# Using Survival Analysis to analyze degree completion

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

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

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

Example of Survival Probability Graph

Example of Survival Probability Graph

Example of Survival Probability Graph

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

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)

Survival analysis - Censoring

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

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

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

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

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

Survival Analysis – SPSS
Analyze Survival Life Tables

Sample Data – Working in SPSS
Analyze Survival Life Tables

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

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

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!

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

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

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.

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)

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

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

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

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.

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

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

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

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

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

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

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

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.

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

Cox Regression Model Results: Example, Gender

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

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

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

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

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