### Discrete - Time Survival Analysis

```Discrete-Time
Survival Analysis
PRESENTED BY
CHIEN-TI LEE
SEPTEMBER 12, 2014
Purpose?
To study the probability (or hazard), of experiencing an
event.
▪ Unlike logistic regression, it takes into account “time” until
the event occurs.
▪ It is also different from continuous-time survival analysis
(e.g., cox regression) in the following ways:
▪ The data are only collected in time intervals (vs. the exact time an
event occurred)
▪ Does not assume hazard-related probability
▪ Probability of hazard: The shape of the survival function over time is the same
for all cases/groups
▪ Can be extended with time-varying covariates, mixture components,
and distal outcomes etc.
Survival Probability vs. Hazard Probability
▪ T stands for the time interval of the event
▪ Survival Probability= S(J)
▪ J = The time interval in which the event occurs
▪ S(J) = P(T > J) means that the probability of surviving beyond time
interval J
▪ Hazard probability= H(J)
▪ H(J) = P(T= J|T ≥ J) means that the probability of the event occurs
in the time interval J, provided it has not occurred prior to j
▪ It is the probability of the event occurring in the interval j among
those at risk in j
▪ S(J) = P(T ≠J|T≥J)P(T≠J-1|T≥ J-1)…P(T≠2 | T≥2) P(T≠1 | T≥1)
=∏ [1- h(k) ]; {k = 1 to a}
▪ ∏ = product of all values in range of series
First you must…Restructure the Dataset
Using the “Data Survival” Command in Mplus to Save Time in Restructuring (See Mplus Manual
7, p.379)
▪ To create variables for discrete-time survival modeling where a binary discrete-time survival
variable represents whether or not a single *non-repeatable* event has occurred in a specific
time period
▪ Here are the rules…
1. If the original variable is missing, the new binary variable is missing
2. If the value of the original variable is *greater than* the cutpoint value, the new binary value
is “1” indicating that the event has occurred
3. If the value of the original variable is less than or equal the cutpoint value, the new binary
value is “0” indicating that the event has not occurred
4. After a discrete-time survival variable for an observation is assigned the value “1”, then
subsequent discrete-time survival variables for that observation are assigned the value of the
missing value flat “ * ”.
Transformation
Before
After
Dep_w1 Dep_w2 Dep_w3 Dep_w4 Dep_w5
Dep_w1 Dep_w2 Dep_w3 Dep_w4 Dep_w5
5
6
5
4
5
1
*
*
*
*
2
3
4
*
3
0
1
*
*
*
2
3
*
*
*
0
1
*
*
*
0
0
1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
2
2
3
3
0
0
0
1
*
0
2
1
2
*
0
0
0
0
*
0
1
1
2
3
0
0
0
0
1
▪ In the context of survival analysis, missing data usually refers to
the event times for some subject that are unknown to the
researcher (aka: censoring).
▪ Right Censoring
▪ When a subject has not experienced the event at the end of the
observation
▪ Left Censoring
▪ When a subject has experienced the event before you began the
observation
▪ It is a very rare phenomenon
▪ The focus of survival analysis is about what happens when risk
exposure begins
Truncation
▪ Left truncation often arises when patient information, such as time of diagnosis, is
gathered retrospectively.
▪ For example, in a study of disease mortality where the outcome of interest is survival
from the time of diagnosis, many patients may not have been enrolled in the study until
several months or years after their diagnosis. Those patients, by virtue of having survived
to the time of enrollment, could not have had an event between diagnosis and the study
enrollment, and therefore they should be removed from the risk set between those two
time points. To leave them in the risk set would bias the survival estimates.
▪ Right truncation happens when the individuals whose event time are less than some
truncation threshold.
▪ For example, the experiment wants to study the effect of smoking before college. Your question
for a group of participants, “when do you start smoking”, can effectively truncate the
participants who start smoking after going to college.
Mplus Syntax – Example 6.20
TITLE: this is an example of a *continuous-time*
survival analysis using the Cox regression model
DATA: FILE = ex6.20.dat;
VARIABLE:
NAMES = t x tc; ! x is the predictor
SURVIVAL = t (ALL); ! t is the variable that contains time-to-event information
TIMECENSORED = tc (0 = NOT 1 = RIGHT); ! Information about right censoring
ANALYSIS:
BASEHAZARD = OFF; ! Non-parametric baseline hazard function is used
MODEL:
t ON x;
Mplus Syntax – Example 6.19
TITLE: this is an example of a discrete-time survival analysis
DATA: FILE IS ex6.19.dat;
VARIABLE:
NAMES ARE u1-u4 x;
CATEGORICAL = u1-u4;
MISSING = ALL (999);
ANALYSIS: ESTIMATOR = MLR;
MODEL:
f BY [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */;
! This represents a proportional odds assumption where the covariate x has the same influence on u1-u4
f ON x;
[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */; ! Residual variance is fixed at zero (default)
Title: YTP project
Data: file = "C:\Users\cl396\Documents\Research Work\Taiwanese Work\USU\TYP\TEST_SURVIVALDAT.dat";
Variable: names are confirm indthk respect harmony ID urban SUBURBAN AGE gender income pedu relation
ASSESSNE DEP1 DEP2 DEP3 DEP4 DEP5 p1-p5 class;
missing are all (-9999);
usevariables are gender income pedu urban relation confirm indthk respect harmony dep1-dep5;
categorical are dep1-dep5;
DSURVIVAL = dep1-dep5;
Analysis:
estimator = MLR;
starts = 1000 250;
optseed=476295;
processors=8 (starts);
Model:
f by [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */;
f on gender income pedu urban
relation confirm indthk respect harmony;
[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */;
Output: tech1 RESIDUAL;
Plot: type is plot2;
Background
▪ The purpose of this study was to identify patterns of cultural values among Taiwanese
youth and explore the relationship of these value affiliations with emerging depressive
symptoms at later time points.
▪ Previous studies showed for Taiwanese youth, the well-being issue is of particular
significance due to competitive educational environment” (Yi, Wu, Wu , Chang &
Chang, 2009, p. 399).”
▪ However, others have shown that there was no difference with the prevalence of
depression between a Chinese and a U.S. sample (Chen, Rubin, Li, 1995; Stewart et al,
2002).
▪ Academic strain is a cultural norm and therefore does not necessary lead to increased
depressive symptoms
Cultural Values
▪ Social Conformity—meeting parental (scholastic) expectations
▪ Fail to meet parental academic expectations, family tension, which has been linked with
depressive symptoms may increase.
▪ Interdependent thinking—receiving parental approval before making important decisions
▪ Western literature generally suggests that individuals are more likely to feel depressed when
they perceive constraints restricting them from reaching a desire outcome or exerting undue
influence on important, personal decisions
▪ Vertical obedience—conforming to the wishes of parents and authority figures
▪ Chinese adolescents’ attitudes have shifted to the point that many believe that parents do not
have absolute authority and power over them
▪ Chinese parents, compared to western families, still exact more parental influence on their
▪ Harmony maintenance—being socially sensitive and self-restraining to ensure peace in
relationship
▪ From a Chinese cultural perspective, free emotional expressions, particularly negative ones,
may disrupt harmony within the family dynamic
Model Diagram Created by Mplus
Model Results
F
ON
Est.
S.E.
Est./S.E.
p value (2-tailed)
GENDER
0.370
0.159
2.325
0.020
INCOME
0.024
0.028
0.867
0.386
PEDU
0.022
0.054
0.414
0.679
URBAN
0.125
0.170
0.737
0.461
RELATION
-0.883
0.159
-5.538
0.000
CONFIRM
0.122
0.178
0.688
0.492
INDTHK
0.086
0.180
0.477
0.633
RESPECT
0.024
0.166
0.142
0.887
HARMONY
0.247
0.178
1.390
0.164
Thresholds
DEP1\$1
2.949
0.832
3.542
0.000
DEP2\$1
6.708
1.028
6.522
0.000
DEP3\$1
4.804
0.871
5.518
0.000
DEP4\$1
3.522
0.848
4.155
0.000
DEP5\$1
3.607
0.874
4.128
0.000
Estimated Baseline Survival Curve for Emerging
Depressive Symptoms with Covariates
Model Diagram
Social
Conformity
Independent
Thinking
Vertical
Obedience
Dep_w1
Dep_w2
c
f
Dep_w3
Dep_w4
Dep_w6
Harmony
Control
Variables
Mplus Diagram Currently Does Not Support Mixture Analysis
Estimated means of cultural values across class
memberships (N = 2,458).
The 5-class average latent class
probabilities for the most likely latent
class membership were .99, .88, .87, .82,
and .80 respectively, indicating
acceptable prediction of class
membership.
TITLE: YTP project
Data: file = "C:\Users\cl396\Desktop\LPA_5class_n.dat";
variable: names are CONFIRM INDTHK RESPECT HARMONY ID1 URBAN SUBURBAN AGE GENDER INCOME PEDU
RELATION ASSESSNE DEP1_D DEP2_D DEP3_D DEP4_D DEP6_D CPROB1 CPROB2 CPROB3 CPROB4 CPROB5 kc;
missing are all (-9999);
usevariables are PEDU URBAN GENDER INCOME ASSESSNE DEP1_D DEP2_D DEP3_D DEP4_D DEP6_D relation
confirm indthk respect harmony CPROB1 CPROB2 CPROB3 CPROB4 CPROB5;
Categorical are DEP1_D DEP2_D DEP3_D DEP4_D DEP6_D;
DSURVIVAL = DEP1_D DEP2_D DEP3_D DEP4_D DEP6_D;
knownclass= c1(kc=1 kc=2 kc=3 kc=4 kc=5);
classes=c (5);
training = CPROB1 CPROB2 CPROB3 CPROB4 CPROB5 (probabilities);
Analysis:
Type=mixture;
Estimator = MLR;
ALGORITHM = INTEGRATION;
starts = 1000 250;
optseed= 490123;
processors=8 (starts);
Model:
%overall%
f by DEP1_D DEP2_D DEP3_D DEP4_D [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */;
f on gender RELATION urban pedu INCOME ASSESSNE;
[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */;
Output: tech1 LOGRANK;
Plot: type is plot2;
Logrank Outputs
LOGRANK OUTPUT
LOGRANK TEST FOR DEP1_D-DEP6_D COMPARING CLASS 2 AGAINST CLASS 1
Chi-Square Value 17.829
Degrees of Freedom 1
P-value
0.000
Estimated Survival Curve for Emerging Depressive Symptoms
with Covariates Stratified by Class Membership
Estimated survival rate of each class membership
across time points (N = 2,310).
1.00
Summary Table of Chi-square Tests for
Estimated Survival Rate between Each
Pair of Class Membership.
Estimated Survival Curve
0.95
0.90
0.85
Class 1 (3%)
Class 2 (30%)
0.80
Class 3 (36%)
Class 4 (17%)
0.75
Class 5 (13%)
0.70
1
2
3
4
Class 2 vs. Class 1
Class 3 vs. Class 1
Class 4 vs. Class 1
Class 5 vs. Class 1
Class 3 vs. Class 2
Class 4 vs. Class 2
Class 5 vs. Class 2
Class 4 vs. Class 3
Class 5 vs. Class 3
Class 4 vs. Class 5
Chi-Square
17.11
6.95
10.60
2.15
5.45
0.56
11.57
1.47
2.30
5.34
p-value
< .0001
0.008
0.001
0.143
0.020
0.455
0.001
0.226
0.129
0.021
5
Time in Waves
Note: The decreased sample size in Fig 2 is due to the missing value in control variables (e.g., family income, parental
education, family relationships).
Findings
▪ Adolescents in Class 1 (3% of the sample) consists of 42.5 %
females with a mean age of 15.49 (SD = .55) and had a
medium/low mean score across cultural values variables.
▪ It is the class most likely to develop depressive.
▪ Class 4 included 18 % of the sample, 46.7 % of whom were
female, and the average age was 15.39 (SD = .50). Adolescents in
this group had the highest mean scores across domains of
traditional cultural values including interdependent thinking.
▪ This Class most closely resembles that of traditional Eastern societies.
▪ They also had the least likely chance of developing depressive
symptoms over time.
Resources
▪ http://www.ats.ucla.edu/stat/mplus/seminars/DiscreteTimeSurvi
val/default.htm
▪ http://www.statmodel.com/bmuthen/masynSurvival2V31.pdf
▪ http://www.ats.ucla.edu/stat/dae/
```