincinnati Reds, his first year with the team after a playoff-free three year stint with the New York Yankees. Those Reds were led primarily by Mariano Duncan, Chris Sabo, Barry Larkin, and Eric Davis at the plate, along with a solid rookie performance from Hal Morris. The pitching staff was led by a fantastic season by Jose Rijo (3.14 FIP, 148 ERA+) and solid bullpen performances from Randy Myers, Norm Charlton, and Rob Dibble. The Reds opened the season with a nine game winning streak and never looked back, winning the NL West by five games, and then steamrolled through the Pirates in the NLCS and then the Athletics in the World Series.

The next two seasons saw a World Series champion finish under .500 and then a second place finish, and Piniella was out of Cincinnati on to his longest and most famous managing job in Seattle. Despite the fact that Piniella’s Mariners boasted such talents as Randy Johnson, Ken Griffey Jr., Edgar Martinez, Alex Rodriguez, and Ichiro Suzuki, they never reached a World Series. In Piniella’s 10 seasons in Seattle, the Mariners reached the playoffs four times, including three division championships and two division series victories. Piniella also was at the helm for the Mariners 116 win season in 2001, a remarkable feat despite the first round exit that season.

The next stop of Piniella’s career is mostly the butt of jokes, as Sweet Lou moved to the Tampa Bay Devil Rays, managing in his hometown. As the pre-Andrew Friedman era went in Tampa, the Devil Rays under Piniella were terrible teams. In three years of managing, Piniella’s Devil Rays finished in 5th twice and in 4th once, compiling an overall 200-285 record and at least 90 losses in each season.

Piniella’s final position in Chicago began well enough, with two division titles. But the Cubs were swept in the first round each time, only increasing the disappointment of a fanbase entering its second century without a world championship. As the core players in Chicago have aged and albatross contracts have prevented the infusion of new, impact free agent talent into the team, the situation in Chicago deteriorated to the point at which it is today – the Cubs are on pace to lose 96 times in 2010.

No doubt, this is a disappointing end for Piniella, who is known as much for the accomplishments of his teams as he is for his antics on the field. Between his seven playoff appearances, .517 winning percentage, World Championship ring, and most importantly, the longevity and visibility of his career, I have little doubt that Piniella will take a place in Cooperstown. Piniella was a fantastic personality, and the game of baseball will certainly miss Sweet Lou.


Willingham’s Best Season Ends Prematurely

Although it has been partially silenced by the media markets and poor teams in Florida and Washington, Josh Willingham has been one of the most consistently good players in Major League Baseball since 2006. This year, Willingham was on pace to have his best season by WAR and easily his best offensive season by wOBA and wRC+ until a knee injury requiring surgery sidelined him on Thursday. The surgery will force Willingham to miss the rest of the 2010 season.

Willingham has distinguished himself as a solid hitter thanks to above average walk rates and power to go with average contact skills. He has never been below 15 homers in a season since 2006 and has posted a walk rate above 10% every season since 2007. Willingham’s limited defense in the corners prevents him from having a truly star-level impact, but that skillset makes him a 2-3 win player.

Through 450 plate appearances in 2010, Willingham accrued 2.7 WAR, thanks to a career best 137 wRC+. Willingham’s power was down, but an increase in walk rate to 14.9% more than made up for it. ZiPS even saw new heights for Willingham, expecting the power to return and the walk rate to remain high. CHONE isn’t quite as optimstic, projecting a .261/.371/.461 line as opposed to the .263/.370/.500 line projected by ZiPS. With either of those lines, Willingham was looking at a 3.5 WAR season if he could stay healthy.

Unfortunately both for Willingham and his teams, injuries have been a factor in his career. Willingham has only reached 600 plate appearances once in his career, back in 2007. Only twice since 2006 has he reached 500 plate appearances, and the 2010 season marks the second time that he will finish between 400 and 500. Despite this, he has provided great value in these five seasons, delivering his team 12.5 WAR while making only $8.6 million dollars.

Willingham is about to become expensive, as he can probably expect something in the $7-10 million range in his third year of arbitration and then will hit the free agent market before his age 33 season. He should continue to be a solid performer for at least a few more years, but it will be interesting to see how the Nationals, who don’t look like a contender in 2011, handle Willingham. Will they attempt to sign him to an extension? Does he leave in a deadline deal, or does he bring back free agent compensation as Rizzo is attempting with Adam Dunn?

Hopefully, Willingham’s injury doesn’t impact his long term ability. If it doesn’t, whether it’s with Washington or with some other team, Willingham is a good bet to provide value in 2011 and beyond.


Cardinals Acquire Feliz, and are now Worse

Another August deal went down involving the National League Central yesterday, this time involving the St. Louis Cardinals and the Houston Astros. The Astros sent Pedro Feliz north to Missouri, and will receive David Carpenter, a 25 year old reliever who is having a solid season for the Cardinals High-A team.

Carpenter is, simply put, not a loss for St. Louis. Tim McCullough of Future Redbirds wrote of Carpenter last night:

For David, he’s likely in a better place now. He’s got legitimate relief arm but he’s more of a fringe setup type than a closer (and that’s a pretty rosy assessment). The Astros system is barren as opposed to the Cardinals relief system that has guys like Adam Reifer, Eduardo Sanchez, Casey Mulligan, Blake King, Francisco Samuel, etc. in front of him.

A guy whose ceiling is a “fringe setup guy” isn’t much of a cost. Sure, it’s good for Ed Wade that he managed to rid himself of Feliz while adding a living, breathing human being with multiple working body parts, including an arm that can throw a baseball, but that’s not the real story here.

The real story is how Pedro Feliz can possibly fit into the Cardinals plans for success. The Cardinals are now 3.5 games behind Cincinnati, thanks to a six game winning streak by Cincinnati combined by a four game losing streak in St. Louis. The Cardinals are certainly looking to win games now, and there is a very legitimate question as to whether Feliz may actually hurt in that regard.

The Astros decision to sign a marginal player such as Feliz was questionable at the time. He was a no-bat, all-glove third baseman who, at age 35, could see either one or both of those factors drop dramatically. The simultaneous collapse of both sides of Feliz’s game has resulted in possibly the worst player in baseball. Feliz is walking less, his power is down, and his BABIP has fallen to .232. Given Feliz’s advanced age, the decline in his other skills, his career .267 BABIP, and his 4% drop in line drive rate, that BABIP doesn’t appear to be bad luck, it just appears to be a complete lack of Major League ability. Throw in a glove that has gone from elite to poor – his UZR, RZR, and TZ are all far, far below their excellent career marks – and the results is a player worth -1.5 WAR in just over 300 plate appearances, making him the worst position player in baseball this season.

There’s simply no reason to think that Feliz and his completely broken skillset provides any sort of upgrade over Felipe Lopez, or anybody around MLB, for that matter. Tyler Greene likely would’ve been a superior option in house, if only for his glove and the fact that he has shown some life in AAA. Aaron Miles, despite being the embodiment of the replacement player, is probably better than Feliz. Craig Counsell would have been a better option off of the waiver wire.

By adding Feliz to the roster, the St. Louis Cardinals have simply made themselves worse. He doesn’t appear to be a capable defensive replacement. What little bat he once had has completely evaporated. Feliz may be the worst player in baseball who has received playing time with any sort of regularity. There’s a chance that Feliz reaches some sort of respectability in St. Louis much as there is a chance of any baseball player having a good month, but that certainly doesn’t mean that we should expect it. The Cardinals job of catching the Reds is at least as hard as it was at this time yesterday. The addition of Feliz may have made it harder.


Just Octavio Being Octavio

In case you missed it, the Colorado Rockies and the Los Angeles Dodgers played a late night thriller on the west coast. Huroki Kuroda put together a strong start, with seven strikeouts against only one walk, allowing two earned runs in seven innings. His Rockies counterpart, Jason Hammel, worked his way around four walks and four hits to allow only two runs in six innings, striking out four along the way. The bullpens, including freshly-demoted Dodgers setup man, Jonathan Broxton, kept things scoreless until extra innings. Joe Torre, going for the win at home, went with his new closer, Octavio Dotel, to start out the 10th, hoping that a scoreless top half would be enough of an opportunity for the Dodgers.

Dotel was certainly looking to improve on his first attempt to save a game with the Dodgers, where it took all of two batters faced for Dotel to blow the save. In one sense, Dotel did improve. This time, it took four batters, including two walks, a stolen base, and two wild pitches for Dotel to bring in the game winning run for the opposing team.

To be fair, Dotel is usually a serviceable reliever. He has crazy stuff, even at 36, and he’s a mortal lock to strike out a batter per inning – he struck out two of the first three batters he faced last night. His FIP this season, however, is only 4.12 and his ZiPS projection is only around 3.70, mainly because he just has no idea where the ball is going. Dotel has a career walk rate of 4.06 and a projected walk rate of 4.20. He can be prone to outings like last night’s, in which the control is just completely missing, and even though that can bring the strikeouts, it’s also going to bring the walks, and subsequently, the runs.

The Dodgers have two relievers that are capable of striking out just as many batters as Dotel while walking far fewer in their bullpen already. One is, of course, Broxton, who has a far superior ground ball rate to go along with a walk rate about one batter lower per nine innings. The other is Hong-Chih Kuo, who has an 11.1 K/9 and a 3.0 BB/9 this season, the best walk rate of the trio. Kuo likely won’t keep the ball inside the park like he has so far this season, but he has comparable strikeout abilities to Dotel to go with superior control and a superior ability to draw ground balls.

Ned Coletti and Joe Torre are living in a world where James McDonald (20 K, 4 BB, 0 HR in 17.2 IP with Pittsburgh) and Andrew Lambo are an acceptable price to add a middling reliever to a team six games out of the playoffs and then turn him into the relief ace over two superior pitchers. The Dodgers are now 12 games out of the NL West lead and 8 games out of the Wild Card. I don’t know what the Dodgers’ endgame was with Octavio Dotel, but there’s no doubt that Coletti and the Los Angeles front office missed big on this one.


Cubs Bring Back Pitching For Lee

Although word has been out for much of the day that Derrek Lee was on the way to Atlanta in a waiver deal, the deal wasn’t made official until 5:25 Eastern Time, according to Fox Sports’s Ken Rosenthal. We now know, as well, that the Cubs will be sending cash along with Derrek Lee to the Braves, and the Braves will send back three minor league pitchers. The pitchers are right handers Robinson Lopez and Tyrelle Harris as well as left hander Jeffrey Lorick.

The real get here is Lopez. The 19-year-old from the Dominican Republic just missed out on Baseball America‘s organizational top 10 prospects list for Atlanta and ranked exactly 10th in Baseball Prospectus’s pre-season look at the Braves’ system. Lopez has a fastball that reaches as high as 96, an extremely projectable frame, and at 19, that’s a player with a ton of potential. If his secondary pitches don’t develop, he will probably be limited to the bullpen. To date, he has thrown 110 innings as a starter and 33 as a reliever, and, naturally, Lopez has performed much better in the bullpen. However, I would not limit a player with Lopez’ potential to the bullpen at such an early stage in his career.

Lorick and Harris are exclusively relievers, which inherently makes their value much lower than that of Lopez, as does the fact that both were, at least at some point this season, teammates with Lopez at low-A Rome despite being 22 and 23, respectively. That doesn’t mean that Lorick and Harris don’t have some value, though, as both have performed well in the minors in their careers and have shown the potential to make up the back-end of a bullpen. Harris has struck out 84 batters in 68 minor league innings, and Lorick has struck out 85 in 86 innings. Both have a tendency toward ground balls, as well – Lorick in particular. Both are far enough away from the majors that they are anything but a sure thing, but it’s not like these players are throw-ins. They definitely supplement the value brought in with Lopez.

The Cubs are a team with plenty of money and a large fanbase, so they really shouldn’t need to perform salary dump deals with players like Derrek Lee. Lee was making roughly $3.25 million for the rest of the season, a number that likely has little impact on the Cubs’ finances for this year or in the future. Instead of dumping Lee’s salary and getting little in return, the Cubs were aggressive, and threw cash into the deal to help coax some legitimate value out of this deal. Jim Hendry deserves major credit for this trade, as instead of spending his money on an aging first baseman on a fifth place team, he now has three more solid pitching prospects to load into his system. This is just an excellent overall pickup for Chicago.


Expanded Four Factors: Do It Yourself (1.0)

A few people have asked me to run some calculations on certain players, and given how easy it is to do with the spreadsheet in front of me, I figured that this would be a resource that at least a few people would be interested in having on their own computers.

Again, I would remind you that this is a bit of a work in progress, but here’s version 1.0 of the Expanded Four Factors spreadsheet.

Download The Four Factors Spreadsheet Here (.xls file)

It’s not a terribly complicated spreadsheet to use, but here’s a screenshot and a quick overview.


[Click to enlarge]

In its base form, the chart will show 20 lines of the league average 2010 player. In order to replace the chart values with those of your player of interest, enter his factor values into the yellow (FanGraphs Beige?) cells. You can go ahead and change the value of the average player, too, if you want to see how things were in different seasons, although this won’t affect anything once you’ve entered the value for an individual player in the yellow cells.

Once the factors are entered into the yellow cells, you’ll have an original line for your player, which will appear in line 1. As such, you probably don’t want to edit line 1 at all. Then, after that, you can use lines 2-20 to take a look at lines with different values for the four factors, and you can change one at a time or all four, although making graphs with four different independent variables can be challenging.

Don’t mess with the formulas under the “results” section unless you feel like changing up the methodology in some way, and if you do I highly recommend you save an original copy first (or bookmark this post) as the equation, particularly for HRs, isn’t exactly intuitive.

That’s all I can think of to add here; if you have any questions/comments/mistakes/etc. to point out, you can of course use the comment section, contact me over twitter, or email me at jhmoore AT wisc DOT edu.