          United States Court of Appeals
                      For the First Circuit

No. 12-2280

 RONNIE JONES; RICHARD BECKERS; WALTER R. WASHINGTON; WILLIAM E.
  BRIDGEFORTH; SHAWN N. HARRIS; EUGENE WADE; GEORGE C. DOWNING,
JR.; CLARARISE BRISTOW; MASSACHUSETTS ASSOCIATION OF MINORITY LAW
        ENFORCEMENT OFFICERS; RACHELLE COUCH; KERI HOGAN,

                     Plaintiffs, Appellants,

                                v.

     CITY OF BOSTON, BOSTON POLICE DEPARTMENT; EDWARD DAVIS,
          Commissioner of the Boston Police Department,

                      Defendants, Appellees.



          APPEAL FROM THE UNITED STATES DISTRICT COURT
                FOR THE DISTRICT OF MASSACHUSETTS
       [Hon. George A. O'Toole, Jr., U.S. District Judge]




                              Before
                 Torruella, Howard, and Kayatta,
                         Circuit Judges.


          John F. Adkins, with whom Laura Maslow-Armand, Lawyers'
Committee for Civil Rights and Economic Justice, Doreen M. Rachal,
and Bingham McCutchen LLP were on brief, for appellants.
          Helen G. Litsas, with whom Nicole I. Taub, Staff
Attorney, Office of the Legal Advisor, was on brief, for appellees.
          Joel Z. Eigerman on brief for Jewish Alliance for Law &
Social Action, Boston Society of Vulcans, Community Change, Inc.,
Massachusetts   Law    Reform   Institute,    Union   of   Minority
Neighborhoods, Justice at Work, Inc., The National Workrights
Institute, Blacks in Law Enforcement of America, and NAACP Boston,
amici curiae in support of appellants.
          Stephen Churchill and Lichten & Liss-Riordan, P.C., on
brief for Massachusetts Employment Lawyers Association, Fair
Employment    Project,    American   Civil   Liberties    Union   of
Massachusetts, Union of Minority Neighborhoods, Charles Hamilton
Houston Institute for Race and Justice at Harvard Law School, Civil
Rights Clinic at Howard University School of Law, Fair Housing
Center of Greater Boston, Massachusetts Law Reform Institute,
Justice at Work, Inc., The National Workrights Institute, Blacks in
Law Enforcement of America, and NAACP Boston, amici curiae in
support of appellants.
           Richard Pianka, ATA Litigation Center, and Prasad Sharma
on brief for American Trucking Associations, Inc., amicus curiae in
support of appellees.
           Mark A. de Bernardo, Joseph E. Schuler, and Jackson Lewis
LLP on brief for The Council for Employment Law Equity, amicus
curiae in support of appellees.
           Mark A. de Bernardo, Matthew F. Nieman, and Jackson Lewis
LLP on brief for The Institute for a Drug-Free Workplace, amicus
curiae in support of appellees.
           Peter A. Biagetti and Mintz, Levin, Cohn, Ferris, Glovsky
and Popeo, P.C., on brief for Psychemedics Corporation, amicus
curiae in support of appellees.




                            May 7, 2014
          KAYATTA, Circuit Judge.      In this racial discrimination

case, ten black plaintiffs challenge the Boston Police Department's

drug testing program.   Seven of the plaintiffs are former officers

fired by the department after testing positive for cocaine; the

eighth is a former cadet in the same situation; the ninth continues

to work as an officer after testing positive and undergoing

rehabilitation as an alternative to termination; and the tenth is

a former applicant to the department whose contingent job offer was

revoked after a positive test.   The plaintiffs' principal claim is

that the department's program, which used hair samples to test for

illegal drug use, caused a disparate impact on the basis of race in

violation of Title VII of the Civil Rights Act of 1964.   During the

eight years for which the plaintiffs present data, black officers

and cadets tested positive for cocaine approximately 1.3% of the

time, while white officers and cadets tested positive just under

0.3% of the time.   The plaintiffs deny that they used cocaine,

arguing that the hair test employed by the department generated

false-positive results in processing the type of hair common to

many black individuals. The plaintiffs also press claims under the




                                 -3-
United States Constitution, via 42 U.S.C. § 1983, and under the

Americans with Disabilities Act (ADA).1

          The   district   court   granted   summary   judgment   to   the

department on all claims.    We vacate the grant of summary judgment

with respect to the plaintiffs' Title VII claim, and we also

reverse the district court's denial of their motion for partial

summary judgment on that claim, finding no genuine issue of

material fact that could preclude them from making a threshold,

prima facie showing of disparate impact.       We otherwise affirm the

district court's decision.

                             I. Background

          The facts described in this opinion are not genuinely

disputed, except where otherwise noted.

A. The Department's Drug Testing Program

          Since 1999, officers and cadets in the Boston Police

Department have been subject to annual drug tests using samples of




     1
        In the district court, the plaintiffs originally pursued a
number of other claims. They have not defended on appeal their
claims under the Equal Protection Clause of the U.S. Constitution
or 42 U.S.C. § 1981, so the grant of summary judgment to the
department on those claims stands.      See, e.g., Dialysis Access
Ctr., LLC v. RMS Lifeline, Inc., 638 F.3d 367, 374 n.7 (1st Cir.
2011). The plaintiffs' brief on appeal also does not separately
discuss their claims under Mass. Gen. Laws ch. 93, § 103, and ch.
151B, § 4, and the Massachusetts Declaration of Rights, except to
say that they are governed by the same analysis as their ADA claim.
Because we affirm the grant of summary judgment to the department
on that claim, we also affirm it on the state law claims.


                                   -4-
their    hair.2      Under    a   provision     of   a    collective       bargaining

agreement between the department and the police officers' union

known as Rule 111, the department selected a private company,

Psychmedics       Corporation,     to    analyze     employees'     hair     for    the

presence of chemicals indicating exposure to five substances:

cocaine, marijuana, opiates, PCP, and amphetamines.

            When Psychmedics reported that an individual's test

results    indicated    exposure        to   cocaine,     a    licensed     physician

selected by the department checked to see whether the individual

had been administered "cocaine hydrochloride                      . . . during a

medical procedure."          As an additional exculpatory safeguard, the

individual could elect to have a "safety-net" test of a different

hair sample.       During much of the period in which the plaintiffs

tested positive, the safety-net tests were significantly more

sensitive than the initial tests in detecting the presence of

cocaine and its chemical by-products.

            If an employee tested positive, and was not exonerated by

either the medical review or the safety-net test, the department

terminated    the     employee     unless      he    or   she     agreed    to     seek

rehabilitation for drug abuse and to accept an unpaid suspension of

45 work days while undergoing treatment.                      Before a termination

became final, however, Massachusetts law required the department to


     2
       The drug testing regime underwent changes in 2007, and the
claims at issue here relate only to the period between 1999 and
2006, inclusive.

                                         -5-
provide a written notice of reasons, followed by an evidentiary

hearing at which an employee could argue that there was no just

cause for termination.     Mass. Gen. Laws Ann. ch. 31, § 41.    A

police administrator customarily presided over the pre-termination

hearings.   If the hearing officer found just cause, the department

fired the employee, who could then mount a post-termination appeal

to the Massachusetts Civil Service Commission.3    Mass. Gen. Laws

Ann. ch. 31, § 42.

            The department also used the hair test to screen job

applicants.    After an applicant received a conditional offer of

employment, the applicant was required to pass the hair test before

the offer would become final.

B. Drug Test Results for Officers and Cadets

            A very small percentage of officers and cadets, either

white or black, tested positive for cocaine during the period

covered by this lawsuit.   Of those who did test positive, however,

there were more black employees than white employees even though

over two-thirds of the officers and cadets tested were white.   As

an example, in 2003, an average year during the period: 6 of 529

black officers and cadets tested positive, or 1.1% of that group,



     3
        In fact, six of the plaintiffs jointly pursued a challenge
to their terminations at the Massachusetts Civil Service
Commission. In February 2013, that challenge resulted in an order
of reinstatement, with partial backpay, for five of the plaintiffs,
each of whom continues to seek full compensation and additional
damages in this case.

                                -6-
while 3 of 1260 white officers and cadets tested positive, or 0.2%

of that group.4

            The small absolute number of positive tests relative to

the total number of tests presents opportunities for markedly

different characterizations of any correlation between test results

and the races of the individuals tested.   One could say that black

officers and cadets were more likely than their white colleagues to

test positive by just one percentage point.   Or one could say that

black officers and cadets were five times more likely to test

positive.   Perhaps trying to prove correct Mark Twain's quip about

statistics, the parties wage battle in their briefs with these

unhelpful types of competing characterizations of the numbers.

            Statisticians, by contrast, customarily approach data

such as this more precisely.   They ask whether the outcomes of an

employment practice are correlated with a specified characteristic,

such as race, and, if so, whether the correlation can reasonably be

attributed to random chance.    The customary yardstick for making

this latter determination is called "statistical significance."

            Statisticians employ a number of different methods to

assess statistical significance in a variety of different contexts.

Federal Judicial Center, Reference Manual on Scientific Evidence

251 (3d ed. 2011) (hereinafter "FJC Reference Manual").      In the

approach most relevant here, statisticians may compare outcomes for


     4
         We describe the genesis of these figures below.

                                 -7-
two different groups (e.g., black employees and white employees)

presuming that members of the two groups have the same likelihood

of receiving a given outcome (e.g., a promotion).                See Paul Meier,

Jerome Sacks, and Sandy L. Zabell, What Happened in Hazelwood:

Statistics, Employment Discrimination, and the 80% Rule, 1984 Am.

Bar Found. Res. J. 139, 147 (1984).              Statisticians are well aware

that this assumption of equal opportunity, even if true, does not

mean that the two groups will experience exactly equal outcomes:

random variation will often create differences.                      To assess the

likelihood that an observed difference in outcomes resulted from

mere chance, statisticians calculate the probability of observing

a difference equal to or greater than that which actually occurred,

assuming      equal     opportunity.5     They    call   this    probability   the

"p-value."          FJC Reference Manual at 250.         Statisticians usually

apply       the    label    "statistically     significant"     to   the   observed

differential outcomes if the p-value is less than five percent, see

Fudge v. City of Providence Fire Dep't, 766 F.2d 650, 658 n.8 (1st

Cir. 1985), although they sometimes use a different cut-off, such

as one percent,            FJC Reference Manual at 251-52.

                  Essentially, a finding of statistical significance means

that the data casts serious doubt on the assumption that the


        5
        Because the parties have not raised the issue, we do not
discuss here the distinction between "one-tailed" and "two-tailed"
tests for statistical significance. See Palmer v. Shultz, 815 F.2d
84, 92-96 (D.C. Cir. 1987) (discussing the issue at length).


                                         -8-
disparity   was   caused   by   chance.   When   statisticians   find   a

disparity between racial groups to be statistically significant,

they are willing to reject the hypothesis that members of the

groups truly had an equal chance of receiving the outcome at issue.

Id.

            Statistical significance and p-value are often connected

with a third concept, "standard deviation."6       In disparate impact

cases, standard deviation serves as another way of measuring the

amount by which the observed disparity in outcomes differs from the

average expected result given equal opportunity, e.g., equal rates

of promotion for black and white employees.       A difference of 1.96

standard deviations generally corresponds to a p-value of five

percent, while a difference of three standard deviations generally

corresponds to a p-value of approximately 0.5%.          FJC Reference

Manual at 251 n.101        As the Supreme Court observed in a case

involving allegations of discriminatory jury selection, "[a]s a

general rule . . . , if the difference between the expected value

and the observed number is greater than two or three standard

deviations, then the hypothesis that the jury drawing was random




      6
       We, like other courts, use the term "standard deviation" to
describe a measure that statisticians often label "standard error."
FJC Reference Manual at 251 n.101.

                                   -9-
would be suspect to a social scientist." Castaneda v. Partida, 430

U.S. 482, 496 n.17 (1977).7

              In this case, the parties work with a large sample of

thousands of test results from which calculations of deviations

from an expected random distribution can be made with a high degree

of statistical power.       The parties also appear to have no material

dispute regarding the raw numbers underlying the analysis: the

plaintiffs' brief relies on a table created by the plaintiffs

labeled "Comparison of African-American and White Positive Hair

Test Results Under Four-Fifths Rule," which adopts counts offered

by the department's experts, and the department's brief makes no

effort to disavow those numbers.       The plaintiffs further cite the

department's calculations of the standard deviations associated

with those counts, and the plaintiffs do not appear to challenge

those       calculations.    We   therefore   deem   these   numbers   and

calculations to be undisputed, except to the limited extent that

the department raises methodological objections to the analysis of

the undisputed data, which we address below.

              The undisputed data and calculations are as follows:


        7
       The formulation of "greater than two or three" as expressed
in Castaneda is an example of the uneasy fit between the language
of law and the language of mathematics. Every number greater than
two or three is greater than two.       What might be said more
precisely is that in many situations two standard deviations will
generate an inference of nonrandomness for a social scientist,
while in other situations a higher threshold may be employed,
depending on the researcher's assessment of the confidence needed
before accepting the result.

                                    -10-
           Year                  # Tested/# Positive                Standard
                                                                    Deviation
                             Black                White
1999                     521/15             1491/10              3.43
2000                     537/4              1467/3               1.35
2001                     530/3              1404/3               0.81
2002                     532/15             1375/4               4.41
2003                     529/6              1260/3               2.01
2004                     522/4              1260/4               1.92
2005                     529/3              1289/1               1.43
2006                     522/5              1289/2               1.95
1999 to 2006             4222/55            10,835/30            7.148

              This evidence does not establish that the differences in

outcomes were large. It shows, instead, the extent to which we can

be confident that the differences in outcomes, whether large or

small,      were   not    random.      To   the   extent   the   facts   make   it

appropriate to consider the eight-year aggregate data as a single

sample, we can be almost certain that the difference in outcomes

associated with race over that period cannot be attributed to

chance alone.       Nor can randomness be viewed as other than a very




       8
        The standard deviation of 7.14 for all years combined is
far greater than the average of the standard deviations in the
individual years for the same reason that the odds of a coin
landing on tails in thirty out of forty flips is far less than the
odds of getting three tails in four flips. We discuss below the
department's perfunctory objection that these drug test outcomes,
unlike flips of a fair coin, are not independent events.

                                        -11-
unlikely explanation for results in at least three of the years

viewed in isolation.

C. Alleged False Positives and Racial Bias of Hair Testing

             In addition to presenting a statistical demonstration

that the racial differential in outcomes very likely did not result

from chance, the plaintiffs also sought to prove that differences

in   the   chemical     and     physical    characteristics         of   hair,     also

associated      with    race,    may    have     accounted    for     the   observed

differential      in    outcomes.          The    parties     presented      sharply

conflicting evidence on this claim.              The plaintiffs' experts argue

that hair tests are relatively unreliable and note that the federal

government has refused to authorize hair testing in drug screening

of federal employees and employees of private industries for which

the government regulates drug testing.                The plaintiffs' experts

also opine that black individuals tend to have higher levels of

melanin    in   their    hair,    and   that     melanin     causes      cocaine    and

associated chemicals called cocaine metabolites to bind to hair at

a higher rate.          They assert that cocaine in the form of an

"aerosolized powder," which forms after someone has snorted or

smoked it, will "deposit on any nearby surface including the hair

of non-users."         These deposits, the plaintiffs say, can become

incorporated into the hair in such a way that current hair testing

methods cannot distinguish from the effects of actual drug use.

Such incorporation might be particularly likely where a person has


                                        -12-
undergone    cosmetic      hair   treatments      more   common       in    the   black

community.      The plaintiffs do not claim on appeal, though, that

each of them was exposed to cocaine prior to their tests.

             The department's experts counter that hair testing has

been validated by numerous scientific studies.                 They also dispute

that there is any scientific evidence of racial bias in hair

testing.     In particular, they point to studies showing that the

relative    rates     of    positive    drug    tests    for    black       and    white

individuals remain materially constant across different methods of

testing,    specifically      hair     testing,   urine     testing,        and    blood

testing, and note that the plaintiffs' experts do not claim that

results from urine and blood testing are racially skewed.                           The

department also questions any correlation between positive test

results and melanin levels.             Asserting that many Caucasians and

most Asian-Americans have melanin levels as high or higher than

those of the plaintiffs, the department points out that no Asian-

American in the department has ever tested positive, nor has any

officer    in   the      department's    Drug     Control      Unit    or       Evidence

Management Unit, where officers would be most likely to be exposed

to cocaine on the job.

D. Procedural History

             The plaintiffs initiated this suit in state court in July

2005, and the department removed it to federal court soon after.

Several    years    of     discovery    followed,    culminating           in   summary


                                        -13-
judgment motions from both sides.       The district court granted

summary judgment to the department on all claims on September 28,

2012. We have jurisdiction over this timely appeal of the district

court's final order under 28 U.S.C. § 1291.

                      II. Standard of Review

          We review de novo the district court's grant of summary

judgment. Travers v. Flight Servs. & Sys., Inc., 737 F.3d 144, 146

(1st Cir. 2013). Under Federal Rule of Civil Procedure 56(a),

"[t]he court shall grant summary judgment if the movant shows that

there is no genuine dispute as to any material fact and the movant

is entitled to judgment as a matter of law."

                           III. Analysis

A. Disparate Impact Racial Discrimination

          Title VII prohibits employers from utilizing "employment

practices that cause[] a disparate impact on the basis of race"

unless those practices are justified by business necessity.     42

U.S.C. § 2000e-2(k). Notably, a disparate impact claim can succeed

even where the employer did not intend to discriminate. See, e.g.,

Boston Chapter, N.A.A.C.P., Inc. v. Beecher, 504 F.2d 1017, 1021

(1st Cir. 1974).   This distinguishes the disparate impact cause of

action from the more traditional disparate treatment approach to

proving discrimination. To make a prima facie showing of disparate

impact, a plaintiff starts by "isolating and identifying" the

employment practice being challenged.   Watson v. Fort Worth Bank &


                                -14-
Trust, 487 U.S. 977, 994 (1988).   A plaintiff must then show that

the identified practice "causes a disparate impact on the basis of

race."   42 U.S.C. § 2000e-2(k)(1)(A)(i).    The Supreme Court has

most recently described a prima facie showing of disparate impact

as "essentially a threshold showing of a significant statistical

disparity . . . and nothing more."     Ricci v. DeStefano, 557 U.S.

557, 587 (2009).   See also Fudge v. City of Providence Fire Dep't,

766 F.2d 650, 658 n.8 (1st Cir. 1985) (holding that a prima facie

case of disparate impact can be established where "statistical

tests sufficiently diminish chance as a likely explanation").

          1.   There is no genuine dispute that there is a
               statistically  significant   correlation  between
               outcomes of the department's drug testing program
               and race.

          In the district court, and in their opening brief on

appeal, the plaintiffs made clear that the employment practice they

challenge is "the Hair Test," defined by the common elements of the

drug tests used by the department between 1999 and 2006, inclusive.

The department does not dispute that this practice constitutes a

"particular employment practice" as required by the statute.     42

U.S.C. § 2000e-2(k)(1)(A)(i).

          Having identified the challenged employment practice, the

plaintiffs presented evidence that the results of this practice had

a statistically significant correlation with race.        As their

threshold for statistical significance, the plaintiffs chose a p-

value of five percent, or 1.96 standard deviations, the threshold

                                -15-
most commonly used by social scientists.    Most federal courts have

also settled on this threshold in analyzing statistical showings of

disparate impact.9   Because the department does not challenge this

convention, we accept it here without ruling on its general

applicability.

          Using the five percent threshold, the plaintiffs showed

that, in at least three of the eight years during the relevant

period, the differential between positive results for black and

white employees was statistically significant.    Moreover, when the

data from the eight years is aggregated, the distribution in test

results for black employees deviated by more than seven standard

deviations from the expected norm.         The department does not

meaningfully challenge the raw math behind these calculations of



     9
        See Tabor v. Hilti, Inc., 703 F.3d 1206, 1223 (10th Cir.
2013) (describing a statistical significance threshold of "two or
three standard deviations"); Chin v. Port Auth. of N.Y. & N.J., 685
F.3d 135, 145, 153-54 (2d Cir. 2012) ("[S]tatistical significance
at the five-percent level is generally sufficient . . . ."); Stagi
v. Nat'l R.R. Passenger Corp., 391 F. App'x 133, 140, 144-45 (3d
Cir. 2010) (holding that the threshold is generally "a probability
level at or below 0.05, or at 2 to 3 standard deviations or
greater"); Paige v. California, 233 F. App'x 646, 648 (9th Cir.
2007) (accepting 1.96 standard deviations as the threshold for
statistical significance); Adams v. Ameritech Servs., Inc., 231
F.3d 414, 424 (7th Cir. 2000) ("Two standard deviations is normally
enough to show that it is extremely unlikely (that is, there is
less than a 5% probability) that the disparity is due to chance,
giving rise to a reasonable inference that the hiring was not race-
neutral . . . ."); Anderson v. Zubieta, 180 F.3d 329, 340 (D.C.
Cir. 1999) (indicating that "disparities . . . exceed[ing] 1.96
standard deviations under a two-tailed test of statistical
significance" are sufficient to establish a prima facie of
disparate impact).

                                -16-
statistical significance.         Instead, the department raises three

methodological objections.

              First, the department's lawyers claim that the employees

who opted to avoid termination in the wake of a positive test

result   by     undergoing    drug    rehabilitation   or    resigning     were

"correctly identified as using illicit drugs" and therefore must be

excluded      from    the   plaintiffs'      statistical    analysis.       The

department's own experts provide no support for this argument

penned by counsel, nor do counsel venture to explain how altering

the raw numbers in this way would produce any material difference

in the plaintiffs' ultimate statistical results.                This argument

also lacks any logical foundation that we can identify without the

benefit of expert testimony.            The plaintiffs identify as the

challenged      employment     practice,      and   therefore    subject     to

statistical analysis, the test used to identify which officers have

used drugs, i.e., the test used to identify which officers will

have to choose between termination and a suspension/rehabilitation

regimen.      The plaintiffs must show, then, that this selection

process produces identifications that are not randomly distributed

by   race.      The   accuracy   of    that    identification    process,    as

determined ex post, is a different matter, perhaps relevant to the

business necessity defense as discussed below, but not relevant to




                                      -17-
the    statistical    showing   of      a    disparate     impact   in     the

identifications themselves.10

            Second, the department argues that plaintiff Clararise

Bristow cannot rely on the statistical analysis of the outcomes of

hair testing for department employees because she was not an

employee.   Rather, she received a conditional offer of employment,

but that offer was retracted when she failed a drug test.                While

this difference is material to our analysis of Bristow's due

process claim, her status as a tested applicant rather than a

tested officer, as described by the department, is immaterial to

her disparate impact claim.      She took the same hair test as that

given to the incumbent officers.        The only difference cited by the

department--that she was not eligible for rehab when she failed the

test--is not a difference that would plausibly affect application

of the hair test to her.    Given that the department claims no other

respect in which Bristow is not similarly situated to those other

people who were tested, there is no basis for precluding her from

relying on the plaintiffs' statistical analysis regarding the

impact of the test she took.

            Third,   the   department       objects   to   the   plaintiffs'

aggregation of data from the first eight years of the drug testing



      10
        For the same reason, we reject the department's argument
that its evidence of the reliability of hair testing, if accepted
by the factfinder, would preclude the plaintiffs from even making
a prima facie showing of a statistically significant disparity.

                                  -18-
program.   Some form of aggregation (albeit certainly not covering

all eight years) may be necessary to sustain the claims of those

plaintiffs who tested positive in a year in which the disparity was

not statistically significant when looking at the data from that

year alone.   The department says that any aggregation was improper

because many of the same individuals were tested in different

years, cutting against any assumption that the test results for the

different years were independent events.           After not raising this

argument in its memorandum supporting its motion for summary

judgment in the district court, the department now relies solely on

a vague, one-sentence footnote from an expert report, which offers

no analysis of the actual magnitude and effect of the claimed lack

of independence in year-to-year results.           Whatever the merit of

this argument,11 the department did not sufficiently develop it in

the district court to rely on it now.

           Apart   from   floating   on   appeal    the   three   foregoing

arguments that we find inadequately supported and preserved, the

department raises no other basis for questioning the existence of

a statistically significant correlation between race and drug test

results for department employees.     While an amicus brief questions



     11
         A demonstration that the statistical analysis was skewed
by a lack of independence in the year-to-year samples would be
complex, implicating such factors as layoff and hiring practices
and the probability that a person who tested negative in one year
will test negative in a later year, as compared to the probability
of a negative result for someone first tested in the later year.

                                 -19-
whether the statistical analysis considered and addressed possible

confounding variables (such as, for example, age) the department's

experts make no attempt to explain away the differentials on such

grounds, nor does the department argue that such an explanation

could negate a prima facie showing of disparate impact.            Rather,

the department's rebuttal to the plaintiffs' prima facie showing

rests on the argument adopted by the district court: even a showing

of a statistically significant disparity is insufficient if the

size of the impact is not sufficiently large, or "practically

significant," as measured by the so-called four-fifths rule.              We

discuss that argument in the next section of this opinion.

           2.    Title VII does not require plaintiffs to prove that
                 the   observed    differential   is    "practically
                 significant" in order to establish a prima facie
                 case of disparate impact.

           We turn now to the department's argument, adopted by the

district court, that even a statistically significant racial skew

in outcomes does not constitute a disparate impact unless the

racial differential is also sufficiently large, or "practically

significant."    The department correctly points out that, with a

large enough set of data, even very small differences can be

statistically significant.     See FJC Reference Manual at 252.           For

example, if you were to flip a coin a million times, and the coin

were to land on tails exactly 50.1% of the time, the deviation from

the   expected   result   of   50%    tails   and   50%   heads   would   be

statistically significant, even though it amounts to just one flip

                                     -20-
per     thousand.        Recognizing         this     possibility,          statisticians

acknowledge that not all statistically significant results are

practically       significant,         meaning      "practically          meaningful     or

important."       E.g., Xitao Fan, Statistical Significance and Effect

Size in Education Research: Two Sides of a Coin, 94 J. Educ. Res.

275, 277 (2001).          According to the Federal Judicial Center's

reference      manual     on     scientific      evidence,        "[w]hen         practical

significance        is   lacking--when        the     size   of       a     disparity     is

negligible--there        is     no     reason    to    worry      about       statistical

significance."       FJC Reference Manual at 252.

              The department therefore argues that courts in disparate

impact    cases     should      ask    not   simply     whether       a     disparity     is

nonrandom, but also whether it is sufficiently large.                            Under this

view, liability may not be justified, for example, where a program

grants promotions to 9.1% of black employees and 9.9% of white

employees, even if the imbalance is statistically significant. Cf.

Waisome v. Port Auth. of N.Y. & N.J., 948 F.2d 1370, 1376 (2d Cir.

1991) (finding no disparate impact where, "though the disparity was

found    to    be    statistically        significant,       it       was    of     limited

magnitude").

              As a gauge for measuring practical significance, the

department     proposes        the    "four-fifths      rule,"    a       rule    of   thumb

developed by the Equal Employment Opportunity Commission (EEOC).

The four-fifths rule provides that where an employment practice


                                          -21-
results in a "selection rate" for any racial group less than four-

fifths of the "selection rate" for another group, these statistics

"will generally be regarded by [f]ederal enforcement agencies as

evidence of" disparate impact.      29 C.F.R. § 1607.4(D).12     For

example, if an employer hires 14% of black applicants and 20% of

white applicants, the four-fifths rule would indicate a disparate

impact, because fourteen is less than four-fifths of twenty.

            The district court largely adopted the department's

position.    The court concluded that a statistically significant

imbalance does not automatically constitute disparate impact where




     12
          The regulation provides in relevant part:

             A selection rate for any race, sex, or ethnic
             group which is less than four-fifths (4/5) (or
             eighty percent) of the rate for the group with
             the highest rate will generally be regarded by
             the Federal enforcement agencies as evidence of
             adverse impact, while a greater than four-fifths
             rate will generally not be regarded by Federal
             enforcement agencies as evidence of adverse
             impact. Smaller differences in selection rate may
             nevertheless constitute adverse impact, where
             they are significant in both statistical and
             practical terms or where a user's actions have
             discouraged applicants disproportionately on
             grounds of race, sex, or ethnic group. Greater
             differences in selection rate may not constitute
             adverse impact where the differences are based on
             small   numbers   and   are   not   statistically
             significant, or where special recruiting or other
             programs cause the pool of minority or female
             candidates to be atypical of the normal pool of
             applicants from that group. . . .
     29 C.F.R. § 1607.4(D).

                                -22-
practical significance is lacking, relying on the four-fifths rule

as a measure of practical significance.

             In   advocating        for    the    adoption     of     a    practical

significance      requirement,      the     department     does     not   press   any

argument based on the text of Title VII.                   Indeed, the statutory

language provides little aid in deciding whether a nonrandom

difference is enough to make a prima facie showing of disparate

impact, or whether the difference must be large.                      For example,

Merriam Webster's dictionary defines "disparate" by using terms

such as "fundamentally different" and "markedly distinct," yet it

also lists as a synonym "different."                 Webster's New Collegiate

Dictionary     329     (8th   ed.    1977);      Merriam-Webster's        Collegiate

Dictionary 360 (11th ed. 2003) (offering the same definitions and

synonym). The sparse and divided case law from other circuits also

fails to offer any clear answers.13

             Several    factors     nevertheless      do    favor    the    district

court's conclusion that the size of a race-based differential in

outcomes matters, in some manner, in assessing disparate impact

claims.   Of understandable importance to the district court, the

EEOC's guidelines are reasonably read as interpreting Title VII to




     13
        The various approaches of the other circuits is reflected
in the cases cited in footnote 9 above, as well as Waisome, 948
F.2d at 1376, and Apsley v. Boeing Co., 691 F.3d 1184, 1200-01
(10th Cir. 2012).


                                          -23-
include a practical significance requirement.14                     While the agency's

four-fifths rule itself has several significant weaknesses, which

we discuss below, the regulation establishing the rule shows that

the commission views practical significance, along with statistical

significance, as relevant in identifying a disparate impact.                         29

C.F.R. § 1607.4(D).               Similarly, the regulation provides that

disparities           failing    to     satisfy       the      four-fifths    rule   may

nevertheless           constitute       disparate       impact     "where     they   are

significant in both statistical and practical terms."                        Id.

                   Second, very small impacts are unlikely to be the product

of intentional discrimination.                 While proof of a disparate impact

claim        requires     no    proof    of    intentional       discrimination,     the

disparate impact theory nevertheless serves, in part, to root out

hidden intentional discrimination.                      See Richard Primus, Equal

Protection and Disparate Impact, 117 Harv. L. Rev. 493, 498-99,

520-21 (2003).           In a case in which a racial disparity is so small

as   to       be     nearly    imperceptible         without    detailed     statistical

analysis, the likelihood that the disparity reveals a hidden intent

to discriminate is correspondingly small.                        Moreover, efforts to

eliminate small impacts may prove counterproductive due to the



        14
        Because "Congress, in enacting Title VII, did not confer
upon the EEOC authority to promulgate rules or regulations," the
agency's guidelines receive weight only to the extent of their
"power to persuade." E.E.O.C. v. Arabian Am. Oil Co., 499 U.S.
244, 257 (1991).


                                              -24-
difficulty    of    concluding    with   confidence   that   an    alternative

practice will truly lessen the already small effect.

             Acknowledging    the    foregoing    arguments       favoring   a

requirement that a difference in results associated with race be

practically significant and not only statistically significant, we

also confront powerful pragmatic arguments against adopting such a

requirement.       To begin, the concept of practical significance is

impossible to define in even a remotely precise manner.                We are

aware of no test generally accepted by statisticians that we might

employ to gauge practical significance (as we employ, for example,

the notion that a p-value less than five percent provides good

reason to presume that a difference in outcomes is not the result

of chance).    With no objective measure of practical significance,

the label may mean that simply the person applying it views a

disparity as substantial enough that a plaintiff ought to be able

to sue over it.       Courts would find it difficult to apply such an

elusive, know-it-when-you-see-it standard, let alone instruct a

jury on how to do so, and parties may find it impossible to predict

results.

             This case illustrates these difficulties.            In trying to

find a measure of practical significance, the district court turned

to the four-fifths rule.         Although the four-fifths rule may serve

as a helpful benchmark in certain circumstances, both the Supreme

Court and the EEOC have emphasized that courts should not treat the


                                     -25-
rule as generally decisive. See Watson v. Fort Worth Bank & Trust,

487 U.S. 977, 995 (1988) (noting that the rule "has been criticized

on technical grounds . . . and has not provided more than a rule of

thumb for courts"); 44 Fed. Reg. 11996-01 (explaining that the rule

was "not intended as a legal definition" and was "not intended to

be controlling in all circumstances").     We previously rejected

reliance on the four-fifths rule by a plaintiff in a case in which

the sample size was small, describing the rule as "not an accurate

test of discriminatory impact."   Fudge v. City of Providence Fire

Dep't, 766 F.2d 650, 658 n.10 (1st Cir. 1985).      And our sister

circuits have both minimized the importance of four-fifths rule and

criticized it directly.   See, e.g., Stagi v. Nat'l R.R. Passenger

Corp., 391 F. App'x 133, 138 (3d Cir. 2010) (unpublished) ("[T]he

'four-fifths rule' has come under substantial criticism, and has

not been particularly persuasive."); Clady v. Los Angeles Cnty.,

770 F.2d 1421, 1428 (9th Cir. 1985) ("[T]he 80 percent rule has

been sharply criticized by courts and commentators.").

          The four-fifths rule can lead to anomalous results.    As

an illustration, imagine that a police department demographically

similar to the Boston Police Department--with approximately 500

black officers and 1200 white officers--implements a policy leading

to the termination of 90 black officers and no white officers.   If

the "selection rate" is taken to be the rate at which employees

survived termination, cf. EEOC v. Joint Apprenticeship Comm. of


                               -26-
Joint        Bd.    of   Elec.   Indus.,   164    F.3d   89,    97   (2d   Cir.   1998)

(selection rate is the rate at which applicants pass a hiring

requirement), the four-fifths rule detects no disparate impact: 82%

of black employees survived, which is more than four-fifths of

100%, the rate at which white employees survived.                     Yet the policy

in   this          hypothetical    illustration      undoubtedly       has    a    very

significant and disproportionate effect on black officers.

                   This illustration highlights several flaws in the four-

fifths rule.             First, to apply the rule in cases involving the

selection of current employees for employment consequences such as

termination, courts must resolve the rule's ambiguity regarding

whether the "selection rate" is the rate at which employees were

selected for termination or the rate at which employees survived

termination.15           This choice can be decisive.          In the above example,

if a court took the "selection rate" as the rate at which employees

were fired, the four-fifths rule would indicate a disparate impact,

because the 0% firing rate for white employees is less than four-

fifths of the 18% firing rate for black employees.                    Construing the

four-fifths rule in this manner, however, would lead to a different


        15
         The same dilemma can occur in cases involving hiring.
Consider an employer who considers only job applicants who have
never been convicted of a crime: the four-fifths rule does not
specify whether the "selection rate" should be the rate at which
applicants are excluded from consideration, or the rate at they are
included. See Green v. Missouri Pac. R.R. Co., 523 F.2d 1290, 1295
(8th Cir. 1975) (choosing the former approach, before the four-
fifths rule was promulgated).


                                           -27-
problem: the rule would detect a disparate impact even if just one

employee were fired (a 0/1200 firing rate for white employees would

be   less   than   four-fifths   of    a     1/500   firing   rate   for   black

employees), a result that seems clearly incorrect.

            Second, and relatedly, the consequences of the four-

fifths rule vary in a seemingly arbitrary way depending on the

magnitude of the selection rates at issue.             In the example above,

the policy leads to the firing of 90 black officers, or 18% of the

population of black employees, but this disparity is not actionable

under the four-fifths rule. Yet, if the police department provided

a raise to just 1% of its white employees (12 of 1200 employees)

and 0.6% of its black employees (3 of 500 black employees), this

would qualify as actionable disparate impact under the four-fifths

rule, even though vastly fewer black employees were affected (and

less severely) than in the original scenario.

            Conversely, the four-fifths rule makes no distinction

between an employment practice that leads to the firing of one of

nine black employees and a practice that leads to the firing of 100

of 900 black employees.      In either case, the same percentage of

black employees is affected. Yet, the larger sample permits a much

stronger inference that a disparity is non-random and can be

expected to persist through future uses of the practice.

            Notwithstanding these limitations, the four-fifths rule

may serve important needs in guiding the exercise of agency


                                      -28-
discretion, or in serving as a helpful rule of thumb for employers

not wanting to perform more expansive statistical examinations.16

The rule itself has some practical utility.              There is simply

nothing in that utility, however, to justify affording decisive

weight to the rule to negate or establish proof of disparate impact

in a Title VII case.       Having previously rejected a plaintiff's

reliance   on   the   four-fifths   rule   where   a   small   sample   size

precluded a showing of statistical significance, Fudge, 766 F.2d at

658 n.10, we reject here the defendant's reliance on the four-

fifths rule to parry a proper statistical proof of a nonrandom

distribution in a case with a large sample size.

           Our rejection of the four-fifths rule as suitable to

trump a showing of statistical significance leaves us with no

statute, regulation, or case law proposing any other mathematical

measure of practical significance. Nor as a matter of theory would

we expect to find any single measure of the size of the impact to

determine its practical significance.        To fully assess practical

significance, one must consider the qualitative nature and weight

of the impact.    See, e.g., Steve Goodman, The Dirty Dozen: Twelve


     16
        The fact that the four-fifths rule is only a rule of thumb
that does not always work does not mean that it can never provide
evidence of a nonrandom disparity. Thus, in Ricci v. DeStefano,
557 U.S. 557 (2009), the Supreme Court could cite pass rates of
37.5% for black and Hispanic candidates and 64% for white
candidates as supporting what all parties conceded was a disparate
impact. Id. at 586-87. Nothing in that citation supported use of
the test (again described as a "rule of thumb") to trump a more
scientific calculation of the actual statistical deviation.

                                    -29-
P-Value Misconceptions, 45 Seminars in Hematology 135, 136-37

(2008). For example, just a small percentage increase in mild side

effects     might   not   render      a    drug     unsafe   as     compared     to   an

alternative drug, but the same percentage increase in fatalities

could well justify the elimination of the drug.                             So, too, an

employment    practice     that       provides      one   part    of    a    multi-part

promotion test has less practical impact on each employee than

does, for example, a practice that calls for the labeling and

firing of police officers as illegal drug users.

             Ultimately, we find any theoretical benefits of inquiring

as to practical significance outweighed by the difficulty of doing

so in practice in any principled and predictable manner.                              We

therefore    conclude     that    a    plaintiff's        failure      to   demonstrate

practical significance cannot preclude that plaintiff from relying

on competent evidence of statistical significance to establish a

prima facie case of disparate impact.

             Our confidence in rejecting a practical significance

requirement is bolstered by the fact that two other requirements to

be met by a plaintiff in a Title VII disparate impact case

indirectly secure most of the advantages that might be gained were

it possible to fashion a principled and predictable direct test of

practical significance.          First, the very need to show statistical

significance will eliminate small impacts as fodder for litigation

in   many    instances    because         proving    that    a    small      impact   is


                                          -30-
statistically significant generally requires large sample sizes,

which are often unavailable. See, e.g., Fudge, 766 F.2d at 657-59.

Second,   even     in   cases   like    this   one,    in   which   the    data   is

available, the subsequent steps required to successfully recover on

a disparate impact theory offer an additional safeguard.                           An

employer may rebut a prima facie case of disparate impact by

showing that its use of the challenged practice is "job related for

the position in question and consistent with business necessity,"

42 U.S.C. § 2000e-2(k)(1)(A)(i), and a plaintiff can prevail in the

face of demonstrated business necessity only by proving a failure

to   adopt    an    alternative        practice   that      would   satisfy       the

department's       legitimate    business      needs   "without     a     similarly

undesirable racial effect." Albemarle Paper Co. v. Moody, 422 U.S.

405, 425 (1979).        See 42 U.S.C. § 2000e-2(k)(1)(c) (adopting case

law prior to June 4, 1989, "with respect to the concept of

'alternative employment practice'").

             Proving that an alternative practice will not have the

impact identified by a plaintiff when that impact is small leaves

little margin for error and will often require extensive data.                     A

plaintiff who subjects a defendant's job-related practice to the

sensitivity of a large sample analysis can fairly be required to

show through statistical evidence, and with equal confidence, that

the proffered alternative practice will have a smaller impact,




                                        -31-
except where the alternative is self-evidently incapable of causing

a differential (e.g., a random selection tool).

            In this manner, the statute as designed by Congress

effectively assigns case-specific practical significance to the

size of the impact: as the size of the impact increases, so too

does the ease of demonstrating an alternative practice that reduces

the impact.    And it is fitting that this relationship exists most

robustly only where the challenged practice can be justified by

business necessity.    Where such necessity does not exist, most of

the reasons favoring some requirement of practical significance

disappear.      In other words, if a practice fails to serve a

sufficient business need, why retain it merely because the number

of people harmed is small?

            Because we have rejected both the department's limited

challenge to the plaintiffs' showing of statistical significance

and   the    department's   advocacy     of   a   practical   significance

requirement, we see no remaining issue of fact that could permit a

reasonable jury to reject the plaintiffs' prima facie proof of

disparate impact.      We therefore reverse the district court's

decision to deny partial summary judgment to the plaintiffs on that

component of their Title VII disparate impact case.




                                  -32-
             3.   We decline to decide in the first instance whether
                  the drug testing program is "job-related . . . and
                  consistent with business necessity" and whether the
                  plaintiffs have offered an adequate alternative.

             Once a plaintiff has made a prima facie showing of a

disparate impact, the burden shifts to the employer to show that

"the challenged practice is job related for the position in

question and consistent with business necessity."             42 U.S.C. §

2000e-2(k)(1)(A)(i).       If the employer makes such a showing, a

plaintiff has one final path to success, by proving the existence

of    an   "alternative   employment   practice,"   42   U.S.C.   §   2000e-

2(k)(1)(A)(ii), defined in case law as a different "test or

selection device[], without a similarly undesirable racial effect,"

which "also serve[s] the employer's legitimate interest . . . .,"

Albemarle Paper Co. v. Moody, 422 U.S. 405, 425 (1975).                If a

plaintiff makes such a showing, and the employer "refuses to adopt

such alternative employment practice," then the plaintiff prevails.

42 U.S.C. § 2000e-2(k)(1)(A)(ii).

             The department invites us to consider on this appeal

whether it has established that its hair testing program satisfies

the    business   necessity    defense    under   the    disparate    impact

provisions of Title VII.      Supreme Court decisions illustrate that




                                   -33-
the defense has two main components.17           First, the department must

show that its program aims to measure a characteristic that

constitutes an "important element[] of work behavior."            Albemarle

Paper Co., 422 U.S. at 431; see also Dothard v. Rawlinson, 433 U.S.

321,     331    (1977)   (holding   that    an   employer   satisfied   this

requirement by showing that the challenged practice measured a

characteristic "essential to effective job performance").           Second,

the department must show that the outcomes of the drug testing

program are "predictive of or significantly correlated with" the

characteristic described above.        Albemarle Paper Co., 422 U.S. at

431.18

               Here, the plaintiffs have not disputed that abstention

from illegal drug use is an important element of police officer

behavior. They have admitted that the department has a "legitimate

purpose of ensuring a drug-free workplace."             What remains to be

determined, then, is whether the results of the department's drug

testing regime are "predictive of or significantly correlated with"


       17
         According to the "Purposes" section of the 1991 statute
that added the disparate impact provision to Title VII, Congress
aimed to "codify the concepts of 'business necessity' and 'job
related' enunciated by the Supreme Court in Griggs . . . and in
other Supreme Court decisions prior to Wards Cove Packing Co.
[decided in 1989]." Pub L. No. 102-166, § 3(2), 105 Stat. 1071,
1071 (1991).

       18
        Our articulation of this two-part test is substantially the
same as the three-part inquiry adopted by the Ninth Circuit. See
Association of Mexican-Am. Educators v. California, 231 F.3d 572,
585 (9th Cir. 2000) (en banc).

                                     -34-
drug use.     The plaintiffs have asserted that hair testing is not

"sufficiently reliable to be job-related and justified by business

necessity."    But they have presented little if any evidence that

could allow a jury to conclude that the drug test is so unreliable

that its results have no significant correlation with drug use.

Indeed, even their own evidence, if believed, would offer cause to

question the accuracy of only some of the reported results, without

indicating that there is a relatively large number of false

positives compared to the size of the police force.          On the other

hand, the department, not the plaintiffs, carries the burden to

prove that the program's results are significantly correlated with

actual drug use.

            In their reply brief, the plaintiffs argue that, for the

purposes of proving the business necessity defense, the department

must separately defend more than a dozen different versions of the

test used over eight years.     Such an argument flies in the face of

the plaintiffs' position that multiple years of test results are

properly    aggregated   as   arising    from   a   single   practice   in

establishing a prima facie disparate impact claim.           See Fudge v.

City of Providence Fire Dep't, 766 F.2d 650, 657 (1st Cir. 1985).

We reject the plaintiffs' about-face as both unpreserved and

unfairly inconsistent with their assumption that they were all

subjected to a single challenged practice.




                                  -35-
          Given that this case has already spanned many years (as

did the post-termination administrative process), we are tempted to

accept the department's invitation to assess whether genuine issues

of material fact remain concerning its business necessity defense.

In view of the size of the record, though, and the fact that the

district court judge who has presided over this case has not yet

parsed that record to assess business necessity or its rejoinder,

we decline to do so in the first instance.          Federal appellate

courts have discretion in deciding whether to take up questions not

considered below, but they generally should not do so.          Singleton

v. Wulff, 428 U.S. 106, 120 (1976).       We see no reason to depart

from that general practice here.19

          With   the   business   necessity   question   left   open   for

further consideration, we have no occasion to consider whether the

plaintiffs' evidence will prove sufficient to show that "the

employer refuses to adopt an available alternative employment

practice that has less disparate impact and serves the employer's

legitimate needs."     Ricci v. DeStefano, 557 U.S. 557, 578 (2009)

(citing 42 U.S.C. §§ 2000e-2(k)(1)(A)(ii) and (C)).       We reiterate,

however, our statement above concerning the manner in which the



     19
         In declining to decide the issues in the first instance,
we do not suggest that the district court must reopen the record to
allow further discovery or expert reports.      The district court
retains its customary discretion to manage the case, and we expect
that it will give due weight to the fact that each party has
already had ample time to put its best foot forward.

                                  -36-
plaintiffs must prove that any such alternative practice would

produce a smaller racial disparity in outcomes than does the

department's current system.

B. Due Process

          The plaintiffs contend that the drug testing program not

only caused a disparate impact but also violated their rights under

the Fourteenth Amendment of the United States Constitution.      The

Due Process Clause of the Fourteenth Amendment prohibits states

from "depriv[ing] any person of life, liberty, or property, without

due process of law."     The plaintiffs contend that they were not

afforded sufficient process in connection with their terminations

(or other adverse action against them), and seek damages under 42

U.S.C. § 1983.

          1.     Nine of the plaintiffs had a constitutionally-
                 protected interest in their employment.

          When a public employee can be fired only for cause under

state law, that employee has a property interest in continued

employment.    Gilbert v. Homar, 520 U.S. 924, 928-29 (1997).   Here,

the department concedes that the seven fired police officers had a

property interest in their employment.     The department also does

not dispute that the fired cadet had a property interest in her

employment.    And the department does not dispute that Rachelle

Couch, who signed the settlement agreement and continues to work

for the department, was deprived of a property interest, presumably

the approximately two months of pay she lost during her suspension.

                                -37-
              The   department   asserts,   however,   that   plaintiff

Clararise Bristow had no interest entitling her to due process. We

agree.    Bristow was an applicant for a position at the department

who in December 2002 received a letter conditionally offering her

the job.    The letter read: "If you successfully pass the medical

examination and hair drug testing components of the screening

process, you will be tendered a final offer of employment."

            This circuit has not decided whether a contingent offer

of employment can create a property interest under the Due Process

Clause. It is clear, however, that the interest created by a

conditional job offer can be no stronger than that created by an

unconditional job offer, and that this interest in turn rises no

higher than that possessed by someone who has recently begun work

in the position. Here, this logic leads to the conclusion that

Bristow was not entitled to due process.

            In Massachusetts, public workers begin their employment

with a six-month probationary period during which they do not have

the protection from termination without just cause afforded to

tenured employees.20   See Mass. Gen. Laws Ann. ch. 31, § 34; Costa

v. Bd. of Selectmen, 377 Mass. 853, 859-60 (1979).            Thus, had


     20
         We do not consider whether the six-month probationary
period could have been or was supplanted under the terms of a
collective bargaining agreement.   See Mass. Gen. Laws Ann. ch.
150E, § 7. The plaintiffs do not assert that Bristow would have
been covered by a collective bargaining agreement that preempted
the probationary period, nor have they submitted evidence that
could support such an argument.

                                  -38-
Bristow received an unconditional offer of employment, and indeed

begun work as an officer, her job would have terminable with or

without cause for six months.              We have previously observed that

probationary employees in Massachusetts do not have a property

interest in their continued employment.                See Brennan v. Hendrigan,

888 F.2d 189, 195 (1st Cir. 1989); see also Dasey v Anderson, 304

F.3d 148, 156-61 (1st Cir. 2002).               Consequently, even had Bristow

begun to work, she would have had no cognizable property interest

in continued employment during the entirety of her probationary

period.   A fortiori, having not begun work, Bristow also had no

cognizable property interest based on the job offer alone. We make

no comment on whether the conditional offer of a job not subject to

a probation period might, on other facts, be sufficient to grant a

due process right to its recipient.

           2.        The department provided sufficient process.

           The       Supreme   Court     has    held   that    "[t]he   fundamental

requirement of due process is the opportunity to be heard at a

meaningful time and in a meaningful manner."                  Mathews v. Eldridge,

424 U.S. 319, 333 (1976) (internal quotation marks omitted).

Although the precise contours of this guarantee vary depending on

the circumstances, public employees are ordinarily entitled to

notice of the reasons for a proposed termination, an explanation of

the evidence supporting those reasons, and an opportunity to give

their   side    of    the   story   at    a    pre-termination     hearing.    See


                                         -39-
Cleveland Bd. of Educ. v. Loudermill, 470 U.S. 532, 546 (1985);

Calderón-Garnier v. Rodríguez, 578 F.3d 33, 38 (1st Cir. 2009).

When a state employee in the ordinary course terminates another

employee who has a property interest in his or her job, the state

normally cannot satisfy due process solely through post-termination

process.   See Loudermill, 470 U.S. at 542; Cotnoir v. Univ. of

Maine Sys., 35 F.3d 6, 12 (1st Cir. 1994) ("Where an employee is

fired in violation of his due process rights, the availability of

post-termination grievance procedures will not ordinarily cure the

violation.").    As   the   Supreme   Court   has   explained,   a   pre-

termination hearing should provide "a meaningful opportunity to

invoke the discretion of the decisionmaker," both as to the facts

supporting the termination and as to its broader appropriateness.

Loudermill, 470 U.S. at 543.

           Here, there is no dispute that employees subject to

termination for a positive drug test were offered an opportunity to

contest their impending firing at a pre-termination disciplinary

hearing. Although the plaintiffs claim that some of those who were

fired never had such a hearing, they concede that all were offered

one.   And, indeed, Massachusetts state law mandates that public

employees, before being fired, receive notice of the reasons for

their termination and the opportunity to protest their firing at an

evidentiary hearing. Mass. Gen. Laws ch. 31, § 41. This statutory

process allowed employees to present their side of the story,


                                -40-
through both testimony and evidence, and even to cross-examine

adverse witnesses.    The plaintiffs lodge only one complaint about

the process, claiming that some of the fired employees were barred

from presenting evidence of drug tests taken by the employees

outside of the department's program. In one case, the department's

chief hearing officer refused to accept results from an outside

hair test in part because there was not sufficient information

regarding the reliability of the test.      She also refused in at

least two cases to accept results from a urine test, calling urine

tests and hair tests "apples and oranges," presumably because urine

tests have a much shorter window of detection.

          While we do not necessarily agree that the evidence was

unworthy of consideration, we find no constitutional violation in

its exclusion.    Even in criminal trials, the state has some leeway

in crafting and applying reasonable evidentiary rules.   See United

States v. Scheffer, 523 U.S. 303, 308 (1998); Montana v. Egelhoff,

518 U.S. 37, 43 (1996).    The Supreme Court has ruled, for example,

that a state did not violate the Due Process Clause where it barred

a defendant accused of drug use from presenting a polygraph test

indicating that he truthfully denied the charge. See Scheffer, 523

U.S. at 312.     The department here was entitled to make a similar

determination as to evidence that was possibly exculpatory but

arguably unreliable or irrelevant.

          Even if exclusion of the evidence rose to the level of a


                                 -41-
constitutional violation, that error would have been adequately

addressed through the extensive civil service appeals process, in

which terminated employees were permitted to present evidence of

outside drug tests. This process involved full hearings before the

impartial Massachusetts Civil Service Commission. Although we have

explained that some pre-termination process is required before the

state fires an employee, an extensive post-termination appeal

system lessens the need for an elaborate pre-termination process.

See Mard v. Town of Amherst, 350 F.3d 184, 192 (1st Cir. 2003)

(finding   a   "limited"      pre-termination    hearing    constitutionally

sufficient     given   "the     availability    of   more   rigorous   post-

deprivation      procedures").       Together,       the     pre-termination

disciplinary hearings and the post-termination appeals process

easily satisfied the state's obligation to provide due process.21

           Finally, the district court correctly granted summary

judgment on the due process claim of plaintiff Rachelle Couch, the

employee who chose an unpaid suspension rather than termination.

     21
         The plaintiffs complain that the civil service appeals
process took a long time, and indeed it did, as much as ten years
for some.    Yet, the Civil Service Commission explained in an
opinion applying to many of the plaintiffs that "the unusual delay
in bringing these appeals to hearing is due, in significant part,
to the decisions of the [employees]," for example in asking for
continuances. We cannot find a constitutional flaw in delay where
the plaintiffs contributed significantly to that delay and do not
attempt to demonstrate that any delay would have occurred absent
their own decisions. The plaintiffs also say that two of the fired
employees "were not a part of the Civil Service Commission
proceedings," but because the plaintiffs do not provide any further
explanation we must assume that this lack of participation was
voluntary.

                                     -42-
Our discussion above demonstrates that the department would have

fulfilled its constitutional obligation to Couch if it had simply

fired her, following its normal termination procedures.            Instead,

the department gave Couch a choice, albeit a very difficult one.

The injection of additional choices cannot convert a constitutional

process into an unconstitutional one.

C. Americans with Disabilities Act

          The plaintiffs next claim that the department fired them

(or subjected them to other adverse employment action) on account

of an erroneous perception that they were drug addicts, thereby

violating the Americans with Disabilities Act.          The ADA protects

individuals who have a "disability," defined as a "physical or

mental impairment that substantially limits one or more major life

activities."   42 U.S.C. § 12102(1)(A).         The Act also covers those

who are regarded as having a physical or mental impairment, so long

as the perceived impairment is not "transitory and minor."               42

U.S.C. § 12102(1)(c), (3).

          Individuals who are recovering from an addiction to drugs

may be disabled in the meaning of the ADA, as the statute aims to

protect them from the stigma associated with their addiction.          See

42 U.S.C. § 12114(a), (b); cf. Bailey v. Georgia-Pac. Corp., 306

F.3d 1162, 1167 (1st Cir. 2002) (holding that "alcoholism is an

impairment" under the ADA).     Similarly, the ADA protects those who

are   erroneously   perceived     as     drug    addicts.     42     U.S.C.


                                  -43-
§ 12114(b)(3), (a).     Importantly, though, the statute explicitly

excludes from protection any individual who is currently using

drugs, whether addicted or not, when the employer acts on the basis

of such use.     42 U.S.C. § 12114(a).        All of this means, in a

nutshell, that in order to survive summary judgment dismissing

their claim under the ADA, the plaintiffs must provide a factual

basis upon which the jury could find that the department fired them

either because they were addicts or because it perceived they were

addicts, rather than because, as a result of the drug tests, it

believed them to be currently using illegal drugs.          See Raytheon

Co. v. Hernandez, 540 U.S. 44, 52-53 (2003) (holding that, in a

disparate   treatment   claim   under   the   ADA   such   as   this   one,

plaintiffs must show that their perceived disability "actually

motivated the employer's decision").22

            The plaintiffs made no such showing.       To the contrary,

the evidence is that the department trained its efforts at directly

identifying users, whether addicts or not. And if the test results

may have caused the department to form an erroneous view of any

plaintiff, that view--to the extent it motivated termination--was

that the plaintiff was a drug user, not that the plaintiff was an

addict.   Nor did the department accept any defense that an officer



     22
        Although the distinction may be subtle, the plaintiffs do
not allege a disparate impact claim under the ADA, even if one
might have been available. See Raytheon, 540 U.S. at 52 (making
clear that disparate impact claims can be pursued under the ADA).

                                 -44-
used illegal drugs only once, or was otherwise not addicted.

Indeed, the very existence of the rehabilitation program shows that

the department was willing to retain employees who were addicted as

long as they are not users.           On this record, no jury could

reasonably    conclude   that   the   department   was   motivated   by   a

perception that plaintiffs were addicted to drugs.          See Lopez v.

Pac. Mar. Ass'n, 657 F.3d 762, 764 (9th Cir. 2011) (affirming grant

of summary judgment to an employer on an ADA claim because "the

triggering event for purposes of [rejection of a job application]

is a failed drug test, not an applicant's drug addiction"); Salley

v. Circuit City Stores, Inc., 160 F.3d 977, 981 (3d Cir. 1998)

(holding that no jury could find that the plaintiff was fired for

his drug addiction rather than misconduct, whether or not the

misconduct was caused by addiction).

D. Failure to Train and Supervise

             The plaintiffs also claim that the department violated

their constitutional rights due to its failure to sufficiently

train and supervise the staff carrying out the program.               The

failure to train theory offers a way for plaintiffs to hold a

municipality liable under 42 U.S.C. § 1983 for the acts of its

employees.     See Hayden v. Grayson, 134 F.3d 449, 456 (1st Cir.

1998).   Typically, a plaintiff first identifies a constitutional

violation (often related to police misconduct) and then attempts to

show that the violation was the direct result of poor training or


                                  -45-
supervision of municipal employees, stemming from "deliberate

indifference to the rights of persons with whom the [employees]

come into contact."    Id.     To prevail, the plaintiff must show that

the constitutional violation had a "direct causal link" to the

deficiency in training.         Canton v. Harris, 489 U.S. 378, 385

(1989).

           Here, the plaintiffs focus extensively on evidence of

poor training but fail to produce any evidence that would allow a

jury to reasonably conclude that this poor training caused a

constitutional violation.       Indeed, the plaintiffs fail to clearly

identify in their brief any constitutional provision allegedly

violated as a result of poor training.              At best, the plaintiffs

simply postulate without analysis violations of the Due Process

Clause and Equal Protection Clause.

           As to the first, we have explained above that the

plaintiffs were not deprived of liberty or property without due

process of law.     Consequently, there can be no liability for such

a violation, whether on a failure to train theory or otherwise.

           As to the second, and even assuming that plaintiffs'

evidence of disparate impact under Title VII would have sufficed to

survive   summary   judgment    on   an     equal   protection   claim,23   the

plaintiffs fail to point to any evidence indicating that training



     23
        The plaintiffs raised such a claim below but declined to
pursue it on appeal.

                                     -46-
so poor as to constitute actionable indifference caused them to

test positive.       We therefore affirm the grant of summary judgment

to the department on the failure to train claim.

                                IV. Conclusion

            The plaintiffs have proven beyond reasonable dispute a

prima facie case of disparate impact under Title VII, while the

department has proffered an uncontested legitimate need to identify

those few of its members who use illegal drugs.              What remains to be

assessed by the district court is whether the department's drug

testing    program    advances    that    goal   and,   if    so,   whether   the

plaintiffs can carry their burden of proving a failure to adopt an

available alternative that meets the department's legitimate needs

while reducing the disparate impact on black employees of the

department.

            For these reasons, we vacate the district court's grant

of summary judgment on the Title VII claims; we reverse the

district   court's     denial    of   partial    summary     judgment   for   all

plaintiffs on the question of whether they have proved a prima

facie case of disparate impact under Title VII; we otherwise affirm

the district court opinion; and we remand for further proceedings

consistent with this opinion.            The district court will decide at

the time of final judgment whether costs of this appeal are to be




                                      -47-
shifted in favor of a finally prevailing party under any applicable

statute.

           So ordered.




                               -48-
