Review the analysis from the standpoint of how many patients survive over the seven-year time period that the clinical trial covered.
Module 12
Chapter 11
Introduction to
Survival Analysis
Learning Objectives
• Identify applications with time-to-event
outcomes
• Construct a life table using the actuarial
approach
• Determine the assumptions of survival
analysis
• Interpret the Cox proportional hazards
regression
• Interpret a hazard ratio
Survival Analysis
• Outcome is time to event.
– Time to heart attack, cancer remission, death
• Measure whether person has event or
not.
– (Yes/No) and time to event
• Estimate “survival time.”
• Determine factors associated with longer
survival.
Issues with Time to Event Data
• Times are positive (often skewed).
• Incomplete follow-up information
– Some participants enroll late.
– Some participants drop out.
– Study ends.
• Censoring
– Measure follow-up time and not time to
event.
– We know survival time > follow-up time.
Experiences of n = 10 Participants
Experiences of Same n = 10
Participants, Time Projected to Zero
Is the Following Different?
Survival Curve – Survival Function
Survival Curve with 95% CI
1.0
0.9
0.8
Survival Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
5
10
15
Time, Years
20
25
Estimating the Survival Function
• There are many parametric approaches
(which make certain assumptions about
survival times).
• We focus on two nonparametric
approaches.
– Actuarial or life-table approach
– Kaplan–Meier approach
Example 11.2.
Estimating the Survival Function
(1 of 2)
• Participants are 65 years and older,
followed for up to 24 years until they
die, the study ends, or they drop out.
• n = 20 participants are enrolled over a
5-year period.
Example 11.2.
Estimating the Survival Function
(2 of 2)
• Year of death or year of last contact
– Years of death: 3, 14, 1, 23, 5, 17
– Years of last contact: 24, 11, 19, 24, 13, 2,
18, 17, 24, 21, 12, 10, 6, 9
Notation
Nt = number of participants who are eventfree and considered at risk during
interval
Dt = number who suffer event during interval
Ct = number censored during interval
qt = proportion suffering event during interval
pt = proportion surviving interval
St = proportion surviving past interval
Example 11.2.
Life Table
Example 11.2.
Life Table—Actuarial Approach
Example 11.2. Life Table—Kaplan–
Meier Approach
Example 11.2.
Survival Function
Comparing Survival Curves
• Log-rank test to compare survival in two
or more independent groups.
• Chi-square test that compares the
observed numbers of events to what
would be expected if the groups had
equal survival
Example 11.3.
Comparing Survival
• Clinical trial to compare two treatments for
advanced gastric cancer
• n = 20 participants with stage IV cancer
are randomly assigned to receive
chemotherapy before surgery or
chemotherapy after surgery.
• Primary outcome is death.
• Participants are followed for up to 48
months following enrollment.
RCT to Compare Two Treatments for
Advanced Gastric Cancer
Module 13
Chapter 11
The Survival
Curve
Learning Objectives
• Evaluate the data for survival analysis if it
is normally distributed
• Evaluate the statistical tests used in
survival analysis
• Perform and interpret the log-rank test
• Evaluate the concept of censored survival
data
Log-Rank Test
H0: Two survival curves are identical
H1: Two survival curves are not identical
Test statistic: χ =
2
(O jt − E jt )
2
E jt
Reject H0 if c2 > c2,df where df = k – 1
and k = number of comparison groups.
RCT to Compare Two Treatments for
Advanced Gastric Cancer
Example 11.3.
Log-Rank Test (1 of 2)
H0: Two survival curves are identical
H1: Two survival curves are not identical
Test statistic:
χ2 =
(O jt − E jt ) 2
E jt
(6 − 2.620) 2 (3 − 6.380) 2
=
+
= 6.151
2.620
6.380
Example 11.3.
Log-Rank Test (2 of 2)
• Reject H0 if c2 ≥ 3.84.
• Reject H0 because 6.151 > 3.84. We have
statistical evidence that two survival
curves are not identical.
Comparing Survival Curves
H0: Two survival curves are equal
c2 Test with df=1. Reject H0 if c2 > 3.84
c2 = 6.151. Reject H0.
Cox Proportional Hazards
Regression (1 of 2)
• Model
h(t) = h0(t) exp (b1X1 + b2X2 + … + bpXp)
• Where h(t) = hazard at time t (risk of
failure at time t),
h0(t) = baseline hazard,
Xi are predictors,
bi are regression coefficients.
Cox Proportional Hazards
Regression (2 of 2)
• Model
ln(h(t)/h0(t)) = b1X1 + b2X2 + … + bpXp
• exp(bi) = hazard ratios
Example 11.5.
Cox Proportional Hazards Regression
(1 of 3)
• Framingham Study
–
–
–
–
Outcome = all-cause mortality
N = 5180 participants ≥ 45 years
10-year follow-up
Analysis with Cox proportional hazards
regression
Example 11.5.
Cox Proportional Hazards Regression
(2 of 3)
Age
Male sex
bi
p
0.11149 0.0001
0.67958 0.0001
HR
1.118
1.973
Example 11.5.
Cox Proportional Hazards Regression
(3 of 3)
• Multivariable model
bi
p
HR (95% CI)
Age
0.11691
0.0001
1.12 (1.11 – 1.14)
Male sex
0.40359
0.0001
1.50 (1.22 – 1.85)
SBP
0.11691
0.0001
1.02 (1.01 – 1.02)
Current
smoker
0.40359
0.0001
2.16 (1.76 – 2.64)
Total chol.
0.40359
0.0001
1.00 (0.99 – 1.00)
Diabetes
0.40359
0.0001
0.82 (0.62 – 1.08)
Critical Thinking Assignment (125 points)
Using the Clinical Trial on breast cancer dataset. Perform a Kaplan-Meier Analysis or the Log-Rank Test
to determine the survival curve for the breast cancer survivors.
H0 The risk of 50% of the participants dying from breast cancer will occur within five years. (Null
Hypothesis)
H1 The risk of 50% of the participants dying from breast cancer does not occur within five years.
(Alternative Hypothesis)
Ensure to submit the following requirements for the assignment:
• Review the analysis from the standpoint of how many patients survive over the seven-year time
period that the clinical trial covered.
• Present your findings as a Survival Time chart in a Word document, with a title page, introduction
explaining why you would conduct a survival analysis, a discussion where you interpret the
meaning of the survival analysis, and a conclusion should be included.
• Your submission should be 4 pages to discuss and display your findings.
• Provide support for your statements with in-text citations from a minimum of 4 scholarly, peerreviewed articles. One of these sources may be from the class readings, textbook, or lectures, but
the others must be external. The Saudi Digital Library is a good place to find these sources and
should be your primary resource for conducting research.
• Follow APA 7th edition and Saudi Electronic University writing standards.
Clinical_Trial_breast_cancer_Student_Data_Set (1)
SUBJECTID age
Alive Survival_length
1118
39.2
1
0.410284168
1230
53.5
0
0.512328767
1056
35.4
0
0.575342466
1146
50.2
1
0.775342466
1057
44.2
0
0.860273973
1102
57.5
0
0.868493151
1172
44.6
1
0.893150685
1049
50.4
0
0.942465753
1168
44.6
0
0.945205479
1199
37.7
0
1.304109589
1206
43
0
1.317808219
1136
60
0
1.345205479
1174
49.1
0
1.353424658
1045
49.6
0
1.468493151
1034
46.6
0
1.520547945
1129
52.5
0
1.676712329
1054
42.8
0
1.684931507
1061
54.2
0
1.802739726
1050
63.2
0
1.816438356
1234
38.1
1
1.901369863
1156
65.4
0
1.928767123
1043
38.9
0
2.04109589
1169
44.2
0
2.071232877
1194
33.8
0
2.219178082
1236
38.6
0
2.279452055
1062
34.6
0
2.380821918
1101
42.8
0
2.58630137
1209
43.4
1
2.657534247
1196
35.4
1
2.739726027
1142
28.8
0
2.769863014
1235
64.1
1
2.810958904
1237
46.5
1
2.824657534
1224
61.1
1
2.871232877
1197
44.7
1
2.876712329
1233
27.9
1
2.947945205
1225
47.8
1
3.054794521
1229
50.8
1
3.060273973
1096
43
0
3.082191781
1198
59.5
1
3.084931507
1222
55.5
1
3.145205479
1002
37.8
0
3.16
1239
48.6
1
3.164383562
1211
48.1
1
3.175342466
1106
50.4
0
3.221917808
1154
65.5
1
3.260273973
1161
56.8
1170
1
3.260273973
44.4
1
3.260273973
1232
50
1
3.279452055
1166
54.4
1
3.353424658
1220
43.1
1
3.361643836
1191
45.9
1
3.391780822
1228
53.3
1
1210
40.6
1
3.44109589
1144
56.3
1
3.454794521
1001
38.7
0
3.46
1226
42.1
1
3.484931507
3.42739726
1065
55.2
1
3.487671233
1163
38.9
1
3.506849315
1221
52.5
1
3.506849315
1152
51.3
1
3.526027397
1180
31.5
1
3.542465753
1216
34.4
1
3.567123288
1223
59.8
1
3.569863014
1204
39.3
1
3.580821918
1189
68.8
1
3.6
1188
38.3
1
3.610958904
1185
49.8
1
3.621917808
1200
43.8
1
3.630136986
1138
35.1
1
3.64109589
1165
39.7
1
3.643835616
1214
47
1
3.657534247
1175
64.3
1
3.682191781
1109
52.5
1
3.695890411
1181
34.7
1
3.717808219
1207
36.3
1
3.717808219
1130
39.1
1
3.767123288
1218
47.8
1
3.778082192
1091
54.6
0
3.791780822
1098
62.3
0
3.821917808
1123
61.9
1
3.876712329
1107
58.4
1
3.890410959
1193
59.6
1
3.904109589
1116
34.6
1
3.909589041
1113
53.5
1
3.923287671
1202
53.5
1
3.947945205
1187
47.1
1
3.983561644
1179
49.6
1
3.989041096
1183
53.8
1
3.989041096
1114
44.8
1
4.016438356
1162
56.4
1
4.02739726
1122
59.7
1
4.057534247
1089
61
1
4.068493151
1121
41.5
1
4.104109589
1201
50.1
1
4.128767123
1112
46.1
1
4.136986301
1177
40.8
1
4.15890411
1157
41.4
1
4.167123288
1149
59.2
0
4.202739726
1151
50.2
1
4.230136986
1083
51.9
1
4.273972603
1171
51
1
4.284931507
1124
47.6
1
4.293150685
1147
59.3
1
4.295890411
1158
61.8
1
4.331506849
1070
36.8
1
1135
58.9
1
4.37260274
1134
43.1
1
4.380821918
1125
44.9
1
4.424657534
1155
42.7
1
4.424657534
1150
51.5
1
4.430136986
1060
67.2
1
4.443835616
1164
60.1
1
4.449315068
1086
41.5
1
4.501369863
4.37260274
1090
51.5
1
4.509589041
1148
50.8
1
4.509589041
1110
56.9
1
4.526027397
1058
54.7
1
4.534246575
1141
49.2
1
4.553424658
1066
57.4
1
4.575342466
1041
46.6
0
1139
46.3
1
1128
33.2
1
4.61369863
1137
29.4
1
4.679452055
1132
51.4
1
4.709589041
1071
38.5
1
4.712328767
1073
49.2
1
4.726027397
1055
52.8
1
4.761643836
4.6
4.6
1084
34.5
1
4.838356164
1099
50.4
1
4.846575342
1092
44.1
1
4.887671233
1097
44.5
1
4.890410959
1047
26.7
1
4.928767123
1095
43.6
1
4.936986301
1115
51.8
1
4.936986301
1037
51.1
1
4.947945205
1111
50.6
1
4.953424658
1082
46.6
1
4.964383562
1117
49
1
5.024657534
1038
37.8
1
5.049315068
1063
52.2
1
5.101369863
1103
50.2
1
5.167123288
1075
33.5
1
5.180821918
1085
43.9
1
5.191780822
1100
63.3
1
5.221917808
1048
52.3
1
5.224657534
1022
51.4
1
5.271232877
1077
41.7
1
5.276712329
1042
58.1
1
5.282191781
1078
31.3
1
5.312328767
1033
59.1
1
5.320547945
1069
57.2
1
5.334246575
1088
53.1
1
5.389041096
1074
36.5
1
5.42739726
1029
63.6
1
5.542465753
1039
31
1
5.561643836
1015
42.2
1
5.58630137
1021
48.4
1
1010
41.5
1
5.89
1026
44.4
1
5.912328767
1019
52.2
1
6.02739726
1030
52.5
1
6.052054795
5.64109589
1008
64.5
1
1011
40.8
1
1009
40.7
1
6.45
1016
35.5
1
6.463013699
1013
49.4
1
6.528767123
1003
49.8
1
6.54
1012
53.8
1
6.643835616
6.41
6.41
1
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