AKA Event History Analysis

Survival/Failure Analysis (AKA Event History Analysis) T & F Chapter 11

Data Example 1. A medical doctor wished to compare the efficacy of two drugs for treating a sometimes fatal illness. Two groups of patients with the disease were identified. One group was given Drug A. The other group was given Drug B.

The age of the patient at the time of drug administration was recorded.

The patients were then monitored by a special team of patient observers.

The age of the patient at time of death was recorded and the survival duration after having taken the drug computed. Several of the patients lived for many years.

The study was terminated when the last patient in the two groups died – more than 60 years after the beginning of the study. (The original researcher died while waiting for the last patient to die. The original researcher’s grandchildren were available to continue the analyses.)

The grandchildren used the Mann-Whitney U-test to compare survival times between the two groups. (The U-test was used because survival times are notoriously positively skewed.)

This is the appropriate way to compare the efficacy of the two drugs.

Problems:

1) The long amount of our time it will take to observe survival times of all patients.

2) What to do about persons who get “lost” – from whom you lose contact. These patients give you incomplete data.

Because of Problem 1 above, we typically do NOT wait until every participant in our research has died before analyzing.

Instead we define a Window of Observation, and observe participants only when that window is open.

Survival Analysis – 1 Printed on 10/26/2016 Window of Observation

The problem is that we don’t have an infinite period of time to wait until everyone quits or dies. Plus, it may be the case that we lose contact with people so for some people we won’t know how long they survived regardless of the length of the window of observation.

The window of observation is the specific time period in which participant survival is recorded.

At some time, we begin recording whether or not each person is surviving or not. At some later time, we quit monitoring each patient.

Because the window is of finite duration, this necessarily results in incomplete information on some participants.

Of particular importance is the fact some will still be alive/working when we quit observing.

This means that we won’t have accurate survival times for some people.

Medical literature

Two treatments for a disease are given. We attempt to record 1) Whether or not each patient died – the dichotomous outcome – and 2) how long each patient survived until death – the continuous outcome.

Group A given Drug A. Group B given Drug B.

Turnover literature

Persons are hired by an organization into two different buildings. We attempt to record 1) Whether or not each employee quits before retirement and 2) how long each employee is employed before quitting.

Building A: Kill and Debone chickens Building B: Cook the chicken carcasses

Survival Analysis – 2 Printed on 10/26/2016 Overview of Types of cases in survival analysis

------|------|------

Ideal Cases – each starting time and ending time is known.

Right Censored Cases: Cases whose ending times (time of termination/death are unknown. These are the most common problem cases.

The above cases are still employed/surviving at the time monitoring ends.

?????????????????????????????? The above case is lost to follow-up (quit answering phone, left state, etc.)

Left Censored Cases: Cases whose starting times are unknown.

We will not include such cases in the analyses conducted here.

Cases whose starting times and ending times are unknown, Fagettaboutit – these are not analyzable.

???? ????

Survival Analysis – 3 Printed on 10/26/2016 Incorrect Analysis 1: Use death/quit rates as a proxy for survival

Assuming that persons with long survival times will be less likely to die within the window of observation, we could use death or quit rates as an indicator of survival time. We could use logistic regression to assess the relation of death or quit rates to independent variables. (Use linear regression in a pinch praying that the God of statistics won’t strike you down). Problem – it’s possible to create situations in which distributions of survival times are different even though proportions of outcomes are identical. Consider the following . . . Assume we’re dealing with employment. In the figures, each arrow represents duration of employment for a person. The horizontal axis is time. The vertical line at the left represents the time at which the window of observation opened. The vertical line at the right represents the time at which the window closed. The -> of the arrow represents death/termination. Group A – Termination Rate = 100%

Group B – Termination Rate = 100%

Clearly, Group A has longer average employment times, but both have the exact same proportion of turnovers – 100% in this example.

So, comparison of death/quit rates may certainly give us an inaccurate picture of the differences between the groups.

Survival Analysis – 4 Printed on 10/26/2016 Incorrect Analysis 2 – Analyze only the durations within the window of observation. Ignore the deaths/turnovers.

Group A. Average Survival time =

Group B. Average Survival time =

In the example above, the two groups have equal (ultimate) death rates but different survival times just within the window of observation – In Group A all subjects had “time to die.” In Group B, two subjects were still living when the window of observation closed. In this case, analysis of survival times within the finite window of observation will give an incorrect picture of the lack of difference between the groups.

Each type of incomplete analysis ignores the other aspect of the complete dependent variable. We need a method of analysis that takes into account both aspects.

Survival Analysis – 5 Printed on 10/26/2016 Survival analysis is an analytic technique that combines both aspects.Survival Analysis (also called Event History Analysis)

An analytic technique that models both survival times and proportions of deaths / quits.

3 separate techniques available in SPSS – Life Table, Kaplan-Meier, Cox Regression

Key concepts common to all techniques

1. Survival function – most important one of all of them

A plot of proportion surviving from time 0 up to a given time vs. time

A cumulative plot.

Generally decreasing curve, since proportion surviving can only remain constant or decrease across time.

Separate curves for separate groups

The curve represents both aspects of survival.

1) The height of the curve at a point represents the proportion not dying. So, the depth of the curve from 100% represents the proportion of persons who have died/terminated.

2) The curve also represents duration of stay/life (how far the curve has progressed to the right from t=0). It is a two-dimensional representation of the two aspects of survival – survival rates and length of life/employment.

Survival Analysis – 6 Printed on 10/26/2016 Comparing survival rates between groups.

The vertical axis represents proportion of survivals or turnovers.

Within a vertical slice at any point, turnover rates up to a particular time can be compared. In the following, we see that Group B had lower survival/higher turnover at the indicated time period.

Comparing Average survival times between two groups.

Within a horizontal slice at any point, average survival times can be compared.

In the following, we see that for group A, average time to reach 70% turnover was longer than it was for Group B.

Survival Analysis – 7 Printed on 10/26/2016 2. Hazard function

A plot of proportion dying/leaving at time intervals computed from those who had survived to that time period.

Among those who have survived until time, t, the hazard function, gives the proportion who will die.

Not a cumulative plot.

Hazard function for human mortality – Highest at young age and at high age.

3. Cumulative Hazard.

A plot of proportion of the whole sample dying/turning over up to a particular time.

A cumulative plot – the inverse of the survival plot.

Survival Analysis – 8 Printed on 10/26/2016 Three general types of Survival Analysis

1. Life Tables analysis.

The window of observation is cut up into n equal-length intervals.

Proportions of persons surviving/dying within each interval are computed.

This is the original method.

Useful for analysis of one group or for comparison of a few groups defined by levels of a single categorical factor.

Can’t incorporate quantitative predictors.

Can’t incorporate more than 2 qualitative predictors in SPSS.

Cannot analyze interactions of 2 or more predictors.

2. Kaplan-Meier analysis.

Event-based. Rather than defining intervals based on time, intervals are defined based on occurrence of death/termination. Each death/termination marks the end of one interval and the beginning of a subsequent interval.

Can’t incorporate quantitative predictors.

Can’t incorporate more than 2 qualitative predictors in SPSS.

Cannot analyze interactions of 2 or more predictors.

Survival Analysis – 9 Printed on 10/26/2016 3. Cox Proportional Hazards Regression (Cox Regression)

A very general, procedure.

Based on a specific mathematical model of survival developed by Cox.

Estimates hazard probabilities for whole sample.

Then estimates ratios of hazards to this overall hazard function for groups/persons with different values of IV’s

As implemented in SPSS, output and analyses look at lot like logistic regression.

Can incorporate quantitative predictors.

Can incorporate multiple qualitative and quantitative factors.

Can incorporate interactions.

Survival Analysis – 10 Printed on 10/26/2016 Based on Tabachnick Table 11.1, p. 515 Analyzed using SPSS Life Tables

Suppose the efficacy of Drug 0 is being compared with that of Drug 1. Each was formulated to prolong life of patients with a usually terminal form of cancer. Seven patients were given Drug 0 and five were given Drug 1. Patients were observed for up to 12 months. After 12 months, the window of observation closed and the results were entered into SPSS. So this problem is analogous to a turnover problem in organizational research with two groups of employees treated differently.

The SPSS syntax to invoke the analysis.

SAVE OUTFILE='G:\MdbT\P595\P595AL07-Survival analysis\TAndFDancingData.sav' /COMPRESSED. SURVIVAL TABLE=months BY drug(0 1) /INTERVAL=THRU 12 BY 1 /STATUS=outcome(1) /PRINT=TABLE /PLOTS (SURVIVAL)=months BY drug.

Survival Analysis – 11 Printed on 10/26/2016 Survival Analysis [DataSet0] G:\MdbT\P595\P595AL07-Survival analysis\TAndFDancingData.sav Survival Variable: months

Life Table Std. Error of Cumulati Cumulati ve ve Number Proportio Proportio Withdraw Number Proportio n n Std. Error Number ing Number of n Proportio Surviving Surviving of Std. Error First-order Interval Entering during Exposed Terminal Terminati n at End of at End of Probabilit Probabilit Hazard of Hazard Controls Start Time Interval Interval to Risk Events ng Surviving Interval Interval y Density y Density Rate Rate drug 0 0 7 0 7.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 1 7 0 7.000 1 .14 .86 .86 .13 .143 .132 .15 .15 2 6 0 6.000 2 .33 .67 .57 .19 .286 .171 .40 .28 3 4 0 4.000 1 .25 .75 .43 .19 .143 .132 .29 .28 4 3 0 3.000 1 .33 .67 .29 .17 .143 .132 .40 .39 5 2 0 2.000 1 .50 .50 .14 .13 .143 .132 .67 .63 6 1 0 1.000 0 .00 1.00 .14 .13 .000 .000 .00 .00 7 1 0 1.000 0 .00 1.00 .14 .13 .000 .000 .00 .00 8 1 0 1.000 0 .00 1.00 .14 .13 .000 .000 .00 .00 9 1 0 1.000 0 .00 1.00 .14 .13 .000 .000 .00 .00 10 1 0 1.000 0 .00 1.00 .14 .13 .000 .000 .00 .00 11 1 0 1.000 1 1.00 .00 .00 .00 .143 .132 2.00 .00 1 0 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 1 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 2 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 3 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 4 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 5 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 6 5 0 5.000 0 .00 1.00 1.00 .00 .000 .000 .00 .00 7 5 0 5.000 1 .20 .80 .80 .18 .200 .179 .22 .22 8 4 0 4.000 1 .25 .75 .60 .22 .200 .179 .29 .28 9 3 0 3.000 0 .00 1.00 .60 .22 .000 .000 .00 .00 10 3 0 3.000 2 .67 .33 .20 .18 .400 .219 1.00 .61 11 1 0 1.000 0 .00 1.00 .20 .18 .000 .000 .00 .00 12 1 1 .500 0 .00 1.00 .20 .18 .000 .000 .00 .00

The results suggest that survival is significantly longer with Drug 1 – the top (orange) curve.

Survival Analysis – 12 Printed on 10/26/2016 Tabachnick Table 11.1, p. 511 Analyzed using SPSS Kaplan-Meier Analyze  Survival  Kaplan-Meier . . .

KM months BY drug /STATUS=outcome(1) /PRINT TABLE MEAN /PLOT SURVIVAL /TEST LOGRANK BRESLOW TARONE /COMPARE OVERALL POOLED.

Survival Analysis – 13 Printed on 10/26/2016 Kaplan-Meier [DataSet2] G:\MdbT\InClassDatasets\Survival(T&Bp511).sav Case Processing Summary drug Total N Censored N of Events N Percent 0 7 7 0 .0% 1 5 4 1 20.0% Overall 12 11 1 8.3%

Survival Table Cumulative Proportion Surviving at the Time drug Time Status Estimate Std. Error N of Cumulative Events N of Remaining Cases 0 1 1.000 1 .857 .132 1 6 2 2.000 1 . . 2 5 3 2.000 1 .571 .187 3 4 4 3.000 1 .429 .187 4 3 5 4.000 1 .286 .171 5 2 6 5.000 1 .143 .132 6 1 7 11.000 1 .000 .000 7 0 1 1 7.000 1 .800 .179 1 4 2 8.000 1 .600 .219 2 3 3 10.000 1 . . 3 2 4 10.000 1 .200 .179 4 1 5 12.000 0 . . 4 0

Means and Medians for Survival Time Meana Median 95% Confidence Interval 95% Confidence Interval drug Estimate Std. Error Lower Bound Upper Bound Estimate Std. Error Lower Bound Upper Bound 0 4.000 1.272 1.506 6.494 3.000 1.309 .434 5.566 1 9.400 .780 7.872 10.928 10.000 .894 8.247 11.753 Overall 6.250 1.081 4.131 8.369 5.000 2.598 .000 10.092 a. Estimation is limited to the largest survival time if it is censored.

Overall Comparisons

Chi-Square df Sig. Log Rank 3.747 1 .053 (Mantel-Cox) Breslow 4.926 1 .026 (Generalized Wilcoxon) Tarone-Ware 4.522 1 .033 Test of equality of survival distributions for the different levels of drug.

As was the case with the analysis using the LIFE TABLES procedure, the results support the conclusion that survival is significantly longer with Drug 1.

Survival Analysis – 14 Printed on 10/26/2016 Tabachnick Table 11.1, p. 511 – start here on 11/6/17 Analyzed using SPSS Cox Regression

The program will not produce a survival curve for a group of cases defined by the value of a variable unless that variable is a categorical variable. (Reminds me of the RCMDR Factor issue.)

For that reason, I told the program that drug is a categorical variable so that survival curves for each value of drug could be obtained.

Since drug is a dichotomy, the analysis could be done without labeling it categorical, but in that case the survival curves for each value of drug could not have been generated.

Survival Analysis – 15 Printed on 10/26/2016 The left panel would yield 1 plot The right panel yields a plot for each value of drug.

COXREG months /STATUS=outcome(1) /PATTERN BY drug /CONTRAST (drug)=Indicator(1) /METHOD=ENTER drug /PLOT SURVIVAL /CRITERIA=PIN(.05) POUT(.10) ITERATE(20).

Cox Regression

[DataSet2] G:\MdbT\InClassDatasets\Survival(T&Bp511).sav

Case Processing Summary N Percent Cases available in analysis Eventa 11 91.7% Censored 1 8.3% Total 12 100.0% Cases dropped Cases with missing values 0 .0% Cases with negative time 0 .0% Censored cases before the 0 .0% earliest event in a stratum Total 0 .0% Total 12 100.0% a. Dependent Variable: months

Categorical Variable Codingsb Frequency (1) druga 0 7 0 1 5 1 a. Indicator Parameter Coding b. Category variable: drug

Survival Analysis – 16 Printed on 10/26/2016 Block 0: Beginning Block

Omnibus Tests of Model Coefficients -2 Log Likelihood 40.740

Block 1: Method = Enter

Omnibus Tests of Model Coefficientsa Overall (score) Change From Previous Step Change From Previous Block -2 Log Likelihood Chi-square df Sig. Chi-square df Sig. Chi-square df Sig. 37.394 3.469 1 .063 3.346 1 .067 3.346 1 .067 a. Beginning Block Number 1. Method = Enter

Variables in the Equation

B SE Wald df Sig. Exp(B) drug -1.176 .658 3.192 1 .074 .309

Covariate Means and Pattern Values Pattern Mean 1 2 drug .417 .000 1.000

I strongly recommend that you create a plot such as the one immediately above by hand to make sure you understand the Cox Regression results. I do it every time I use this procedure.

Survival Analysis – 17 Printed on 10/26/2016 COXREG plots are plots of predicted survival, not actual survival. In this sense, they’re like the tables and plots of estimated marginal means from GLM. I usually report observed survival functions, using Kaplan- Meier, rather than these predicted survival functions. However, these are certainly useful in situations in which you want to show what survival should be for specific groups at specific times controlling for the other variables in the equation.

Survival Analysis – 18 Printed on 10/26/2016 Real Life Example: Turnover at a local Manufacturing Plant

1. Effect of Friends and/or family at the plant

In this study, turnover at a local manufacturing plant was studied. On the application blank, applicants were asked to indicate whether or not they had friends or family already working at the plant.

Some did not respond to this question. They’re included in the analysis. A screen shot of the data editor The variable, wsfr2, represents whether or not the applicant had friends at the company.

wsfr2 = 0.50 means yes. wsfr2 = -0.50 means no. wsfr2 = 0.15 means no info.

Wsfr2 was created to deal with missing values in a special way. The fact that the values are fractional has no bearing on the analyses. They could just as well have been 0, 1, 2 or 1, 2, 3.

Having said that, because the LIFE TABLES procedures requires integer values of each factor, I’ll skip it here.

Kaplan-Meier analysis is shown

Some of SPSS’s procedures are written so that a grouping variable, can have any set of values. K-M is one of them.

K-M allows you to simply specify the name of the factor, and the program figures out how many groups are implied by the values of the factor.

That’s good unless you have a grouping variable with some incidental values representing unique cases or groups of cases – cases you wish to be excluded from the analysis.

Survival Analysis – 19 Printed on 10/26/2016 KM dos BY wsfr2 /STATUS=status(1) /PRINT TABLE MEAN /PLOT SURVIVAL /TEST LOGRANK BRESLOW TARONE /COMPARE OVERALL POOLED .

Survival Analysis – 20 Printed on 10/26/2016 Kaplan-Meier

[DataSet3] G:\MdbR\1TurnoverArticle\TurnoverArticleDataset061005.sav Large table was deleted.

Case Processing Summary wsfr2 Whether F/F at company N of Censored for whole sample analyses Total N Events N Percent -.50 423 174 249 58.9% .15 Whole sample missing value 100 40 60 60.0% .50 778 220 558 71.7% Overall 1301 434 867 66.6%

Means and Medians for Survival Time Meana Median 95% Confidence Interval 95% Confidence Interval wsfr2 Whether F/F at company Estimat Std. Lower Upper Estimat Std. Lower Upper for whole sample analyses e Error Bound Bound e Error Bound Bound -.50 610.597 25.559 560.500 660.693 667.000 . . . .15 Whole sample missing value 579.795 49.233 483.299 676.291 528.000 151.013 232.014 823.986 .50 769.900 18.559 733.524 806.277 . . . . Overall 706.965 15.009 677.548 736.383 . . . . a. Estimation is limited to the largest survival time if it is censored.

Note that there is no estimate of median survival for the 0.50 group. I’m not absolutely sure why, but I believe it’s because more than 50% of the persons in that group were still on the job at the end of the observation window. For that reason, a median was not computable.

Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 25.344 2 .000 Breslow (Generalized Wilcoxon) 25.325 2 .000 Tarone-Ware 25.004 2 .000 Test of equality of survival distributions for the different levels of wsfr2 Whether F/F at company for whole sample analyses.

Clearly there are significant differences in overall survival between the groups.

Survival Analysis – 21 Printed on 10/26/2016

The data strongly suggest that applicants who had friends or family at the company had higher survival rates at all times up to 1100 days (about 3 years).

For example, at the end of 1 year survival (leftmost arrow in the above figure) rate of those with friends and family was about 70% while that for those who said they did not have friends or family at the organization was about 60%.

By two years (middle arrow), the rate of retention of those with was about 68% while the rate of those without had decreased to 50%.

The fact that the curve for those for whom no information was available was between the other two curves suggests that those employees for whom no information was available were a mixture of some who did have friends and family and those who did not.

Survival Analysis – 22 Printed on 10/26/2016 Same analysis using SPSS Cox Regression

Analyze  Survival  Cox Regression . . .

Survival Analysis – 23 Printed on 10/26/2016 COXREG dos /STATUS=status(1) /PATTERN BY wsfr2 /CONTRAST (wsfr2)=Indicator /METHOD=ENTER wsfr2 /PLOT SURVIVAL /CRITERIA=PIN(.05) POUT(.10) ITERATE(20).

Cox Regression Case Processing Summary N Percent Cases available in analysis Eventa 434 33.4% Censored 867 66.6% Total 1301 100.0% Cases dropped Cases with missing values 0 0.0% Cases with negative time 0 0.0% Censored cases before the 0 0.0% earliest event in a stratum Total 0 0.0% Total 1301 100.0% a. Dependent Variable: dos

Survival Analysis – 24 Printed on 10/26/2016 Categorical Variable Codingsa Frequency (1) (2) wsfr2b -.50=-.50 423 1 0 .15=Whole sample missing value 100 0 1 .50=.50 778 0 0 a. Category variable: wsfr2 (Whether F/F at company for whole sample analyses) b. Indicator Parameter Coding

Block 0: Beginning Block Omnibus Tests of Model Coefficients -2 Log Likelihood 5871.672

Block 1: Method = Enter Omnibus Tests of Model Coefficientsa -2 Log Overall (score) Change From Previous Step Change From Previous Block Likelihood Chi-square df Sig. Chi-square df Sig. Chi-square df Sig. 5847.172 25.290 2 .000 24.499 2 .000 24.499 2 .000 a. Beginning Block Number 1. Method = Enter

Variables in the Equation B SE Wald df Sig. Exp(B) wsfr2 24.812 2 .000 wsfr2(1) .489 .102 23.230 1 .000 1.631 wsfr2(2) .425 .172 6.104 1 .013 1.529

Recall that the sign of each coefficient is relative to “Termination”.

WSFR2(1) compares the proportion terminating in the -.50 group to the proportion in the +.50 group.

Since the coefficient is +.489, this says that the -.50 group has larger probability of terminating than the .50 group.

Same for the wsfr2(2) – The no response group has greater probability of terminating than the +.50 group.

Covariate Means and Pattern Values Pattern Mean 1 2 3 wsfr2(1) .325 1.000 .000 .000 wsfr2(2) .077 .000 1.000 .000

Survival Analysis – 25 Printed on 10/26/2016 Survival Analysis – 26 Printed on 10/26/2016 Using Survival Analysis to score and validate selection test questions. An I/O consulting firm gave a 30-question pre-employment questionnaire to 1000+ employees of a local company. Each question had from one to five alternatives. The consulting company wanted to identify questions that predicted long tenure with the organization. (They would have preferred to identify questions that predicted high performance, but it was not possible to get good performance data. Don’t get me started on why organizations don’t gather good performance data.)

In order to identify responses associated with long tenure, a survival analysis was conducted for each question. A few of the analyses are presented below.

For each survival function, each curve is the survival function of persons who made a particular response to the item. I picked only those for which the difference in survival curves was significant or approached significance. Question 1 Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 5.382 2 .068 Breslow (Generalized Wilcoxon) 4.307 2 .116 Tarone-Ware 4.756 2 .093

Survival Analysis – 27 Printed on 10/26/2016 Question 2 Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 7.647 4 .105 Breslow (Generalized Wilcoxon) 6.950 4 .139 Tarone-Ware 7.298 4 .121

Survival Analysis – 28 Printed on 10/26/2016 Question 3 Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 5.070 3 .167 Breslow (Generalized Wilcoxon) 5.525 3 .137 Tarone-Ware 5.493 3 .139 Test of equality of survival distributions for the different levels of GenQ4 Gen Q4 L:I prefer a job that / S: How often you experience conflict with a co- worker?.

Survival Analysis – 29 Printed on 10/26/2016 Question 4

Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 7.753 4 .101 Breslow (Generalized Wilcoxon) 6.762 4 .149 Tarone-Ware 7.439 4 .114 Test of equality of survival distributions for the different levels of GenQ3 Gen Q3 L: Recieved safety training? / S: You are asked to do more physically demanding work than you were hired to do because someone out sick, how do you react?.

Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 10.971 4 .027 Breslow (Generalized Wilcoxon) 9.931 4 .042 Tarone-Ware 10.597 4 .031 Test of equality of survival distributions for the different levels of GenQ2 Gen Q2 L: Your team in disagreement over who will clean the floor. What method is fair?/ S: Recent supervisor rate dependability?.

Overall Comparisons Chi-Square df Sig. Log Rank (Mantel-Cox) 8.052 3 .045 Breslow (Generalized Wilcoxon) 12.729 3 .005 Tarone-Ware 10.614 3 .014 Test of equality of survival distributions for the different levels of GenQ1 GenQ1 L: Which strategies inspire a team and help be more effective?/ S:Your team in disagreement over who will clean the floor. What method is fair?.

Survival Analysis – 32 Printed on 10/26/2016 Creation of an overall Tenure Index

Thirty questions were evaluated in the above fashion.

After examination of the individual survival curves for the 30 questions, those for which significant differences in survival between responses were identified by examining the survival analysis for each question as shown above.

Finally, an overall index was calculated, using syntax like the following . . .

In this particular case, the response associated with long survival added 1 to the index.

The response associated with short survival subtracted 1 from the index.

Tenure Scale Computation

Compute genshort=0. if ((genq1=3 or genq1=4)) genqshort=genqshort + 1. if ((genq1=1 or genq1=2)) genqshort=genqshort - 1. if ((genq2=3 or genq2=4)) genshort=genshort + 1. if ((genq2=1 or genq2=2 or genq2=5)) genshort=genshort - 1. if ((genq6=3)) genshort=genshort + 1. if ((genq6=1 or genq6=2)) genshort=genshort - 1. if ((genq12=1)) genshort=genshort + 1. if ((genq12=3)) genshort=genshort - 1. if ((genq13=1)) genshort=genshort + 1. if ((genq13=2 or genq13=3 or genq13=4)) genshort=genshort - 1. if ((genq21=1 or genq21=3)) genshort=genshort + 1. if ((genq21=2)) genshort=genshort - 1.

Survival Analysis – 33 Printed on 10/26/2016 Validity of the Tenure Index

The following is not based on the scale above but on a similar scale.

The median score on the scale was determined to be -14, Group 0 was all employees with an index value less than or equal to -14 – persons who generally responded with the “short tenure” answer.

Group 1 was all employees with an index value greater than -14 – persons who generally responded with the “long tenure” answer.

The graph indicates that those in Group 1, with large values of the index, had a nearly 70% retention rate after 50 months.

Those in Group 0 had a 40% retention rate after the same length of time.

The implication of this analysis would be to recommend to the company to use the scale in hiring of employees, giving preference to those with higher scores on the scale.

Remember that these responses were obtained at time of application. The effect lasted for 4 years.

Potential problems

The above curve was based on the same sample that was used to select the questions. So clearly there is capitalization on chance. The scale should be tested on a different sample. That is the results need to be cross validated.

Survival Analysis – 34 Printed on 10/26/2016

Multivariate Analysis using Cox Regression

Turnover as a function of 1) friends at the organization (wsfr2) and 2) sex of the employee, and 3) ethnic group of the employee (neth)

COXREG dos /STATUS=status(1) /PATTERN BY wsfr2 /CONTRAST (neth)=Indicator(1) /CONTRAST (wsfr2)=Indicator /METHOD=ENTER wsfr2 nsex neth /PLOT SURVIVAL /CRITERIA=PIN(.05) POUT(.10) ITERATE(20).

Wsfr2 -0.50 does not have friends at company 0.15 no info on whether has friends 0.50 friends at the company

Nsex 1 Female 2 Male

Neth 1 Employee is White 2 Employee is Black

Survival Analysis – 35 Printed on 10/26/2016 3 Employee is American Indian or Asian or Hispanic

Survival Analysis – 36 Printed on 10/26/2016 Survival Analysis – 37 Printed on 10/26/2016 Cox Regression

[DataSet1] G:\MDBR\1TurnoverArticle\TurnoverArticleDataset061005.sav

Case Processing Summary

N Percent

Eventa 434 33.4%

Cases available in analysis Censored 867 66.6%

Total 1301 100.0%

Cases with missing values 0 0.0%

Cases with negative time 0 0.0%

Cases dropped Censored cases before the earliest 0 0.0% event in a stratum

Total 0 0.0%

Total 1301 100.0%

a. Dependent Variable: dos Days of service: termdate-effdate or 3/1/1-effdate or 12/31/4-effdate

Categorical Variable Codingsa,c

Frequency (1) (2)

-.50=-.50 423 1 0

.15=Whole sample wsfr2b 100 0 1 missing value

.50=.50 778 0 0

1.00=White 903 0 0

2.00=Black 324 1 0 nethb 3.00=Am 74 0 1 Ind,Asian,Hisp

a. Category variable: wsfr2 (Whether F/F at company for whole sample analyses)

b. Indicator Parameter Coding

c. Category variable: neth (1=White, 2=Black, 3=Am Ind,Asian, Hisp)

Survival Analysis – 38 Printed on 10/26/2016 No interactions were included in this analysis.Block 0: Beginning Block

Omnibus Tests of

Model Coefficients

-2 Log Likelihood

5871.672

Omnibus Tests of Model Coefficientsa

-2 Log Likelihood Overall (score) Change From Previous Step Change From Previous Block

Chi-square df Sig. Chi-square df Sig. Chi-square df Sig.

5827.342 42.322 5 .000 44.330 5 .000 44.330 5 .000

a. Beginning Block Number 1. Method = Enter

Variables in the Equation

B SE Wald df Sig. Exp(B)

wsfr2 22.427 2 .000

wsfr2(1) .464 .102 20.763 1 .000 1.590

wsfr2(2) .421 .173 5.969 1 .015 1.524

nsex -.223 .100 4.952 1 .026 .800

neth 10.799 2 .005

neth(1) .088 .109 .657 1 .417 1.092

neth(2) -.908 .295 9.490 1 .002 .403

Covariate Means and Pattern Values

Mean Pattern

wsfr2(1) .325 1.000 .000 .000

wsfr2(2) .077 .000 1.000 .000

nsex 1.421 1.421 1.421 1.421

neth(1) .249 .249 .249 .249

neth(2) .057 .057 .057 .057

Survival Analysis – 39 Printed on 10/26/2016 The Kaplan-Meier Curve, for comparison . . .

Survival Analysis – 40 Printed on 10/26/2016 Testing for Interactions in Cox Regression

1. The interaction of Friends and Nsex

Omnibus Tests of Model Coefficientsa -2 Log Overall (score) Change From Previous Step Change From Previous Likelihood Block Chi- df Sig. Chi- df Sig. Chi- df Sig. square square square 5824.879 44.989 7 .000 46.792 7 .000 46.792 7 .000 a. Beginning Block Number 1. Method = Enter

Variables in the Equation B SE Wald df Sig. Exp(B) wsfr2 3.022 2 .221 wsfr2(1) .429 .306 1.975 1 .160 1.536 wsfr2(2) -.333 .530 .394 1 .530 .717 nsex -.282 .138 4.158 1 .041 .754 neth 10.964 2 .004 neth(1) .097 .109 .800 1 .371 1.102 neth(2) -.907 .295 9.464 1 .002 .404 nsex*ws 2.517 2 .284 fr2 nsex*ws .023 .213 .011 1 .915 1.023 fr2(1) nsex*ws .541 .347 2.424 1 .119 1.717 fr2(2)

So no significant interaction means that the effect of having friends is the same for Females as it is for Males Survival Analysis – 41 Printed on 10/26/2016 Survival Analysis – 42 Printed on 10/26/2016 2. The interaction of Friends and Neth

Omnibus Tests of Model Coefficientsa -2 Log Overall (score) Change From Previous Step Change From Previous Likelihood Block Chi- df Sig. Chi- df Sig. Chi- df Sig. square square square 5820.584 49.194 9 .000 51.088 9 .000 51.088 9 .000 a. Beginning Block Number 1. Method = Enter

Variables in the Equation B SE Wald df Sig. Exp(B) wsfr2 27.320 2 .000 wsfr2(1) .599 .121 24.386 1 .000 1.820 wsfr2(2) .623 .209 8.846 1 .003 1.864 nsex -.224 .100 4.973 1 .026 .799 neth 6.603 2 .037 neth(1) .298 .150 3.934 1 .047 1.347 neth(2) -.465 .344 1.835 1 .176 .628 neth*wsfr2 6.377 4 .173 neth(1)*wsfr2(1) -.392 .230 2.906 1 .088 .675 neth(2)*wsfr2(1) -1.222 .791 2.385 1 .123 .295 neth(1)*wsfr2(2) -.534 .378 1.995 1 .158 .586 neth(2)*wsfr2(2) -1.093 1.075 1.035 1 .309 .335

Again, the lack of a significant interaction means that the effect of Friends is the same for each ethnic group.

What the heck? What about the interaction of nsex and neth?

Block 1: Method = Enter Omnibus Tests of Model Coefficientsa -2 Log Likelihood Overall (score) Change From Previous Step Change From Previous Block Chi-square df Sig. Chi-square df Sig. Chi-square df Sig. 5827.298 42.395 7 .000 44.373 7 .000 44.373 7 .000 a. Beginning Block Number 1. Method = Enter

Variables in the Equation B SE Wald df Sig. Exp(B) wsfr2 22.438 2 .000 wsfr2(1) .465 .102 20.791 1 .000 1.591 wsfr2(2) .421 .173 5.941 1 .015 1.523 nsex -.216 .118 3.365 1 .067 .806 neth .888 2 .642 neth(1) .112 .327 .117 1 .732 1.118 neth(2) -.739 .884 .700 1 .403 .477 neth*nsex .043 2 .979 neth(1)*nsex -.018 .234 .006 1 .940 .982 neth(2)*nsex -.125 .624 .040 1 .842 .883

Survival Analysis – 43 Printed on 10/26/2016 Comparing Turnover in two plants

A company was interested in determining the causes of turnover in two of its plants.

Plant A: One part of the preparation of food for sale to retailers is undertaken. Plant B: A different part of the preparation of food for sale to retailer is undertaken.

The two plants hire from the same pool of employees. Each plant is managed by a different person.

The overall “survival” of employees in the two plants, reploc=1 and reploc=2, is as follows . . . filter off. compute reploc = newloc. value labels reploc 1 "A" 2 "B". filter by useme. KM dayswrkd by reploc /STATUS=termed(1)/PRINT MEAN /PLOT SURVIVAL /TEST LOGRANK BRESLOW TARONE /COMPARE OVERALL POOLED. Kaplan-Meier [DataSet1] G:\MDBR\???\AllEmployeesNN041025.sav

reploc Total N N of Events Censored

N Percent

1.00 A 310 126 184 59.4%

2.00 B 837 285 552 65.9%

Overall 1147 411 736 64.2%

Means and Medians for Survival Time

reploc Meana Median

Estimate Std. Error 95% Confidence Interval Estimate Std. Error 95% Confidence Interval

Lower Bound Upper Bound Lower Bound Upper Bound

1.00 A 355.796 16.345 323.760 387.832 377.000 39.815 298.962 455.038

2.00 B 424.911 9.197 406.884 442.938 559.000 . . .

Overall 407.357 8.081 391.519 423.195 489.000 33.040 424.242 553.758

a. Estimation is limited to the largest survival time if it is censored.

Overall Comparisons

Chi-Square df Sig.

Log Rank 13.633 1 .000

(Mantel-Cox)

Breslow 10.203 1 .001

(Generalized

Wilcoxon)

Tarone-Ware 11.880 1 .001

Test of equality of survival distributions for the different levels of reploc.

Survival Analysis – 44 Printed on 10/26/2016 Survival Analysis – 45 Printed on 10/26/2016 filter off.

Clearly, employee “retention/survival” is best in Plant B – reploc = 2.

The manager of Plant A was pretty defensive.

Survival Analysis – 46 Printed on 10/26/2016 Are these differences in survival rates the same for the different ethnic groups employed by the company?

Perhaps the differences between buildings are due to the fact that the different buildings have different proportions of ethnic groups neweth * reploc Crosstabulation reploc 1.00 A 2.00 B Total neweth .00 White or Black Count 130 219 349 % within reploc 41.4% 25.7% 29.9% 1.00 Hispanic Count 184 634 818 % within reploc 58.6% 74.3% 70.1% Total Count 314 853 1167 % within reploc 100.0% 100.0% 100.0% coupled with the fact that the different ethnic groups have different survival rates . . .

These differences suggest that the difference in survival between buildings might be a side-effect of the difference in proportion of Hispanics in the two buildings combined with the difference in survival between Hispanics vs. White/Black,

The way to resolve this issue is to perform a multivariate analysis, assessing the Plant effect while controlling for the Ethnic Group effect..

This can only be done with Cox Regression.

Survival Analysis – 47 Printed on 10/26/2016 Multivariate analysis joint effect of plant and ethnic group.

filter off. filter by useme. COXREG dayswrkd /STATUS=termed(1) /METHOD=ENTER reploc neweth /CRITERIA=PIN(.05) POUT(.10) ITERATE(20).

Cox Regression

N Percent Cases available in analysis Eventa 411 33.9% Censored 736 60.7% Total 1147 94.6% Cases dropped Cases with missing values 65 5.4% Cases with negative time 0 0.0% Censored cases before the earliest event in a 0 0.0% stratum Total 65 5.4% Total 1212 100.0% a. Dependent Variable: dayswrkd

Block 0: Beginning Block

Omnibus Tests of Model Coefficients -2 Log Likelihood 5312.092

Omnibus Tests of Model Coefficientsa Overall (score) Change From Previous Step Change From Previous Block -2 Log Chi- Chi- Chi- Likelihood square df Sig. square df Sig. square df Sig. 5222.145 101.652 2 .000 89.947 2 .000 89.947 2 .000 a. Beginning Block Number 1. Method = Enter

Variables in the Equation B SE Wald df Sig. Exp(B) reploc -.159 .111 2.072 1 .150 .853 neweth -.916 .102 80.462 1 .000 .400

Covariate Means Mean reploc 1.730 neweth .697 filter off. So, when controlling for differences in ethnic groups, no difference in survival (turnover) between the two buildings was found. The manager of Building A was very happy with this result. Survival Analysis of a phenomenon with a positive outcome

Survival Analysis – 48 Printed on 10/26/2016 PEG vs. PEGJ Example

The data for this example compared two methods of feeding trauma patients, one using a percutaneous esophagogastrojejunostomy (PEGJ) and the other using percutaneous esophagogastrostomy (PEG). It was hoped that the data would show that the PEGJ technique would provide continuous uninterrupted nutrition with greater consistency than with PEG. Time to reach a nutrition goal was the continuous dependent variable. Patients were observed for 14 days. Whether or not a patient reached the goal was the status. Reaching the goal was the +1 state. A patient who had not reached the goal in 14 days, was treated as a censored case. Group=1 is the PEGJ group. Group=2 is the PEG group.

NUTRSD NUTRGOAL DAYSGOAL GOALIN14 GROUP ISS AGE 02/15/98 02/16/98 1 1 1 29 43 01/10/98 01/12/98 2 1 1 5 88 02/14/98 02/18/98 4 1 1 29 37 02/02/98 02/06/98 4 1 1 27 36 01/10/98 01/13/98 3 1 1 13 92 01/09/98 . 15 0 2 19 73 01/02/98 01/04/98 2 1 2 26 42 01/20/98 01/22/98 2 1 2 36 55 03/18/98 . 5 1 1 27 23 02/04/98 02/06/98 2 1 2 13 72 01/23/98 . 15 0 2 10 45 02/01/98 02/02/98 1 1 1 22 59 02/20/98 02/21/98 1 1 1 17 54 02/03/98 02/04/98 1 1 2 14 78 03/31/98 04/02/98 2 1 2 18 30 04/13/98 04/15/98 2 1 2 27 49 05/08/98 05/09/98 1 1 2 9 22 04/14/98 04/20/98 6 1 2 9 60 05/27/98 05/28/98 1 1 1 17 27 05/13/98 . 15 0 2 29 95 05/07/98 05/16/98 9 1 2 25 31 04/16/98 04/17/98 1 1 2 32 31 03/23/98 03/25/98 2 1 2 20 41 04/07/98 04/08/98 1 1 2 16 29 03/29/98 03/30/98 1 1 1 25 24 04/30/98 05/01/98 1 1 2 29 52 05/05/98 05/08/98 3 1 2 38 79 05/28/98 05/30/98 2 1 1 4 76 06/08/98 06/10/98 2 1 2 16 70 05/27/98 05/28/98 1 1 1 9 27 04/27/98 04/29/98 2 1 1 22 87 04/10/98 04/11/98 1 1 1 27 36 02/26/98 03/04/98 6 1 1 25 54 03/27/98 03/28/98 1 1 1 29 22 04/17/98 04/18/98 1 1 1 22 22 02/25/98 03/05/98 8 1 1 25 79 03/18/98 03/19/98 1 1 1 25 56 01/28/98 01/29/98 1 1 1 17 66 03/23/98 03/24/98 1 1 1 16 20 04/29/98 05/03/98 4 1 1 26 22 07/19/98 08/02/98 14 1 2 34 33 08/13/98 08/15/98 2 1 1 25 49 08/25/98 . 15 0 2 26 77 10/06/98 10/07/98 1 1 2 34 19 09/10/98 09/11/98 1 1 2 27 36 08/14/98 08/15/98 1 1 1 30 35 08/25/98 08/27/98 2 1 2 27 29 09/20/98 09/21/98 1 1 2 36 62 09/29/98 10/01/98 2 1 2 17 19 10/09/98 . 15 0 2 38 74

Survival Analysis – 49 Printed on 10/26/2016 NUTRSD NUTRGOAL DAYSGOAL GOALIN14 GROUP ISS AGE 10/02/98 10/03/98 1 1 1 10 40 08/26/98 09/04/98 9 1 2 18 48 08/19/98 08/21/98 2 1 1 18 31 08/03/98 08/04/98 1 1 1 41 46 08/25/98 08/28/98 3 1 2 24 37 09/17/98 . 15 0 2 26 75 07/02/98 . 15 0 1 19 28 08/03/98 08/05/98 2 1 2 13 52 07/15/98 07/17/98 2 1 2 38 71 07/27/98 08/01/98 5 1 2 34 33 04/30/98 05/02/98 2 1 2 4 61 05/29/98 05/30/98 1 1 1 29 58 05/16/98 05/18/98 2 1 2 19 42 06/20/98 06/23/98 3 1 1 25 19 08/30/98 . 15 0 1 25 70 04/30/98 05/02/98 2 1 2 43 33 07/01/98 07/02/98 1 1 1 43 79 09/29/98 . 15 0 2 17 18 05/28/98 06/08/98 11 1 2 36 57 07/15/98 07/16/98 1 1 2 27 59 08/11/98 08/12/98 1 1 1 19 43 10/12/98 10/13/98 1 1 1 36 18 08/24/98 08/25/98 1 1 1 20 84 10/22/98 . 15 0 1 25 17 10/08/98 10/09/98 1 1 2 25 20 10/06/98 . 15 0 2 17 31 07/30/98 08/02/98 3 1 1 22 26 04/16/98 04/17/98 1 1 1 38 18 10/08/98 10/09/98 1 1 1 25 34 08/19/98 08/21/98 2 1 1 34 22 03/20/98 03/21/98 1 1 1 25 48 06/20/98 06/21/98 1 1 1 11 45 07/30/98 07/31/98 1 1 1 25 33 09/07/98 . 15 0 2 36 28 07/17/98 07/18/98 1 1 1 22 62 09/15/98 09/17/98 2 1 2 20 47 07/07/98 07/08/98 1 1 1 33 27 10/01/98 10/02/98 1 1 2 25 33 09/11/98 09/12/98 1 1 1 41 31

Specifying the analysis using Life Tables . . .

Survival Analysis – 50 Printed on 10/26/2016 The output of LIFE TABLES SURVIVAL TABLE=DAYSGOAL BY GROUP(1 2) /INTERVAL=THRU 15 BY 1 /STATUS=GOALIN14(1) /PRINT=TABLE /PLOTS ( SURVIVAL)=DAYSGOAL BY GROUP .

Survival Analysis

G:\MdbT\P595\P595AL07-Survival analysis\PEGPEGJData.sav

Survival Variable: DAYSGOAL

Life Table

Std. Error of Cum ulative Cum ulative Nu m ber Proporti on Proportion Std. E rror Std. Num ber With drawin Num b er Num ber of Proportion Su rviving at Surviving at of Error of Interval Start Enterin g g durin g Exposed to T erm inal T erm inatin Proportion End of End of Prob ability Proba bility Hazard Hazard First-o rder Control s T im e Interval Interval Risk Events g Su rviving Interval Interval Density Den sity Rate Rate GROUP 1 .000 46 0 46.000 0 .00 1.00 1 .00 .00 .0 00 .000 .00 .00 1.000 46 0 46.000 2 8 .61 .39 .39 .07 .6 09 .072 .88 .15 2.000 18 0 18.000 6 .33 .67 .26 .06 .1 30 .050 .40 .16 3.000 12 0 12.000 3 .25 .75 .20 .06 .0 65 .036 .29 .16 4.000 9 0 9.000 3 .33 .67 .13 .05 .0 65 .036 .40 .23 5.000 6 0 6.000 1 .17 .83 .11 .05 .0 22 .022 .18 .18 6.000 5 0 5.000 1 .20 .80 .09 .04 .0 22 .022 .22 .22 7.000 4 0 4.000 0 .00 1.00 .09 .04 .0 00 .000 .00 .00 8.000 4 0 4.000 1 .25 .75 .07 .04 .0 22 .022 .29 .28 9.000 3 0 3.000 0 .00 1.00 .07 .04 .0 00 .000 .00 .00 10.000 3 0 3.000 0 .00 1.00 .07 .04 .0 00 .000 .00 .00 11.000 3 0 3.000 0 .00 1.00 .07 .04 .0 00 .000 .00 .00 12.000 3 0 3.000 0 .00 1.00 .07 .04 .0 00 .000 .00 .00 13.000 3 0 3.000 0 .00 1.00 .07 .04 .0 00 .000 .00 .00 14.000 3 0 3.000 0 .00 1.00 .07 .04 .0 00 .000 .00 .00 2 .000 43 0 43.000 0 .00 1.00 1 .00 .00 .0 00 .000 .00 .00 1.000 43 0 43.000 1 1 .26 .74 .74 .07 .2 56 .067 .29 .09 2.000 32 0 32.000 1 5 .47 .53 .40 .07 .3 49 .073 .61 .15 3.000 17 0 17.000 2 .12 .88 .35 .07 .0 47 .032 .13 .09 4.000 15 0 15.000 0 .00 1.00 .35 .07 .0 00 .000 .00 .00 5.000 15 0 15.000 1 .07 .93 .33 .07 .0 23 .023 .07 .07 6.000 14 0 14.000 1 .07 .93 .30 .07 .0 23 .023 .07 .07 7.000 13 0 13.000 0 .00 1.00 .30 .07 .0 00 .000 .00 .00 8.000 13 0 13.000 0 .00 1.00 .30 .07 .0 00 .000 .00 .00 9.000 13 0 13.000 2 .15 .85 .26 .07 .0 47 .032 .17 .12 10.000 11 0 11.000 0 .00 1.00 .26 .07 .0 00 .000 .00 .00 11.000 11 0 11.000 1 .09 .91 .23 .06 .0 23 .023 .10 .10 12.000 10 0 10.000 0 .00 1.00 .23 .06 .0 00 .000 .00 .00 13.000 10 0 10.000 0 .00 1.00 .23 .06 .0 00 .000 .00 .00 14.000 10 0 10.000 1 .10 .90 .21 .06 .0 23 .023 .11 .11

Median Surv ival Time

First-orde r Co ntro ls M e d T im e G RO UP 1 1 .8 2 2 2 .7 0

Survival Analysis – 51 Printed on 10/26/2016 First-order Control: GROUP

These data are strange because the “event” is something that is sought after - reaching a feeding goal, rather than something that is to be avoided - death or termination. So for these data, lower "survival" is preferred, since the "event" is not death, but reaching a nutrition goal. The sooner a patient reached the nutrition goal the better. Thus, the investigators hoped that patients in the PEJ condition would reach those goals faster, leading to lower "survival" curves. In this case, survival should be called "Failure to reach feeding goal."

Survival Analysis – 52 Printed on 10/26/2016 Analysis of the same data using Kaplan-Meier

KM DAYSGOAL BY GROUP /STATUS=GOALIN14(1) /PRINT TABLE MEAN /PLOT SURVIVAL HAZARD /TEST LOGRANK BRESLOW TARONE /COMPARE OVERALL POOLED .

Kaplan-Meier

Cen so red G RO UP T o ta l N N o f E ven ts N P e rce nt 1 4 6 43 3 6 .5 % 2 4 3 34 9 20 .9% O veral l 8 9 77 12 13 .5%

Means and Medians for Survival Time

a M ea n M ed ia n 9 5% Co nfid en ce In te rva l 95 % Con fi de n ce Interval G RO UP E sti m ate S td . Error Lo we r Bo un d Up p er Bo u nd E stim a te S td . E rro r L o wer B o un d Upp e r B ou n d 1 2 .7 17 .52 7 1 .6 8 5 3 .7 5 0 1.0 0 0 . . . 2 5 .4 88 .85 7 3 .8 0 8 7 .1 6 9 2.0 0 0 .21 4 1 .5 81 2 .4 19 O vera ll 4 .0 56 .51 7 3 .0 4 3 5 .0 6 9 2.0 0 0 .21 1 1 .5 87 2 .4 13

a . Esti m a tio n is lim ited to th e larg est survival tim e if it is cen so red.

Survival Analysis – 53 Printed on 10/26/2016 Ov erall Comparisons

Ch i-S q u a re d f S i g. L o g Ra n k (M a n te l-Co x) 8 .4 7 9 1 .0 0 4 B re sl o w (G e n e ra l ize d 9 .5 8 8 1 .0 0 2 Wi lco xo n ) T a ro n e -Wa re 9 .3 0 6 1 .0 0 2

T e st of e q u a li ty o f su rviva l d istrib utio n s fo r th e d iffe re n t l e ve ls o f G RO UP .

Survival Analysis – 54 Printed on 10/26/2016 The same analysis using Cox Regression

Survival Analysis – 55 Printed on 10/26/2016 COXREG DAYSGOAL /STATUS=GOALIN14(1) /PATTERN BY GROUP /CONTRAST (GROUP)=Indicator(1) /METHOD=ENTER GROUP /PLOT SURVIVAL HAZARD /CRITERIA=PIN(.05) POUT(.10) ITERATE(20) .

Cox Regression

N Pe rcent Ca se s a va ilab le E ven ta 77 8 6.5% in ana lysis Cen so red 12 1 3.5% T otal 89 10 0 .0 % Ca se s d ro pp ed Cases with m issin g val ue s 0 .0% Cases with ne g ative tim e 0 .0% Cen so red case s b e fo re the e a rliest e ve nt i n a 0 .0% stra tu m T otal 0 .0%

T o ta l 89 10 0 .0 %

a . Depe n de nt V aria bl e: DA YS GO A L

Categorical Variable Codingsb

Fre qu ency (1 ) G RO UP a 1 4 6 0 2 4 3 1

a . In dicato r Pa ra m eter Co d in g

b . Ca te go ry va riab le : G RO UP

Block 0: Beginning Block

Omnibus Tests of Model Coefficients

-2 Lo g Likeli ho od 6 18 .2 81

Omnibus Tests of Model Coefficientsa,b

O ve ra ll (sco re ) Ch ange Fro m P re vio us S te p Ch a ng e From P revio u s B lock -2 Lo g L ike lih o od Chi -sq ua re d f S ig . Ch i-sq u are df S ig . Ch i-squ a re d f S ig. 61 2 .895 5 .4 4 8 1 .0 2 0 5.38 5 1 .0 2 0 5 .3 85 1 .0 20

a . B e ginn in g Bl ock Nu m be r 0, in itia l L og Li kelih oo d fu nction: -2 L o g likeliho od : 61 8 .2 81

b . B e ginn in g Bl ock Nu m be r 1. M eth od = E nter

Survival Analysis – 56 Printed on 10/26/2016 Variables in the Equation

B S E Wa ld d f S ig. E xp(B ) G RO UP -.5 42 .2 35 5 .3 3 2 1 .0 2 1 .58 2

Cov ariate Means and Pattern Values

P atte rn M ea n 1 2 G RO UP .48 3 .00 0 1 .0 0 0

The above graph presents predicted proportions. They are analogous to plots of y-hats vs. predictors in a regression analysis.

When you perform a Cox-regression analysis, you may also have to run a Kaplan-Meier analysis just for the observed survival curves the K-M procedure produces.

Survival Analysis – 57 Printed on 10/26/2016 Survival Analysis – 58 Printed on 10/26/2016