LoveJ-SurvivalAnalysis to analyse degreee completion

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. http://wpfau.blogspot.com/2011/08/safe-withdrawal-rates-and-life.html
Example of Survival Probability Graph

5. http://faculty.tamucc.edu/sfriday/wordpress/?p=1358
Example of Survival Probability Graph

6. http://www.statcan.gc.ca/daily-quotidien/000216/dq000216b-eng.htm
Example of Survival Probability Graph

7. • 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
SURVIVAL ANALYSIS HISTORY
http://articles.chicagotribune.com/2011-08-18/news/ct-met-meier-obit-20110818_1_clinical-trials-research-experimental-treatment

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
o Study participant can’t be located
o or lives beyond the end of the study
o or drop outs before the study is completed
o or is still enrolled
o An observation with incomplete information
o Don’t have to handle these individuals as “missing”
o Do have to follow rules with respect to censored data
o # of censored should be small relative to non-censored
o Censored and non-censored population should be similar (Kaplan-Meier)

9. Censored Event Total
2 3 5
Outcome data
Student 1
Student 2
Student 3
Student 4
Student 5
1 2 3 4 5 6 7 8 9 10 11 12
Time in Terms
Dropped out after5 terms
"Survived"- still enrolledat
the endof the studyperiod
terms enrolled Graduation_status
Student 1 5 0
Student 2 9 1
Student 3 14 0
Student 4 7 1
Student 5 8 1
SURVIVAL ANALYSIS - CENSORING

10. SURVIVAL ANALYSIS - CENSORING
Consequences of mishandling or ignoring censored data:
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
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

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

12.  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
SURVIVAL ANALYSIS - OVERVIEW

13. SURVIVAL ANALYSIS – SPSS DATA LAYOUT
Basic student data
• Time variable – terms enrolled
• Event status – graduation status
terms_enrolled graduate_status gender 1st_term_gpa
Student 1 5 0 1 3.4
Student 2 9 1 0 4.0
Student 3 14 0 1 2.9
Student 4 7 1 1 3.9
Student 5 8 1 0 3.1
Group into
categories
Censored
indicator
Binary or
dummy
variables

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
SPSS
• Analyze
• Survival
• Life Tables

16. SAMPLE DATA – WORKING IN SPSS
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
# withdrawing
during interval =
censored
# exposed to risk:
# entering interval
minus ½ censored
# terminal
events = #
graduated
Proportion
Terminating: #
Terminal events ÷ #
exposed to risk:
example Term 10 =
38 ÷ 829.5 = .05
Proportion
surviving = 1
– proportion
terminating
Probability Density =
Estimated probability of
graduating in interval
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 = (829.5 - 38)
÷ 884 = 0.90

19. SURVIVAL FUNCTION GRAPH PRODUCED BY SPSS
The proportion of the cohort that has survived (still enrolled) at any
term
There is a 90%
probability of surviving
to the end of 10th
term.
Surviving = remaining
enrolled!
Each step of the
curve represents an
event

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
SPSS
• 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
Median Survival Time = Time at which
50% of the original cohorts have not-
survived (graduated)
Hazard Rate =
Instantaneous failure rate.
% chance of graduating
given not having graduated
at start of interval

24. SURVIVAL ANALYSIS: HAZARD RATIO
 Hazard Ratio = ratio of the
hazard rates.
 At 12th term, Hazard ratio =
1.63 / 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
Interval
Start Time
Number
Entering
Interval
Number of
Terminal
Events
Hazard
Rate
0 586 0 .00
1 586 0 .00
2 586 0 .00
3 586 0 .00
4 585 0 .00
5 584 0 .00
6 584 0 .00
7 583 0 .00
8 583 0 .00
9 583 38 .07
10 545 22 .04
11 523 73 .15
12 450 404 1.63
13 46 15 .41
14 28 11 .49
15 17 17 .00
0 298 0 .00
1 298 0 .00
2 298 0 .00
3 298 0 .00
4 298 0 .00
5 298 0 .00
6 298 1 .00
7 296 0 .00
8 296 1 .00
9 295 10 .03
10 285 16 .06
11 268 46 .19
12 222 183 1.41
13 38 18 .62
14 20 6 .36
15 13 13 .00
Life Table - Hazard Rate Column
First-order Controls
Gender Female
Male

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 = 0.951 x 0.978
= 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. • Measures influence of explanatory variables
• Most used Survival analysis method
• Only time independent variables are appropriate
• Assumptions: Hazards are proportional
COX REGRESSION (PROPORTIONAL HAZARDS)

34. COX REGRESSION, CHECKING PROPORTIONAL HAZARDS
ASSUMPTION
Repeat for
each factor!
SPSS
• Analyze
• Survival
• Cox Regression

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. NON-PELL GRANT
RECIPIENT
Tip: To edit the
default chart,
click on the
chart until the
“Chart Editor”
opens
Per Kaplan-
Meier
Estimation,
Pell-Grant
Student curve is
not equal to
non-Pell Grant
students curve,
highly significant
at p < .001

40. 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. 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
SUMMARY

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

43. Contact Info: jlove@ponet.ucla.edu
Thank you!
REFERENCES
Dunn, S. (2002). Kaplan-Meier Survival Probability Estimates. Retrieved from
http://vassarstats.net/survival.html
Harris, S. (2009). Additional Regression techniques, October 2009, Retrieved from
http://www.edshare.soton.ac.uk/id/document/9437
Newell, J. & Hyun, S. (2011). Survival Probabilities With and Without the Use of Censored
Failure Times Retrieved from
https://www.uscupstate.edu/uploadedFiles/Academics/Undergraduate_Research/Reseach_Jo
urnal/2011_007_ARTICLE_NEWELL_HYUN.pdf
Singh, R., Mukhopadhyay, K. (2011). Survival analysis in clinical trials: Basics and must know
areas, Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3227332/t
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