## Abstract

A variety of literature addresses the question of how the age distribution of deaths changes over time as life expectancy increases. However, corresponding terms such as *extension*, *compression*, or *rectangularization* are sometimes defined only vaguely, and statistics used to detect certain scenarios can be misleading. The matter is further complicated because mixed scenarios can prevail, and the considered age range can have an impact on observed mortality patterns. In this article, we establish a unique classification framework for realized mortality scenarios that allows for the detection of both pure and mixed scenarios. Our framework determines whether changes of the deaths curve over time show elements of extension or contraction; compression or decompression; left- or right-shifting mortality; and concentration or diffusion. The framework not only can test the presence of a particular scenario but also can assign a unique scenario to any observed mortality evolution. Furthermore, it can detect different mortality scenarios for different age ranges in the same population. We also present a methodology for the implementation of our classification framework and apply it to mortality data for U.S. females.

## Introduction

Mortality evolutions—that is, realized changes in mortality rates—have been analyzed extensively in the last decades. These analyses typically examine changes in the distribution of lifetimes and hence go far beyond determining trends in the evolution of life expectancy. In this sense, changes in aggregated statistics such as life expectancy are simply a consequence of the underlying change of the age distribution of deaths.

A wide range of literature addresses the question of how realized mortality has changed over time and how patterns of past developments—which we also call *mortality scenarios*—can be described and classified. In this context, different terms have been created: for example, rectangularization, compression, extension, expansion, and shifting mortality. These terms have been helpful in the analysis of historical mortality evolution patterns. Their definitions are, however, mostly intuitive, which can lead to ambiguity. For instance, Fries (1980) defined *rectangularization* as the convergence of the survival curve to a theoretical but not completely reachable final state, where everybody dies at the same age. Many authors have adopted this definition (see, e.g., Cheung et al. 2005; Kannisto 2000; Manton and Tolley 1991). However, as we show in the following section, this definition can be misleading. Similarly intuitive but difficult to verify from observed mortality patterns is the Debón et al. (2011:5) definition of compression as a “state in which mortality from exogenous causes is eliminated and the remaining variability in the age at death is caused by genetic factors.” Thus, a precise and feasible definition for each mortality scenario is necessary to test its occurrence in practice.

Furthermore, different authors have defined certain scenarios in different ways. In contrast to the aforementioned intuitive definition in Debón et al. (2011), many authors have used certain statistics of the deaths curve—that is, the age distribution of deaths—to define compression. According to Kannisto (2001), (old-age) compression can be observed if the modal age at death *M* (i.e., the age with the largest number of deaths) increases and *SD*(*M*+) (i.e., the standard deviation of the distribution of deaths above *M*) simultaneously decreases. Other authors have (at least implicitly) applied this definition (e.g., Cheung and Robine 2007; Ouellette and Bourbeau 2011). Wilmoth and Horiuchi (1999), on the other hand, identified compression by a shrinking interquartile range (*IQR*)—that is, the length of the age range between the 25th and the 75th percentile of the distribution of deaths. Analogously, Kannisto (2000) used the so-called *C*α-statistics—the shortest age range in which α % of all deaths occur. Thatcher et al. (2010) observed compression if the slope parameter in a logistic mortality model increased with time. We show in the following section that different definitions of compression do not always yield the same results.

Scenario definitions can also be critical when they rely on observations for a rather small age range only. For instance, when analyzing the evolutions of *M* and *SD*(*M*+), one completely ignores the mortality evolution for all ages less than *M*. As we show in the following section, if *M* increases and *SD*(*M*+) decreases, compression need not be present for the whole age range under consideration.

The distinction between different scenarios is also not always clear. For instance, Wilmoth (2000:1117) stated that rectangularization should be “best thought of as ‘compression of mortality.’” For Myers and Manton (1984), compression and rectangularization also seem to be the same scenario. Others, like Nusselder and Mackenbach (1996), see rectangularization as a special case of compression in which the life expectancy increases with time. A similar issue exists for definitions of extension, expansion, and shifting mortality. For example, Debón et al. (2011) used the terms *expansion* and *shifting mortality* but did not explain the differences between them. Others have defined the three terms differently. For example, Wilmoth and Horiuchi (1999) used the term *expansion* if the force of mortality decreased faster for older ages than for younger ages. Bongaarts (2005:36), on the other hand, explained the scenario of shifting mortality as a result of “delays in the timing of deaths”; that is, the force of mortality curve exhibits simply a shift in age. Cheung et al. (2005) used the term *longevity extension* for a scenario in which longevity beyond the modal age at death increases.

Sometimes, one scenario is defined by the absence of some other scenario. For instance, Canudas-Romo (2008:1198) regarded shifting mortality as a scenario in which “the compression of mortality has stopped.” Obviously, such a definition implies that these scenarios—namely, shifting mortality and compression—are mutually exclusive, which rules out mixed scenarios by definition. As we show later, elements of different mortality scenarios can often be present at the same time. Therefore, analyses that focus solely on testing for one particular scenario—for example, compression—can never provide a comprehensive insight into the mortality evolution.

In this article, we address these issues and establish a unique classification framework for mortality scenarios. The framework is based on observed changes in the deaths curve for the age range under consideration. We build on existing concepts such as shifting mortality, extension, and compression, and then combine these concepts into a framework that particularly allows for the detection of mixed scenarios of mortality change. We provide precise definitions of scenarios and show how they can be identified.

Furthermore, our framework is applicable to any age range from some starting age to the age at which everybody has died. Thus, the age range can be chosen such that it best suits the question at hand. We show that different scenarios might prevail for different age ranges and that our framework can identify this. For instance, sometimes scenarios can be observed in which more and more deaths are shifted from younger to older ages but deaths become more and more evenly spread at the older ages. Such a scenario might be thought of as compression on the age range starting at 0 but quite the opposite on the age range starting at 60 (see the following section). We also provide a possible methodology for implementing our framework and show its practical applicability in an example.

## Typical Issues with Scenario Definitions and Statistics

In this section, we identify and discuss some shortcomings of existing approaches for the classification of mortality scenarios. These shortcomings motivate a need for a new classification framework, as developed in the following section.

### Imprecise Mortality Scenario Definitions

*Mortality scenarios* describe patterns in the evolution of mortality over time—that is, a process of change. However, in the literature, we find several imprecise mortality scenario definitions. One example is the attempt to define the process of change solely by some (only theoretical and hence unreachable) final stage. This is the case when *rectangularization* is defined as a process in which the survival curve approaches a rectangular form. However, a rectangular form can be reached via different routes.

This is illustrated by the left panel of Fig. 1, which shows a hypothetical albeit not unrealistic evolution of deaths curves *d*(*x*) over time.^{1} Assume that at some point in time, mortality in a population follows the curve labeled State 1. At some later point, it follows State 2 and so on until it reaches State 5. Without using any formal definition, one would intuitively conclude that some scenario of compression takes place between States 3 and 5. Between States 1 and 3, however, a scenario that is somewhat the opposite of compression can be observed.

If, however, one looks at the corresponding survival curves *l*(*x*) (right panel of Fig. 1), one might intuitively conclude that with every step, the shape becomes more rectangular. Therefore, one might identify the whole transition from State 1 to State 5 as rectangularization, which is sometimes seen as a special case of compression. This clearly contradicts the observation that between States 1 and 3, the opposite of compression prevails.

Thus, the definition of a mortality scenario by some theoretical final state that is being approached will not always lead to a correct result.

### Misleading or Insufficient Statistics

Of course, a reduction of complexity by looking at some key statistics of deaths or survival curves rather than at the whole curves is desirable. On the other hand, this approach always leads to a loss of information. Therefore, one should very carefully identify statistics that preserve the part of the information that is of interest. Unfortunately, for some statistics that are frequently being used to describe patterns of mortality changes, this is not the case (at least if they are not analyzed together with additional statistics). In this subsection, we will explain this point.

Returning to Fig. 1, observe that the modal age at death *M* increases from state to state starting with 83 years in State 1 and reaching 104 years in State 5. At the same time, *SD*(*M*+) decreases from state to state starting at 7.62 in State 1 and ending at 2.71 in State 5. Following, for example, Robine et al. (2008), this would mean that compression prevails throughout the process and, in particular, also between States 1 and 3, which is inconsistent with the intuition from looking at the left panel of Fig. 1.

Sometimes, different statistics designed to measure the same thing can lead to contradicting results. For example, compression is often defined by a reduction of the *IQR* and/or a *C*α-statistic (see Wilmoth and Horiuchi 1999; Kannisto 2000). Figure 2 shows two scenarios of mortality evolution in which the structures of the mortality distributions changed considerably from State 1 to State 2, with clear characteristics of mortality improvement and compression. However, *IQR* remains unchanged in the left panel of the figure, whereas *C*50 remains at the same value in the right panel. Thus, neither *IQR* nor *C*50 alone is always able to identify compression. Such issues can always occur when changes of the entire deaths curve are identified using statistics that take into account only parts or certain points of the deaths curve.

### Ignoring Mixed Scenarios

Next, we show that it may not be appropriate to define a certain mortality scenario as the opposite of some other scenario or, more generally, that mixed scenarios should be allowed for. Hence, more than one dimension is required to get a full picture of a mortality scenario.

A classical example is the relationship between shifting mortality^{2} (or extension) and compression. The left panel of Fig. 3 shows a mixed scenario in which (in the transition from State 1 to State 2) shifting mortality and compression seem to coexist. Therefore, identification of one scenario should not rule out the other. Analogously, in the right panel of Fig. 3, neither shifting mortality nor compression can be observed. Thus, rejection of one scenario does not imply that the other scenario prevails. Thus, clearly it is not suitable to consider compression and shifting mortality as disjoint categories. This again shows the need for a more sophisticated classification system that combines different concepts of compression, shifting mortality, and so forth in the form of mixed scenarios.

### Effect of Age Range

Different types of mortality evolution can occur in different age ranges. Myers and Manton (1984) compared the survival curve starting at age 0 with the survival curve starting at age 65 for U.S. females and males between 1962 and 1979. They observed a clear tendency toward rectangularization for the entire age range but not at the older ages. If one is interested primarily in a certain age range (e.g., old-age mortality), one should therefore consider only the corresponding part of the mortality curve.

However, when restricting the age range, undesired effects may occur whenever statistics are being used that depend on the number of people being alive at the beginning of the considered age range: for example, *d*(*M*), the number of deaths at age *M*. Assume that one is interested in the age range starting at age 65. If between two points in time, younger-age mortality decreased, then more people would reach age 65. Even if older-age mortality did not change at all, *d*(*M*) would increase (with *M* remaining unchanged), suggesting a change in old-age mortality. And if a change in old-age mortality actually occurred, the change in *d*(*M*) would be affected by both the change in old-age mortality of interest and a change in younger-age mortality *not* of interest. These undesired effects can be eliminated by normalizing the population sizes such that at all considered points in time, the number of people alive at the beginning of the considered age range is the same (e.g., *l*(65) = 1).

The left panel of Fig. 4 shows some mortality evolution over the entire age range. Here, clearly compression toward higher ages can be observed. If one is interested only in the age range 65+, one might intuitively look at the respective age range of the left panel of Fig. 4 (i.e., without normalizing), which displays signs of compression. However, in the normalized curves (right panel of Fig. 4), the deaths curve of State 2 looks less dense than for State 1, which is an indication against compression.

## A New Classification Framework for Mortality Scenarios

In the previous section, we identified shortcomings of existing approaches for the classification of mortality evolutions. We now propose a new framework in which unique mortality scenarios are defined based on observable changes in the shape of the deaths curve. In this section, we introduce the intellectual concept of the framework. In the next section, we describe a methodology that can be applied to estimate the statistics used in our framework and to identify trends and trend changes in these statistics.

Our framework combines and uniquely defines four concepts for the change of mortality over time that are well known from the literature: (1) shifting mortality (see, e.g., Canudas-Romo 2008), (2) longevity extension (see, e.g., Rossi et al. 2013), (3) compression of mortality (see, e.g., Myers and Manton 1984), and (4) concentration of mortality (see, e.g., Kannisto 2001). As we will show, only a combined look at all four dimensions—which automatically allows us to consider both pure and mixed scenarios—gives a full picture of the considered mortality evolutions.

Our classification framework can be applied for any age range that includes the right tail of the deaths curve. Depending on the question at hand, the age range could start, for example, at 0, some juvenile age, or retirement age. In particular, it is possible that the classification framework identifies different mortality scenarios for different age ranges (see the upcoming section on the application of our classification framework for an example).

For any given age range, we use four key characteristics of the death curve, each corresponding to one of the aforementioned concepts. Significant changes in one or several characteristics over time mean that the deaths curve has changed. Conversely, if these four characteristics remain unaltered, changes in a deaths curve are regarded as immaterial. We will show that these four characteristics are sufficient to distinguish between a great variety of deaths curves and to uniquely classify mortality scenarios. The four characteristics are as follows:

The position of a deaths curve’s peak is measured by the modal age at death

*M*and describes general shifts in the distribution of deaths. Because the shape of a deaths curve typically changes over time, a pure shift of the entire deaths curve will rarely occur, and therefore we consider its center*M*as a reference point. An increase in*M*indicates right-shifting mortality, and a decrease in*M*implies left-shifting mortality. In this section, we assume that the modal age at death can be determined uniquely.^{3}The support of a deaths curve is determined by its upper bound, which we refer to as

*UB*.^{4}We denote the respective changes of*UB*as*extension*(if*UB*increases over time) and*contraction*(if*UB*decreases over time). Estimating*UB*in practice involves some ambiguity; see the following section for more details.The degree of inequality in the distribution of deaths, which we denote by

*DoI*, is the least obvious of the four key characteristics. However, Fig. 5 shows two deaths curves that are significantly different, although the other three statistics of our framework coincide. Therefore, an additional statistic related to the shape of the curve is required. The deaths curve of State 2 is almost 0 up to age 50, while State 1 shows a somewhat more balanced distribution of deaths over all ages.*DoI*is designed to pick up such differences by measuring the equality/inequality of the distribution of deaths over the whole age range. Intuitively, a low value of*DoI*indicates that deaths are rather equally distributed over the whole age range considered and vice versa. We use the terms*compression*/*decompression*if*DoI*increases/decreases; see the following section for more details.The height of the peak of a deaths curve is given by

*d*(*M*). This component addresses the evolution of a deaths curve at and close to its center,*M*. An increase in*d*(*M*) is referred to as*concentration*and indicates that the distribution of deaths becomes more concentrated around*M*. The counterpart to concentration is what we refer to as*diffusion*, and it is observed if*d*(*M*) decreases. Similar to*DoI*,*d*(*M*) can also be seen as an indicator for the equality/inequality of the distribution of deaths. A large*d*(*M*) implies that many deaths are concentrated at and around*M*. However,*d*(*M*) is a more local measure for a small region around*M*, whereas*DoI*measures the equality/inequality of the distribution of deaths over the whole age range.

Of course, each of the four components can remain unchanged over time. In this case, the respective component is referred to as *neutral*. Thus, every component can attain three states.^{5}

Two of the four aforementioned statistics (*UB* and *M*) primarily determine the position of the deaths curve, and the other two (*d*(*M*) and *DoI*) primarily describe its shape. We believe that these four characteristics provide a good trade-off between granularity and complexity. The four components are summarized in Table 1.

In principle, any combination of the three states for each component is possible, implying that we can classify both pure and mixed scenarios, which was one of the requirements we outlined earlier. In a pure scenario, only one component of the scenario vector is different from neutral. For instance, the vector (*neutral*, *extension*, *neutral*, *neutral*) denotes a pure extension scenario. On the other hand, a vector such as (*neutral*, *extension*, *compression*, *neutral*) describes a mixed scenario, which contains elements of both extension and compression. In total, 3^{4} = 81 mortality scenarios are possible, which might seem unfeasible at first glance. However, many scenarios will hardly be observed in practice—for example, (*left-shifting mortality*, *extension*, *compression*, *diffusion*). Those scenarios are nevertheless part of our classification framework to make sure that there are no unclassifiable evolutions and that classifications are unique.

## Methodology for the Implementation of the Classification Framework

The application of the classification framework introduced in the preceding section involves two main steps. First, the four statistics need to be estimated from deaths curves for each year in the observation period. A reasonable estimator for each of the statistics is proposed in the following subsection. Thereafter, trends in the resulting time series need to be analyzed to determine the prevailing states in each of the four scenario components, as we address in a later subsection. Obviously, various different estimators and methods could be used in both steps, and thus the specific estimators and methods described in this section are only one possible implementation.

### Estimation of Statistics

We now explain how we calculate the four statistics from the deaths curve in any given year. Both, raw or smoothed deaths curves can be used in principle. In our application later in the article, we explain why we prefer using smoothed data.

*M*, we use the following estimator by Kannisto (2001):

*x*

_{d_max}is the age for which the largest number of deaths is observed. As a byproduct, the

*height of a deaths curve’s peak*(

*d*(

*M*)), can then be estimated by the number of deaths at age

*x*

_{d_max}:

*upper bound of a deaths curve’s support*(

*UB*), we use the age at the α percentile of the distribution of deaths,

*x*

_{α}, plus an estimate for the remaining life expectancy at that age. Thus, the estimator for

*UB*is

This approach builds on Rossi et al. (2013), who proposed using the 90th percentile of the distribution of deaths as an approximation for the highest attainable ages. We prefer our combined estimator because it is considerably less biased. In our application, we set α = 99 %. For the populations we analyzed, this choice provides a reasonable compromise between cutting off only a small part of the distribution of deaths and stability in the statistic’s evolution over time. For smaller (sub)populations, however, smaller values for α might be more appropriate.

*degree of inequality*(

*DoI*) in the distribution of deaths needs to take into account the whole age range. Therefore, statistics such as

*SD*(

*M*+),

*IQR*, or

*C*α—which, as explained earlier, are commonly used to measure compression —are not feasible. An intuitive alternative is the area between the actual deaths curve and a hypothetical flat deaths curve

*d*

_{flat}(

*x*) as illustrated in Fig. 6. Using discrete data, this area can be approximated by summing the absolute differences in the numbers of deaths between the two deaths curves. Thus, we estimate

*DoI*as

*x*

_{0}is the starting age of the deaths curve, and $c=UB\u2212x0+12\u2219lx0UB\u2212x0$ is a scaling factor such that

*DoI*assumes its maximum value of 1 if all people die at the same age. The minimum value of

*DoI*is 0 if deaths are uniformly distributed over all ages—that is, if

*d*(

*x*) =

*d*

_{flat}(

*x*) holds for all

*x*.

Note that the dependence of *DoI* on *UB* is uncritical in our framework given that we are interested only in changes of *DoI* over time. A potential misestimation/bias of *UB* would affect *DoI* in the same way for each point in time. Further, changes in *UB* over time do not automatically imply changes in *DoI*. For instance, if *UB* increases while the deaths curve’s shape does not change materially, the slight changes of *d*(*x*), *d*_{flat}(*x*), and the scaling factor *c* would basically cancel each other.

As mentioned earlier, alternative estimators could be used for the four statistics. In particular, an extensive literature explores measuring *UB*, which is sometimes referred to as *maximum lifespan* (see, e.g., Finch and Pike 1996) or *finite lifespan* (see Fries 1980). Alternative estimators for *UB* can be found in, for example, Cheung and Robine (2007), Fries (1980), and Wilmoth (1997). As alternative measures for *DoI*, one could consider the variance in the number of deaths, the Gini index, as proposed by Debón et al. (2011); or entropy, as originally proposed by Demetrius (1974) and adopted by Keyfitz (1985) and Wilmoth and Horiuchi (1999). These statistics also consider the whole age range as required. However, the Gini index and entropy are defined on the survival curve, which makes them less intuitive in our deaths curve–based framework.

### Determination of Prevailing States

After the four statistics are estimated for each year in the observation period (see an example of the resulting time series in Fig. 7), the trends prevailing at each point in time need to be determined. We now introduce a possible methodology that we found to be suitable for all data sets we analyzed. However, a different methodology or modifications of our methodology—for example, with respect to the significance levels in the different tests—could be used and might be advisable for certain applications.

#### Elimination of Outliers

Potential outliers should be eliminated because they are irrelevant with respect to long-term trends and can significantly blur the trend analysis. Such outliers are typically caused by extreme events, such as the Spanish flu pandemic. To detect whether a data point is an outlier, we fit a linear regression to the 10 adjacent data points. The sample variance of the residuals (assumed to be normally distributed) can then be used to derive a 99 % prediction interval for the data point under consideration. If the data point lies outside the prediction interval, it is considered an outlier.

#### Determination of Trends, Trend Changes, and Jumps

To determine trends in the four statistics, we fit piecewise linear trends to the respective time series. Most of the time, mortality evolves rather steadily over time, and hence the piecewise linear trends should connect continuously. However, jumps can occur in case of extreme changes—such as the fall of the Soviet Union or a world war—or changes in data processing methods. Thus, at every data point of a time series under consideration, the previous linear trend can persist, a new trend can commence starting at the end point of the previous trend (change in slope), or a new trend can commence at some other level (jump and change in slope). The following methodology first determines which of the three possibilities is the most likely for each data point and then analyzes how many changes in slope and jumps are most suitable to describe the structural patterns in the entire time series and where they should occur.

To identify *candidate* data points for trend changes—that is, changes in slope with or without jumps—we first perform a preliminary analysis. We carry out three fits for every possible combination of three data points^{6}: (1) a straight regression line to the data from the first to the third data point, (2) a continuous regression line to the data from the first to the third data point with a change in slope at the second data point, and (3) two straight regression lines (to the data from the first to the second and from the second to the third data point, respectively) that allow for a jump at the second data point. A set of Chow tests (see Chow 1960) is used to determine which trend evolution is most likely for the second data point, under the assumption that adjacent trend changes are located at the first and third data point or that these data points are the first or last data points of the entire time series. In the first Chow test (significance level of 1 %), the null hypothesis of one persistent trend—that is, no change at the second data point—is tested against a continuous change in slope. The result of the test (the new null hypothesis) is then tested against a jump in a second Chow test. The results of the Chow tests usually depend on the choice for the first and third data points. Thus, whether a data point is a candidate for a trend change (and if so, of which kind) depends on the position of the neighboring trend changes.

After the preliminary analysis, we use the following main algorithm to identify the number and locations of trend changes that result in an optimal fit^{7}:

*Step 1:*We commence by fitting a straight regression line to the entire time series. This is the case of no trend change (i.e., the number of possible trend changes*n*is 0).*Step 2:*The number of possible trend changes is increased from*n*to*n*+ 1.*Step 3:*We determine the sample variances of the residuals from the fit with*n*trend changes. They will be required as variance estimators in Step 5. The sample variances are to be computed separately for each period with constant trend. We use a regime-switch argument here to justify that the variance can change when the trend changes and thereby allow for heteroscedasticity, as can be observed in Fig. 7, for example.*Step 4:*Building on the preliminary analysis, we determine all feasible combinations of*n*+ 1 candidate data points for trend changes. The preliminary analysis also indicates for each candidate data point whether the trend change would be a change in slope with or without jumps. If there is no feasible combination, the fit with*n*trend changes is the overall optimal fit, and the algorithm terminates.*Step 5:*For each feasible combination of trend changes from Step 4, we fit a piecewise linear trend curve to the data (and allow for discontinuities only when the type of the potential trend change is a change with jump). To account for heteroscedasticity, we use the sample variances from Step 3 as weights.*Step 6:*The optimal trend change positions (and thus also the trend change types) for*n*+ 1 trend changes are determined by comparing the fits from Step 5 by the Akaike information criterion (AIC) (Akaike 1973). The number of parameters is two (initial intercept and slope) plus*n*+ 1 for the trend change positions plus*n*+ 1 for the changes in slope plus one for every jump (i.e., the new intercepts after the jumps).*Step 7:*Finally, we compare the optimal fits with*n*and with*n*+ 1 trend changes to assess the contribution of the additional trend change to the time series representation. To this end, we use another Chow test (again with significance level 1 %). Because the original test by Chow considers only one trend change versus none, we use an extended version of the test. The test statistic remains unchanged, but the number of parameters increases (one for each trend change position, each intercept, and each slope). Note that for*n*≥ 2, we can also account for heteroscedasticity in this test by applying variance estimates from the optimal fit with*n*– 1 trend changes as weights. The null hypothesis in the Chow test is the case of*n*trend changes. Thus, the additional trend change is accepted only if it significantly improves the fit, which is in line with our intention of determining long-term trends. If the null hypothesis stands, the time series can be adequately described by*n*trend changes, and the algorithm terminates. If the additional trend change significantly improves the fit, we return to Step 2.

#### Testing for Increasing, Decreasing, or Neutral Statistics

Finally, we have to determine whether the resulting trend curve (see the lines in Fig. 7) should be considered increasing, decreasing, or neutral in the context of our framework. For each period with constant trend, we use an *F* test with a significance level of 10 % to analyze whether the slope of the trend is significantly different from 0. If the slope is not significantly different from 0, the state *neutral* is assigned. Otherwise, we consider the statistic as increasing (decreasing) if the slope is positive (negative) during the corresponding period. This definition implies that the state *neutral* is assigned not only if the slope is clearly close to 0 but also if the uncertainty in the underlying data is too large to identify a significant trend.

## Application of the Classification Framework

In this section, we apply our classification framework to the mortality evolution of females in the United States.^{8} We derive log mortality rates ln(*m*(*x*, *t*)) for ages 0 to 109 from the deaths and exposure data in the Human Mortality Database (HMD) for years 1933–2013. For each calendar year, these log mortality rates are then smoothed and extrapolated using *P* splines, allowing us to derive normalized and smoothed deaths curves. We prefer this approach over using the raw deaths curves from the HMD for several reasons: (1) potential disturbing effects resulting from birth cohorts of different sizes are eliminated; (2) random effects in the data, which might lead to double peaks in the deaths curve, are significantly reduced; (3) the potential effect of age misspecifications in the raw data, particularly with respect to estimating *UB*, is reduced; and (4) the time series for the four statistics exhibit fewer random fluctuations and are thus easier to analyze.

We consider deaths curves covering different age ranges as discussed earlier. The curves *d*_{10}(*x*, *t*) start with a fixed radix at age 10 and thus exclude effects from infant mortality, whereas the *d*_{60}(*x*, *t*) curves allow for an analysis of mortality at typical retirement ages. Fig. 7 displays the four components of our classification framework for both starting ages along with the respective piecewise linear trend lines.^{9}

By definition, the curves for the modal age at death *M* coincide for both starting ages. From a theoretical perspective, the same holds for *UB*. However, the chosen estimator yields slight differences for the different starting ages. Because the two sets of data points would be difficult to distinguish, and the resulting scenarios for this component are the same for both starting ages, we display *UB* only for starting age 10.

From Fig. 7, we note that our framework identifies several trend changes for each of the statistics and both starting ages. Such trend changes can mean that (1) the direction of the trend changes (e.g., from increasing to neutral or decreasing), or (2) only the intensity of the trend (i.e., the slope of the trend line) changes significantly while its direction remains unchanged. For example, the first two trend changes in *M* are changes in the direction of the trend, whereas the subsequent trend changes (except the last one) concern only the intensity of the increase (i.e., the pace of the right shift in mortality). Thus, a trend change does not inevitably lead to a change in the scenario vector. Moreover, as mentioned earlier, trends can change with or without a jump in the absolute level of the statistic. In our example, such jumps occur for all statistics except *d*(*M*)_{60}.

The direction of each trend as well as the position of the trend changes and their types—that is, a change in slope with or without an upward/downward jump—are summarized in Fig. 8. This representation allows for an easy visual assessment of the scenario vector at each point in time. For instance, in the year 2010, for both starting ages, the scenario vector is (0, +, +, –): that is, the scenario is neutral with respect to shifting mortality and exhibits extension, compression, and diffusion at the same time.

By comparing Fig. 7 with Fig. 8, we find some periods with seemingly increasing (decreasing) trends in Fig. 7 but a classification as neutral in Fig. 8. One such example is the first trend for *UB*. Here, the underlying data has a relatively strong variance, and therefore the seemingly increasing trend is not significant, as explained earlier.

The results of our analysis particularly underline the need for combining different concepts of mortality change in one framework given that we observe mixed scenarios over almost the whole observation period. There are even periods in which all four indicators change—for example, between 1973 and 1982 for starting age 10. During this period, we simultaneously observe right-shifting mortality and extension (i.e., both the mode and the upper bound of the deaths curve move to the right) combined with a compression of the whole curve and an increase of the concentration around the mode. In contrast, pure scenarios seem to be very rare: only for starting age 60 and years 1941 to 1948 do we find a scenario of pure diffusion.

Furthermore, the results show that each of our four components is relevant in the sense that no component can be explained by the others. For instance, as one would expect, *M* and *UB* increase over the observation period in general: we observe right-shifting mortality and extension. However, particularly for *UB*, we observe the opposite trend (i.e., contraction) for some periods (e.g., the 1990s), and thus these two statistics do not move in the same direction throughout the entire observation period. This also holds for *DoI* and *d*(*M*), although they also frequently follow the same trend. For example, after 2006, *d*(*M*) decreases for both starting ages, while *DoI* increases for both starting ages: we observe diffusion and compression at the same time.

The results also highlight the importance of choosing a suitable age range. For both *DoI* and *d*(*M*), we find several periods in which the trends differ by starting age. For instance, between 1975 and 1990, we observe compression for starting age 10 but decompression for starting age 60.

## Conclusion

In this article, we explain why many existing approaches to classify patterns of mortality evolution have four major shortcomings: (1) mortality scenario definitions are often imprecise and intuitive rather than rigorous; (2) some frequently used statistics are not always sufficient to identify the respective scenarios; (3) mixed scenarios are usually not accounted for; and (4) the effect of the considered age range is often ignored.

We propose a new framework for classifying patterns of mortality evolution. Our approach is based on changes of the deaths curve and uses four statistics that should be considered simultaneously. Each mortality scenario then consists of four components: (1) the deaths curve can exhibit a right shift or a left shift or be neutral in that respect; (2) the deaths curve can exhibit extension or contraction or be neutral in that respect; (3) the deaths curve can exhibit compression or decompression or be neutral in that respect; or (4) the deaths curve can exhibit concentration or diffusion or be neutral in that respect. This approach overcomes the shortcomings of previous approaches: each mortality evolution is uniquely and precisely classified; by considering all four components simultaneously, mixed scenarios are automatically detected; and the framework is applicable to different age ranges.

For some of the statistics used, the estimation is not straightforward. Beyond an introduction of the intellectual concept of the framework, we therefore also introduce a methodology that can be used to estimate the statistics and determine trends and trend changes in the data. We apply our approach to data for U.S. females, illustrating that the structure of the change in mortality can be quickly assessed and well understood. We further demonstrate empirically that none of the four components can be explained by the other three and that results can significantly differ for different age ranges.

The purpose of our framework is a classification of realized mortality evolutions. In this sense, it is purely descriptive: it does not provide explanations for observed trends and trend changes. It seems obvious that any research that intends to provide such explanations or seeks to explore a link between determinants of mortality and observed patterns of mortality change needs as a prerequisite a common understanding of which pattern of mortality change has been observed in which situation. Our methodology can provide this and hence serves as a basis for such research. In particular, the detected trend changes can be an indication when and how demographic changes have occurred. Similarly, by applying our framework to different populations, time and structure of differences in their demographic evolutions can be detected, which again can serve as a basis for research on the causes.

If a mortality model is to be calibrated to historical data, our framework can also be used to identify suitable time spans (e.g., without major trend breaks). Further, the framework can be applied for testing whether existing mortality projections are consistent with observed trends in the most recent history.

## Notes

^{1}

All deaths curves in this article are scaled such that the areas underneath the curves each integrate to 1. Thus, the corresponding survival curves start with a radix of 1. Also note that all examples in the second and third sections of this article are based on hypothetical illustrative curves that are, however, reasonable given that overall mortality improves and life expectancy increases.

^{2}

As noted in the Introduction, the terms *expansion*, *extension*, and *shifting mortality* coexist in the literature. We consider *expansion* and *extension* to be the same, and use the term *extension* for that. We consider *shifting mortality* to be a different phenomenon, as explained in the next section.

^{3}

The peak might not be unique in only rather theoretical scenarios—for example, because of multiple peaks of the same height or a plateau. In such a case, one might use a suitable alternative to *M* or modify the framework to include additional statistics.

^{4}

In theory, *UB* can exist only if the probability of death reaches 1 for some age. If the probability of death remains below 1 for all ages, any age could be reached in principle. Research by several authors (see, e.g., Gampe 2010) has indicated that probabilities of death typically flatten out at very old ages, possibly somewhere near 0.5. Thus, the population surviving up to such ages would get halved every year; but if the initial population was large enough, there would be a few survivors up to any age. Therefore, one could argue that *UB* does not exist in theory, which is, however, irrelevant for our application.

^{5}

If a distinction between different intensities of increase or decrease is desired, more than three states can be considered or additional information about the slope of the respective trend line (see the section on methodology) can be added.

^{6}

If the time series has *k* data points, we consider all *k* × (*k* – 1) × (*k* – 2) / 6 possible triples.

^{7}

The presentation of the algorithm aims for a clear presentation of and distinction between the steps involved and does not pay attention to computing efficiency.

^{8}

We also applied the framework to several other populations, such as Sweden, Japan, and West Germany. In all cases, the framework yielded reasonable and informative results. For the sake of brevity, however, we show the results for only one population. We chose U.S. females for illustration because the variety of different observed scenarios was the largest. See Genz (2017) for an application of our framework to a larger number of countries and a comparison of the respective mortality patterns.

^{9}

We also considered the starting ages 0 (i.e., the complete age range) and 30 in order to exclude effects of young adult’s mortality, such as accidents. The observed scenarios for starting ages 0, 10, and 30 are quite similar.

## References

*Population: An English*