Transcript Using Regression Analysis to Assess Potential Effect Modifiers and

Using Regression Analysis to
Assess Potential Effect Modifiers
and Confounders
Y (blood pressure reduction)
X1 (drug : 0  placebo 1  active)
X 2 ( gender : 0  female 1  male)
Question:
• Does the effect of the drug on Mean BPR
depend on the gender of the patient?
• It turns out that we can build a model to
address this question.
E (Y )   0  1 X 1   2 X 2  3 X 1 X 2
Implications of this model
• By specializing the model to the females and then to the
males we can see:
If X 2  0 ( females); E (Y )   0  1 X 1
and so 1  (  0  1 )   0 is the effect the drug for females
If X 2  1 (males ); E (Y )  (  0   2 )  ( 1   3 ) X 1
and so 1   3 is the effect the drug for males
So then  3  ( 1   3 )  1
• Measures the difference between the 2 drug effects
• i.e. whether gender is an effect modifier
Sometimes a table can aid in understanding
the implications of a model
X 1  0 and X 2  0
E (Y )   0
X 1  0 and X 2  1
E (Y )   0   2
Difference
Middle  Top
2
X 1  1 and X 2  0
E (Y )   0  1
X 1  1 and X 2  1
E (Y )   0  1   2   3
Difference
Middle Top
 2  3
Effect For Females
Right  Left
1
Effect For Males
Right  Left
1   3
Effect Modificati on
Right  Left
Middle  Top
3
Assess effect modification first
• If gender is a modifier, its assessment as a
confounder is rarely relevant.
• If there is evidence that 3  0 then one
should present the gender specific estimates of
the drug effect (together with their SEs and
maybe CIs too)
• No further testing of the components of this
model is typically required.
• Since we know that 3  0 we then know that
the drug has an effect and that the effect
DEPENDS on the gender of the patient.
What if gender is not an effect
modifier?
Then E (Y )   0  1 X 1   2 X 2
• And we can then assess whether gender
is a confounder by comparing
E (Y )   0  
adj
1
X1  2 X 2
• With E (Y )   0   X 1
cr
1
In other words:
• By studying the context, using confidence
intervals and other epidemiological ideas
Is 
adj
1
different from 
cr
1
?
What if the potential modifier/confounder is
‘continuous’? Say: age
• Now look at:
E (Y )   0  1 X 1   2 X 2  3 X 1 X 2
• As 2 straight lines in age
E (Y )   0   2 (age)
(on placebo )
E (Y )   0  1  (  2   3 )( age) (on active)
• And so the drug effect is the difference:
1  3 (age)
6
4
2
0
3
4
5
6
7
ld
Fitted values
Fitted values
8
Age specific drug effect
• From the previous graph:
• Compare the vertical difference between the red
line and the blue line when ld = 3 with the
vertical difference when ld =7.
• For example, red minus blue (‘drug’ effect) is
about -3 (when ld=3) and a bit more than 2
(when ld=7)
• The next graph shows the ‘drug’ effect (de)
versus ld demonstrating that ld is a modifier.
3
4
5
6
ld
7
8
-4
-2
0
de
2
4
The next graph shows:
• A ‘Drug’ effect adjusted for ld
• Notice that the 2 lines are parallel and that
the ‘drug’ effect (red line minus blue line) is
the same for any value of ld
• This fixed difference is the ‘adjusted’ drug
effect
4
3
3.5
2.5
3
4
5
6
7
ld
Fitted values
Fitted values
8
‘Crude’ drug effect
• In the next graph, the 2 lines are horizontal
(to emphasize that the effect is NOT
adjusted for ld)
• In this illustration, the adjusted effect and
crude effect are nearly the same and
hence there is no evidence of confounding
• Remember, though, that if we had
demonstrated modification, we would not
even address the issue of confounding
4
3
3.5
2.5
3
4
5
6
7
ld
Fitted values
Fitted values
8