1. Using Survival Analysis to analyze degree completion
Janice Love
University of California, Los Angeles
Office of Academic Planning & Budget
CAIR 2014

2. Agenda Survival Analysis History & Background Overview
Survival Analysis example using SPSS
Results of Survival Analysis

3. Survival Analysis Background
Definition
A statistical method for studying the time to an event. The term “survival” suggests that the event of interest is death but the technique is useful for other types of events.
Alternative terminology
Event analysis, Time series analysis, Time-to-event analysis
Survival analysis –studies involving time to death (biomedical sciences)
Reliability theory / Reliability analysis (engineering)
Duration analysis / Duration modeling (economics)
Event history analysis (Sociology)
Uses
Clinical trials
Cohort studies

4. Example of Survival Probability Graph

5. Example of Survival Probability Graph

6. Example of Survival Probability Graph

7. Survival Analysis History
Unknown – been around for a few hundred years
Techniques developed in medical / biological sciences
World War II –military vehicles (reliability and failure time analysis)
The Kaplan-Meier Estimator was introduced with the publication of NONPARAMETRIC ESTIMATION FROM INCOMPLETE OBSERVATIONS – E. L. Kaplan / Paul Meier, 1958
Cited 34,000 times as of 2011

8. Survival analysis - Overview
A set of statistical methods where the outcome variable is the time until the occurrence of an event of interest
Follows cohort over specified time period with focus on an event
Useful when the rate of the occurrence of the event varies over time
Differs from other statistical methods: handles censored data (the withdrawal of individuals from the study)
Censored observations :
Individuals who have not experienced “the event” by the end of the study
Right censoring
Study participant can’t be located
or lives beyond the end of the study
or drop outs before the study is completed
or is still enrolled
An observation with incomplete information
Don’t have to handle these individuals as “missing”
Do have to follow rules with respect to censored data
# of censored should be small relative to non-censored
Censored and non-censored population should be similar (Kaplan-Meier)

9. Survival analysis - Censoring

10. Survival analysis - Censoring
Consequences of mishandling or ignoring censored data:
Example
Student cohort, N = 50, event of interest = Graduation
Still enrolled at the end of the study, N = 6
No longer enrolled but did not graduate, N = 4
Options:
Code all 10 as missing
Code 4 as missing, 6 as graduated as of study end
Consequences:
Mean time to degree is over or understated
selection bias risk
Ignoring censored records completely or arbitrarily assigning event dates introduces bias into the results
Inclusion of the censored data produces less bias. Newell/Nyun 2011

11. Survival analysis – handling censored data
Two methods to produce the cumulative probability of survival that the survival graph is based upon:
SPSS Life Table: (Each time period) the effective size of the cohort is reduced by ½ of the censored group
Kaplan-Meier Survival Table: The survival probability estimate for each time period, except the first, is a compound conditional probability

12. Survival analysis - Overview
Data required for analysis:
Clearly defined event: (death, onset of illness, recovery from illness, marriage, birth, mechanical failure, success, job loss, employment, graduation).
Terminal event
Event status (1 = event occurred, 0 = event did not occur)
Time variable = Time measured from the entry of a subject into the study until the defined event. Months, terms, days, years, seconds.
Covariates:
To determine if different groups have different survival times
Gender, age, ethnicity, GPA, treatment, intervention
Regression models

13. Survival analysis – SPSS Data layout
Basic student data
Time variable – terms enrolled
Event status – graduation status
Binary or dummy variables
Censored indicator
Group into categories

14. Cohort Description
Undergraduates, one division
Fall 2006, Fall 2007 entering freshmen, N = 884
Respondents to 2008 UCUES* survey
Freshmen admits (transfers excluded)
1st term gpa >= 3.0
Censored = 10 or 1.1%
Explanatory variables available: gender, URM status, domestic-foreign status, Pell Grant recipient status, hours worked (survey), double/triple major
* UCUES = University of California Undergraduate Survey

15. Survival Analysis – SPSS
Analyze
Survival
Life Tables

16. Sample Data – Working in SPSS
Analyze
Survival
Life Tables

17. Survival Analysis – Life Table produced by SPSS
primary output of the survival analysis procedure
Intervals = terms. count is from admit term
Count of still enrolled students at start of term

18. Survival Analysis – Life Table produced by SPSS
primary output of the survival analysis procedure
# exposed to risk: # entering interval minus ½ censored
# terminal events = # graduated
Probability Density = Estimated probability of graduating in interval
# withdrawing during interval = censored
Proportion Terminating: # Terminal events ÷ # exposed to risk: example Term 10 = 38 ÷ = .05
Hazard Rate = Instantaneous failure rate. % chance of graduating given not having graduated at start of interval
Cumul. Surviving = cumulative % of those surviving at end of interval = ( ) ÷ 884 = 0.90
Proportion surviving = 1 – proportion terminating

19. Survival Function Graph Produced by SPSS
The proportion of the cohort that has survived (still enrolled) at any term
Each step of the curve represents an event
There is a 90% probability of surviving to the end of 10th term.
Surviving = remaining enrolled!

20. Function & One minus a function
y = x2
y = 1-x2
y = x+1
y = 1- (x+1)

21. One Minus survival function
There is a 10% probability of not-surviving to the end of 10th term.
Not surviving = graduating!!

22. Survival Analysis: SPSS, with Covariate Factor = Gender
Analyze
Survival
Life Tables
SURVIVAL TABLE=Terms_enrolled BY Gender(1 2)
/INTERVAL=THRU 15 BY 1
/STATUS=graduated(1)
/PRINT=TABLE
/PLOTS (SURVIVAL OMS)=Terms_enrolled BY Gender.

23. Survival Analysis – SPSS, Life Table by gender
Hazard Rate = Instantaneous failure rate. % chance of graduating given not having graduated at start of interval
Median Survival Time = Time at which 50% of the original cohorts have not-survived (graduated)

24. Survival Analysis: Hazard Ratio
Hazard Ratio = ratio of the hazard rates.
At 12th term, Hazard ratio = / 1.41 = 1.16, females are 16% more likely to graduate in the 12th term than males
At 13th term, Hazard ratio = .41 / .62 = .66, females are 34% less likely to graduate in the 13th term than males

25. Survival functions - SPSS Factor = gender
Survival Pattern: SPSS will produce a different colored line for each of the factor’s values

26. Survival Analysis: Kaplan-meier Method
Assumptions
Censored individual – student who has not experienced the event (graduated) by the end of the study, e.g. they are no longer enrolled
Check for differences between censored and non-censored groups
Cohorts should behave similarly – groups entering at different times should be similar
Avoid “selection bias” in data

27. Survival functions – SPSS, Kaplan_meier Factor = gender
KM Terms_enrolled BY Gender
/STATUS=graduated(1)
/PRINT TABLE MEAN
/PLOT SURVIVAL
/TEST LOGRANK BRESLOW TARONE
/COMPARE OVERALL POOLED.

28. Kaplan-Meier Survival TAble
This is an example of the survival table produced by the Kaplan-Meier procedure.
Kaplan-Meier Survival Probability Estimate calculation example:
Interval 4: Cumulative Proportion Surviving =
# remaining / # at risk =
[(# at start of interval - (# censored + # of events)]
÷ [# at start of interval - # of events] =
[(46 – (2 + 1)] ÷ [(46 – 2)] = 43 ÷ 44 = 0.978
Interval 5: Cumulative Proportion Surviving =
[(43 – (2 + 2)] ÷ (43 – 2) = 39 ÷ 41 = x = 0.930
Kaplan-Meier Survival Table: The survival probability estimate for each time period, except the first, is a compound conditional probability

29. In this way the fudging is kept conceptual, systematic, and automatic.
Kaplan & Meier, 1958

30. Kaplan-Meier Results – Gender
Null Hypothesis: Female Curve = Male Curve

31. Kaplan-Meier output Log Rank weights all graduations equally
Breslow gives more weight to earlier graduations
Taron-Ware is mixture of two

32. Kaplan-Meier Results – Gender
Null Hypothesis: Female Curve = Male Curve
Curves not significantly different at p < .05

33. Cox Regression (Proportional hazards)
Measures influence of explanatory variables
Most used Survival analysis method
Only time independent variables are appropriate
Assumptions: Hazards are proportional

34. Cox Regression, Checking proportional hazards assumption
SPSS
Analyze
Survival
Cox Regression
Repeat for each factor!

35. Cox Regression: Use log minus log function to check Proportional Hazards Assumption
Do not use Cox Regression if the curves cross. This means the hazards are not proportional.

36. Cox Regression Model – Example, Gender
SPSS
Analyze
Survival
Cox Regression
(move gender to Covariates box)

37. Cox Regression Model Results: Example, Gender

38. Interpretation of SPSS Cox Regression Results:
The reference category is female because I made that choice for this model
It is not statistically significant at p < 0.05 that females and males have different survival curves
Exp(B) = Hazard ratio: Female vs. Male The null hypothesis is that this ratio = 1.
Hazard Ratio = eB = e-0.04 = 0.961

39. Cox Regression Model Results: Pell Grant Recipients vs
Cox Regression Model Results: Pell Grant Recipients vs. Non-Pell Grant Recipient
Per Kaplan-Meier Estimation, Pell-Grant Student curve is not equal to non-Pell Grant students curve, highly significant at p < .001
Tip: To edit the default chart, click on the chart until the “Chart Editor” opens

40. Cox Regression Model Results: Pell Grant Recipients vs
Cox Regression Model Results: Pell Grant Recipients vs. Non-Pell Grant Recipient
Pell Grant Recipients
1. Work more hours than non-Pell Grant Recipients
2. Pell Grant Recipients with similar GPAs to non-Pell Grant Recipients have attempted 10 more units

41. Summary Survival Analysis provides the following:
Handles both censored data and a time variable
Life table
Graphical representation of trends
Kaplan-Meier survival function estimator
Survival comparison between 2 or more groups
Regression models – relationships between variables and survival times
p value is produced that indicates if difference between curves is significant or not

42. Descriptive power of survival analysis :
Terms Enrolled by 1st Term GPA – Using Survival Graph (K-M) to display data
At end of 12th term:
~ 34% probability of continued enrollment
~ 9% probability of continued enrollment

43. REFERENCES
Dunn, S. (2002). Kaplan-Meier Survival Probability Estimates. Retrieved from
Harris, S. (2009). Additional Regression techniques, October 2009, Retrieved from
Newell, J. & Hyun, S. (2011). Survival Probabilities With and Without the Use of Censored Failure Times Retrieved from
Singh, R., Mukhopadhyay, K. (2011). Survival analysis in clinical trials: Basics and must know areas, Retrieved from
Wiorkowski, J., Moses, A., & Redlinger, L. (2014).The Use of Survival Analysis to Compare Student Cohort Data, Presented at the 2014 Conference of the Association of Institutional Research
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