Penalized Methods as Universal Tools for Data Analysis
Lab 2. Data augmentation – Offset method on summarized data
Consider the association between chocolate consumption and risk of stroke in a
prospective cohort of middle-aged and elderly men (Larsson et al. Neurology, 2012).
The multivariable relative risk of stroke comparing the highest quartile of chocolate
consumption (median 62.9 g/week) with the lowest quartile (median 0 g/week) was
0.83 (95% CI 0.70–0.99). Given that the means must be higher than the medians, the
RRs at issue are for roughly 70g (2.5 oz) per week.
The two-way table corresponding to the multivariable adjusted RR and 95% CI is the
following.
. iri 236 279 46068 45163
|
Exposed
Unexposed |
Total
-----------------+------------------------+-----------Cases |
236
279 |
515
Person-time |
46068
45163 |
91231
-----------------+------------------------+-----------|
|
Incidence rate | .0051229
.0061776 |
.005645
a) Poisson regression on summarized data
Create a dataset from the two-way table above and fit a Poisson regression model
to estimate the exposure-disease relative risk.
clear
input x case py
1 236 46068
0 279 45163
end
. glm case x , lnoffset(py) fam(poisson)
-----------------------------------------------------------------------------|
OIM
case |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x | -.1872204
.0884393
-2.12
0.034
-.3605583
-.0138824
_cons | -5.086822
.0598684
-84.97
0.000
-5.204162
-4.969482
ln(py) |
1 (exposure)
------------------------------------------------------------------------------
. lincom x, eform
-----------------------------------------------------------------------------case |
exp(b)
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------(1) |
.829261
.0733393
-2.12
0.034
.6972869
.9862135
------------------------------------------------------------------------------
The glm command has the eform option to report exponentiated coefficients
Greenland S., Orsini N., IMM, KI, Sept 16-17, 2013
1
b) Poisson regression on augmented data with a null centre prior
Enter the data corresponding to a prior with 0.95 probability on RR between
0.8 and 1.25.
scalar prior_logrr = 0
scalar prior_v =((log(1.25)-log(.8))/(2*invnormal(.975)))^2
gen constant = 1
gen H = log(py)
set obs `=_N+1'
scalar S = 30
scalar A = scalar(S)^2/prior_v
scalar H = log(scalar(A)) - (prior_logrr/scalar(S))
replace case = scalar(A) in l
replace x = 1/scalar(S) in l
replace H = scalar(H) in l
replace constant = 0 in l
. clist
1.
2.
3.
x
1
0
.0333333
case
236
279
69433.65
py
46068
45163
.
constant
1
1
0
H
10.73787
10.71803
11.14813
Fit a Poisson regression model on augmented data. Is the posterior relative risk
similar to information-weighted averaging?
. glm case x constant
, fam(poisson) offset(H) nocons
-----------------------------------------------------------------------------|
OIM
case |
IRR
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
.8896238
.0620893
-1.68
0.094
.7758874
1.020033
constant |
.0059782
.0003298
-92.79
0.000
.0053655
.0066609
H |
1 (offset)
------------------------------------------------------------------------------
Results are not different compared to an inverse variance weighted approach.
. postrri , prior(.8 1.25) data(0.83 0.70 0.99) format(%9.0g)
Posterior median for RR = .8902788
95% posterior limits for RR ( .7763922,
Greenland S., Orsini N., IMM, KI, Sept 16-17, 2013
1.020871)
2
c) Poisson regression on augmented data with a non-null centre prior
The pooled relative risk of stroke for approximately 70 gr per week of
chocolate consumption was 0.707 (95% CI 0.565–0.885). Consider the results
of this meta-analysis to inform the prior and augment the observed data.
clear
input x case py
1 236 46068
0 279 45163
end
Fit a Poisson regression model on augmented data.
. ppoisson case x , exposure(py) prior(x .565 .885)
irr
Penalized poisson regression
No. of obs =
2
Prior _b[x]: Normal(-0.347, 0.013)
-----------------------------------------------------------------------------case |
IRR
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
.7811926
.0547403
-3.52
0.000
.680945
.8961984
_cons |
.0063462
.0003422
-93.84
0.000
.0057097
.0070536
------------------------------------------------------------------------------
Results are negligibly different compared to averaging of the ln(RR) weighted
by their inverse variance (information, precision), which is fixed-effects metaanalysis including the prior data as a study.
. postrri , prior(.565 .885) data(0.83 0.70 0.99)
Posterior median for RR = .7817668
95% posterior limits for RR ( .6815676,
Greenland S., Orsini N., IMM, KI, Sept 16-17, 2013
.8966966)
3