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We live in a time that will likely be remembered by some—unofficially, of course—as the Steroid Era in baseball. Beginning in the 1990s when offensive production and reported uses of steroids, some later confirmed, went way up. With the publishing of the Mitchell Report in 2007, which detailed oodles of allegations against players that they used performance enhancing drugs, including anabolic steroids. Even now, 20ish (gulp) years removed from those halcyon days when flannel was a thing, they’re apparently still around. We still see players testing positive for the use of PEDs, even though everyone knows that there’s a testing program in place.

The thing about being 20 years in is that now, in addition to policing the current game, we have to do the Hall of Fame thing. You can bet that once again, everything will come to a head again in January during Hall of Fame voting season. The past few years, we’ve seen the likes of Rafael Palmeiro, Roger Clemens, Barry Bonds, and Mark McGwire, players who generally have pretty clear Hall of Fame cases, but who have either admitted to or been suspected of PED use. There are the #NeverBonds folks who won’t vote for anyone implicated and the PED agnostics who wave a hand at it all, but then there are some in the middle who try to answer the question “What would he have been like without steroids? If… he did them.”

And yes, in five or six years, the recently released Alex Rodriguez will be joining the Hall of Fame ballot. Absent Rodriguez’s own previous admission that he used PEDs, he would be an easy first-ballot inductee. In fact, if 60 WAR gets you into the conversation for the Hall of Fame, then Rodriguez nearly had two Hall of Fame careers. But… what would that career have looked like without PEDs? What would the 90s have looked like? How much of a difference do steroids actually make.

It’s a maddeningly un-answerable question, which is sad because it’s one of the foundational questions of the era of baseball that we live in. But we don’t know for sure who did and who didn’t, and if they did, what did they take and when. The sort of trail that we’d need to do any sort of responsible analysis of exactly how much steroids mattered.

I’d like to ask the question in a different way. Maybe we’ll never know exactly who did (and didn’t) do what and when. But let’s see if we can establish an upper bound to what effect steroids might have had.

Warning! Gory Mathematical Details Ahead!

We know that steroids don’t actually work by giving a player super powers (you’re thinking of a video game), but instead by allowing a person to build muscle mass more quickly than s/he would otherwise be able to. In fact, anabolic steroids are actually prescribed medically for people who are severely malnourished and need to put on weight quickly. It’s not instant. You still have to work out, but the muscle builds more quickly. But let’s just assume that you take one little shot and voila! you are now Mario having eaten the flower and are ready to throw balls of fire at turtles. And hit the ball for miles.

There are limits to what the human body can do. We know that even in the 90s, no one became The Kid Who Only Hit Homers, but the steroids did something. Right?

Let’s use some of that newfangled StatCast technology that everyone loves to talk about, particularly exit velocity off the bat. Before the widespread availability of StatCast data, people studying the steroid issue primarily relied on looking for spikes in statistics like HR rate or ISO. The nice thing about exit velocity is that it stabilizes very quickly for hitters, much more quickly than something like homerun rate or ISO. That means it’s less prone to random fluctuations in results. You can get random spikes in home run rate that are just that: random noise. That’s less likely to happen with exit velocity. We also know statistically that exit velocity is mostly the hitter’s doing rather than what the pitcher did to him.

Using exit velocity is also nice because if there’s one thing that we expect steroids to be able to do for a hitter, it’s that he should hit the ball harder as a result of the extra muscles. Exit velocity, more than home run rate or any other outcome metric, tells us exactly that. Shame that no one thought to set up the StatCast system in the 1990s.

So, let’s look at the extremes. Statcast data have been public since the beginning of last season. Between 2015 and 2016 (or at least up through last Friday when I looked at the data), with a minimum of 200 batted balls in each year, the biggest increases in average exit velocity have belonged to:

Player

Change in EV (mph)

Danny Espinosa

5.4

Kole Calhoun

3.6

Eric Hosmer

3.4

Victor Martinez

3.2

Kevin Pillar

3.1

Now, let’s do the thing that the internet does whenever someone gets surprisingly better from one season to the next and assume that all of these guys are

(Hi there, lawyers! Just figured I’d put a little disclaimer in here. I have no evidence at all that any of these guys has done anything illegal or against MLB rules, including taking PEDs. In fact, since they are all currently active MLB players, they are being tested like everyone else and none of these guys have tested positive yet, so that’s a pretty good indicator that they’re all on the up and up. Just wanted to say that.

I find it rather silly that everyone jumps to “it must be ‘roids” rather than more obvious conclusions like “he’s healthier this year” [Martinez], “he’s been trending upward the past few years and we all knew it would happen eventually” [Hosmer], “he’s a guy who’s always had some promise and is now in his late 20s where we know guys naturally tend to peak” [Espinosa, Calhoun], or “he had nowhere else to go but up” [Pillar]. Or maybe he just worked out, worked with a hitting coach, and/or worked out a new approach.

So, lawyers, just to be clear, in no way am I suggesting that these guys are)

on steroids.

So, if steroids are all that and a bag of potato chips and are doing what the Internet thinks that they are doing, then someone who took steroids should see something on that order of magnitude, right? Something on like 3.5 mph of extra exit velo (on average)? Coincidentally, that number (3.5 mph) is near to an estimate that physicist Roger Tobin made in estimating what someone who adds 10 percent more muscle mass might expect in terms of added exit velocity. That’s no small feat, but seemingly, we have pretty good evidence that players who are playing clean can do the same thing. And again, the effect of the steroids may not instant.

Let’s stop for a second and peel it back a little further. There are 23 players in the data set (out of 146) who have added at least 2 mph of exit velocity this year. This might have been helped along by the ball being a little bit… more lively this year. But of course, not all 23 of those guys are on steroids. In fact, it’s most likely that none of those 23 hitters are on steroids. In most of those cases, it’s probably just a case where they just… got better. People do that.

So, the big steroid effect might just be something that is roughly around what some small, but non-zero, percent of our sample had happen to them anyway without any chemical help.

Steroid don’t turn someone into a world beater. Yes, it’s using drugs to cut the line when life is handing out breakout seasons. I suppose whether that is “cheating” in some absolute sense is a question of morals and philosophy, but let’s keep the effects in perspective. You can’t take a nobody and make him into an All-Star. Steroids probably do have the ability to enhance performance on the field (more on that in a minute), but they aren’t magic.

To give some idea of what 3.5 mph of exit velocity looks like, looking at last year’s (2015) end-of-season stats, the top five exit velocities in baseball (min 200 batted balls… which eliminated Giancarlo Stanton’s 98.6 mph and Miguel Sano’s 94.8 mph marks) were Miguel Cabrera (94.5 mph), David Ortiz (94.0 mph), Nelson Cruz (93.7 mph), Jose Bautista (93.6 mph), and Mark Trumbo (93.4 mph). That’s a rather nice group of hitters, settling somewhere in the 94ish mph zone. Cabrera and Ortiz are future Hall of Famers (and if he ever figures out how to stay healthy, so is Stanton). Trumbo and Cruz aren’t going to the Hall, although if they had Xeroxed their lines from last year 15 times over, they’d get a Cooperstown invite.

Still, if you take someone with Miguel Cabrera’s 94.5 mph mark from 2015, and assume that he got there through steroids and penalize him 3.5 mph, then his 91.0 mph mark means he started out as Tyler Flowers, Ian Desmond, or Hanley Ramirez (all had 91.0 mph marks last year). Not awful company to be in. Bryce Harper checked in at 91.2, as did Freddie Freeman and Mookie Betts. There are ways to succeed without being at the absolute top of the exit velocity chart, although it’s not like Ian Desmond is going to the Hall of Fame. Chances are that if a player uses steroids to get to the top, then he was already pretty good to begin with.

Let’s not completely dismiss what 3.5 mph of exit velocity is worth either. Alan Nathan has suggested that (outside of Denver), each additional mph of exit velocity is worth about five feet of distance on the ball, if you launch it in the “sweet zone” of a 25-30 degree launch angle. It’s worth a little less on a line drive or a pop up, though you still get some better carry with it and even on ground balls, not surprisingly, harder ground balls are more likely to produce hits. In fact, Dr. Nathan suggests that that average 3.5 mph in exit velocity is roughly equivalent to the effect of having a 5 mph wind at your back all the time.

(Side note: Remember when the Twins admitted that they manipulated the ventilation system at the Metrodome when the Twins were batting late in close games, so that there would be artificial “wind” blowing out behind Kirby Puckett and Kent Hrbek?)

Now the flip side. It’s entirely possible that steroids gave some players a leg up from “pretty good” to “Hall of Fame good.” It doesn’t mean that everyone who got good did so chemically, nor does it mean that everyone who took steroids got some sort of benefit. In fact, if it were that easy, then there would be a whole bunch of Hall of Famers running around. The CDC estimates that about 1 in 20 male high school seniors have used steroids at least once without a doctor’s prescription.

Being able to hit the ball further is great… if you can hit the ball. StatCast has also taught us that it’s not just about hitting the ball, but launching it at the optimal angle. Plus, we’ve known about swings and misses for a while. Hitting a baseball is not about raw muscle. It’s about controlled power. There’s a well-known genre of prospects that have plenty of raw power and no “hit tool.” They generally don’t pan out well.

On top of that, there’s evidence that anabolic steroid users are more susceptible to injuries such as tendon tears. Surprisingly, they’re also more susceptible to mood disorders, such as depression (in the study linked, more than one in five were diagnosable!), and depression can slow down reaction times in a game that traffics in tenths of a second. There are risk factors that go along with steroid use that can sap all of the gains that the extra muscle mass provided. Not everyone will fall prey to them, but a good chunk will.

It’s likely that even in an era of rampant steroid use, the number of players that already had the pre-requisite suite of tools that would have allowed the extra muscle mass to play up (and who could have laid down the requisite neurological pathways to quickly control all that extra mass) would have been limited and some of them would have fallen prey to some of the less-discussed side effects.

Beyond the Hysteria
Let’s take stock of what we have. First off, given a sober look at the science of the matter, the effects that steroids might have on a player’s performance are actually pretty moderate. That’s masked by a certain amount of survivorship bias in who we talk about. There are probably a number of players who used PEDs and were never suspected of it because they never really became anything in MLB (or perhaps even made MLB). Or the steroids didn’t do much for them. We don’t talk about those guys, but we do talk about the players who made a name for themselves. It can give the illusion that everyone who injected themselves gained super powers and that everyone was doing it. The reality is more boring. For those who it “worked” for, if we think of MLB players all standing in a line (or perhaps several amorphous blobs) in order of how “good” they are, steroids were probably just a way to move a few spots up in that line.

Second, we have pretty good evidence that players who are clean can see improvements on the same magnitude as what we might expect from steroid users. That suggests that there might be some players who had breakout years in the 90s and 00s (and entirely clean ones) that put them on par with guys who we later found out were using PEDs. In the absence of information, it’s easy to make the jump to “Well, how else could they have done it?” (Worse, when those players deny the charge, their denials are dismissed as “Well, of course you’d say that.”) Now that we have exit velocity numbers and a rock solid testing program, we can see that it’s entirely plausible that they are telling the truth. For some reason, people have a hard time believing that baseball players can get better. It happens.

Finally, we come to the question of Hall of Fame voting. Again, there are those who look at PED use as a moral issue and come down on the side that even if steroids didn’t actually enhance someone’s performance, the fact that they tried means no vote. There are those who come down on the side that a player should be judged by what he did on the field, no matter how he managed to be able to do that. And then there are those who look at the issue instrumentally. They recognize that a player might have gotten a boost from PEDs, but if he would have been Hall-worthy without them, then he might deserve a vote.

I will leave it to the conscience of the individual readers (and voters) on which path to follow, but for those who follow the third path, I think that in some cases, you have science on your side. We’ll never know exactly who did what and when and how much and if someone did use, we’ll never know exactly how much it helped him, so there’s going to be some guesswork. And yeah, A-Rod is probably the perfect test case for that line of thinking. His performance over the course of his career was so elite that even penalizing him somewhat still leaves a pretty good player.

But even beyond A-Rod, I think we need to look at “the Steroid Era” in a new way. Yes, there were guys who used. Yes, it was a high-octane era for offense. There’s probably some correlation in there. However, I think that people have let their imaginations get away from them. Not all cheaters prospered, and not all who prospered were cheaters.

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pwhite
8/16
As a Hall voter annually faced with this dilemma, let me toss in another level of consideration. The steroid era began with that nebulous time of "something's going on, there's no testing, we're not comfortable with whatever it is and everybody's doing it." Not exactly true but testing came in and a significant portion of "everybody" stopped doing it. A few decided, "screw this, I'm above it all and I'll just keep doing it (with testing in place, officially defined as cheating)." There's a moral line there and one maybe worthy of using in Hall consideration. We still don't know for certain who did what when, but we do know some who clearly thumbed their noses at the system.
pizzacutter
8/16
I suppose I should be thankful for the luxury of not having to file a ballot! I can respect a reasonable and well-thought out argument on either side of the moral issue, and I think you're right in that there's a qualitative difference between those who did it when it wasn't _officially_ against the rules vs. those who did it when it was most definitely against the rules (and they got caught!) There will be a bloc of voters (and probably more than 25 percent of them, which means he doesn't get in) who will be #NeverARod folks based entirely on his admission of use. I'm interested to see if his candidacy fares better than does Bonds who has been suspected of PED use, though he denies any use.
MatternK
8/16
Really great article! Good job and thanks for writing this. I remember collecting all of AROD's rookie cards. Lol He was my favorite player....but...I really hope that AROD is not allowed in the Hall of Fame. He took steroids to better his performance. He knew that's what steroids does for you....why else take it. He knew what he was doing. To overlook the use of steroids by honoring the achievements of their users sends a dangerous message to our youth. We must not let our Youth think it's okay to take steroids. Or, maybe in T-ball leagues across the states, we need to start juicing up our kids with this stuff....
LeafontheWind
8/16
Please won't someone think of the children!
map2history
8/16
the "athletes are role models" argument - its the sports-equivalent of the "No True Scotsman" logical fallacy. The HOF enshrines performance, not philanthropy.
sbnirish77
8/16
The apologist's Bible ... 1. Steroids have no effect upon performance. 2. The effect of steroids can't be quantified. 3. Even if the effect of steroids could be quantified on an individual basis, there is no way to quantify the effect across MLB. 4. Even if we could quantify the effect across MLB, the numbers of users is so small that its not worth worrying about. 5. Even if the numbers of users was large enough to make a difference, both pitchers and hitters were users, so the effect is a wash to the game. 6. Even if steroids did have an effect on the game, isn't it better for the game if we just turn the page and move on? 7. Who Cares? I posted this nearly a decade ago as BP was working through its understanding of steroids. This article is just another step in the journey.
LeafontheWind
8/16
Sounds like a pretty solid list to me. Especially number 7.
MatternK
8/16
Well, the players sure believed it did or they wouldn't have taken it... Hopefully the voters keep on blocking out the cheaters..
lmarighi
8/16
Of course, plenty of players also believe that "Phiten necklaces" help them perform, and that not changing socks keeps a hit streak going. The placebo effect doesn't mean that everything actually boosts performance.
Dodger300
8/16
Seriously, Imarighi? Don't you believe that players should be disqualified from any Hall of Fame consideration based on their beliefs? Shame on you. Actually, any player who ever had sex with a fat, gnarly, gross chick should also be disqualified based upon his beliefs that it will improve performance, as Mark Grace explained to Jim Rome: "A slumpbuster is if a team's in a slump, or if you personally are in a slump, you gotta find the fattest, gnarliest, grossest chick and you just gotta lay the wood to her. And when you do that, you're just gonna have instant success. And it could also be called jumping on a grenade for the team." Results guaranteed to be much quicker and more effective than steroids, or your money back!
buddha
8/17
Did you go find that article and copy/paste, or do you have this saved in a notepad somewhere? I've been reading here for too long to know how many times you've posted this exact thing, but it's way more than two now. We get it buddy...
sbnirish77
8/16
For anyone who still thinks steroids have no effect upon performance listen to Lenny Dykstra's interview with Boomer and Carton from this morning. "You can’t take a nobody and make him into an All-Star." http://newyork.cbslocal.com/2016/08/16/boomer-carton-lenny-dykstra-spends-quality-time-with-the-guys/
Taldan9
8/16
The Russians and state sponsored drugging are in the Olympics if A-Rod is in the HOF.
sclem21
8/16
Loved the Matt Christopher allusion.
pizzacutter
8/17
Sylvester Codmeyer III likes this.
tnt9357
8/17
Hear, hear! I remember finding his books in the library, while looking for the sequels to The White Mountains.
sbnirish77
8/16
I thought making the Hall of Fame was an honor. Why admit someone who brought dishonor to the game? Or who at least made the decision so very difficult by creating the confusion over the very numbers that have have been very important to baseball? The fact that you can't know what to believe in the Steroid Era is the real damage done to baseball. And for that alone any contributor to the confusion shouldn't expect any honor to come their way.
map2history
8/16
By definition, the HOF enshrines baseball's greatest players. If it systematically excludes baseballs greatest players, its something else - the Hall of Some Fame, perhaps, or the Hall of Most of the Best - but its no longer the HOF. If you're going to exclude ARod because he used PEDs, fine, but you'll need to retroactively exclude Dennis Eckersley (noted cocaine user), and depending on how insane one wants to get, you might have to go as far as those who smoked cigarettes and drank coffee (Joe DiMaggio, for instance). If you're going to exclude ARod because of the "moral line" he crossed, again, fine, but only if you're willing to exclude Babe Ruth (violating prohibition), Tris Speaker and Honus Wagner (members of the KKK) and KM Landis (instituting the "color line" in baseball). In ARod's case, his ANONYMOUS 2003 test results were leaked - a clear violation of his rights to privacy. In my view, I think he was dumb to admit using, but he opted in favor of honesty. Ironically, that got him labeled as a "cheater". Then, in 2014, he was linked to Biogensis - he never tested positive for PED use, never had a day in court - and was given a second offense penalty and suspended for the entire season. At the end of the day, he probably would have been a first ballot HOFer anyway. He's fourth all time in HRs, third all time in RBIs, a 14-time All Star, three time MVP, and finished 12th all time in WAR. There is no doubt that those are HOF numbers. We will never know what he would have done without PEDs - perhaps he would have made half the number of All Star appearances, or hit - if the article above is correct - a third less homers (which would put him around 500). It doesn't matter - let's insert the infamous asterisk and be on our way.
newsense
8/16
Russell- I'm surprised you didn't rely on more Gory Math here. Even when steroid use was high, many players didn't use them, so what we should expect to see is an increase in performance variance among players. You should be able to quantify that.
pizzacutter
8/16
I see where you're going with that, but I was more interested in the question on an individual basis (the HOF angle). The problem there is that to go back in time like that, I'd have to go with less reliable stats (HR rate and ISO are the obvious ones, I could probably dream a few more up.) There's going to be some noise built into those. I'm not opposed to doing it, it was just not the angle I wanted to take on this one.
andykaylor
8/16
From the official Hall of Fame rules: "5. Voting: Voting shall be based upon the player's record, playing ability, integrity, sportsmanship, character, and contributions to the team(s) on which the player played." Fully half of the listed criteria have nothing to do with performance. The question for me is how these criteria are to be weighted. Certainly if a player were required to excel in every one of these areas the Hall of Fame would be a very select group indeed. I tend to think that the list should be viewed as contributing factors any of which can promote a player's case. For instance, by statistics alone Kirby Puckett probably didn't have a Hall of Fame resume, but based on "integrity, sportsmanship, character, and contributions to the team" (not to mention playing ability) there's no doubt. So maybe a guy like Alex Rodriguez could be accused of falling short on integrity and sportsmanship (though honestly I have a hard time seeing him that way without applying PED use as a very black-and-white litmus test), but perhaps his playing record is sufficient to overcome that flaw. Now let's talk about Pete Rose. I did a quick calculation (excuse me for not showing the math) and I believe that if he hadn't bet on baseball Pete would have recorded 4256 hits. You'd have a hard time convincing me that anything other than his abrasive personality is keeping Pete out of the Hall of Fame.
Dodger300
8/16
I loved Kirby Puckett, and I believe he belongs in the Hall of Fame. I even named my dog Puckett. However, for you to cite him as an exemplar of integrity and character is perplexing, at best. Puckett was not only charged with sexual assault and false imprisonment, but his wife reported continuous threats and domestic violence, as did his mistress of eighteen years. Along with public urination in shopping center parking lots, no less. Not to mention my personal suspicion, albeit completely unsubstantiated, that he was among the early users of steroids. But as for the rest, Casey Stengel would say, "you can look it up" in this S.I. article by Frank Deford, among many others: http://www.si.com/vault/2003/03/17/8100518/the-rise-and-fall-of-kirby-puckett-the-media-and-the-fans-in-minnesota-turned-the-twins-hall-of-famer-into-a-paragon-of-every-virtueand-that-made-his-human-flaws-when-they-came-to-light-all-the-more-shocking
pizzacutter
8/17
The Puckett reference followed the comparison of the effects of steroids to having a 5 mph wind at your back. As an aside, I recalled the days when the Twins would turn on the air conditioning during close games when the Twins were up to actually give them a little extra "wind" in the Metrodome (click through on the linked story). I went with the first two Twins-iest names I could think of from that era as exemplars.
andykaylor
8/17
I guess that was a terrible example. I wasn't aware of the issues in his personal life. My intention was to make reference to the way he played the game.
sbnirish77
8/16
I hope map2history reads your comment about HOF criteria and realizes the error of his declaration "By definition, the HOF enshrines baseball's greatest players."
map2history
8/16
Three of the five criteria directly relate to playing, unless you are interpreting "contribution to team(s)" as a given player's willingness to carpool to practice or helping to clean up the dugout after games. And, would you honestly argue that Donnie Baseball was kept from the HOF because he didn't build the requisite number of orphanages or didn't help a minimum number of old ladies cross the street? Whether you formally weight the criteria or not, player performance is what gets one even considered. The BWAA has, until recently, been fairly consistent in using performance first, and then weeding out the nearly-great largely on non-performance criteria. Seriously - if non-performance criteria were even remotely important as anything other than a weed-out, why is Ty Cobb enshrined in the HOF?
andykaylor
8/17
Integrity, sportsmanship and character were the three criteria I was claiming were not related to performance. I wouldn't interpret these as referring to any sort of personal saintliness but rather how those three traits are seen in terms of baseball. A lot of the comments here are conflating virtue or vice in the person's personal life with integrity, sportsmanship and character as they relate to baseball. The problem with PED use isn't that it is a moral failing, it's that it undermines the integrity of the game and the integrity of the PED user's participation in the game. As for Don Mattingly, I don't know for sure why he is not a Hall of Famer and Kirby Pucket is (if I may double down on my awkwardly chosen example), but it certainly isn't statistical performance. If you look up Puckett's similarity score on Baseball Reference, you'll find Mattingly at the top of the list. Maybe the voters weight two World Series rings heavier than an MVP award. Maybe they just think there are too many Yankees in Cooperstown already and Mattingly didn't stand out as a Yankee the way Puckett did as a Twin. Or maybe they were recognizing something intangible about the way Kirby Puckett played the game. Ty Cobb is an interesting case because he had what many would consider character issues both on and off the field. I suppose his statistical achievements overwhelm any character shortcomings (and rightly so, I think).
map2history
8/17
I will concede your point on steroid use as possibly indicating a means of undermining the integrity of the game. The problem with integrity, sportsmanship, and character are that they defy any means of substantiation. Great character guys with mediocre stats have no shot of making the HOF. Vice versa, they have a great shot. In ARod's case - as in Bonds' and in Clemmens'- the BWAA are actually over-weighting the importance of non-performance criteria in a way that is unprecedented. No one prior to the "steroid era" has EVER been denied enshrinement because of character, integrity, or sportsmanship. In fact, quite the contrary. In short - your comparison between Puckett and Mattingly is valid in so far as both are (statistically) on the bubble, but it wasn't the fact that Mattingly once got arrested for public urination in a KC mall that kept him out, nor Puckett's alleged abusive behavior, though one could certainly argue that both instances detracted from their integrity, character, and the example they set for young people. Was there any greater crime to the integrity of baseball than Kennesaw Moutain Landis' decision to segregate the game? Yet, would it be a Hall of Fame to exclude the man who saved baseball after the 1919 Black Sox scandal? I revert to my original point - you can exclude him, or ARod, or Bonds, of Clemmens, but then its not a Hall of Fame.
tnt9357
8/17
So why, if a 3.5 mph increase seems like a good estimate of the upper bound for improvement for non-PED users, do you then assume that it's the upper bound for PED users? I'm sympathetic to the idea that PEDs would only move a player a "few places ahead" in line (though those few places are worth tens, and sometimes hundreds of millions of dollars; you can start to understand why some of the most vociferous objections have come from fellow players), but the basis for the study seems flawed, in that it assumes a magnitude that's likely to be greater in reality. What will be interesting is in about two decades, after more players get caught and Statcast has been following them both before and after their positive tests.