The Logic of Biases via Causal Diagrams€¦ · The Logic of Biases via Causal Diagrams Sander...
Transcript of The Logic of Biases via Causal Diagrams€¦ · The Logic of Biases via Causal Diagrams Sander...
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29 marzo 08 Greenland 1
The Logic of Biases via
Causal Diagrams
Sander Greenland
Epidemiology and Statistics
University of California
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Two definitions of bias:
• Epidemiology: Nonrandom differencebetween an estimate and the true value ofthe target parameter; systematic error;invalidity.
• Statistics: Any difference between theaverage value of an estimator and the truevalue of the target parameter (e.g., arelative risk)
There are subtle differences between thetwo; the second definition subsumes otherimportant problems.
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Types of bias
Epi categories (overlapping):
• Confounding (nonrandom exposure)
• Selection bias (nonrandom sampling)
• Bias from measurement error
Further statistical categories (often importantbut overlooked in epidemiology):
• Bias from use of a wrong model form(model-form mis-specification)
• Method invalidity (e.g., stepwise selection)
• Method failure (e.g., sparse-data bias)
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• There are many finer divisions of epi bias,
but they obscure the underlying deductive
logic of the biases
• Logic is about conclusions that could be
drawn regardless of the content
• Logical deduction concerns what must
follow from what is assumed
• Deductions can only be hypotheticals of
the form “If we assume this, we can
deduce that…” (some would say this is all
science can offer beyond data)
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The easiest way to remember the
logic of epidemiologic biases:
Causal diagrams
• Causal diagrams are schematics for
causal explanations (e.g., “Process P may
have caused bias B”) of possible
associations.
• Diagramming a study can reveal many
avenues for bias that are otherwise
overlooked.
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Directed acyclic graphs (DAGs)
and causal diagrams• A directed acyclic graph shows the factors
in the problem linked by arrows only, with
no feedback loops.
• A graph is a causal diagram if the arrows
are interpreted as links in causal chains
• Causal effects of one variable on another
are transmitted by causal sequences,
which are directed (head-tail) paths:
X Y Z means X can affect Z
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Example DAG: A and B can affect
any variable except each other
A B
C
F
E D
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Colliders vs. noncolliders on a path• Paths are closed at colliders: Associations
cannot be transmitted across a collider( C ) on a path unless we stratify(condition) on it or something it affects(such as F in C F).
• Paths are open (unblocked) atnoncolliders: Associations can betransmitted across a noncollider ( Cor C ) on a path unless we docompletely stratify on it.
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Think of associations as signals
flowing through the graph
• A variable can transmit associations along
some open (unblocked) directions but not
along closed (blocked) directions.
• The open and closed directions are
switched by conditioning (stratifying) on
the variable (and may be partially switched
by partially or indirectly conditioning)
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Spot the open and closed
directions for C:
A B
C
F
E D
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Colliders vs. noncolliders on a path
• Associations may be transmitted across a
collider ( C ) on a path if we stratify
(condition) on it or something it affects
(such as F in C F).
• Associations may be transmitted across a
noncollider ( C or C ) on a path if
we do not completely stratify on it.
“(C)” = C unobserved, “[C]” = C conditioned
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Spot the open and closed
directions for C given C:
A B
[C]
F
E D
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Spot the open and closed
directions for C given F:
A B
C
[F]
E D
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Closed and open paths
• Closed (blocked) path: Closed at some
variable within the path, hence cannot
transmit associations.
• Open (unblocked) path: Open at all
variables within the path, hence can
transmit associations.
Conditioning may open some closed paths
and close some open paths
(C) = C unobserved, [C] = C conditioned
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Spot the open and closed paths,
and rank the signal strengths:
A B
C
F
E D
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Size of associations
• The more steps along a given path, the
more attenuated the signal (the weaker
the transmitted association, a.e.), but
• Distance along distinct paths are not
comparable unless all steps (arrows) in
both paths are assigned a size!
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EAC larger than EACD (a.e.), but
can’t say relative to ECD
A B
C
F
E D
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Spot the open and closed paths
given C:
A B
[C]
F
E D
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Spot the open and closed paths
given F:
A B
C
[F]
E D
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“Control” of bias
• Target path: A path that transmits some
of the effect we want to estimate; must be
a directed path from cause to effect.
• Biasing path: Any other open path
between the cause and effect variables.
• By judicious conditioning, we must close
all biasing paths without closing target
paths or opening new biasing paths.
This isn’t always possible with available data
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Confounding
There are many definitions, none universally
accepted. My definition:
• Noncausal association transmitted via
effects on the outcome
This definition appears to correspond best to
the intuitive definitions given since the 19th
century: Confounding is a mixing of the
effect of interest with other effects on the
outcome (Mill, 1843).
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Biasing paths I: Confounding paths
and confounders
• Confounding path: Any path capable of
transmitting confounding
• Confounder: Any variable within a
confounding path
• Without conditioning, all biasing paths in a
DAG are confounding paths,
HOWEVER,
• Upon conditioning other kinds of bias arise
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Confounding paths from E to D:
EACD, ECBD, ECD
A B
C
F
E D
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Confounding paths from E to D
after conditioning on C: EACBD
A B
[C]
F
E D
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Confounding paths from E to D:
EACD, ECBD, ECD, EACBD
A B
C
[F]
E D
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Confounding paths from E to D:
ECD
[A] [B]
C
[F]
E D
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Confounding paths from E to D:
None!
A [B]
[C]
F
E D
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Biasing path from A to B: ACB,
which is not a confounding path!
A B
[C]
F
E D
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Selection Bias
There are many definitions, none universallyaccepted. My definition:
• Noncausal association created bynonrandom selection.
This definition appears to correspond best tothe intuitive definitions given in epi textssince the mid-20th century.
• Confounding and selection bias overlap,but one is not always the other. (Usinggraphs, the distinction is not important.)
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Confounding that is not selection
bias: ECD
C
F
E D
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Selection bias that is not
confounding: Berksonian bias
E D
[S]
T
Uncontrollable biasing path: ESD
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Case-control matching is
Intentional selection bias
We must control the matching factor M
to block the bias induced by matching
M
E
[S] [D]
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M-bias that is both confounding &
selection bias (via EACBD)
A B
C
[S]
E D
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Collider bias: Selection bias and
confounding induced by conditioning
Many variations:
• Beksonian bias
• M-bias
• Confounding produced by control of
intermediates to estimate direct effects, or
by selection affected by intermediates
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E has no direct effect on D, but control of
C or F can make it appear so (via ECBD)
E B
[C]
F
D
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Instrumental variables:
ED = AED/AE or ED = FAED/FAE
A (B)
E
F
D
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Differential measurement error: Can’t tell
direction of bias without further info
A B
(E)
E* D
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Independent nondifferential error:
bias toward the null in typical cases
A C
(E)
E*
D
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Effect-Measure Modification
(Heterogeneity)
Sander Greenland
Epidemiology and Statistics
University of California
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The term “interaction” gets used for
several distinct phenomena:
• Biologic interaction (synergy, antagonism,
coaction): One factor changes the physical
mechanism of action of another.
• “Statistical interaction”: Change in a
measure (of effect or association) upon
change in a third factor.
In the 1970s, few researchers understood
the difference. Many still don’t today (e.g.,
in genetics)
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Solution: Invent new term for
“statistical interaction”…
Effect Modification (Miettinen, 1974)
Unfortunately, the term still suggests
biologic interaction, so Rothman and
Greenland (1998) call it
Effect-Measure Modification (EMM)
Greenland prefers heterogeneity (of effect
or association), already in use in statistics
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Consider the simple case
with no confounding:
C
E D
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Presence & direction of EMM depends
on measure! (Berkson,1958)
C=1 C=0
E=1 E=0 E=1 E=0
D=1 32 20 10 4
N 105 105 105 105
RD: 32-20 = 12 per 105 10-4 = 6 per 105
RR: 32/20 = 1.6 10/4 = 2.5
NOTE: NO CONFOUNDING PRESENT!
105
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Only the RD has a simple relation
to biologic interaction:• If RD changes across strata and there is
no bias, this implies biologic interaction
must be present (known in bioassay since
the 1920s)
• Unfortunately, nearly all epi studies
present RRs only, so confusion remains.
• EMM has no bearing on confounding:
Both require C to be a risk factor given E,
but one can be present with the other
absent!
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Most epi studies have little power
to detect EMM, hence:A literature distortion is created:
• Studies examine only the RR
• All of them fail to detect RR modification
• Hence reviewers conclude there is no RRmodification (that the RR is homogeneous)
BUT, if they had examined only RD instead,
• All of them would fail to detect RDmodification and reviewers would infer thatthe RD is homogeneous!