#1: You Cannot Directly Interpret the Coefficients as Substantive Effects

Regardless of which duration model you estimate,^{Footnote 4} all duration models with covariates are nonlinear in parameters—the same as logits, probits, and count models, among others. Therefore, we cannot directly interpret the magnitude of any βs as we might in a simple additive linear regression model because they are not equivalent to the coefficients’ substantive effects (marginal or otherwise) on y. Instead, we must generate additional quantities to present our model’s substantive results (King, Tomz, and Wittenberg Reference King, Tomz and Wittenberg2000).

Stated simply, you cannot stop at the regression table. Our hypotheses are usually about x’s effect on y. However, in all nonlinear models, β does not tell us about the relationship between x and y but rather between x and y*.^{Footnote 5} To reach a conclusion about y, we need to convert y* back to the quantity we care about, y, through a link function. In a logit, applying the logistic link transforms y* into a new quantity, y (technically, Pr(y=1)), whose values fall between 0 and 1, yielding probabilities.^{Footnote 6} For duration models, the usual link function is exp(y*), which produces a y (≡ t) that must be greater than 0, as time cannot be negative. Without this link function, β represents x’s effect on either the log-hazard (for proportional hazard models) or the log-duration (for accelerated failure time models),^{Footnote 7} neither of which is likely to be the focus of our hypotheses. Therefore, you must transform the βs in some way to glean substantive meaning, as discussed in Maxim #2. How you should transform the βs is directly related to your hypothesis about x’s effect on y—specifically, the way in which you framed your discussion of y, as we discuss further in Maxim #3.

#2: To Generate Substantive Effects, You Will Need to Do “Something” to the Coefficients

There are several ways to substantively interpret duration-model results. Each interpretation is technically correct (see Maxim #3), which is both a blessing and a curse—no matter which strategy you choose, your inferences will be technically sound, but clear-cut standards cannot exist regarding when some strategies perform better than others. Therefore, determining the best strategy for your work entails getting a sense of what the different strategies are, how they relate to one another, and which questions a respective strategy allows you to address most easily.

We loosely group extant interpretation strategies along two dimensions. The first relates to the underlying quantity of interest. The second dimension pertains to whether the quantity is an absolute or a relative quantity. The result is four groupings, shown in table 1.

Table 1 Current Approaches to Substantive Interpretation

There are four noteworthy observations from the table:

1. 1. All interpretation strategies involve either exponentiating the model’s βs or generating a predicted quantity using the βs, for reasons discussed in Maxim #1.

2. 2. Duration models have two families of predicted quantities. These focus on how long until something occurs (i.e., the duration, t) or, equivalently, on the risk that something occurs (i.e., the hazard, h(t)); t and h(t) are the duration model equivalent of OLS’s predicted y $\left( {\hat{y}} \right)$.^{Footnote 8} For exposition purposes, think of hazards as being conceptually similar to probabilities, but also note the two are not usually synonyms.

3. 3. Parametric duration models are built from asymmetric distributions, which means the mean and median survival times will not be equal. Observed survival times tend to be right-skewed, and the implications of computing any skewed variable’s mean versus median apply equally in a duration context. Furthermore, there may be right-censored subjects—subjects that will eventually experience the event of interest but have not experienced it yet when last observed.^{Footnote 9} Typically, right-censored observations fall in the right tail of t’s observed distribution.^{Footnote 10} As a general rule of thumb, the more right-censored subjects there are, the more appealing the median becomes.

4. 4. There are numerous ways to quantify “risk” from a duration model, including the risk of an event occurring (the hazard, h(t)); the probability of an event not occurring (the survivor function, S(t)); the total risk that an event will have occurred (the cumulative hazard function, H(t)); and the probability that an event will have occurred (transition probabilities). Each of these is a different way to express the same underlying concern: how (un-)likely is it that an event will occur by some point in time.

Some of these quantities are easier to compute than others, depending on users’ statistical program of choice. We inventory R and Stata’s respective capabilities in appendix D.

#3: All of the Techniques Are Correct from a Technical Perspective, but Some Make More Sense Than Others

All of these interpretation strategies come from the same underlying model estimates (i.e., the βs and standard errors). Thus, all of these strategies are correct, from a technical perspective,^{Footnote 11} the same way that odds ratios and predicted probabilities are equally correct ways of interpreting logit output. Therefore, you must use other nontechnical criteria to guide decisions about which strategy to employ. We suggest considering two criteria, both relating to your paper’s presentation.

First, consider the extent to which a given interpretation strategy matches your theory and hypotheses. Reference Kropko and HardenKropko and Harden (forthcoming) make this point succinctly: If your hypothesis is framed in terms of durations—for instance, how long until a civil war recurs—then presenting your results in terms of durations logically follows (e.g., mean or median duration). Conversely, if your hypothesis is framed in terms of events (e.g., which factors make a civil war more likely to recur), then presenting the results in terms of risk-based quantities would be a better match. These quantities speak more directly to an event’s occurrence (or lack thereof) by depicting the conditions under which subjects are more likely to “survive”—here, that states remain at peace. Although our first point appears fairly simple and logical, misalignment between framing and interpretation strategies is rampant in political science. Reference Kropko and HardenKropko and Harden’s (forthcoming) meta-analysis of 80 articles using Cox duration models reveals that approximately 33 (41.25%) have predominantly duration frames and another 10 to 14 use both frames equally (12.5%–17.5%). However, none of these articles generate duration-based quantities for interpretation.

Second, consider the relative ease with which a particular post-estimation quantity allows you to present and interpret your results. Some techniques may be more straightforward for your audience to understand than others. We return to this point in our broader discussion of the various quantities’ strengths and weaknesses.

#4: Whatever Technique You Choose, You Will Need p-Values or Confidence Intervals

With duration models, as with other regression models, measures of uncertainty around our predicted quantities improve our ability to make inferences. Standard practice for logit/probit models is to report such measures around both the estimated coefficients and any predicted quantities. However, standard practice for duration models is much less consistent. A review of all articles in the Journal of Politics in 2017 underscores this point. Of these articles, 10 report at least one logit or probit model in the main text, and 9 of those 10 analyses generate a post-estimation quantity^{Footnote 12} with some measure of uncertainty, suggesting that this practice is well internalized.^{Footnote 13} In comparison, four articles estimate duration models, with measures of uncertainty reported less consistently. They are reported for any first differences and hazard ratios but not for survival curves.

This pattern holds more broadly: practitioners inconsistently report measures of uncertainty for duration model post-estimation quantities. We examine all articles in the Journal of Politics, American Journal of Political Science, and American Political Science Review from 2012 to 2016 and assess whether they report (1) a Cox model in the main text, and (2) any post-estimation quantities in the main text.^{Footnote 14} There are 16 such articles,^{Footnote 15} of which four do not report any post-estimation quantities (25%).

Of the remaining 12 articles that do report post-estimation quantities,^{Footnote 16} hazard ratios appear most frequently (9 of 12), each time with a measure of uncertainty. However, hazard ratios have weaknesses stemming from being a measure of relative change, as we elaborate on in the next section. Five of these 12 articles report only hazard ratios (Cox’s “Rel” segment in figure 1), equivalent to a logit analysis reporting and interpreting odds ratios only. Finally, 6 of the 12 articles report a survivor curve (S(t)) and/or hazard rates (h(t)), but only one includes a measure of uncertainty around the quantity.^{Footnote 17}

Figure 1 Meta-Analyses Comparison

Note: “CIs” used as shorthand for “any measure of uncertainty.”

Figure 1 visually depicts the two patterns from our two meta-analyses. The bars for both models should be filled entirely with the darkest gray if all articles report post-estimation quantities with measures of uncertainty. This is clearly not the case for duration models, indicating that the interpretation of duration models differs from similar models. This trend is troubling because without measures of uncertainty, drawing meaningful inferences from post-estimation quantities can be difficult.^{Footnote 18}