Data Set Assignment – Harold Otieno and Karin Tanaka, 05 December 2008, Math 157
Part 1
1. Dickson, E.R., P.M. Grambsch, et al. (1989). “Prognosis in Primary Biliary Cirrhosis:
Model for Decision Making.” Hepatology 10: 1-7.
Markus, B.H., E.R. Dickson, et al. (1989). “Efficacy of Liver Transplantation in
Patients with Primary Biliary Cirrhosis.” N Engl J Med 320: 1709-13.
2. The goal of the study is to create a mathematical model for predicting survival for
patients with primary biliary cirrhosis based on a small number of universally available
measurements to aid in the selection of patients for liver transplantation. A model using
five variables – age, serum bilirubin concentration, serum albumin concentration,
prothrombin time, and a score for the severity of edema – was developed. It performed
as well as the previously accepted model in cross-testing for validity using an
independent set of patients.
3. The data are based on a clinical trial as well as additional patients who declined
participation in the trial. Those patients in the clinical trial were randomized to (a)
D-penicillamine or (2) placebo.
4. Primary biliary cirrhosis patients already enrolled in either of two double-blind,
placebo-controlled, randomized clinical trials at the Mayo Clinic were eligible for the
study. They had previously met established criteria to be involved in these clinical trials
(see Dickson et al. 1985). Of 418 total eligible patients, 312 agreed to participate in the
study. The study was supported by a research grant from the National Institutes of
Health.
5. Due to the selection process described above, no additional exclusion criteria are
mentioned, and of the patients enrolled in the previous clinical trials, none were newly
excluded for this study. The original exclusion criteria included that no patients were
taking additional anti-inflammatory or immunosuppressive medication.
Patients lost to follow-up or who underwent liver transplantation were censored at time
of loss or transplant, and their data were used. However, six of the 112 patients who
declined to participate in the original trial were lost to follow-up within a short period
of time and were not included in the cross-validation.
6. Variables (clinical variables taken at baseline, that is, upon entry to the study)
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id; patient ID
fu.days; number of days between registration and death, transplantation or study
analysis time, whichever came first
status; status = 0 is still alive, status = 1 had a transplant, status = 2 died
(note: “alive” and “transplant” both indicate censored observations)
drug: 1= D-penicillamine, 2=placebo
partic: 1=participated in the randomized clinical trial, 0 = didn’t participate
age; number of days since birth
sex; sex = 0 is male, sex = 1 is female
edema; presence of edema, edema = 0 is no edema and no diuretic therapy for
edema, edema = 0.5 is edema present without diuretics or edema resolved by
diuretics, edema = 1 is edema present despite diuretic therapy
bili; serum bilirubin in mg/dl
chol: serum cholesterol in mg/dl
albumin; albumin in mg/dl
pro.time; prothrombin time in seconds
Note: Only the variables most relevant in understanding the structure of the study and
the final model are given here. A total of 45 prognostic factors were measured in the
study; most are omitted.
Part 2
Summary Statistics.
There were 374 female patients and 44 male patients. 354 patients had no Edema, 44 had
edema without diuretics and 20 had edema despite diuretic therapy. 161 patients died, 232
survived and 25 had a transplant.
Maximum
Minimum
Mean
Medium
SD
Follow-up days 4795
44
1917.782
1730
1104.673
bilirubin
conc.
28 mg/dl
0.3 mg/dl
3.220mg/dl
1.4 mg/dl
4.407506
4.64 mg/dl
1.96 mg/dl
3.49744 mg/dl
3.53 mg/dl
0.4249716
28650 days
9598 days
18628 days
3815.845
albumin conc.
age
18533.35 days
Descriptive Graphics
Box plots of the age of the patients.
Transplant
Alive
16000
14000
12000
20000
15000
10000
15000
20000
18000
25000
25000
20000
Died
From the plot, the median age of patients who survived appears slightly lower than the
median age of patients who died.
Box plots of bilirubin concentration.
Survived
Transplant
10
0
0
0
2
5
5
4
10
6
15
10
8
20
15
12
25
14
Died
From the plot, the median bilirium conc. of patients who survived seems lower than that of
patients who died. This could be a significant variable in prediction.
Box plots of albumin concentration.
Transplant
Survived
2.0
2.5
2.5
2.5
3.0
3.0
3.0
3.5
3.5
3.5
4.0
4.0
4.0
4.5
4.5
Died
From the plot, the median albumin concentration of patients who survived appears higher
than that of patients who died.
Inference:
We did a logistic regression on the data in order to test whether the age, edema, bilirubin
conc., albumin conc., and prothrombin time variables are significant in predicting whether
patients die or survive.
Call:
glm(formula = status ~ age + edema + bilirubin + albumin + prothtime, family =
"binomial")
Deviance Residuals:
Min
1Q Median
3Q
Max
-2.1205 -0.8761 0.5560 0.7649 2.7551
Coefficients:
Estimate
(Intercept)
age
edema
bilirubin
albumin
prothtime
--Signif. codes:
Std. Error
z value
Pr(>|z|)
2.878e+00
1.349e+00
2.134
0.03285 *
-1.431e-04
3.316e-05
-4.316
1.59e-05 ***
-1.793e+00
5.883e-01
-3.047
0.00231 **
-3.047e-01
5.031e-02
-6.057
1.39e-09 ***
3.309e-01
3.016e-01
1.097
0.27256
8.288e-03
7.514e-03
1.103
0.27001
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 554.34 on 415 degrees of freedom
Residual deviance: 426.59 on 410 degrees of freedom
(2 observations deleted due to missingness)
AIC: 438.59
Number of Fisher Scoring iterations: 5
From the associated p.values we reject the null hypothesis that age, presence of edema and
bilirubin concentration, are not significant predictors of survival.
On the other hand we fail to reject the null hypothesis that albumin concentration and
prothrombine time are not significant predictors of survival.
Odds Ratio & Confidence Intervals:
We further calculated the confidence intervals for the odds ratio associated with a 1 unit
increase in the age and bilirubin concentration and concluded that: We are 95% confident
that the odds ratio of survival associated with a one day increase in age is between
0.999792 and 0.9999219. Note than the interval is less than one, so we can claim that the
survival odds decrease. However note that the interval is also very close to one.
We are also 95% confident that the odds ratio associated with a one unit (mg/dl) increase in
bilirubin concentration is between 0.6681066 and 0.8137579. Again since this interval is
lower than one we can claim that the odds of survival go down.