Compliance and Causal Analysis Lecture 2 Randomisationbased methods 1: likelihood estimation

1
Compliance and Causal AnalysisLecture 2
Randomisation-based methods 1 likelihood
estimation

• Ian White
• MRC Biostatistics Unit, Cambridge, UK
• ian.white_at_mrc-bsu.cam.ac.uk

2
Causal analysis in randomised trials

• A randomised controlled trial is used to evaluate
  an intervention
• No confounding because randomised group R is
  independent of all predictors of outcome
• But we get departures from randomised
  intervention actual treatment D differs from
  allocated treatment R
• Want to infer causal effect of treatment D

3
Departures from randomised intervention

• Sometimes called non-compliance
• I prefer departures
• avoids implicit value judgements
• more precise includes both
• non-adherence (randomised to X, clinician
  prescribes X, patient does Y)
• changes in prescribed treatment (randomised to
  X, clinician prescribes Y, patient does Y)

4
Types of departure from randomised intervention

• Switches to other trial treatment
• Changes to non-trial treatment
• Includes changes to nothingin a comparative
  trial
• Departures may be
• all-or-nothing (either always get A or always get
  B)
• or quantitative (e.g. dose changes)
• or time-dependent (e.g. emergency operation)

? FOCUS ON SWITCHES
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Motivating example MASS trial

• Abdominal aortic aneurysms are often fatal if
  they rupture
• May be repaired if detected before rupture
• Reliably detected by ultrasound screening
• MASS trial (Lancet, 2002) 67,800 men were
  randomised to invitation to screening or control
• Main outcome was aneurysm-related mortality

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MASS trial Aneurysm-related mortality
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Intention-to-treat analysis

• Intention-to-treat analysis compares groups as
  randomised, ignoring any departures
• respects the randomisation
• avoids selection bias
• Now the standard analysis and rightly so
• Answers an important pragmatic question e.g. what
  is the public health impact of prescribing X?
• Disadvantage this may be the wrong question!

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Disadvantage of ITT

• Doctor doctor, how much will taking this tablet
  reduce my risk of heart disease?
• I dont know, but prescribing it reduces
  disease risk by 10
• on average
• thats on average over whether you take it or not

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Example MASS trial (ctd)

• Intention-to-treat (ITT) analysis invitation
  to screening reduced aneurysm-related death
• hazard ratio 0.58 (95 CI, 0.42 to 0.78), P0.002
• 20 of invited group didnt attend for screening
  (all-or-nothing non-compliance)
• ITT measures the average benefit of screening in
  invitees
• What is the benefit of screening in attenders?

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Plan for this lecture

• Basic idea for all-or-nothing compliance
• Binary outcome estimation
• Normal outcome
• Simple vs. ML estimation
• Negative weights method
• Back door method

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1. Basic idea for all-or-nothing compliance
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Notation

• Randomise to E (experimental) or S (standard)
  treatment
• S could be nothing / placebo
• Potential treatments are E and S assume everyone
  gets either all E or all S (all-or-nothing
  compliance)
• Ri 1/0 indicates randomisation of ith
  individual to E/S
• Di 1/0 indicates receipt of E/S

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Theoretical framework

• Ri 1/0 indicates randomisation of ith
  individual to E/S
• Di 1/0 indicates actual receipt of E/S
• Di(0) 1/0 indicates receipt of E/S if
  randomised to S observed in S arm,
  counterfactual in E arm
• Di(1) 1/0 indicates receipt of E/S if
  randomised to E observed in E arm,
  counterfactual in S arm
• So actual Di Di(Ri)
• The pair Di(0), Di(1) define an individuals
  compliance type.

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Compliance types

• Always-takers (A) Di(0) Di(1) 1 always take
  E regardless of randomisation
• Compliers (C) Di(0) 0, Di(1) 1 take whatever
  they were allocated to
• Never-takers (N) Di(0) Di(1) 0 never take E
  regardless of randomisation
• Defiers (D) Di(0) 1, Di(1) 0 take the
  opposite of what they were allocated to

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Using compliance types

• Note that compliance-type is a characteristic
  that is inherent to the individual before
  randomisation
• unlike actual compliance, which is affected by
  randomisation
• This means we can meaningfully adjust/stratify by
  compliance-type, but not by actual compliance
• But unfortunately compliance-type is incompletely
  observed requires careful statistical methods
• Compliance types are an example of Principal
  strata (Frangakis Rubin, 2002)

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What do we observe about compliance-types?
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What do we observe about compliance-types?
Simplification if S arm cant get E (no
always-takers and no defiers)
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Defining true treatment effect

• Could be defined in various ways
• Use potential outcomes Y(r,d) outcome if
  randomised to r and received d
• Y(r,d) may depend on r
• e.g. never-takers of a counselling intervention
  might do worse in the E arm (where they refused
  the counselling) than in the S arm (where they
  werent offered the counselling)
• but in many settings it wont (exclusion
  restriction)

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Defining true treatment effect

• Average effect of treatment
• EY(1,1)-Y(0,0)
• Average effect of treatment among the treated
• EY(1,1)-Y(0,0) D(1)1
• For an always-taker, we will observe
• Y(0,1) if we randomise to S
• Y(1,1) if we randomise to E
• but we cannot observe Y(0,0)
• Is there a measure of causal effect that involves
  potentially observable outcomes?

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Complier average causal effect (CACE)

• ITT effect EY(1,D(1)) Y(0,D(0))
• Define average causal effect of treatment
  assignment in compliance-type t as EY(1,D(1))
  Y(0,D(0)) Tt
• Complier Average Causal Effect (CACE)
• EY(1,1) Y(0,0) TC
• also Local Average Treatment Effect (LATE)
  (Angrist et al, 1996)
• Never-taker average causal effect (NACE?)
• EY(1,0) Y(0,0) TN
• Always-taker average causal effect (AACE?)
• EY(1,1) Y(0,1) TA
• CACE measures treatment efficacy but only
  involves potentially observable outcomes (Imbens
  Rubin, 1997)

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CACE (2)

• Ignore defiers (mainly for simplicity)
• Can write ITT wC CACE wN NACE wA AACE
• where wC P(compliance-type C) etc.
• Its often reasonable to assume NACEAACE0, so
  that ITT wC CACE
• Give a simple estimate of the CACE

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CACE (3)

• For binary outcomes, we can define the CACE on
  different scales.
• Risk difference scale
• CACE EY(1,1) TC EY(0,0) TC
• Risk ratio scale
• CACE EY(1,1) TC / EY(0,0) TC
• Odds ratio scale
• CACE OY(1,1) TC / OY(0,0) TC
• where OXEX/(1-EX)

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2. Estimation
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Vitamin A trial(Sommer and Zeger, 1991)

• Vitamin A vs. control in Indonesian children
• randomise villages (24000 children)
• outcome is mortality
• about 20 of villages didnt get their vitamin A
  supply
• this analysis ignores clustering by village

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ITT analysis
ITT odds ratio ? 46/77 0.60
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Observed compliance (Vitamin A arm)
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Inferred compliance (Control arm)
CACE odds ratio ? 12/43 0.28
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Estimation method of subtraction
nNE
if nE nS
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Formally
Hence write down and maximise (log-)likelihood. No
te 1-w, w previous wC, wN
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MLE
In this simple case, pCE is estimated as
(dCE/nCE) pCS is estimated as (dS dNE
nS/nE) / (nS nNE nS/nE) provided both
terms 0 (Ill assume this is true)
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MLE

• Use estimates of pCE and pCS to estimate CACE
• as pCE-pCS on RD scale
• as pCE/pCS on RR scale
• as pCE/(1-pCE) / pCS /(1-pCS) on OR scale
• On RD scale only, can show that MLEs obey CACE
  ITT / (1-w)

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CACE compared with other quantities
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Vitamin A vs. control summary

• ITT 0.38 vs. 0.64, RR0.60
• CACE 0.12 vs. 0.45, RR0.28
• Per-protocol 0.12 vs. 0.64, RR0.19
• As-treated 0.12 vs. 0.77, RR0.16
• On-treatment and as-treated are too extreme
  because of strong selection effect 1.41
  (untreated in treatment arm) vs. 0.64 (untreated
  in control arm)

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Comparison of assumptions

• Per-protocol and as-treated analyses assume
  random non-compliance
• no association between compliance-type and
  outcome, once treatment effect is taken into
  account
• CACE analysis assumes exclusion restriction
• randomisation doesnt affect mean outcome for
  never-takers and always-takers
• no assumption of comparability of different
  compliance-types
• usually much more plausible

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Extensions to CACE model (1)

• Above we had all the S arm getting S
• Easy to allow for S arm possibly getting E
  (Cuzick et al, 1997)
• CACE is again estimable under
• 2 exclusion restrictions NACEAACE0
• Either no defiers or same causal effect in
  defiers as in compliers (DACE-CACE)

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Extensions to CACE model (2)

• Introduce covariates
• Covariates that predict Y improve precision (as
  in ITT analysis)
• Covariates that predict D
• also improve precision (unlike in ITT analysis)
  (Jo, 2002)
• enable estimation of NACE etc. as well as CACE
  (Hirano et al, 2000)

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Extensions to CACE model (3)

• Define g difference in outcome between
  never-takers and compliers after allowing for
  their differences in actual treatment
• selection effect
• difference in counterfactual outcomes
• As-treated analysis assumes g0 (random
  non-compliance)
• CACE and ITT analyses make no assumption about g
• Could estimate g from data leads to CACE
  analysis
• Instead, introduce appropriate prior information
  about g (White, 2005)

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Model for log odds of death (observed risk)
Use informative prior gN(0,s2) for various
values of s
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Vitamin A trial Bayesian CACE analyses
as-treated
CACE
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3. Normal outcome
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Normal outcome

• Can simply modify the method of subtraction
  work with means instead of proportions
• Link to instrumental variables (IV) method
  (lecture 5)

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Model
CACE mCE-mCS
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Likelihood
CACE mCE-mCS
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ML estimation

• No closed form solution
• EM algorithm is easy compliance type as the
  missing data
• usual problems in estimating standard errors
• Newton-Raphson also fairly straightforward
• No directly available software in Stata, but
  gllamm can be used see lecture 6.

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4. Method comparisons
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Comparison of CACE estimators

• Weve looked at
• Simple estimation using ITT wC CACE (for
  difference of means, not risk ratio or odds
  ratio)
• Method of subtraction
• Maximum likelihood
• For binary outcomes, they all give the same
  answer
• For continuous outcomes, ML estimation is
  different (potentially more efficient see later)

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Simple CACE vs. ITT

• They estimate different parameters
• But they test the same null hypothesis
• Significance levels are equal
• obvious from CACE ITT / (1-w)
• binary case explored in detail by Branson and
  Whitehead (2003) significance levels are equal
  when likelihood ratio test is performed

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Vitamin A trial profile likelihoods
Likelihood ratio test has same value for both
models
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.. and quadratic approximation (dotted)
Likelihood ratio test has same value for both
models but Wald test doesnt.
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ML estimation of CACE vs. ITT

• For binary outcome, significance levels are equal
• For Normal outcome, significance levels arent
  equal
• CACE is more efficient whenever theres a
  non-zero selection effect or a non-zero treatment
  effect
• the next slides are thanks to Taeko Becque

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Asymptotic relative efficiencyof CACE vs. ITT
(approximate)
q1 CACE, q2 selection effect, outcome SD 1
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Power of CACE and ITT analyses
Compliance rate 50 Selection
effect-0.5 CACE-0.5 Standard deviation1.5
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Power and compliance rate
Sample size 300 Selection effect-0.5 CACE-0.5
Standard deviation1.5
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Power including covariateweak predictor of
compliance
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Power including covariatestrong predictor of
compliance
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Summary
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5. Negative weights method(Kim and White, 2004)
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Negative weights method

• For simplicity take nEnS
• Recall that we subtracted the number of
  events/people in the NE cell (never-takers
  randomised to E) from the number of events/people
  in the S cell (all randomised to S)
• Can also achieve this by including them in the S
  arm but with a weight -1
• nS/nE in general

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Negative weights in Stata

• Unfortunately many Stata commands are too
  sensible to allow negative weights
• Exceptions are regress, logistic, cox
• Illustration uses MASS data pretending outcome is
  binary

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• . use mass, clear
• . l
• rand screen event n
• 1 1 0 27104
• 1 1 1 43
• 1 0 0 6670
• 1 0 1 22
• 0 0 0 33848
• 0 0 1 113
• . tab rand screen fwn, sum(event) mean
• Means and Number of Observations of
  AAA-related death?
• Invited to Screened?
• screening? 0 1 Total
• -------------------------------------------
• 0 .00332735 . .00332735

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• . CACE via negative weights
• . tab rand fwn
• Invited to
• screening? Freq. Percent Cum.
• -----------------------------------------------
• 0 33,961 50.09 50.09
• 1 33,839 49.91 100.00
• -----------------------------------------------
• Total 67,800 100.00
• . gen w 1
• . replace w -33961/33839 if randgtscreen
• (2 real changes made)
• . l
• rand screen event n w

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• . logistic event rand fwn, coef ITT analysis
• Logistic regression Number
  of obs 67800
• LR
  chi2(1) 12.97
• Prob gt
  chi2 0.0003
• Log likelihood -1229.0537 Pseudo
  R2 0.0052
• --------------------------------------------------
  ---------------------
• event Coef. Std. Err. z Pgtz
  95 Conf. Interval
• -------------------------------------------------
  ---------------------
• rand -.5508119 .1558631 -3.53 0.000
  -.856298 -.2453258
• _cons -5.702247 .094229 -60.51 0.000
  -5.886933 -5.517562
• --------------------------------------------------
  ---------------------
• . logistic event screen iwnw, coef CACE
  analysis
• --------------------------------------------------
  ---------------------
• event Coef. Std. Err. z Pgtz
  95 Conf. Interval
• -------------------------------------------------
  ---------------------

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Bootstrap standard error

• Now we bundle the previous commands into a file
  negwt.ado
• To speed things up, I use only 1/30 of the
  controls

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• . . negwt
• Logistic regression
  Number of obs 1210
• LR
  chi2(1) 15.43
• Prob
  gt chi2 0.0001
• Log likelihood -410.28453
  Pseudo R2 0.0185
• --------------------------------------------------
  ----------------------
• event Coef. Std. Err. z Pgtz
  95 Conf. Interval
• -------------------------------------------------
  ----------------------
• screen -.7461635 .1952982 -3.82 0.000
  -1.128941 -.363386
• _cons -1.787902 .1141955 -15.66 0.000
  -2.011721 -1.564083
• --------------------------------------------------
  ----------------------
• . bootstrap _bscreen, reps(1000) negwt
• --------------------------------------------------
  ----------------------
• Var Reps Observed Bias Std. Err.
  95 Conf. Interval
• -------------------------------------------------
  ----------------------

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Negative weights summary

• Agrees exactly with direct method in this simple
  case
• Easy to generalise e.g. to situations with
  covariates (weights would have to depend on
  covariates)
• Naïve standard errors are too small bootstrap
  needed
• Formal rationale is via unbiased estimating
  equations (Abadie 2002)

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6. Back-door method (Nagelkerke 2000)
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Back door method

• Idea would like to regress Y on D, adjusting for
  U
• Its enough to adjust for E that blocks every
  indirect path from D to Y (back door criterion)
• Approximate E by the residual from a linear
  regression of D on R

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Properties

• Nagelkerke et al showed that the method agrees
  exactly with the instrumental variables method
  for a linear model
• For non-linear models it only approximately
  agrees with the method of subtraction
• Not clear whether standard errors are adequate or
  whether bootstrapping is needed
• Easy to generalise to more complex settings

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Back door method for MASS

• . reg screen rand fwn
• Source SS df MS
  Number of obs 67800
• -------------------------------------------
  F( 1, 67798) .
• Model 10908.799 1 10908.799
  Prob gt F 0.0000
• Residual 5368.59021 67798 .079185082
  R-squared 0.6702
• -------------------------------------------
  Adj R-squared 0.6702
• Total 16277.3892 67799 .240083028
  Root MSE .2814
• --------------------------------------------------
  ----------------------------
• screen Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• rand .80224 .0021614 371.16
  0.000 .7980037 .8064764
• _cons 5.55e-17 .001527 0.00
  1.000 -.0029929 .0029929
• --------------------------------------------------
  ----------------------------
• . predict E, residual
• . tab rand E fwn

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Back door method results

• . logistic event screen E fwn
• Logistic regression
  Number of obs 67800
• LR
  chi2(2) 20.04
• Prob
  gt chi2 0.0000
• Log likelihood -1225.5162
  Pseudo R2 0.0081
• --------------------------------------------------
  ----------------------
• event Odds Ratio Std. Err. z Pgtz
  95 Conf. Interval
• -------------------------------------------------
  ----------------------
• screen .4738008 .094594 -3.74 0.000
  .3203717 .7007087
• E 1.015178 .295373 0.05 0.959
  .5739573 1.795582
• --------------------------------------------------
  ----------------------
• Note tiny effect of E no evidence of selection
  in these data

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Method comparison

• . logistic event rand fwn ITT analysis
• --------------------------------------------------
  ----------------------------
• event Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• rand .5764816 .0898522 -3.53
  0.000 .4247315 .7824496
• --------------------------------------------------
  ----------------------------
• . logistic event screen fwn As-treated
  analysis
• --------------------------------------------------
  ----------------------------
• event Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• screen .476156 .0834625 -4.23
  0.000 .3377127 .6713534
• --------------------------------------------------
  ----------------------------
• . logistic event screen E fwn Approximate
  CACE analysis
• --------------------------------------------------
  ----------------------------
• event Odds Ratio Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------

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Summary

• Weve explored a variety of methods for
  all-or-nothing treatment switches
• randomise to E or S everyone gets all E or all S
• In lecture 4, we will extend to much more complex
  patterns of switching
• e.g. get E just for 3 months, then S
• Another problem is where some participants get no
  treatment at all (or a treatment other than E/S)
• ITT difference depends on 2 effects (E vs.
  nothing, S vs. nothing)
• Walter (2006) adapted the compliance-type
  approach but assumed equality between some
  compliance-types