Pesquisa Regressão Logística
Example 39.3: Logistic Modeling with Categorical
Predictors
Consider a study of the analgesic
effects of treatments on elderly patients with neuralgia. Two test treatments
and a placebo are compared. The response variable is whether the patient
reported pain or not. Researchers recorded age and gender of the patients and
the duration of complaint before the treatment began. The data, consisting of
60 patients, are contained in the data set Neuralgia.
Data Neuralgia; input Treatment $ Sex $ Age Duration Pain $ @@; datalines; P F 68 1 No B M 74 16 No P F 67 30 No P M 66 26 Yes B F 67 28 No B F 77 16 No A F 71 12 No B F 72 50 No B F 76 9 Yes A M 71 17 Yes A F 63 27 No A F 69 18 Yes B F 66 12 No A M 62 42 No P F 64 1 Yes A F 64 17 No P M 74 4 No A F 72 25 No P M 70 1 Yes B M 66 19 No B M 59 29 No A F 64 30 No A M 70 28 No A M 69 1 NoB F 78 1 No P M 83 1 Yes B F 69 42 No
B M 75 30 Yes P M 77 29 Yes P F 79 20 Yes A M 70 12 No A F 69 12 No B F 65 14 No B M 70 1 No B M 67 23 No A M 76 25 Yes P M 78 12 Yes B M 77 1 Yes B F 69 24 No P M 66 4 Yes P F 65 29 No P M 60 26 Yes A M 78 15 Yes B M 75 21 Yes A F 67 11 No P F 72 27 No P F 70 13 Yes A M 75 6 Yes B F 65 7 No P F 68 27 Yes P M 68 11 Yes P M 67 17 Yes B M 70 22 No A M 65 15 No P F 67 1 Yes A M 67 10 No P F 72 11 Yes A F 74 1 No B M 80 21 Yes A F 69 3 No ;
The
data set Neuralgia contains
five variables: Treatment, Sex, Age, Duration, and Pain. The last
variable, Pain, is
the response variable. A specification of Pain=Yes indicates
there was pain, and Pain=No indicates no pain. The
variable Treatment is
a categorical variable with three levels: A and B represent the two test
treatments, and P represents the placebo treatment. The gender of the patients
is given by the categorical variable Sex. The
variable Age is
the age of the patients, in years, when treatment began. The duration of
complaint, in months, before the treatment began is given by the variable Duration. The following
statements use the LOGISTIC procedure to fit a two-way logit with interaction
model for the effect of Treatment and Sex, with Age and Duration as
covariates. The categorical variables Treatment and Sex are declared
in the CLASS statement.
proc logistic data=Neuralgia;
class Treatment Sex;model Pain= Treatment Sex Treatment*Sex Age Duration / expb;
run;
In
this analysis, PROC LOGISTIC models the probability of no pain (Pain=No). By default,
effect coding is used to represent the CLASS variables. Two dummy variables are
created for Treatment and one
for Sex, as
shown in Output 39.3.1.
Output
39.3.1: Effect Coding of CLASS Variables
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PROC LOGISTIC displays a table of the Type III
analysis of effects based on the Wald test (Output
39.3.2). Note that the Treatment*Sex interaction and the duration of complaint
are not statistically significant (p=0.9318 and p=0.8752, respectively). This indicates that there is no evidence that the
treatments affect pain differently in men and women, and no evidence that the
pain outcome is related to the duration of pain.
Output
39.3.2: Wald Tests of Individual Effects
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Parameter estimates are displayed in Output
39.3.3. The Exp(Est) column
contains the exponentiated parameter estimates. These values may, but do not
necessarily, represent odds ratios for the corresponding variables. For
continuous explanatory variables, the Exp(Est) value corresponds to the odds
ratio for a unit increase of the corresponding variable. For CLASS variables
using the effect coding, the Exp(Est) values have no direct interpretation as a
comparison of levels. However, when the reference coding is used, the Exp(Est)
values represent the odds ratio between the corresponding level and the last
level. Following the parameter estimates table, PROC LOGISTIC displays the odds
ratio estimates for those variables that are not involved in any interaction
terms. If the variable is a CLASS variable, the odds ratio estimate comparing
each level with the last level is computed regardless of the coding scheme. In
this analysis, since the model contains the Treatment*Sex interaction term, the odds ratios for Treatment and Sex were not computed. The odds ratio estimates
for Age and Duration are precisely the values given in the
Exp(Est) column in the parameter estimates table.
Output
39.3.3: Parameter Estimates with Effect Coding
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The following PROC LOGISTIC statements
illustrate the use of forward selection on the data set Neuralgia to identify the effects that
differentiate the two Pain responses.
The option SELECTION=FORWARD is
specified to carry out the forward selection. Although it is the default, the
option RULE=SINGLE is explicitly specified to select one effect in each step where
the selection must maintain model hierarchy. The term Treatment|Sex@2 illustrates another way to specify main effects
and two-way interaction as is available in other procedures such as PROC GLM. (Note
that, in this case, the "@2" is unnecessary because no interactions
besides the two-way interaction are possible).
proc logistic data=Neuralgia;class Treatment Sex;
model Pain=Treatment|Sex@2 Age Duration/selection=forward rule=single expb; run;
Results
of the forward selection process are summarized in Output 39.3.4. The
variable Treatment is selected
first, followed by Age and then Sex. The results are
consistent with the previous analysis (Output 39.3.2) in which
the Treatment*Sex interaction
and Duration are not
statistically significant.
Output 39.3.4: Effects Selected into the Model
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Output
39.3.5 shows
the Type III analysis of effects, the parameter estimates, and the odds ratio
estimates for the selected model. All three variables, Treatment, Age, and Sex, are statistically significant at the 0.05 level
(p=0.0011, p=0.0011, and p=0.0143, respectively). Since the selected model
does not contain the Treatment*Sex interaction, odds ratios for Treatment and Sex are computed. The estimated odds ratio is
24.022 for treatment A versus placebo, 41.528 for Treatment B versus placebo,
and 6.194 for female patients versus male patients. Note that these odds ratio
estimates are not the same as the corresponding values in the Exp(Est) column
in the parameter estimates table because effect coding was used. From Output
39.3.5, it is
evident that both Treatment A and Treatment B are better than the placebo in
reducing pain; females tend to have better improvement than males; and younger
patients are faring better than older patients.
Output 39.3.5: Type
III Effects and Parameter Estimates with Effect Coding
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Finally, PROC LOGISTIC is invoked to refit the
previously selected model using reference coding for the CLASS variables. Two
CONTRAST statments are specified. The one labeled 'Pairwise' specifies three
rows in the contrast matrix, L, for all the pairwise comparisons between the
three levels of Treatment. The contrast labeled 'Female vs Male'
compares female to male patients. The option ESTIMATE=EXP is specified in both
CONTRAST statements to exponentiate the estimates of
proc logistic data=Neuralgia; class Treatment Sex /param=ref;model Pain= Treatment Sex age;
contrast 'Pairwise' Treatment 1 0 -1, Treatment 0 1 -1, Treatment 1 -1 0 / estimate=exp; contrast 'Female vs Male' Sex 1 -1 / estimate=exp; run;
Output 39.3.6: Reference
Coding of CLASS Variables
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The reference coding is shown in Output 39.3.6. The Type III analysis of effects, the parameter
estimates for the reference coding, and the odds ratio estimates are displayed
in Output 39.3.7. Although the parameter estimates are different
(because of the different parameterizations), the "Type III Analysis of
Effects" table and the "Odds Ratio" table remain the same as
in Output 39.3.5. With effect coding, the treatment A parameter
estimate (0.8772) estimates the effect of treatment A compared to the average
effect of treatments A, B, and placebo. The treatment A estimate (3.1790) under
the reference coding estimates the difference in effect of treatment A and the
placebo treatment.
Output 39.3.7: Type III Effects and Parameter Estimates with Reference
Coding
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Output 39.3.8 contains two tables: the "Contrast Test
Results" table and the "Contrast Rows Estimation and Testing
Results" table. The former contains the overall Wald test for
each CONTRAST statement. Although three rows are specifed in the 'Pairwise'
CONTRAST statement, there are only two degrees of freedom, and the Wald test
result is identical to the Type III analysis of Treatment in Output 39.3.7. The
latter table contains estimates and tests of individual contrast rows. The
estimates for the first two rows of the 'Pairwise' CONTRAST statement are the
same as those given in the "Odds Ratio Estimates" table (in Output 39.3.7). Both
treatments A and B are highly effective over placebo in reducing pain. The
third row estimates the odds ratio comparing A to B. The 95% confidence
interval for this odds ratio is (0.0932, 3.5889), indicating that the pain
reduction effects of these two test treatments are not that different. Again,
the 'Female vs Male' contrast shows that female patients fared better in
obtaining relief from pain than male patients.
Output 39.3.8: Results of CONTRAST Statements
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Copyright © 1999 by
SAS Institute Inc., Cary, NC, USA. All rights reserved.
Data
Neuralgia;
input Treatment $ Sex $ Age Duration Pain
$ @@;
datalines;
P
F 68 1
No B M
74 16 No
P F 67
30 No
P
M 66 26
Yes B F
67 28 No
B F 77
16 No
A
F 71 12
No B F
72 50 No
B F 76
9 Yes
A
M 71 17
Yes A F
63 27 No
A F 69
18 Yes
B
F 66 12
No A M
62 42 No
P F 64
1 Yes
A
F 64 17
No P M
74 4 No
A F 72
25 No
P
M 70 1
Yes B M
66 19 No
B M 59
29 No
A
F 64 30
No A M
70 28 No
A M 69
1 No
B
F 78 1
No P M
83 1 Yes B
F 69 42 No
B
M 75 30
Yes P M
77 29 Yes P
F 79 20 Yes
A
M 70 12
No A F
69 12 No
B F 65
14 No
B
M 70 1
No B M
67 23 No
A M 76
25 Yes
P
M 78 12
Yes B M
77 1 Yes B
F 69 24 No
P
M 66 4
Yes P F
65 29 No
P M 60
26 Yes
A
M 78 15
Yes B M
75 21 Yes A
F 67 11 No
P
F 72 27
No P F 70 13 Yes
A M
75 6 Yes
B
F 65 7
No P F
68 27 Yes P
M 68 11 Yes
P
M 67 17
Yes B M
70 22 No
A M 65
15 No
P
F 67 1
Yes A M
67 10 No
P F 72
11 Yes
A
F 74 1
No B M
80 21 Yes A
F 69 3 No
;
proc logistic data=Neuralgia;
class Treatment Sex;
model Pain= Treatment Sex Treatment*Sex
Age Duration / expb;
run;
Interpretamos que Dor Yes = 0 (melhor interpretar como analgesia=0)
Dor No = 1 (Analgesia)
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