Ok here are my 2 cents.
Sometimes, the target variable is a rare event, like fraud. In this case, using logistic regression will have significant sample bias due to insufficient event data. Oversampling is a common method due to its simplicity.
However, model calibration is required when scores are used for decisions (this is your case) – however nothing need to be done if the model is only for rank ordering (bear in mind the probabilities will be inflated but order still the same).
Parameter and odds ratio estimates of the covariates (and their confidence limits) are unaffected by this type of sampling (or oversampling), so no weighting is needed. However, the intercept estimate is affected by the sampling, so any computation that is based on the full set of parameter estimates is incorrect.
Suppose the true model is: ln(y/(1-y))=b0+b1*x. When using oversampling, the b1′ is consistent with the true model, however, b0′ is not equal to bo.
There are generally two ways to do that:
- weighted logistic regression,
- simply adding offset.
I am going to explain the offset version only as per your question.
Let’s create some dummy data where the true relationship between your DP (y) and your IV (iv) is ln(y/(1-y)) = -6+2iv
data dummy_data;
do j=1 to 1000;
iv=rannor(10000); *independent variable;
p=1/(1+exp(-(-6+2*iv))); * event probability;
y=ranbin(10000,1,p); * independent variable 1/0;
drop j;
output;
end;
run;
and let’s see your event rate:
proc freq data=dummy_data;
tables y;
run;
Cumulative Cumulative
y Frequency Percent Frequency Percent
------------------------------------------------------
0 979 97.90 979 97.90
1 21 2.10 1000 100.00
Similar to your problem the event rate is p=0.0210, in other words very rare
Let’s use poc logistic to estimate parameters
proc logistic data=dummy_data;
model y(event="1")=iv;
run;
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -5.4337 0.4874 124.3027 <.0001
iv 1 1.8356 0.2776 43.7116 <.0001
Logistic result is quite close to the real model however basic assumption will not hold as you already know.
Now let’s oversample the original dataset by selecting all event cases and non-event cases with p=0.2
data oversampling;
set dummy_data;
if y=1 then output;
if y=0 then do;
if ranuni(10000)<1/20 then output;
end;
run;
proc freq data=oversampling;
tables y;
run;
Cumulative Cumulative
y Frequency Percent Frequency Percent
------------------------------------------------------
0 54 72.00 54 72.00
1 21 28.00 75 100.00
Your event rate has jumped (magically) from 2.1% to 28%. Let’s run proc logistic again.
proc logistic data=oversampling;
model y(event="1")=iv;
run;
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -2.9836 0.6982 18.2622 <.0001
iv 1 2.0068 0.5139 15.2519 <.0001
As you can see the iv estimate still close to the real value but your intercept has changed from -5.43 to -2.98 which is very different from our true value of -6.
Here is where the offset plays its part. The offset is the log of the ratio between known population and sample event probabilities and adjust the intercept based on the true distribution of events rather than the sample distribution (the oversampling dataset).
Offset = log(0.28)/(1-0.28)*(0.0210)/(1-0.0210) = 2.897548
So your intercept adjustment will be intercept = -2.9836-2.897548= -5.88115 which is quite close to the real value.
Or using the offset option in proc logistic:
data oversampling_with_offset;
set oversampling;
off= log((0.28/(1-0.28))*((1-0.0210)/0.0210)) ;
run;
proc logistic data=oversampling_with_offset;
model y(event="1")=iv / offset=off;
run;
Standard Wald
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -5.8811 0.6982 70.9582 <.0001
iv 1 2.0068 0.5138 15.2518 <.0001
off 1 1.0000 0 . .
From here all your estimates are correctly adjusted and analysis & interpretation should be carried out as normal.
Hope its help.
