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Kaplan-Meier Product-Limit Estimation
Survival analysis using variable length survival times
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
KEY CONCEPTS*****
Kaplan-Meier Product-Limit Survival Analysis
Limitations in using life table analysisFixed v. variable length survival timesKaplan-Meier procedure for estimating cumulative survival time probabilitiesAdvantages of using censored cases in survival analysis
vis-a-vis dismissing them as missing dataCalculation of cumulative survival timesSE of cumulative survival timesNumber remainingCumulative eventsMean survival time, SE, and confidence intervalMedian survival time, SE, and confidence intervalSurvival functionLog-Rank Test (Mantel-Haenszel Test)Breslow TestTarone-Ware TestProcedures for testing differences in survival functions when levels of a second nonmetric variable are used as strata
Pooled over strataFor each stratumPairwise for each strataPairwise over each stratum
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
2
Bibliography
Cox D. R. The analysis of exponentially distributed life-times with two types of failure. J. of the Royal Statistical Society, 1959, 21, 411-42.1
Gehan, E. A. A generalized Wilcoxon test for comparing arbitrarily singly-censored data. Biometrica, 1965, 52, 203-223.
Gehan, E. A. A generalized two-sample Wilcoxon test for doubly-censored data. Biometrica, 52,1965, 650-653.
Gross A. J. & Clark V. A. Survival Distributions: Reliability Applications in the Medical Sciences. Wiley 1975
Kaplan, E. L. & Meier, P. Nonparametric estimation from incomplete observations. J.of the American Statistical Association, 53, 1958, 457-48.
Lee E. T. & Desu M. M. A computer program for comparing K samples with right-censored data. Computer Programs in Biomedicine, 1972, 2, 315-321.
Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports, 1966, 50, 163-170.
Peto R. & Peto J. Asymptotically efficient rank invariant procedures. J. of the Royal Statistical Society,1972, 135, 185-207.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Lecture Overview
The problem of working with fixed length survival intervals versus variable length survival times
The Kaplan-Meier survival function
Case studies using the Kaplan-Meier procedure: disposition of cases on a criminal docket
Problem 1: Survival of criminal cases from filing to disposition
Problem 2: Survival of criminal cases from filing to disposition as a function of type of counsel
Problem 3: Survival of criminal cases from filing to disposition as a function of pre-arrest status
Problem 4: Survival of criminal cases from filing to disposition as a function of pre-arrest status, holding type of counsel constant
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
4
The Problem with Fixed Length Time Intervals in Survival Analysis
Life table analysis works with fixed length time intervals.
Cases are sorted into time intervals such as:
Interval 1: 1 month up to 2 monthsInterval 2: 2 months up to 3 months, etc.
If 26 cases are recorded in interval 1 …
The exact survival time of any case is unknown.
We only know that it is somewhere between 1 up to 2 months, the best guess is 1.5 months
The Problem
Information is lost when cases are categorized in fixed intervals of time.
The wider the intervals, the more information that is lost.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Kaplan-Meier Product-Limit Estimation
Dependency technique
Independent variable
Variable length measures of time
Dependent variable
Survival: coded as
0 = uncensored case, terminal event has occurred
1 = censored case, terminal event has not occurred
A useful technique when
The number of cases is small but representative
And the exact survival times are known.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Kaplan-Meier Product-Limit Estimation(Kaplan, E. L. & Meier, P. Nonparametric estimation from incomplete observations. J. of
the American Statistical Association, 53, 1958, 457-481)
S(t) = t i=1 [ (n - i) / (n – i + 1) ]Ci
S (t) = estimated survival function at time t
t i=1 = denotes the multiplication of the survival times across all cases less than or equal to t (the geometric mean)
t = time, e.g. days, weeks, months, etc.
n = total number of cases in the sample
i = the number of cases surviving up to time t
Ci = a constant such that …
0 = uncensored case, or terminal case1 = censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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An Example of Kaplan Meier Probabilities(C = status: 0 = uncensored, 1 = censored)
S (t) = t i=1 [ (n - i) / (n – i + 1) ]Ci
( = pi, symbol for multiplication)
Time Status Prior # Surviving
Number Remaining
Cumulative Survival
0 - 65 65 1.00005 0 65 64 1.00006 0 64 63 1.00007 1 63 62 0.98418 1 62 61 0.9683
10 0 61 60 0.968312 1 60 59 0.9522
S(0) = [ (65 - 0) / (65 – 0 + 1) ] 0 = (65/66) 0 = 1.0
S(5) = S(0) [ (65 - 1) / (65 – 1 + 1) ] 0 = (1.0) (1.0) = 1.0
S(6) = S(5) [ (65 - 2) / (65 – 2 + 1) ] 0 = (1.0) (1.0) = 1.0
S(7) = S(6) [ (65 - 3) / (65 – 3 + 1) ] 1 = (1.0) (0.9841) 1 =0.9841
S(8) = S(7) [ (65 - 4) / (65 – 4 + 1) ] 1 = (0.9841) (0.9839) 1 = 0.9683
S(10) = S(8) [ (65 - 5) / (65 – 5 + 1) ] 0 = (0.9683) (1.0) 0 = 0.9683
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Plot of the Kaplan-Meier ProbabilitiesSurvival Function
Probability of Survival
1.0 1.00
0.99
0.98 0.9841
0.97 0.9683
0.96
0.9522
0.01
0.00 1 2 3 4 5 6 7 8 9 10 11 12
Time in Days
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
9
The Advantage of Using Censored Cases
A censored case
A case for which the terminal event has not happened as yet …
Or may never happen.
May also included lost cases, missing dataor a subject who quits the study.
The advantage of including censored cases in the analysis
If there are many censored cases …
And they are excluded from the study …
The survival probabilities will be too low, an error of underestimation.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
10
Problem 1 Survival Rate of Cases on a
Criminal Docket
A random sample of 65 cases was drawn from the historical docket of a criminal district court.
Population:All cases filed in the
last 5 years
Sample N = 65
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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An Example Survival Rate of Cases on a Criminal Docket (cont.)
Number of days on the criminal docket
Mean = 383 days (conventional mean)
Median = 161 days (conventional median)
Minimum = 5 days
Maximum = 1775 days
Status of Cases
Cases disposed = 29 (uncensored cases)
Cases still on the docket = 36 (censored cases)
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
12
The Variables in the Case Disposition Database
Case Offender case number
Time Number of days from case filing to the current date if still on the docket or to the date the case was disposed
Censored The status of the case: 0 = case disposed, 1 = censored case, still on the docket
Days Time in days from offense to arrest
Counsel 0 = retained, 1 = court appointed
Jail-Tm Jail time: number of predisposition days in jail
Pre-Stat Pre arrest status: offender's status at the time of arrest: 1 = on bond or ROR, 2 = on probation, 3 = other
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
13
A Look at the Case Disposition Database
Case Time Cen-sored
Days Coun-sel
Jail_Tm
Pre_Stat
1 15 1 54 0 111 1
2 8 1 40 0 166 1
3 642 0 51 0 132 1
4 46 0 42 0 61 2
5 127 1 48 0 36 2
… … … … … … …
… … … … … … …
63 110 1 23 1 178 3
64 13 1 28 1 77 1
65 7 1 35 0 67 2
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
14
Calculation of the Probability Of Surviving for t Number of Days
S(t) = t i=1 [ (n - i) / (n – i + 1) ]Ci
Probability of a case being disposed of in 10 days
S(10) = S(8) [ (65 - 5) / (65 – 5 + 1) ]0 = 0.9836
Probability of a case being disposed of in 25 days
S (25) = S (23) [ (65 - 10) / (65 – 10 + 1) ]0
= (0.9836) (55/56) = 0.9660
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
15
Mean and Median Survival Times
Mean survival time
Mean = 809.97 days
This is not an arithmetic mean, which would be 383 days
This is the area under the survival curve for the uncensored cases, the cases that have been disposed of by the court
Median survival time
Median = 730 days
This is not the conventional median that would be 161 days
It is the time on the docket associated with the first case to have a cumulative survival probability 0.5
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
16
Standard Error of the Cumulative Probability
SE (tk) = S (tk) { (di) / [ni (ni – di)] }
SE (tk) = Estimated cumulative probability at time (t) of the event (k)
di = Number of events at time t
ni = Number of cases surviving prior to time t. Cases not terminated or censored
Examples:
Standard error at t = 10 days
SE(10) = 0.9836 (1) / [ 61 (61-1)] = 0.0163
Standard error at t = 25 days
SE(25) = 0.966 (1) / [61 (61-1)] + (1) / [56(56-1)]
= 0.0236
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
17
SPSS Results for Problem 1
Kaplan-Meier
Survival Analysis for TIME
Time Status Cumulative Standard Cumulative Number Survival Error Events Remaining
5.00 1.00 0 64 6.00 1.00 0 63 7.00 1.00 0 62 8.00 1.00 0 61 10.00 .00 .9836 .0163 1 60 12.00 1.00 1 59 13.00 1.00 1 58 15.00 1.00 1 57 23.00 1.00 1 56 25.00 .00 .9660 .0236 2 55 26.00 1.00 2 54 29.00 .00 .9482 .0292 3 53 30.00 1.00 3 52 39.00 .00 .9299 .0338 4 51 44.00 1.00 4 50 46.00 .00 .9113 .0379 5 49 47.00 .00 .8927 .0415 6 48 48.00 1.00 6 47 50.00 .00 7 46 50.00 .00 .8547 .0476 8 45 51.00 .00 9 44 51.00 .00 .8167 .0525 10 43 54.00 .00 .7978 .0546 11 42 60.00 .00 .7788 .0565 12 41 63.00 .00 .7598 .0583 13 40 64.00 .00 .7408 .0598 14 39 65.00 .00 .7218 .0612 15 38 66.00 .00 .7028 .0625 16 37 68.00 .00 .6838 .0636 17 36 110.00 1.00 17 35 127.00 1.00 17 34 136.00 .00 .6637 .0649 18 33 161.00 .00 .6436 .0659 19 32 167.00 1.00 19 31 228.00 1.00 19 30 237.00 1.00 19 29 253.00 .00 .6214 .0673 20 28 280.00 .00 .5992 .0685 21 27 297.00 .00 .5770 .0694 22 26 305.00 1.00 22 25 322.00 .00 .5539 .0704 23 24 339.00 1.00 23 23 389.00 1.00 23 22 439.00 1.00 23 21 456.00 1.00 23 20 499.00 1.00 23 19 551.00 1.00 23 18 589.00 1.00 23 17_
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
18
SPSS Results for Problem 1 (cont.)
592.00 1.00 23 16 624.00 .00 .5193 .0740 24 15 660.00 1.00 24 14 730.00 .00 .4822 .0775 25 13 815.00 1.00 25 12 836.00 .00 .4420 .0808 26 11 838.00 1.00 26 10 875.00 1.00 26 9 994.00 .00 .3929 .0854 27 8 1024.00 .00 .3438 .0877 28 7 1106.00 1.00 28 6 1264.00 1.00 28 5 1350.00 .00 .2750 .0933 29 4 1367.00 1.00 29 3 1536.00 1.00 29 2 1549.00 1.00 29 1 1775.00 1.00 29 0
Number of Cases: 65
Censored: 36( 55.38%), cases still on the docket
Events: 29, cases that were disposed
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
19
SPSS Results for Problem 1 (cont.)
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 809.97 111.94 ( 590.57, 1029.36 )
(Limited to 1775.0 )
Median: 730.00 355.81 ( 32.61, 1427.39 )
Survival Function
Survival Function
TIME
200010000-1000
Cum Survival
1.2
1.0
.8
.6
.4
.2
Survival Function
Censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
20
Problem 2Survival on the Docket as a Function of Type of Counsel
(Retained = 0, Court Appointed = 1)
CounselStatus
TotalsUncensored Censored
Retained 20 28 48
Appointed 9 8 17
Totals 29 36 65
In this analysis the subjects will be divided into two groups.
Statistical tests will be run to determine if the two survival functions differ significantly.
H1: Cases with retained counsel will remain on the docket significantly longer.
H0: There will be no significant difference in the survival functions of the two groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
21
Significance of the Difference in Survival as a Function of Counsel
(Test probabilities are in parentheses)
Counsel Mean(Median)
Log-Rank
Breslow Tarone-Ware
Retained 891(836)
1.93(0.1645)
1.74(0.1874)
1.84(0.1747)
Appointed 382(253)
Interpretation
The null hypothesis is accepted.
There is no significant difference between the survival functions of the counsel two groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
22
Tests of Significance of the Differences In Survival Functions of Multiple Groups
The Three Tests
Log-Rank (Mantel-Haenszel Test)
Breslow Generalized Wilcoxon Test
Tarone-Ware Test
The Equation
U = wi ( Di – Ei )
wi = weight
Di = Number of terminal events observed
Ei = Number of terminal events expected: number at risk cases & terminations at each event time (t)
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
23
Tests of Significance Between Group Survival Functions (cont.)
The three statistical tests differ in the weighting factor they use (wI).
Log-Rank Test
All cases weighted equally.
Least conservative of the three tests
Breslow Test
wI = Number of cases at risk at event time (t). Earlier events weighted more heavily.
Most conservative of the three tests
Tarone-Ware Test
wI = Square root of the number of cases at risk at event time (t). Weighs earlier cases less heavily than the Breslow Test does.
Mid-conservative of the three tests.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
24
The Statistical Power of the Three Tests( 1 - )
Log-Rank Test
More powerful than the Breslow test if …
The mortality (number of terminated cases) of the groups is proportional, i.e. the mortality of the groups differ by a constant multiplier.
Breslow Test
More powerful than the Log-rank test if the mortality of the groups is not proportional.
The power of the Breslow Test declines as the number of censored cases increases
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
25
SPSS Results for Problem 2
Retained counsel group (0) Factor COUNSEL = .00
Time Status Cumulative Standard Cumulative Number Survival Error Events Remaining
5.00 1.00 0 47 6.00 1.00 0 46 7.00 1.00 0 45 8.00 1.00 0 44 12.00 1.00 0 43 15.00 1.00 0 42 23.00 1.00 0 41 29.00 .00 .9756 .0241 1 40 30.00 1.00 1 39 39.00 .00 .9506 .0341 2 38 44.00 1.00 2 37 46.00 .00 .9249 .0417 3 36 48.00 1.00 3 35 50.00 .00 4 34 50.00 .00 .8721 .0535 5 33 51.00 .00 6 32 51.00 .00 .8192 .0620 7 31 54.00 .00 .7928 .0654 8 30 60.00 .00 .7663 .0683 9 29 64.00 .00 .7399 .0709 10 28 66.00 .00 .7135 .0731 11 27 127.00 1.00 11 26 161.00 .00 .6861 .0753 12 25 167.00 1.00 12 24 228.00 1.00 12 23 237.00 1.00 12 22 280.00 .00 .6549 .0780 13 21 297.00 .00 .6237 .0803 14 20 305.00 1.00 14 19 339.00 1.00 14 18 456.00 1.00 14 17 551.00 1.00 14 16 589.00 1.00 14 15 624.00 .00 .5821 .0850 15 14 660.00 1.00 15 13 730.00 .00 .5373 .0895 16 12 836.00 .00 .4926 .0926 17 11 838.00 1.00 17 10 875.00 1.00 17 9 994.00 .00 .4378 .0971 18 8 1024.00 .00 .3831 .0992 19 7 1106.00 1.00 19 6 1264.00 1.00 19 5 1350.00 .00 .3065 .1049 20 4 1367.00 1.00 20 3
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
26
SPSS Results for Problem 2 (cont.)
1536.00 1.00 20 21549.00 1.00 20 11775.00 1.00 20 0
Number of Cases: 48 Censored: 28 ( 58.33%) Events: 20
Mean and median survival times for retained counsel group
Survival Time Standard Error 95% Confidence Interval
Mean: 891.00 127.87 ( 640.38, 1141.62 ) (Limited to 1775.0 ) Median: 836.00 237.43 ( 370.65, 1301.35 )
Court appointed group (1)
Factor COUNSEL = 1.00
Time Status Cumulative Standard Cumulative Number Survival Error Events Remaining
10.00 .00 .9412 .0571 1 16 13.00 1.00 1 15 25.00 .00 .8784 .0807 2 14 26.00 1.00 2 13 47.00 .00 .8109 .0988 3 12 63.00 .00 .7433 .1113 4 11 65.00 .00 .6757 .1200 5 10 68.00 .00 .6081 .1256 6 9 110.00 1.00 6 8 136.00 .00 .5321 .1309 7 7 253.00 .00 .4561 .1324 8 6 322.00 .00 .3801 .1304 9 5 389.00 1.00 9 4 439.00 1.00 9 3 499.00 1.00 9 2 592.00 1.00 9 1 815.00 1.00 9 0
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
27
SPSS Results for Problem 2 (cont.)
Number of Cases: 17 Censored: 8 ( 47.06%) Events: 9
Mean and median survival time for the court appointed group
Survival Time Standard Error 95% Confidence Interval
Mean: 382.40 92.18 ( 201.72, 563.07 ) (Limited to 815.00 ) Median: 253.00 147.49 ( .00, 542.08 )
Summary of events Total Number Number Percent Events Censored Censored
COUNSEL .00 48 20 28 58.33 COUNSEL 1.00 17 9 8 47.06
Overall 65 29 36 55.38
Tests for the significance of difference between the two counsel groups, retained versus court appointed Test Statistics for Equality of Survival Distributions for COUNSEL
Statistic df Significance
Log Rank 1.93 1 .1645 Breslow 1.74 1 .1874 Tarone-Ware 1.84 1 .1747
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
28
SPSS Results for Problem 2 (cont.)
The survival functions of the two counsel groups
Survival Functions
TIME
200010000-1000
1.1
1.0
.9
.8
.7
.6
.5
.4
.3
COUNSEL
1.00
1.00-censored
.00
.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
29
Problem 3Survival on the Docket as a Function of
Pre-Arrest Status(1 = bond/ROR, 2 = probation, 3 = other status)
Pre-ArrestStatus
Status Totals
Uncensored Censored
Bond/ROR 10 12 22
Probation 10 12 22
Other 9 12 21
Totals 29 36 65
In this analysis the subjects will be divided into three groups.
Statistical tests will be run to determine if the three survival functions differ significantly.
H1: There will be significantly different survival times on the docket as a function of pre-arrest status.
H0: There will be no significant difference among the survival functions of the three groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
30
Statistical Tests for Significant Differences among Groups Defined by Pre-Arrest
Status
(Test probabilities are in parentheses)
Pre-ArrestStatus
Mean(Median)
Log-Rank
Breslow Tarone-Ware
Bond/ROR
622(253)
Probation 702(322)
3.01(0.2217)
5.19(0.0745)
14.52(0.1042)
Other 954(1024)
Interpretation
The null hypothesis is accepted
There are no significant differences among the survival functions of the three groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
31
SPSS Results for Problem 3
Bond/ROR group
Factor PRE_STAT = 1.00
Time Status Cumulative Standard Cumulative Number Survival Error Events Remaining
8.00 1.00 0 21 10.00 .00 .9524 .0465 1 20 12.00 1.00 1 19 13.00 1.00 1 18 15.00 1.00 1 17 23.00 1.00 1 16 39.00 .00 .8929 .0722 2 15 50.00 .00 .8333 .0886 3 14 51.00 .00 .7738 .1003 4 13 54.00 .00 .7143 .1088 5 12 60.00 .00 .6548 .1149 6 11 66.00 .00 .5952 .1189 7 10 68.00 .00 .5357 .1210 8 9 228.00 1.00 8 8 253.00 .00 .4688 .1230 9 7 305.00 1.00 9 6 339.00 1.00 9 5 499.00 1.00 9 4 551.00 1.00 9 3 624.00 .00 .3125 .1517 10 2 1106.00 1.00 10 1 1549.00 1.00 10 0
Number of Cases: 22 Censored: 12 ( 54.55%) Events: 10
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 622.08 185.41 ( 258.68, 985.47 ) (Limited to 1549.0 ) Median: 253.00 242.74 ( .00, 728.77 )
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
32
SPSS Results for Problem 3 (cont.)
Probation group Factor PRE_STAT = 2.00
Time Status Cumulative Standard Cumulative Number Survival Error Events Remaining
5.00 1.00 0 21 7.00 1.00 0 20 25.00 .00 .9500 .0487 1 19 26.00 1.00 1 18 29.00 .00 .8972 .0689 2 17 30.00 1.00 2 16 44.00 1.00 2 15 46.00 .00 .8374 .0865 3 14 47.00 .00 .7776 .0988 4 13 51.00 .00 .7178 .1078 5 12 64.00 .00 .6580 .1142 6 11 65.00 .00 .5981 .1185 7 10 127.00 1.00 7 9 136.00 .00 .5317 .1225 8 8 237.00 1.00 8 7 322.00 .00 .4557 .1264 9 6 439.00 1.00 9 5 456.00 1.00 9 4 660.00 1.00 9 3 730.00 .00 .3038 .1500 10 2 875.00 1.00 10 1 1775.00 1.00 10 0
Number of Cases: 22 Censored: 12 ( 54.55%) Events: 10
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 702.78 211.41 ( 288.41, 1117.15 ) (Limited to 1775.0 ) Median: 322.00 285.60 ( .00, 881.77 )
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
33
SPSS Results for Problem 3 (cont.)
Other status group
Factor PRE_STAT = 3.00
Time Status Cumulative Standard Cumulative Number Survival Error Events Remaining
6.00 1.00 0 20 48.00 1.00 0 19 50.00 .00 .9474 .0512 1 18 63.00 .00 .8947 .0704 2 17 110.00 1.00 2 16 161.00 .00 .8388 .0854 3 15 167.00 1.00 3 14 280.00 .00 .7789 .0981 4 13 297.00 .00 .7190 .1073 5 12 389.00 1.00 5 11 589.00 1.00 5 10 592.00 1.00 5 9 815.00 1.00 5 8 836.00 .00 .6291 .1260 6 7 838.00 1.00 6 6 994.00 .00 .5243 .1421 7 5 1024.00 .00 .4194 .1474 8 4 1264.00 1.00 8 3 1350.00 .00 .2796 .1506 9 2 1367.00 1.00 9 1 1536.00 1.00 9 0
Number of Cases: 21 Censored: 12 ( 57.14%) Events: 9
Mean and median survival times Survival Time Standard Error 95% Confidence Interval
Mean: 954.45 136.95 ( 686.03, 1222.88 ) (Limited to 1536.0 ) Median: 1024.00 132.11 ( 765.06, 1282.94 )_
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
34
SPSS Results for Problem 3 (cont.)
Tests of the significance of differences among the three groups Total Number Number Percent Events Censored Censored
PRE_STAT 1.00 22 10 12 54.55 PRE_STAT 2.00 22 10 12 54.55 PRE_STAT 3.00 21 9 12 57.14
Overall 65 29 36 55.38
Test Statistics for Equality of Survival Distributions for PRE_STAT
Statistic df Significance
Log Rank 3.01 2 .2217 Breslow 5.19 2 .0745 Tarone-Ware 4.52 2 .1042
Survial functions of the three groups
Survival Functions
TIME
200010000-1000
1.2
1.0
.8
.6
.4
.2
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
35
Problem 4Survival on the Docket as a Function of
Pre-Arrest Status Holding Type of Counsel Constant
Counsel Pre_ArrestStatus
Status Totals
Uncensored Censored
Bond/ROR 7 10 17
Retained Probation 5 10 15
Other 8 8 16
Subtotals (20) (28) (48)
Bond/ROR 3 2 5
Appointed Probation 5 2 7
Other 1 4 5
Subtotals (9) (8) (17)
Totals 29 36 65
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
36
Different Ways of Comparing the Survival Functions of Groups While Controlling
For a Categorical Factor
Pooling the data over strata
Examines the overall difference among the 3 pre-arrest status groups, pooling type of counsel (strata) over each pre-arrest group.
For each stratum (two analyses)
Examines overall differences among the 3 pre-arrest status groups for cases with retained counsel, and again for the cases with appointed counsel.
Pairwise over strata
Examines differences among all combinations of the 3 pre-arrest status groups, while pooling type of counsel (strata) over each pre-arrest group
Pairwise for each stratum (two analyses)
Compares differences among all possible combinations of the 3 pre-arrest status groups for cases with retained counsel, and again for the cases with appointed counsel.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
37
Problem 4.1Pre-Arrest Status Survival Data Pooled
Over Type of Counsel
Q Are there overall differences among pre-arrest status groups pooling type of counsel over each group?
Counsel Pre-Arrest Status
Bond/ROR Probation Other
Retained
Appointed
a1 a2 a3
If these tests are significant, there are differences between two or more of the groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
38
SPSS Results When the Data Are Pooled Over Counsel
Group 1: Retained counsel, offenders on bond/ROR
Strata COUNSEL = .00 Factor PRE_STAT = 1.00
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 692.67 212.91 ( 275.37, 1109.96 ) (Limited to 1549.0 ) Median: 624.00 380.81 ( .00, 1370.40 )
Group 2: Retained counsel, offenders on probation
Strata COUNSEL = .00 Factor PRE_STAT = 2.00
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 938.94 269.84 ( 410.05, 1467.83 ) (Limited to 1775.0 ) Median: 730.00 618.96 ( .00, 1943.17 )
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
39
SPSS Results When the Data Are Pooled Over Counsel (cont.)
Groups 3: Retained counsel, other status
Strata COUNSEL = .00 Factor PRE_STAT = 3.00
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 942.70 150.82 ( 647.11, 1238.30 ) (Limited to 1536.0 ) Median: 1024.00 137.28 ( 754.94, 1293.06 )
Group 4: Court appointed counsel, on bond/ROR
Strata COUNSEL = 1.00 Factor PRE_STAT = 1.00
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 220.67 94.17 ( 36.09, 405.24 ) (Limited to 499.00 ) Median: 253.00 102.86 ( 51.40, 454.60 )
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
40
SPSS Results When the Data Are Pooled Over Counsel (cont.)
Group 5: Court appointed counsel, on probation
Strata COUNSEL = 1.00 Factor PRE_STAT = 2.00
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 176.54 63.30 ( 52.48, 300.60 ) (Limited to 439.00 ) Median: 136.00 50.64 ( 36.74, 235.26 )
Group 6: Court appointed counsel, other status
Strata COUNSEL = 1.00 Factor PRE_STAT = 3.00
Mean and median survival times
Survival Time Standard Error 95% Confidence Interval
Mean: 664.60 134.52 ( 400.94, 928.26 ) (Limited to 815.00 ) Median: . . ( . , . )
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
41
SPSS Results When the Data Are Pooled Over Counsel (cont.)
Summary of Results for the Six Groups
Total Number Number Percent Events Censored Censored
COUNSEL .00 48 20 28 58.33 PRE_STAT 1.00 17 7 10 58.82 PRE_STAT 2.00 15 5 10 66.67 PRE_STAT 3.00 16 8 8 50.00
COUNSEL 1.00 17 9 8 47.06 PRE_STAT 1.00 5 3 2 40.00 PRE_STAT 2.00 7 5 2 28.57 PRE_STAT 3.00 5 1 4 80.00
Overall 65 29 36 55.38
Tests of Significance for Differences Among Groups
Test Statistics for Equality of Survival Distributions for PRE_STAT Adjusted for COUNSEL
Statistic df Significance
Log Rank 2.55 2 .2792 Breslow 3.20 2 .2017 Tarone-Ware 3.06 2 .2167
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
42
SPSS Results When the Data Are Pooled Over Counsel (cont.)
Survival Functions of the Three Pre-Arrest Status Groups Pooled Over Type of Counsel
1. Survival function for retained counsel
Survival Functions
COUNSEL = .00
TIME
200010000-1000
1.2
1.0
.8
.6
.4
.2
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
43
SPSS Results When the Data Are Pooled Over Counsel (cont.)
2. Survival function for appointed counsel
Survival Functions
COUNSEL = 1.00
TIME
10008006004002000
1.2
1.0
.8
.6
.4
.2
0.0
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
44
Mean & Median Survival Times When Data Are Pooled Over Counsel
*****(medians are in italics)
CounselPre-Arrest Status
Bond/ROR Probation Other
Retained 692.67
(624.00)
938.94
(730.00)
942.70
(1024.00)
Appointed 220.67
(253.00)
176.54
(136.00)
664.60
(NA) *
Interpretation
The three pre-arrest status groups do not have significantly different survival times when the data are pooled over type of counsel.
* NA The survival function is constant and has no median.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
45
Problem 4.2Pre-Arrest Status Survival Data for
Each Type of Counsel
Q Are there overall differences among the pre-arrest status groups who had retained counsel? And again, for those who had appointed counsel?
Counsel Pre-Arrest Status
Bond/ROR Probation Other
Retained a10 a20 a30
Counsel Pre-Arrest Status
Bond/ROR Probation Other
Appointed a11 a21 a31
If these tests are significant, there are differences between two or more of the groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
46
Problem 4.2SPSS Results for the Pre-Arrest Status
Groups for Each Type of Counsel
The mean and median survival times for each pre-arrest status / counsel group are identical to those presented in the SPSS results in Problem 4.1. Therefore they are not repeated in the results of this analysis.
Tests of significance for differences among pre-arrest status groups …
For offenders with retained counsel, no significant differences were found among pre-arrest status groups
Test Statistics for Equality of Survival Distributions for PRE_STAT For COUNSEL = .00
Statistic df Significance
Log Rank .72 2 .6964 Breslow 2.13 2 .3451 Tarone-Ware 1.54 2 .4620
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
47
Problem 4.2 SPSS Results for the Pre-Arrest Status Groups for Each Type of Counsel (cont.)
For offenders with appointed counsel, no significant differences were found among pre-arrest status groups
Test Statistics for Equality of Survival Distributions for PRE_STAT For COUNSEL = 1.00
Statistic df Significance
Log Rank 3.01 2 .2222 Breslow 2.29 2 .3175 Tarone-Ware 2.66 2 .2649
Survival functions
For offenders with retained counsel
Survival Functions
COUNSEL = .00
TIME
200010000-1000
1.2
1.0
.8
.6
.4
.2
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
48
Problem 4.2 SPSS Results for the Pre-Arrest Status Groups for Each Type of Counsel (cont.)
For offenders with appointed counsel
Survival Functions
COUNSEL = 1.00
TIME
10008006004002000
Cum Survival
1.2
1.0
.8
.6
.4
.2
0.0
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
49
Problem 4.3 Pre-Arrest Survival Data Pooled Over Type
of Counsel: Pairwise Over Strata
Q Are there differences among the various combinations of pre-arrest status groups pooling type of counsel pairwise over each group?
Counsel Pre-Arrest Status
Bond/ROR Probation Other
Retained
Appointed
a1 a2 a3
Three possible comparisons between groups:
(a1 v a2 ) (a1 v a3 ) (a2 v a3 )
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
50
Problem 4.3SPSS Results Pre-Arrest Survival Data Pooled Pairwise Over Type of Counsel
The mean and median survival times for each pre-arrest status / counsel group are identical to those presented in the SPSS results in Problem 4.1. Therefore they are not repeated in the results of this analysis.
Tests of significance for differences between pairs of pre-arrest status groups …
Log-Rank Test (Mantel-Haenszel Test)
Log Rank Statistic and (Significance) Adjusted for COUNSEL
Factor 1.00 2.00
2.00 .06 ( .8056)
3.00 1.94 2.22 ( .1632) ( .1363)
Interpretation No significant differences among the pairs of pre-arrest status groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
51
Problem 4.3 SPSS Results Pre-Arrest Survival Data Pooled Pairwise Over Type of Counsel (cont.)
Breslow Test
Breslow Statistic and (Significance) Adjusted for COUNSEL
Factor 1.00 2.00
2.00 .09 ( .7652)
3.00 2.66 2.40 ( .1027) ( .1210)
Interpretation No significant differences among the pairs of pre-arrest status groups.
Tarone-Ware Test
Tarone-Ware Statistic and (Significance) Adjusted for COUNSEL
Factor 1.00 2.00
2.00 .10 ( .7527)_
3.00 2.43 2.40 ( .1191) ( .1213)
Interpretation No significant differences among the pairs of pre-arrest status groups.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
52
Problem 4.3 SPSS Results Pre-Arrest Survival Data Pooled Pairwise Over Type of Counsel (cont.)
Survival Functions
For offenders with retained counsel
Survival Functions
COUNSEL = .00
TIME
200010000-1000
1.2
1.0
.8
.6
.4
.2
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Problem 4.3 SPSS Results Pre-Arrest Survival Data Pooled Pairwise Over Type of Counsel (cont.)
For offenders with appointed counsel
Survival Functions
COUNSEL = 1.00
TIME
10008006004002000
1.2
1.0
.8
.6
.4
.2
0.0
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Problem 4.4Pre-Arrest Survival Data for Each Type of
Counsel, Pairwise for Each Stratum
Q Are there differences among all possible combinations of pre-arrest status groups who had retained counsel? And again, for those who had appointed counsel?
Two separate analyses …
For Retained Counsel
Counsel Pre-Arrest Status
Bond/ROR Probation Other
Retained a10 a20 a30
Possible comparisons
(a10 v a20) (a10 v a30) (a20 v a30)
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Problem 4.4 Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each Stratum (cont.)
For Appointed Counsel
Counsel Pre-Arrest Status
Bond/ROR Probation Other
Appointed a11 a21 a31
Possible comparisons
(a11 v a21) (a11 v a31) (a21 v a31)
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
56
Problem 4.4SPSS Results Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each
Stratum
The mean and median survival times for each pre-arrest status / counsel group are identical to those presented in the SPSS results in Problem 4.1. Therefore they are not repeated in the results of this analysis.
Tests of significance for differences between pairs of pre-arrest status groups
For the retained counsel group
Log-Rank Test (Mantel-Haenszel Test)
Log Rank Statistic and (Significance) For COUNSEL = .00
Factor 1.00 2.00
2.00 .21 ( .6431)
3.00 .76 .35 ( .3823) ( .5563)
Interpretation No significant differences.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
57
Problem 4.4 SPSS Results Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each Stratum (cont.)
Breslow Test
Breslow Statistic and (Significance) For COUNSEL = .00
Factor 1.00 2.00
2.00 .10 ( .7539)
3.00 2.01 1.24 ( .1562) ( .2646)
Interpretation No significant differences.
Tarone-Ware Test
Tarone-Ware Statistic and (Significance) For COUNSEL = .00
Factor 1.00 2.00
2.00 .16 ( .6864)
3.00 1.48 .82 ( .2237) ( .3641)
_
Interpretation No significant differences.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
58
Problem 4.4 SPSS Results Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each Stratum (cont.)
For the appointed counsel group
Log-Rank Test (Mantel-Haenszel Test)
Log Rank Statistic and (Significance) For COUNSEL = 1.00
Factor 1.00 2.00
2.00 .03 ( .8538)
3.00 1.73 2.90 ( .1879) ( .0885)
Interpretation No significant differences.
Breslow Test Breslow Statistic and (Significance) For COUNSEL = 1.00
Factor 1.00 2.00
2.00 .00 (1.0000)
3.00 1.40 2.31 ( .2373) ( .1288)
Interpretation No significant differences.
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
59
Problem 4.4 SPSS Results Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each Stratum (cont.)
Tarone-Ware Test
Tarone-Ware Statistic and (Significance) For COUNSEL = 1.00
Factor 1.00 2.00
2.00 .01 ( .9183)
3.00 1.56 2.61 ( .2114) ( .1064)
Interpretation No significant differences.
Survival Functions
For offenders with retained counsel
Survival Functions
COUNSEL = .00
TIME
200010000-1000
1.2
1.0
.8
.6
.4
.2
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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Problem 4.4 SPSS Results Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each Stratum (cont.)
For offenders with appointed counsel
Survival Functions
COUNSEL = 1.00
TIME
10008006004002000
Cum Survival
1.2
1.0
.8
.6
.4
.2
0.0
PRE_STAT
3.00
3.00-censored
2.00
2.00-censored
1.00
1.00-censored
Kaplan-Meier Product-Limit Estimation: Charles M. Friel Ph.D., Criminal Justice Center, Sam Houston State University
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