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## Survival models for SEM

### Highlights

- Structural equation models with survival outcomes
- Latent predictors of survival outcomes
- Path models, growth curve models, and more
- Multilevel models — random intercepts and random coefficients
- Survival outcomes with other outcomes

- Right-censoring
- Left-truncation
- Parametric models — exponential, loglogistic, Weibull, lognormal, and gamma survival distributions
- Support for complex survey data
- Marginal predictions and marginal means

Survival-time outcomes measure durations such as time to death, length of hospital stay, time to recurrence of a particular type of cancer, or length of time living in a city. We can fit survival models to evaluate the effects of covariates on these survival times.

For example, we might be interested in high school students who
are at risk of dropping out of school. We can model the number
of months from the start of ninth grade until dropout as a function
of grade point average, attendance, and eighth-grade achievement
test scores. We can fit either a parametric or a semiparametric
model with Stata's **streg** or **stcox** commands.

For more complex models, we can use **gsem** to model
survival-time outcomes as part of a larger structural equation
model. **gsem** fits multilevel structural equation models and
structural equation models with binary, ordinal, count, and other
types of outcomes. What does this mean for modeling
survival-time outcomes? Let's consider some extensions to our
model for time to dropout.

Suppose that at the beginning of ninth grade, each at-risk
student in our sample answered a series of questions regarding
the support each receives from parents, teachers,
counselors, and peers. The responses to these questions measure
the student's perceived level of support, a latent concept that
we believe affects time to dropout. **gsem** allows us to include
a latent variable as a predictor of time to dropout.

Perhaps we believe that time to dropout is also affected by
unobserved characteristics of the school such as an administrator's
abilities to identify early warning signs of dropout. With
**gsem**, we can include school-level random effects in our model.

Say that we are also interested in modeling a count outcome, the
number of times a student has been in trouble at school. We
believe this outcome is also related to the student's perceived
level of support. With **gsem**, we can simultaneously fit a
Weibull model for time to dropout and a Poisson model for number
of offenses at school, and the two models can be tied together
through the student's perceived level of support. We can also
include number of offenses as a predictor of time to dropout.

In other words, **gsem** allows us to extend parametric models for
survival-time outcomes to include latent variables, to include
multiple levels of random effects, and to include
multiple outcomes.

Now, let's consider another type of survival-time outcome.

We want to analyze survival times of nursing home residents. We have censored data; thankfully, not all the residents have died yet.

We posit that survival times are determined by age, depression level, and overall health. Both depression and overall health are latent traits. We have four variables that measure aspects of depression and four variables that measure aspects of overall health. We fit a Weibull model for survival time along with the measurement models for depression and overall health.

We can create our model using Stata's SEM Builder:

Or we can type the command

. gsem (surv_time <- age Depress Health, family(weibull, fail(death))) (Depress -> dep1 dep2 dep3 dep4) (Health -> hlth1 hlth3 hlth3 hlth4), variance(Depress@1 Health@1)

The results are

Coef. Std. Err. z P>|z| [95% Conf. Interval] | ||

surv_time <- | ||

age | .1078273 .0120192 8.97 0.000 .0842701 .1313844 | |

Depress | 1.2204 .1143472 10.67 0.000 .9962836 1.444516 | |

Health | -1.782012 .1526019 -11.68 0.000 -2.081106 -1.482918 | |

_cons | -6.949501 .8724378 -7.97 0.000 -8.659448 -5.239554 | |

dep1 <- | ||

Depress | 1.038601 .0409685 25.35 0.000 .9583046 1.118898 | |

_cons | .091441 .0510287 1.79 0.073 -.0085734 .1914553 | |

dep2 <- | ||

Depress | .5079995 .0381469 13.32 0.000 .433233 .582766 | |

_cons | .0439719 .039877 1.10 0.270 -.0341856 .1221295 | |

dep3 <- | ||

Depress | .7126734 .0370931 19.21 0.000 .6399723 .7853746 | |

_cons | .0856129 .0421749 2.03 0.042 .0029516 .1682743 | |

dep4 <- | ||

Depress | 1.248634 .055827 22.37 0.000 1.139215 1.358053 | |

_cons | .0591629 .0665563 0.89 0.374 -.0712852 .1896109 | |

hlth1 <- | ||

Health | 1.062984 .0414027 25.67 0.000 .9818361 1.144132 | |

_cons | -.0424677 .0523266 -0.81 0.417 -.1450259 .0600906 | |

hlth3 <- | ||

Health | .4933346 .038082 12.95 0.000 .4186953 .5679738 | |

_cons | -.0357306 .0398651 -0.90 0.370 -.1138647 .0424035 | |

hlth3 <- | ||

Health | .7293404 .0362297 20.13 0.000 .6583315 .8003493 | |

_cons | .0165731 .0419967 0.39 0.693 -.0657389 .09888515 | |

hlth4 <- | ||

Health | 1.273872 .0550442 23.14 0.000 1.165988 1.381757 | |

_cons | -.0487405 .0668903 -0.73 0.466 -.179843 .0823621 | |

surv_time | ||

/ln_p | -.5314218 .0716854 -7.41 0.000 -.6719227 -.390921 | |

var(Depress) | 1 (constrained) | |

var(Health) | 1 (constrained) | |

cov(Health, | ||

Depress) | .0411 .0495383 0.83 0.407 -.0559932 .1381932 | |

var(e.dep1) | .2232991 .0293288 .1726185 .2888595 | |

var(e.dep2) | .5370324 .0357861 .4712802 .6119583 | |

var(e.dep3) | .3814731 .0282049 .3300113 .4409598 | |

var(e.dep4) | .6558288 .0570452 .5530335 .7777311 | |

var(e.hlth1) | .2397716 .0264884 .1930913 .297737 | |

var(e.hlth2) | .5513769 .0362948 .4846381 .6273062 | |

var(e.hlth3) | .350238 .0257087 .3033068 .404431 | |

var(e.hlth4) | .6153669 .0515441 .5221991 .7251571 | |

**gsem** reports coefficients. Because we fit a Weibull
model for **surv_time**, exponentiated coefficients can be interpreted as
hazard ratios and are reported by **estat eform**.

.estat eform surv_time

surv_time | exp(b) Std. Err. z P>|z| [95% Conf. Interval] | |

age | 1.113855 .0133876 8.97 0.000 1.087923 1.140406 | |

Depress | 3.388543 .3874702 10.67 0.000 2.708198 4.2398 | |

Health | .1682992 .0256828 -11.68 0.000 .1247921 .2269745 | |

_cons | .0009591 .0008368 -7.97 0.000 .0001735 .0053026 | |

**Depress** and **age** have hazard ratios greater than 1; being more
depressed and being older both correspond to increased hazard and thus
decreased survival times. **Health** has a hazard ratio less than 1; better
overall health corresponds to increased survival times.

We use **margins** to compute marginal mean survival times for
ages ranging from 70 to 85.

.margins, at(age=(70(3)85)) predict(mu outcome(surv_time)) noatlegendAdjusted predictions Number of obs = 500 Model VCE : OIM Expression : Marginal predicted mean (analysis time when record ends), predict(mu outcome(surv_time))

Delta-method | ||

Margin Std. Err. z P>|z| [95% Conf. Interval] | ||

_at | ||

1 | 5.263968 1.965496 2.68 0.007 1.411666 9.116271 | |

2 | 3.035966 1.111428 2.73 0.006 .8576069 5.214325 | |

3 | 1.750977 .6397585 2.74 0.006 .4970736 3.004881 | |

4 | 1.009867 .3749436 2.69 0.007 .2749909 1.744743 | |

5 | .5824353 .2233669 2.61 0.009 .1446443 1.020226 | |

6 | .3359165 .1348604 2.49 0.013 .0715949 .600238 | |

We can plot these marginal means using **marginsplot**.

As expected, the predicted marginal mean survival time decreases with age.

You can read more about Stata's SEM features and see several worked examples in *Stata Structural Equation Modeling Reference Manual*.