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Observational Studies
Based on Rosenbaum (2002)
David Madigan
Rosenbaum, P.R. (2002). Observational Studies (2nd edition). Springer
Introduction•A empirical study in which:
•Examples:
•smoking and heart disease
•vitamin C and cancer survival
•DES and vaginal cancer
“The objective is to elucidate cause-and-effect relationships in which it is
not feasible to use controlled experimentation”
•aspirin and mortality
•cocaine and birthweight
•diet and mortality
Asthma Study
• Have data on 2,000 kids
• What is the effect of tobacco experimentation on asthma?
Sex Male
Female
Ethnicity
African American
Asian
Hispanic
Other
White
Smoking at Home Yes
No
Tobacco Experimentation
Yes
No
Asthma Self-Diagnosis
Yes
No
Asthma ISAAC Yes
No
Cameron and Pauling Vitamin C
•Gave Vitamin C to 100 terminally ill cancer patients
•For each patient found 10 controls matched for age, gender, cancer site, and tumor type
•Vitamin C patients survived four times longer than controls
•Later randomized study found no effect of vitamin C
•Turns out the control group was formed from patients already dead…
LESSONS: - observational studies are tricky- randomized study is the gold standard
why?
Why does randomization work?
•The two groups are comparable at baseline
•Could do a better job manually matching patients on 18 characteristics listed, but no guarantees for other characteristics
•Randomization did a good job without being told what the 18 characteristics were
•Chance assignment could create some imbalances but the statistical methods account for this properly
The Hypothesis of No Treatment Effect
• In a randomized experiment, can test this hypothesis essentially without making any assumptions at all
• “no effect” formally means for each patient the outcome would have been the same regardless of treatment assignment
• Test statistic, e.g., proportion (D|TT)-proportion(D|PCI)
TT DTT DPCI
L
PCI
L
TT DPCI
D
TT LPCI
L
TT DPCI
D
PCI
L
TT L
PCI
D
TT DTT LPCI
L
PCI
D
TT DPCI
L
TT L
PCI
D
PCI
D
TT LTT L
P=1/6
observed
Estimates, etc.
• Note: the probability distribution needed for the test is known, not assumed or modeled
• Randomized experiment provides unbiased estimator of the average treatment effect
• Internal versus external validity• Confidence intervals by inverting tests• Partially ordered outcomes, censoring,
multivariate outcomes, etc.
Overt Bias in Observational Studies
“An observational study is biased if treatment and control groups differ prior to treatment in ways that matter for the
outcome under study”
Overt bias: a bias that can be seen in the dataHidden bias: involves factors not in the data
Can adjust for overt bias…
Overt BiasM units, j=1,…,M jx
covariate vector
jZtreatment (assume binary 0 or 1). j =Pr(Zj=1)
M
j
zj
zjMM
jjzZzZ1
11 )1(),,Pr(
unknown
An OS is free of hidden bias if the j’s are known to depend only on the ’s (i.e., )
(so two units with same x have same prob of getting the treatment)
jx
)( jj x
unknown
Stratifying on x• Suppose can group units into
strata with identical x’s. Then:
• Conditional on all ’s are equally likely…just like in a uniform randomized experiment
S
s
mns
ms
ssszZ1
)1()Pr(
i sis zm Z
Stratifying on the Propensity Score
• Obviously exact matching not always possible
• Idea: form strata comprising units with the same ’s ( i.e. could have )
• Problem: don’t know the ’s• Solution: estimate them (logistic
regression, SVM, decision tree, etc.)• Form strata containing units with “similar”
probability of treatment
)()( sjsisjsi xxxx but
Matched Analysis Using a model with 29 covariates to predict VHA use, we were able to obtain an accuracy of 88 percent (receiver-operating-characteristic curve, 0.88) and to match 2265 (91.1 percent) of the VHA patients to Medicare patients. Before matching, 16 of the 29 covariates had a standardized difference larger than 10 percent, whereas after matching, all standardized differences were less than 5 percent
Conclusions VHA patients had more coexisting conditions than
Medicare patients. Nevertheless, we found no significant difference in mortality between VHA and Medicare patients, a result that suggests a similar quality of care for acute myocardial infarction.
What about hidden bias?
• Sensitivity analysis!• Consider two units j and k with the same x. hidden bias they may not have the same
• Consider this inequality:
• Sensitivity analysis will consider various ’s
)1(
)1(1
jk
kj
An equivalent latent variable model
for two units j and k with the same x:
10,)(1
log
jjjj
j uux
)}(exp{)1(
)1(
kjjk
kj uu
between –1 and 1
)exp()1(
)1()exp(
jk
kj
so the model implies the previous inequality with
(implication goes the other way too)
)exp(
Matched Pairs • Strata of size 2, one gets the treatment,
one doesn’t
• If =0, every unit has the same chance of treatment
• Standard test statistic for matched pairs is:
21
)exp()exp(
)exp(
)exp()exp(
)exp()Pr(
21
2
1 21
1
ss z
ss
s
zS
s ss
s
uu
u
uu
uzZ
S
s isisis ZcdrZtT
1
2
1
),(
rank of 21 ss rr otherwise 0 and if 211 1 sss rrc
sum of the ranks for pairs in which treated unit > control unit
Wilcoxonrank sum test
More on Matched Pairs
• No hidden bias => know the null distribution of T because sth pair contributes ds with prob ½ and 0 with prob ½
• with hidden bias, the sth pair contributes ds with prob:
and zero with prob 1-ps
• so null distribution of T is unknown…
S
s isisis ZcdrZtT
1
2
1
),(
)exp()exp(
)exp()exp(
21
2211
ss
sssss uu
ucucp
Even More on Matched Pairs
• The P-value we are after is • Lower bound on P-value:
where T- is the sum of S quantities, the sth one being ds with prob and 0 otherwise
• Upper bound likewise using • This directly provides bounds on P-
values for fixed
)Pr( obsTT
)Pr( obsTT
• easy to see that:
11
1sp
sp
sp
sp
sp
Smoking & Lung Cancer Example
• Hammond (1964) paired 36,975 heavy smokers to non-smokers. Matched on age, race, plus 16 other factors Minimu
mMaximu
m1 < 0.0001 < 0.0001
2 < 0.0001 < 0.0001
3 < 0.0001 < 0.0001
4 < 0.0001 0.0036
5 < 0.0001 0.03
6 < 0.0001 0.1
Asthma Study
• Need a of three to make the effect of tobacco experimentation on asthma become non-significant
