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I am trying to interpret the results of a competing risks survival analysis. I am interested in understanding the effect of the variable party on the risk of a protest. Note that party is a binary variable that takes a value of 1 if a party is present in a given country-year observation and 0 otherwise.

Estimates: 
                         data mean  est        L95%       U95%       se         exp(est)   L95%       U95%     
shape                           NA   1.44e+00   1.16e+00   1.78e+00   1.57e-01         NA         NA         NA
scale                           NA   5.11e+02   8.70e+01   3.00e+03   4.61e+02         NA         NA         NA
party                     7.63e-01   2.86e-01  -2.40e-01   8.11e-01   2.68e-01   1.33e+00   7.86e-01   2.25e+00
party_growth              1.58e-02  -3.77e+00  -9.20e+00   1.65e+00   2.77e+00   2.30e-02   1.01e-04   5.23e+00
gdp_1k                    2.56e+00  -3.23e-02  -8.78e-02   2.32e-02   2.83e-02   9.68e-01   9.16e-01   1.02e+00
chgdpen_fearonlaitin      1.80e-02   3.89e+00  -9.60e-01   8.74e+00   2.47e+00   4.88e+01   3.83e-01   6.22e+03
Oil_fearonlaitin          1.63e-01   1.16e-01  -4.26e-01   6.58e-01   2.77e-01   1.12e+00   6.53e-01   1.93e+00
postcoldwarlag            1.25e-01   3.00e-01  -2.66e-01   8.66e-01   2.89e-01   1.35e+00   7.67e-01   2.38e+00
civiliandictatorshiplag   4.89e-01   5.55e-01  -8.34e-02   1.19e+00   3.26e-01   1.74e+00   9.20e-01   3.30e+00
militarydictatorshiplag   3.87e-01   6.21e-01   2.29e-02   1.22e+00   3.05e-01   1.86e+00   1.02e+00   3.38e+00
communist                 1.55e-01   2.49e-01  -3.86e-01   8.84e-01   3.24e-01   1.28e+00   6.80e-01   2.42e+00
lpopl1_fearonlaitin       8.93e+00  -9.88e-02  -2.37e-01   3.92e-02   7.04e-02   9.06e-01   7.89e-01   1.04e+00
ethfrac_fearonlaitin      4.46e-01  -3.19e-01  -1.10e+00   4.61e-01   3.98e-01   7.27e-01   3.33e-01   1.58e+00
relfrac_fearonlaitin      3.55e-01   3.20e-01  -7.02e-01   1.34e+00   5.22e-01   1.38e+00   4.96e-01   3.83e+00
age                       5.51e+01  -3.67e-02  -5.80e-02  -1.53e-02   1.09e-02   9.64e-01   9.44e-01   9.85e-01

N = 3774,  Events: 56,  Censored: 3718
Total time at risk: 3794
Log-likelihood = -262.1239, df = 15
AIC = 554.2477

Does a positive coefficient on party imply that a covariate lowers the risk of a protest? That is what I have read, but my thought was that a positive coefficient in survival analysis would imply that a covariate is associated with increased hazard and decreased survival times.

For context, these results are from the replication of a study published in a paper. Here is the code the authors used to generate the results:

* REVOLT, CIVIL WAR, AND SUCCESSION ENTRY LEFT OUT BECAUSE IT IS A PERFECT PREDICTOR
streg  party  party_growth gdp_1k chgdpen_fearonlaitin Oil_fearonlaitin postcoldwarlag civiliandictatorshiplag militarydictatorshiplag communist lpopl1_fearonlaitin ethfrac_fearonlaitin relfrac_fearonlaitin  age, distribution(weibull) time
stcurve, hazard
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  • $\begingroup$ We can't tell unless you provide the actual form of the model. It looks like it might be an accelerated-failure-time parametric survival model, which can be modeled in different ways that have different implications for interpreting coefficients. This particular model is likely to be extremely overfit. With only 56 events, you probably shouldn't be trying to estimate more than 3 or 4 coefficients; you are estimating over 12. $\endgroup$
    – EdM
    Commented Aug 26, 2023 at 16:57
  • $\begingroup$ Thanks for the response. I have added the code above, which I believe is for an accelerated failure-time metric. Does that help? $\endgroup$
    – w5698
    Commented Aug 26, 2023 at 18:13
  • $\begingroup$ I was just reading an article that notes the following for AFT model results: "If the coefficient is positive, then the exp(coefficient) will be >1, which will decelerate the event time (increase the mean/median survival time). Similarly, a negative coefficient will reduce the mean/median survival time (accelerate the event time)." Thus, the positive coefficient on party increases the survival time. If the coefficient were instead negative, that would mean that party decreases the survival time. Does that seem like a reasonable interpretation to you? $\endgroup$
    – w5698
    Commented Aug 26, 2023 at 18:22

1 Answer 1

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This is one of several confusions that can arise from different parameterizations of survival models.

The general form of a parameterized accelerated failure time model can be written as

$$\log T = X' \beta + \sigma W, $$

where $T$ is time to event, $X$ is a vector of covariates, $\beta$ is the corresponding vector of regression coefficients, $W$ is an underlying standard probability distribution (maximum extreme value for a Weibull model), and $\sigma$ determines the width of the distribution of (log) event times around $X \beta$. That's the form used by Frank Harrell in Regresssion Modeling Strategies.

In that form, a positive coefficient $\beta$ means that an increase in the corresponding covariate value increases the time to event, other things being equal. That's consistent with the quote in your comment:

If the coefficient is positive, then the exp(coefficient) will be >1, which will decelerate the event time (increase the mean/median survival time). Similarly, a negative coefficient will reduce the mean/median survival time (accelerate the event time).

That is an opposite interpretation from that of a coefficient in a Cox proportional hazard regression, in which case a positive coefficient means a greater hazard of event (shorter survival) as the corresponding covariate value increases.

It's possible, however, to parameterize an accelerated failure time model in the opposite direction, as Germán Rodríguez does in his course notes:

$$\log T = -X' \beta + \sigma W.$$

In that form, a positive regression coefficient $\beta$ means a decrease in the time to event as the corresponding covariate value increases. That's the same general direction as you might expect from a Cox model coefficient, and perhaps more intuitively consistent with the "accelerated" part of "accelerated failure time."

Thus you have to know the details of the software parameterization of a survival model to know how to interpret the sign of a regression coefficient from an accelerated failure time model.

In particular, you must be careful when you compare different software systems, as they might be using opposite parameterizations in this sense. The code you cite seems to be from Stata, while the list of coefficient estimates seems to come from some R survival package. I'm not sure that they parameterize in the same way. The simplest way to check is to fit with both software systems some simple data having a known association of a covariate with outcome, and comparing what you get.

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  • $\begingroup$ Thanks for the incredibly detailed comment. This is very helpful. I ran the models in both R and Stata, and I received the same results. I ran another set of regression models with a different independent variable (a continuous as opposed to dichotomous variable), and the coefficient is negative. This means, in other words, that the variable is associated with a reduced survival time (i.e., accelerate failure)? I guess the trouble is that I have been thinking about survival analysis in the Cox PH framework. $\endgroup$
    – w5698
    Commented Aug 27, 2023 at 0:06

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