Stat 921 Notes 17
Reading: Chapters 5.1-5.3
I. Further Notes on Sensitivity Analysis
Gastwirth, Krieger and Rosenbaum (2000), “Asymptotic
separability in sensitivity analysis,” Journal of the Royal
Statistical Society, Series B, 62, 545-555 develops
sensitivity analysis for a full matching. This would be a
good topic for a final project.
II. Dilated Treatment Effect Model
Effects that Vary from Person to Person: The effect of a
treatment may vary from one person to the next. One
person may benefit or suffer greatly from treatment while
another person may experience little or no effect. In other
words, the effect of the treatment on the ith person in
stratum s, rTsi  rCsi , may not be constant but may change
with i and s.
Example: Thun (1993) reported data from an observational
study concerning 20 highly exposed workers at a cadmium
production plant in Denver and 29 workers at a nearby
hospital where cadmium exposure was unlikely The
outcome is a measure of kidney dysfunction, namely the
level of a protein,   2 microglobulin, found in urine. We
are first going to analyze this observational study under the
assumption that it is free of hidden bias, i.e., it can be
analyzed as a randomized experiment, and then will
consider sensitivity analysis in the next notes.
hospital=c(116.5,209.6,83.2,134.1,564.6,81.4,120,173.1,110.4,135.46,199.1,
113.7,305,256.8,250,159.3,311.4,255.7,225.5,177.5,253.8,95.8,213.3,375.9,
142,246.6,337.5,242.2,221.8);
cadmium=c(200.8,2803,891.7,10208,2302,122,97.5,328.1,700,488,67632,24
288,211,512.5,1144,389.2,172.8,18836,33679,107143);
boxplot(hospital,cadmium,names=c("Hospital","Cadmium"),ylab="Beta-2
microglobulin");
The additive treatment effect model does not appear
reasonable – the cadmium outcomes are larger and more
dispersed.
Multiplicative Treatment Effect Model:
rTi   rCi
For   1 , treated outcomes will be larger and more
dispersed than control outcomes.
Multiplicative treatment effect model is an additive
treatment effect model on the log scale:
log(rTi )  log(rCi )  log 
boxplot(log(hospital),log(cadmium),names=c("Hospital","Cadmium"),ylab=
"Log(Beta-2 microglobulin)");
The log of the cadmium outcomes are still larger and more
dispersed than the log of the hospital outcomes so a
multiplicative treatment effect model does not appear to
hold.
Dilated Treatment Effect Model (Chapter 5.3):
Model for treated outcomes that are both higher and more
dispersed than control outcomes.
The treatment has a dilated effect if
rTi  rCi  (rCi ), i  1, , N for some nonnegative,
nondecreasing function () .
With a dilated effect, the effect of the treatment is
nonnegative and is larger, or at least no smaller, when the
response that would have been observed under control, rCi ,
is higher.
Examples of dilated effects:
1. No effect (   0 in additive treatment effect model) and
positive effects in additive treatment effect model (   0 ).
Here, (rCi )   .
2. Multiplicative treatment effect rTi   rCi with   1.
Here (rCi )  (  1)rCi .
3. Linear effect. rTi     rCi with   0,   1 . Here,
(rCi )    (  1)rCi .
4. Suppose (rCi )  0 for r  rCi and (r )  0 for
rCi  r with () nondecreasing; then individuals who
would exhibit low responses under the control, r  rCi , are
not susceptible to the treatment, but the treatment does
affect other individuals with rCi  r .
Dispersion under dilated treatment effects:
When the treatment effect is dilated, the potential outcomes
under treatment are not only higher than the potential
outcomes under control, but also more dispersed. A
common way to measure dispersion is by the difference in
two order statistics, such as the range, which is the between
the maximum and the minimum, or the interquartile range,
which is the difference between the lower and upper
quartiles. Let rT ( m) , rC ( m) denote the m th order statistics of
the potential outcomes under treatment and control
respectively. Note that because larger control potential
outcomes , rCi , entail larger treated potential outcomes,
rTi  rCi  (rCi ) , when the effect is dilated, it follows that
the rCi ’s and rTi ’s are ordered in the same way, so that
rT ( m )  rC ( m )  (rC ( m ) ) .
When the effect is dilated, the order statistics of potential
outcomes under treatment are farther apart than the order
statistics of the potential outcomes under control; that is for
every k  j , we have
rT ( k )  rT ( j )  [rC ( k )  (rC ( k ) )]  [rC ( j )  (rC ( j ) )]  rC ( k )  rC ( j )
because (rC ( k ) )  (rC ( j ) ) .
Fix a k and let   rC ( k ) be the k / N quantile of the potential
outcomes under control, and consider drawing inferences
about the effect of the treatment at this quantile, (  ) .
Inferences about dilated effects:
The logic of inference about an additive effect requires
some changes before it can be applied to a dilated effect.
Under the additive effect model, rTi  rCi   ,under the null
hypothesis H 0 :   0 , the adjusted outcome Ri  Zi 0 ,
equals the potential outcome under the control
rCi  Ri  Zi 0 and also rTi  rCi  0 ; we can use these
values of rCi , rTi to find the null hypothesis distribution of
any test statistic. However, under the dilated effect model,
(0)
the adjusted outcome Ri  Zi (  ) does not equal Ri and
continues to depend on the treatment assignment through
Ri  Zi (  )  rCi  Zi {(rCi )  (  )} .
Although the magnitudes of the adjusted outcomes are not
equal to the magnitudes of the outcomes under control,
there is a sense in which they have the correct sign. More
precisely, the adjusted outcome, Ri  Zi (  ) is above
 just when rCi is above  . This provides a basis for
exact randomization inference for (  ) . Write
sign(a)  1 if a  0 , sign( a)  0 if a  0 and sign(a)  1
if a  0 .
Proposition 1: If the treatment has a dilated effect, for
i  1, , N ,
sign( Ri  Zi (  )   )  sign(rCi   ) .
Proof: Recall that () is nonnegative and nondecreasing.
It follows that if rCi   , then rTi  rCi  (rCi )    (  ) ,
so that Ri  Zi (  )   . Similarly, if rCi   , then
rTi  rCi  (rCi )    (  ) , so that Ri  Zi (  )   .
Finally if rCi   , then rTi  rCi  (rCi )    (  ) so that
Ri  Zi (  )   .
Testing hypotheses about (  ) . Under the model of a
dilated effect, for fixed k , consider testing the null
hypothesis (  )  0 . Calculate the adjusted outcomes
Ai  Ri  Zi 0 and let A( N )  A( N 1)   A(1) be their
order statistics. If the null hypothesis is true, then
proposition 1 implies   A( k ) . Let qi  1 if Ai  A( k ) and
qi  0 if Ai  A( k ) . Again, by proposition 1, if the null
(0)
hypothesis is true, qi  1 if Yi   and qi  0 if rCi   .
N
Let q   qi .
i 1
N
Consider the test statistic T ( 0 )   Z i qi . Under the null
i 1
hypothesis (  )  0 , the test statistic T ( 0 ) is the number
of treated subjects whose outcomes under control, rCi ,
would have equalled or exceeded   rC ( k ) . Under the null
hypothesis (  )  0 , T ( 0 ) has a hypergeometric
distribution,
 q  N  q 
 

j  m  j 

P(T ( 0 )  j ) 
, j  0, , m
N
 
(1.1)
 
m 
For a one-sided test, H 0 : (  )  0 vs. H a : (  )  0 , we
reject for large values of T ( 0 ) and for a one-sided test,
H 0 : (  )  0 vs. H a : (  )  0 , we reject for small
values of T ( 0 ) .
The test statistic T ( 0 ) is a monotone decreasing function
of  0 .A confidence interval for (  ) can be formed by
inverting the hypothesis test.
A point estimate for (  ) can be found using the HodgesLehmann estimate.
q
E
(
T
(

)
|



)

m
 E0 .
0
0
From (1.1), we have
N
Since T ( 0 ) is a monotone decreasing function of  0 , we
have that the Hodges-Lehmann estimate is
inf  0 : E0  T ( 0 )  sup  0 : E0  T ( 0 )
ˆ(  ) 
HL
2




R code:
# Test of dilated treatment effect
# See Notes 5
dilated.treateffect.test.func=function(Delta0,treated,control,k,alternative="hi
gher",returntype="pval"){
# Create vectors for Ri and Zi, and find total number in experiment and
# number of treated subjects
Ri=c(treated,control);
Zi=c(rep(1,length(treated)),rep(0,length(control)));
N=length(Ri);
m=length(treated);
# Calculate adjusted responses and rho=r_{C(k)}
A=Ri-Zi*Delta0;
sorted.A=sort(A);
rho=sorted.A[k];
# q=1 if adjusted response>=rho, 0 otherwise
q=(A>=rho);
qpos=sum(q);
# Test statistic = # of assigned to treatment units with q=1
teststat.obs=sum(q*Zi);
# For returning the p-value,
# p-value computed using hypergeometric distribution, see Notes 5
if(returntype=="pval"& alternative=="lower"){
returnval=phyper(teststat.obs,qpos,N-qpos,m);
}
if(alternative=="higher"){
returnval=1-phyper(teststat.obs-1,qpos,N-qpos,m);
}
# For returning the test statistic minus its expected value
if(returntype=="teststat.minusev"){
returnval=teststat.obs-m*qpos/N;
}
returnval;
}
# Search for endpoints of lower and upper .025 confidence intervals;
k=25;
pval.Delta0.func=function(Delta0,treated,control,k,alternative){
dilated.treateffect.test.func(Delta0,treated,control,k,alternative)-.025;
}
upper.ci.limit=uniroot(pval.Delta0.func,c(10000,10000),treated=cadmium,control=hospital,k=k,alternative="lower")$r
oot;
lower.ci.limit=uniroot(pval.Delta0.func,c(10000,10000),treated=cadmium,control=hospital,k=k,alternative="higher")$
root;
# Find Hodges Lehmann estimate
hlest=uniroot(dilated.treateffect.test.func,c(10000,10000),treated=cadmium,control=hospital,k=k,returntype="teststat.m
inusev");
Inferences for dilated treatment effect in study of effects of
cadmium exposure
k
12 (lower quartile)
25 (median)
38 (upper quartile)
Hodges-Lehmann
Estimate
91.0
490.4
18582.1
95% CI for
(  )
[-20.0, 377.6]
[85.9, 2643.7]
[832.6, 67389.8]
The treatment effect of cadmium exposure for subjects in
the upper quartile of the potential outcomes under control
distribution is estimated to be much larger than for subjects
at the median, being about 38 times larger. The treatment
effect for subjects in the lower quartile is estimated to be
about five times smaller than at the median, and it is even
plausible that there is no effect for subjects in the lower
quartile.