kaplan meier estimation

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


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.


3

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


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.


5

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.


6

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


7

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


8

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


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.


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


11

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)


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


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


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


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


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


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_


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


19


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


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.


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.


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)


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.


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


25


Retained counsel group (0) Factor COUNSEL = .00


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


26


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


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


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


27



Mean and median survival time for the court appointed group


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


28


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


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.


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.


31


Bond/ROR group

Factor PRE_STAT = 1.00


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






32


Probation group Factor PRE_STAT = 2.00


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






33


Other status group

Factor PRE_STAT = 3.00


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


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 )_


34


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



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


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


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.


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.


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




Group 2: Retained counsel, offenders on probation






39

SPSS Results When the Data Are Pooled Over Counsel (cont.)

Groups 3: Retained counsel, other status





Group 4: Court appointed counsel, on bond/ROR

Strata COUNSEL = 1.00 Factor PRE_STAT = 1.00





40


Group 5: Court appointed counsel, on probation





Group 6: Court appointed counsel, other status




Mean: 664.60 134.52 ( 400.94, 928.26 ) (Limited to 815.00 ) Median: . . ( . , . )


41


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




42


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


43


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


44

Mean & Median Survival Times When Data Are Pooled Over Counsel

*****(medians are in italics)

CounselPre-Arrest Status


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.


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?



Retained a10 a20 a30



Appointed a11 a21 a31

If these tests are significant, there are differences between two or more of the groups.


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


Log Rank .72 2 .6964 Breslow 2.13 2 .3451 Tarone-Ware 1.54 2 .4620


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



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


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


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?



Retained

Appointed

a1 a2 a3

Three possible comparisons between groups:

(a1 v a2 ) (a1 v a3 ) (a2 v a3 )


50

Problem 4.3SPSS Results Pre-Arrest Survival Data Pooled Pairwise Over Type of Counsel


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.


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)


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)



52


Survival Functions


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


53



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


54

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



Retained a10 a20 a30

Possible comparisons

(a10 v a20) (a10 v a30) (a20 v a30)


55

Problem 4.4 Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each Stratum (cont.)

For Appointed Counsel



Appointed a11 a21 a31

Possible comparisons

(a11 v a21) (a11 v a31) (a21 v a31)


56

Problem 4.4SPSS Results Pre-Arrest Survival Data for Each Type of Counsel, Pairwise for Each

Stratum


Tests of significance for differences between pairs of pre-arrest status groups

For the retained counsel group


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.


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)


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)

_



58


For the appointed counsel group


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)


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)



59


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)


Survival Functions


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


60



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


61

kaplan meier estimation

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