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, , 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, 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(200.8,2803,891.7,10208,2302,122,97.5,328.1,700,488,67632,24288,211,512.5,1144,389.2,172.8,18836,33679,107143);
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:
For , 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:
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 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, , is higher.
Examples of dilated effects:
1. No effect ( in additive treatment effect model) and positive effects in additive treatment effect model (). Here, .
2. Multiplicative treatment effect with . Here .
3. Linear effect. with . Here, .
4. Suppose for and for with nondecreasing; then individuals who would exhibit low responses under the control, , are not susceptible to the treatment, but the treatment does affect other individuals with .
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 denote the th order statistics of the potential outcomes under treatment and control respectively. Note that because larger control potential outcomes ,, entail larger treated potential outcomes, , when the effect is dilated, it follows that the ’s and ’s are ordered in the same way, so that .
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 , we have
because .
Fix a and let be the 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, ,under the null hypothesis , the adjusted outcome , equals the potential outcome under the control and also ; we can use these values of to find the null hypothesis distribution of any test statistic. However, under the dilated effect model, the adjusted outcome does not equal and continues to depend on the treatment assignment through .
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, is above just when is above . This provides a basis for exact randomization inference for . Write if , if and if .
Proposition 1: If the treatment has a dilated effect, for ,
.
Proof: Recall that is nonnegative and nondecreasing. It follows that if , then , so that . Similarly, if , then , so that . Finally if , then so that .
Testing hypotheses about . Under the model of a dilated effect, for fixed , consider testing the null hypothesis . Calculate the adjusted outcomes and let be their order statistics. If the null hypothesis is true, then proposition 1 implies . Let if and if . Again, by proposition 1, if the null hypothesis is true, if and if . Let .
Consider the test statistic . Under the null hypothesis , the test statistic is the number of treated subjects whose outcomes under control, , would have equalled or exceeded . Under the null hypothesis , has a hypergeometric distribution,
(1.1)
For a one-sided test, vs. , we reject for large values of and for a one-sided test, vs. , we reject for small values of .
The test statistic is a monotone decreasing function of .A confidence interval for can be formed by inverting the hypothesis test.
A point estimate for can be found using the Hodges-Lehmann estimate.
From (1.1), we have . Since is a monotone decreasing function of , we have that the Hodges-Lehmann estimate is
R code:
# Test of dilated treatment effect
# See Notes 5
dilated.treateffect.test.func=function(Delta0,treated,control,k,alternative="higher",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")$root;
lower.ci.limit=uniroot(pval.Delta0.func,c(-10000,10000),treated=cadmium,control=hospital,k=k,alternative="higher")$root;
# Find Hodges Lehmann estimate
hlest=uniroot(dilated.treateffect.test.func,c(-10000,10000),treated=cadmium,control=hospital,k=k,returntype="teststat.minusev");
Inferences for dilated treatment effect in study of effects of cadmium exposure
/ Hodges-Lehmann Estimate / 95% CI for12 (lower quartile) / 91.0 / [-20.0, 377.6]
25 (median) / 490.4 / [85.9, 2643.7]
38 (upper quartile) / 18582.1 / [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.