1 General Purpose
Poisson regression cannot only be used
for counted rates but also for binary outcome variables. Poisson
regression of binary outcome data is different from logistic
regression, because it uses a log instead of logit (log odds)
transformed dependent variable. It tends to provide better
statistics.
2 Schematic Overview of Type of Data File
3 Primary Scientific Question
Can Poisson regression be used to
estimate the presence of an illness. Presence means a rate of 1,
absence means a rate of 0. If each patient is measured within the
same period of time, no weighting variable has to be added to the
model. Rates of 0 or 1, do, after all, do exist in practice. We
will see how this approach performs as compared to the logistic
regression, traditionally, used for binary outcomes. The data file
is below.
4 Data Example
In 52 patients with parallel-groups of
two different treatments the presence or not of torsades de pointes
was measured. The first ten patients of the data file given below.
The entire data file is entitled chapter47poissonbinary, and is in
extras.springer.com. We will start by opening the data file in
SPSS.
|
Treat
|
Presence of torsade de pointes.
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
|
,00
|
1,00
|
5 Data Analysis, Binary Logistic Regression
First, we will perform a traditional
binary logistic regression with torsade de pointes as outcome and
treatment modality as predictor.
For analysis the statistical model
Binary Logistic Regression in the module Regression is
required.
Command:
-
Analyze….Regression….Binary Logistic….Dependent: torsade….Covariates: treatment….click OK.
Variables in the equation
|
B
|
S.E.
|
Wald
|
df
|
Sig.
|
Exp(B)
|
||
|---|---|---|---|---|---|---|---|
|
Step 1a
|
VAR00001
|
1,224
|
,626
|
3,819
|
1
|
,051
|
3,400
|
|
Constant
|
−,125
|
,354
|
,125
|
1
|
,724
|
,882
|
|
The above table shows that the
treatment is not statistically significant. A Poisson regression
will performed subsequently.
6 Data Analysis, Poisson Regression
For analysis the module Generalized
Linear Models is required. It consists of two submodules:
Generalized Linear Models and Generalized Estimation Models. The
first submodule covers many statistical models like gamma
regression (Chap. 30), Tweedie regression (Chap.
31), Poisson regression (Chaps.
21 and the current chapter), and the
analysis of data files with both paired continuous outcomes and
predictors (Chap. 3). The second is for analyzing
paired binary outcomes (Chap. 42).
Command:
-
Analyze....Generalized Linear Models....Generalized Linear Models ….mark Custom….Distribution: Poisson ….Link Function: Log….Response: Dependent Variable: torsade…. Predictors: Factors: treat....click Model....click Main Effect: enter “treat”…..click Estimation: mark Robust Tests….click OK.
Parameter estimates
|
95 % Wald confidence interval
|
Hypothesis test
|
||||||
|---|---|---|---|---|---|---|---|
|
Parameter
|
B
|
Std. error
|
Lower
|
Upper
|
Wald chi-square
|
df
|
Sig.
|
|
(Intercept)
|
−,288
|
,1291
|
−.541
|
−,035
|
4,966
|
1
|
,026
|
|
[VAR00001=,00]
|
−,470
|
,2282
|
−,917
|
−,023
|
4,241
|
1
|
,039
|
|
[VAR00001 = 1,00]
|
0a
|
||||||
|
(Scale)
|
1b
|
||||||
The above table shows the results of
the Poisson regression. The predictor treatment modality is
statistically significant at p = 0.039. According to the Poisson
model the treatment modality is a significant predictor of torsades
de pointes.
7 Graphical Analysis
We will check with a 3-dimensional
graph of the data if this result is in agreement with the data as
observed.
Command:
-
Graphs….Legacy Dialog….3-D Bar: X-Axis mark: Groups of Cases, Z-Axis mark: Groups of Cases…Define 3-D Bar: X Category Axis: treatment, Z Category Axis: torsade….OK.
The above graph shows that in the
0-treatment (placebo) group the number of patients with torsades de
pointe is virtually equal to that of the patients without. However,
in the 1-treatment group the number is considerably smaller. The
treatment seems to be efficacious.
8 Conclusion
Poisson regression is different from
linear en logistic regression, because it uses a log transformed
dependent variable. For the analysis of yes/no rates Poisson
regression is very sensitive and probably better than standard
regression methods. The methodology is explained.
9 Note
More background, theoretical and
mathematical information about Poisson regression is given in
Statistics applied to clinical studies 5th edition, Chap. 23,
Springer Heidelberg Germany, 2012, from the same authors.