/* This program uses the GENMOD procedure
in SAS to fit loglinear models to the
Wisconsin driver data.
*/
DATA SET1;
INFILE 'drivall.dat';
/* This program code is stored in the file
INPUT AGE
driverw.sas
SEX
D
V
R
Y;
IF(Y EQ 0) THEN Y=Y+1.0E-20;
The data are stored in the file
drivall.dat
LABEL
*/
AGE = AGE GROUP
D = DRIVER GROUP
V = VIOLATION STATUS
R = RESIDENTIAL AREA
/* Enter the data. To prevent PROC GENMOD
Y = COUNT;
from deleting combinations of
run;
categories for which the observed count
is zero, add a small amount
1.0e-20
to each zero count */
871
872
proc format; value sex 1 = 'Male'
2 = 'Female';
value age 1 = '16-36'
2 = '36-55'
3 = 'over 55';
value d
1 = 'Disease'
2 = 'Control';
value v
1 = 'Some'
2 = 'None';
value r
1 = '> 150000'
2 = '39-150000'
3 = '10-39000'
/*
Conditional analysis of the
association between disease status
and traffic violations */
proc freq data=set1;
tables age*sex*d*v
/ chisq nopercent
nocol norow cmh relrisk
out=set2;
weight Y;
run;
4 = '< 10000'
5 = 'rural';
run;
873
proc print data=set2;
run;
874
/* Fit the model for conditional
/* Delete the v*sex*age interaction
independence of D and V given any
of
D and V given AGE and SEX
*/
combination of AGE and SEX
categories */
proc genmod data=set2 order=internal;
class age sex d v;
proc genmod data=set2 order=internal;
model count = d|sex|age v|sex v|age /
class age sex d v;
dist = poisson link=log
model count = d|sex|age v|sex|age /
maxit=50 covb itprint obstats ;
dist = poisson link=log
make 'parmest' out=est2;
maxit=50 covb itprint obstats ;
make 'parmest' out=est1;
make 'obstats' out=obstat2;
run;
make 'obstats' out=obstat1;
run;
proc print data=est2; run;
875
876
/*
Analyze the V*D*S marginal table */
proc freq data=set1;
tables sex*d*v
/ chisq nopercent
nocol norow cmh relrisk out=set3;
/* Analyze the V*D marginal table */
weight Y;
run;
proc freq data=set1;
tables d*v
/* Fit a no three factor interaction
/ chisq nopercent
nocol norow cmh relrisk;
model
*/
weight Y;
proc genmod data=set3 order=internal;
run;
class sex d v;
model count = d|sex v|sex
v|d /
dist = poisson link=log
maxit=50 covb itprint obstats ;
run;
877
878
/*
Analyze the V*D*A marginal table */
The FREQ Procedure
proc freq data=set1;
tables age*d*v
Summary Statistics for D by V
/ chisq nopercent
Controlling for AGE and SEX
nocol norow cmh relrisk out=set4;
Cochran-Mantel-Haenszel Statistics
weight Y;
Alternative Hypothesis
run;
/* Fit a no 3-factor interaction model */
proc genmod data=set4 order=internal;
Value
Prob
1
Nonzero Correlation
1
2.1067
0.1467
2
Row Mean Scores Differ
1
2.1067
0.1467
3
General Association
1
2.1067
0.1467
Estimates of the Common Relative Risk (Row1/Row2)
class age d v;
model count = d|age v|age
DF
Type of Study
v|d /
dist = poisson link=log
maxit=50 covb itprint obstats ;
Method
Value
95% Conf. Limits
Case-Control
M-H
1.2877
0.9166
1.8092
(Odds Ratio)
Logit
1.1902
0.8364
1.6937
run;
880
879
The GENMOD Procedure
Model Information
Distribution
Poisson
Link Function
Homogeneity of the Odds Ratios
Chi-Square
Observations Used
COUNT
Frequency
24
8.7473
DF
Pr > ChiSq
Log
Dependent Variable
Breslow-Day Test for
Class Level Information
5
0.1196
Class
881
Levels
Values
AGE
3
1 2 3
SEX
2
1 2
D
2
1 2
V
2
1 2
882
Analysis Of Parameter Estimates
Standard
AGE*SEX
1
1
1
-0.5189
0.1387
AGE*SEX
Confidence Limits
AGE*SEX
1
2
0
0.0000
0.0000
0.000
2
1
1
-0.4027
0.1482
-0.693
AGE*SEX
4.9950
AGE*SEX
0.4520
2
2
0
0.0000
0.0000
0.000
0.000
Wald 95%
Parameter
DF
Estimate
Error
Intercept
1
4.8208
0.0888
4.6467
D
1
1
0.2184
0.1192
-0.0152
D
2
0
0.0000
0.0000
0.0000
SEX
1
1
0.3373
0.1151
0.1116
SEX
2
0
0.0000
0.0000
0.0000
SEX*D
1
1
1
1.1244
0.1446
0.8410
-0.790
3
1
0
0.0000
0.0000
AGE*SEX
0.0000
AGE*SEX*D
0.5629
3
2
0
0.0000
0.0000
0.000
1
1
1
1
-0.5757
0.3239
-1.210
AGE*SEX*D
0.0000
AGE*SEX*D
1.4078
1
1
2
0
0.0000
0.0000
0.000
1
2
1
0
0.0000
0.0000
0.000
SEX*D
1
2
0
0.0000
0.0000
0.0000
AGE*SEX*D
0.0000
1
2
2
0
0.0000
0.0000
0.000
SEX*D
2
1
0
0.0000
0.0000
0.0000
2
1
1
1
0.2871
0.2185
-0.141
2
0.0000
AGE*SEX*D
0.0000
AGE*SEX*D
2
1
2
0
0.0000
0.0000
0.000
1.1915
AGE*SEX*D
0.7964
AGE*SEX*D
2
2
1
0
0.0000
0.0000
0.000
2
2
2
0
0.0000
0.0000
0.000
0.0000
AGE*SEX*D
-2.6155
AGE*SEX*D
3
1
1
0
0.0000
0.0000
0.000
3
1
2
0
0.0000
0.0000
0.000
3
2
1
0
0.0000
0.0000
0.000
3
2
2
0
0.0000
0.0000
0.000
1
1
-3.7452
0.1923
-4.122
2
0
0.0000
0.0000
0.000
SEX*D
2
0
0.0000
0.0000
0.0000
AGE
1
1
0.9884
0.1036
0.7853
AGE
2
1
0.5795
0.1107
0.3625
AGE
3
0
0.0000
0.0000
0.0000
AGE*D
1
1
1
-3.1226
0.2587
-3.6297
AGE*D
1
2
0
0.0000
0.0000
0.0000
AGE*D
2
1
1
-1.4370
0.1824
-1.7944
AGE*D
2
2
0
0.0000
0.0000
0.0000
AGE*D
3
1
0
0.0000
0.0000
0.0000
AGE*D
3
2
0
0.0000
0.0000
0.0000
0.0000
AGE*SEX*D
-1.0796
AGE*SEX*D
0.0000
V
0.0000
V
0.0000
883
884
SEX*V
1
1
1
1.2380
0.1658
0.9131
1.5629
SEX*V
1
2
0
0.0000
0.0000
0.0000
Obs
0.0000
SEX*V
2
1
0
0.0000
0.0000
0.0000
0.0000
SEX*V
2
2
0
0.0000
0.0000
0.0000
1
0.0000
8
1
1
1
8.63105
26.36895
Count
SEX
GROUP
STATUS
Pred
AGE*V
1
1
1
1.3904
0.1578
1.0812
2
1.6996
27
1
1
2
AGE*V
1
2
0
0.0000
0.0000
0.0000
3
0.0000
94
1
2
1
90.99597
AGE*V
2
1
1
0.4882
0.1732
0.1486
4
0.8277
275
1
2
2
278.00404
AGE*V
2
2
0
0.0000
0.0000
0.0000
5
0.0000
1
2
1
1
1.73367
AGE*V
3
1
0
0.0000
0.0000
0.0000
6
0.0000
19
2
1
2
18.26634
AGE*V
3
2
Scale
0
0.0000
0.0000
0.0000
7
0.0000
30
2
2
1
31.63948
0
1.0000
0.0000
1.0000
8
1.0000
335
2
2
2
333.36068
885
886
Total Sample Size = 2800
Model Information
Data Set
WORK.SET3
Distribution
Criteria For Assessing Goodness Of Fit
Poisson
Link Function
Log
Dependent Variable
COUNT
Observations Used
Frequency
8
Class Level Information
Class
Levels
Values
SEX
2
1 2
D
2
1 2
V
2
1 2
Criterion
DF
Value
Value/DF
Deviance
Scaled Deviance
1
0.2899
0.2899
1
0.2899
0.2899
Pearson Chi-Square
1
0.2995
0.2995
Scaled Pearson X2
1
0.2995
0.2995
Log Likelihood
14935.9024
888
887
Standard
Parameter
DF
Intercept
Estimate
Error
Wald 95%
Confidence Limits
1
6.5216
0.0382
6.4467
6.5964
D
1
1
-1.0521
0.0741
-1.1973
-0.9069
D
2
0
0.0000
0.0000
0.0000
0.0000
SEX
1
1
-0.0198
0.0540
-0.1257
0.0861
SEX
2
0
0.0000
0.0000
0.0000
0.0000
SEX*D
1
1
1
1.3926
0.0885
1.2190
1.5661
Obs
SEX*D
1
2
0
0.0000
0.0000
0.0000
0.0000
1
SEX*D
2
1
0
0.0000
0.0000
0.0000
SEX*D
2
2
0
0.0000
0.0000
0.0000
V
1
1
-2.7757
0.1459
V
2
0
0.0000
SEX*V
1
1
1
SEX*V
1
2
SEX*V
2
1
SEX*V
2
2
D*V
1
1
D*V
1
D*V
D*V
Scale
Observation Statistics
DRIVER
VIOLATION
SEX
GROUP
STATUS
102
1
1
1
103.34706
0.0000
2
938
1
1
2
936.65296
0.0000
3
127
1
2
1
125.65297
-3.0616
-2.4898
4
665
1
2
2
666.34704
0.0000
0.0000
0.0000
5
10
2
1
1
8.65302
1.1074
0.1653
0.7835
1.4313
6
236
2
1
2
237.34705
0
0.0000
0.0000
0.0000
0.0000
7
41
2
2
1
42.34713
0
0.0000
0.0000
0.0000
0.0000
8
681
2
2
2
679.65296
0
0.0000
0.0000
0.0000
0.0000
1
-0.5359
0.1332
-0.7969
-0.2750
2
0
0.0000
0.0000
0.0000
0.0000
2
1
0
0.0000
0.0000
0.0000
0.0000
2
2
0
0.0000
0.0000
0.0000
0.0000
0
1.0000
0.0000
1.0000
1.0000
889
Count
Pred
890
Total Sample Size = 2800
Parameter Estimates
The GENMOD Procedure
Standard
Class Level Information
Class
Levels
Values
AGE
3
1 2 3
D
2
1 2
V
2
1 2
Criteria For Assessing Goodness Of Fit
Criterion
DF
Value
Value/DF
Deviance
2
3.1816
1.5908
Scaled Deviance
2
3.1816
1.5908
Pearson Chi-Square
2
3.0433
1.5217
Scaled Pearson X2
2
3.0433
1.5217
Log Likelihood
Wald 9
Parameter
DF
Estimate
Error
Confiden
Intercept
1
5.7062
0.0569
5.5946
D
1
1
0.9998
0.0663
0.8699
D
2
0
0.0000
0.0000
0.0000
AGE
1
1
0.7136
0.0694
0.5775
AGE
2
1
0.3608
0.0737
0.2163
AGE
3
0
0.0000
0.0000
0.0000
AGE*D
1
1
1
-3.6794
0.1574
-3.9880
AGE*D
1
2
0
0.0000
0.0000
0.0000
AGE*D
2
1
1
-1.3154
0.0968
-1.5051
AGE*D
2
2
0
0.0000
0.0000
0.0000
AGE*D
3
1
0
0.0000
0.0000
0.0000
AGE*D
3
2
0
0.0000
0.0000
0.0000
V
1
1
-3.0477
0.1793
-3.3991
V
2
0
0.0000
0.0000
0.0000
14075.4288
892
891
Observation Statistics
Frequency
AGE
DRIVER
VIOLATION
GROUP
GROUP
STATUS
9
1
1
1
46
1
1
2
124
1
2
1
610
1
2
2
5
43
2
1
1
0.0000
6
310
2
1
2
0.0000
7
29
2
2
1
0.0000
0.0000
8
436
2
2
2
1.0000
1.0000
9
60
3
1
1
10
818
3
1
2
11
15
3
2
1
12
300
3
2
2
AGE*V
1
1
1
1.4162
0.1935
1.0369
1.7956
Observation
AGE*V
1
2
0
0.0000
0.0000
0.0000
0.0000
AGE*V
2
1
1
0.4958
0.1782
0.1466
0.8450
1
AGE*V
2
2
0
0.0000
0.0000
0.0000
0.0000
2
AGE*V
3
1
0
0.0000
0.0000
0.0000
0.0000
3
AGE*V
3
2
0
0.0000
0.0000
0.0000
0.0000
4
D*V
1
1
1
0.4481
0.1660
0.1227
0.7735
D*V
1
2
0
0.0000
0.0000
0.0000
D*V
2
1
0
0.0000
0.0000
0.0000
D*V
2
2
0
0.0000
0.0000
0
1.0000
0.0000
Scale
893
Count
894
# This is an illustration of using
# the glm function in
Splus
to fit
# log-linear models to the Wisconsin
# Drivers data.
#
This file is stored as
#
Change the age (a), sex (s),
#
disease status (d), traffic violation (v)
#
into factors
cvdg$a <- as.factor(cvdg$a)
driverw.ssc
cvdg$s <- as.factor(cvdg$s)
cvdg$v <- as.factor(cvdg$v)
# The data are in the file drivall.txt
cvdg$d <- as.factor(cvdg$d)
# Enter the data into a data frame and
#
cvd <- read.table("drivall.dat",
col.names=c("a","s","d","v","r","y"))
#
Compute total counts summing
#
across the levels of r
Print the table
cvdg
# Use the
glm
function to fit the
# log-linear model for conditional
# independence of v and d given any
# combination of levels for a and s.
cvdg<-aggregate(cvd, list(a=cvd$a,
s=cvd$s, v=cvd$v, d=cvd$d), FUN=sum)
options(contrasts=c("contr.treatment",
"contr.poly"))
895
896
cvd2.step <- step(cvd2, scope= ~ . ^4,
scale=1, trace=T)
cvd2 <- glm(y ~ v + d + s + a + v*s +
v*a + v*s*a + s*a + d*s + d*a + d*s*a,
family=poisson, data=cvdg)
#
# After reviewing the results of the previous
# steps obtain results for the final model
Print the results
goftests(cvdg$y, cvd2$fit, cvdg$y, cvd2$df)
summary(cvd2, correlation=F)
step
v*a + s*a + d*s + d*a + d*s*a,
family=poisson, data=cvdg, x=T)
#
Use the
function to search for
#
a better
#
a lower bound as the null model and an
#
upper bound as the four factor interaction
#
model.
#
steps in the search.
model.
cvd3 <- glm(y ~ v + d + s + a + v*s +
The scope function sets
trace=T shows results for various
It uses the AIC criterion
#
to determine which terms should be added to
#
or deleted from the model.
#
attention to hierarchical models.
It restricts
summary(cvd3, correlation=F)
cvd3$fit
cvd3$parms
goftests(cvdg$y, cvd3$fit, cvdg$y, cvd3$df)
anova(cvd3, test="Chisq")
cvd3$cov <- solve(t(cvd3$x)%*%
diag(cvd3$weight)%*%cvd3$x)
cvd3$cov
897
898
# This is an illustration of using
# the glm function in
Splus
to fit
# log-linear models to the Wisconsin
# Drivers data.
#
#
Change the age (a), sex (s),
#
disease status (d), traffic violation (v)
#
into factors
cvdg$a <- as.factor(cvdg$a)
This file is stored as
cvdg$s <- as.factor(cvdg$s)
driverw.ssc
cvdg$v <- as.factor(cvdg$v)
cvdg$d <- as.factor(cvdg$d)
# The data are in the file drivall.txt
# Enter the data into a data frame and
# Print the table
cvd <- read.table("drivall.dat",
cvdg
col.names=c("a","s","d","v","r","y"))
#
Compute total counts summing
#
across the levels of r
a s v d a1 s2 d3 v4
1 1 1 1 1
y
5
5
5 15
8
2 2 1 1 1 10
5
5
5 15
40
3 3 1 1 1 15
54
5
5
5 15
5 10
5
5 15
1
5 2 2 1 1 10 10
5
5 15
3
6 3 2 1 1 15 10
5
5 15
6
cvdg<-aggregate(cvd, list(a=cvd$a,
s=cvd$s, v=cvd$v, d=cvd$d), FUN=sum)
r
5
4 1 2 1 1
899
900
# Use the
glm
function to fit the
# log-linear model for conditional
7 1 1 2 1
5
5
5 10 15
8 2 1 2 1 10
5
5 10 15 245
9 3 1 2 1 15
5
5 10 15 666
10 1 2 2 1
5 10
5 10 15
19
11 2 2 2 1 10 10
5 10 15
65
12 3 2 2 1 15 10
5 10 15 152
13 1 1 1 2
5
5 10
5 15
94
14 2 1 1 2 10
5 10
5 15
18
15 3 1 1 2 15
5 10
5 15
15
5 10 10
5 15
30
17 2 2 1 2 10 10 10
5 15
11
18 3 2 1 2 15 10 10
5 15
0
16 1 2 1 2
19 1 1 2 2
# independence of v and d given any
27
# combination of levels for a and s.
options(contrasts=c("contr.treatment",
"contr.poly"))
cvd2 <- glm(y ~ v + d + s + a + v*s +
v*a + v*s*a + s*a + d*s + d*a + d*s*a,
family=poisson, data=cvdg)
#
Print the results
goftests(cvdg$y, cvd2$fit, cvdg$y, cvd2$df)
5
5 10 10 15 275
20 2 1 2 2 10
5 10 10 15 217
21 3 1 2 2 15
5 10 10 15 173
Pearson test =
5 10 10 10 15 335
Degrees of freedom =
22 1 2 2 2
23 2 2 2 2 10 10 10 10 15 219
p-value =
24 3 2 2 2 15 10 10 10 15 127
Deviance test =
df =
p-value =
901
10.66
6
0.1
13.11
6
0.04
902
summary(cvd2, correlation=F)
Call: glm(formula = y ~ v + d + s + a + v*s + v*a
d:s
+ v*s*a + s*a + d*s + d*a + d*s*a,
family = poisson, data = cvdg)
Deviance Residuals:
0.548716497
0.2898641
1.89301313
da2 -2.548352249
0.1975995 -12.89655475
da3 -3.698257832
0.1949089 -18.97428563
Max
v:sa2 -0.415060477
0.3775628
-1.09931504
-2.31245 -0.136422 0.00550854 0.146847 1.399646
v:sa3 -0.008519217
Min
1Q
Median
3Q
Coefficients:
Value Std. Error
(Intercept)
2.178905997
0.1894732
0.4830368
-0.01763679
sa2d
0.862758772
0.3329249
2.59145172
sa3d
0.575684805
0.3239304
1.77718685
t value
11.49980958
v
1.085454204
0.1145215
9.47816751
d
2.355448583
0.1768656
13.31772832
s -1.702429853
0.3399423
-5.00799657
a2
1.280197382
0.2339384
5.47236983
a3
1.823207341
0.2251231
8.09871177
v:s
1.349855505
0.2195407
6.14854433
va2
0.989667676
0.1803359
5.48791123
va3
1.412649998
0.1697051
8.32414471
sa2 -0.595201978
0.4660114
-1.27722635
sa3 -1.097806340
0.5468260
-2.00759720
(Dispersion Parameter for Poisson family
taken to be 1 )
Null Deviance: 3839.919 on 23 df
Residual Deviance: 13.10699 on 6 df
Number of Fisher Scoring Iterations: 4
904
903
Start:
#
Use the
step
function to search for
#
a better
#
a lower bound as the null model and an
model.
The scope function sets
#
upper bound as the four factor interaction
#
model.
Single term deletions
Model:
y ~ v + d + s + a + v*s + v*a + v*s*a
+ s*a + d*s + d*a + d*s*a
trace=T shows results for various
#
steps in the search.
#
to determine which terms should be added to
It uses the AIC criterion
#
or deleted from the model.
#
attention to hierarchical models.
It restricts
cvd2.step <- step(cvd2, scope= ~ . ^4,
scale=1, trace=T)
AIC= 49.107
Df Sum of Sq
<none>
Cp
v:s:a
2
1.288553 11.92078 43.92078
d:s:a
2
6.864393 17.49662 49.49662
Single term additions
Df Sum of Sq
<none>
v:d
905
RSS
10.63223 46.63223
RSS
Cp
10.63223 46.63223
1
2.110434
8.52180 46.52180
906
Step:
# After reviewing the results of the previous
AIC= 46.3509
# steps obtain results for the final model
Single term deletions
cvd3 <- glm(y ~ v + d + s + a + v*s +
v*a + s*a + d*s + d*a + d*s*a,
Model:
family=poisson, data=cvdg, x=T)
y ~ v + d + s + a + v:s + v:a + s:a +
d:s + d:a + d:s:a
Df Sum of Sq
<none>
summary(cvd3, correlation=F)
RSS
Cp
11.27280
43.2728
97.1632
v:s
1
55.89040 67.16320
v:a
2
83.03208 94.30488 122.3049
d:s:a
2
6.86525 18.13805
goftests(cvdg$y, cvd3$fit, cvdg$y, cvd3$df)
anova(cvd3, test="Chisq")
Call: glm(formula = y ~ v + d + s + a + v*s +
46.1380
v*a + s*a + d*s + d*a + d*s*a,
family = poisson, data = cvdg, x = T)
Single term additions
Df Sum of Sq
<none>
RSS
Deviance Residuals:
Cp
Min
11.27280 43.27280
v:d
1
1.729600
9.54319 43.54319
v:s:a
2
1.295216
9.97758 45.97758
1Q
Median
3Q
-2.421475 -0.1967607 -0.01235606 0.2698361
Max
1.125939
907
Coefficients:
(Intercept)
v
d
s
a2
a3
v:s
va2
va3
sa2
sa3
d:s
da2
da3
sa2d
sa3d
908
cvd3$fit
Value Std. Error
t value
2.1553668 0.1877495 11.480016
1.1168202 0.1084668 10.296428
2.3554486 0.1768634 13.317895
-1.6051295 0.3153001 -5.090800
1.3533819 0.2228837
6.072144
1.8383183 0.2192511
8.384535
1.2380025 0.1655972
7.475988
0.9022461 0.1593530
5.661932
1.3904023 0.1576925
8.817175
-0.9789585 0.3119324 -3.138368
-1.0945409 0.2947236 -3.713788
0.5487165 0.2898466
1.893127
-2.5483522 0.1975963 -12.896762
-3.6982578 0.1949062 -18.974554
0.8627588 0.3329042
2.591612
0.5756849 0.3239111
1.777293
(Dispersion Parameter for Poisson family
taken to be 1 )
Null Deviance: 3839.919 on 23 degrees of freedom
Residual Deviance: 14.35092 on 8 degrees of freedom
1
2
6
7
909
4
5
8
9
10
3.647401 26.36894 251.5936 665.7456 18.26634
11
12
13
14
15
65.479 154.3526 90.99599 27.54566 14.16644
16
17
18
19
20
31.63937 8.526955 2.931771 278.004 207.4543
21
22
23
24
173.8336 333.3607 221.4731 124.0683
goftests(cvdg$y, cvd3$fit, cvdg$y, cvd3$df)
Pearson test =
Degrees of freedom =
p-value =
Deviance test =
df =
p-value =
Number of Fisher Scoring Iterations: 3
3
8.631056 33.40644 54.25445 1.733664 2.521013
11.3
8
0.19
14.35
8
0.07
910
anova(cvd3, test="Chisq")
Analysis of Deviance Table
Poisson model
Response: y
Terms added sequentially
Df Deviance Resid.
NULL
v 1 2061.160
d 1
18.586
s 1 271.006
a 2 104.791
v:s 1
40.487
v:a 2
57.061
s:a 2 158.850
d:s 1 259.640
d:a 2 847.342
d:s:a 2
6.645
(first to last)
Df Resid. Dev
Pr(Chi)
23
3839.919
22
1778.759 0.00000000
21
1760.173 0.00001624
20
1489.166 0.00000000
18
1384.375 0.00000000
17
1343.889 0.00000000
15
1286.827 0.00000000
13
1127.978 0.00000000
12
868.338 0.00000000
10
20.996 0.00000000
8
14.351 0.03606229
cvd3$cov <- solve(t(cvd3$x)%*%
diag(cvd3$weight)%*%cvd3$x)
911