APC curvature plots: Displaying nonlinear age-period-cohort patterns on Lexis plots

“This term should not be confused with ‘local curvature,’ also referred to as ‘contrast’ and ‘second-order difference’ (Acosta et al. 2019b; Clayton and Schifflers 1987; Holford 1991; Pullum 1980; Tango and Kurashina 1987; Tarone and Chu 1996), which are used to measure the change in the slope of…”

“Consistent with this idea, the hand-drawn contours indicating cohort mortality improvement patterns presented in the work of Kermack, McKendrick, and McKinlay (1934) have been recognized as a pioneering example of research demonstrating that long-term mortality change tends to follow the birth cohort dimension (Finch…”

“While there are infinitely many ways to partition the linear effects into APC components, the curvature components have the same shape and magnitude regardless of the parameterization used to fit the model (Clayton and Schifflers 1987; Holford 1983, 1991).”

“Here, we used mortality data for certain countries drawn from the Human Mortality Database (HMD 2019) to compare not only the curvature ridges of the hump, but the additional features of their changes in magnitude and spread over time and across different countries.”

“While there are infinitely many ways to partition the linear effects into APC components, the curvature components have the same shape and magnitude regardless of the parameterization used to fit the model (Clayton and Schifflers 1987; Holford 1983, 1991).”

“Several arithmetic and statistical models have been proposed to identify deviation from trends, such as the Median Polish technique (Keyes and Li 2010; Selvin 2001; Tukey 1977) and the Detrended Age-Period-Cohort models (dAPC) (Carstensen 2007; Clayton and Schifflers 1987; Holford 1983).”

“Such curvature would 3 Note that the term ‘curvature’ used here refers to deviations from linear effects, as originally proposed by Holford and used in the APC literature (Clayton and Schifflers 1987; Holford 1983, 1991, 2005; Tango and Kurashina 1987).”

“Given this limitation, the use of graphical analyses is often regarded as a more transparent approach to identify APC effects than the use of statistical modeling (Murphy 2010; Preston and Wang 2006; Willets 2004).”

“The question of whether it is possible to independently isolate these age, period, and cohort (APC) effects on temporal changes in population phenomena has been vigorously debated since the first half of the 20th century (Keyes et al. 2010; Murphy 2010).”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“…that focus instead on identifying divergence from secular trends (also known as curvature3) in each of the three temporal dimensions (Holford 1983; Rodgers 1982; Tango and Kurashina 1987) are more effective and less polemical than those focused on identifying the dominant patterns of change.”

“…presented in the work of Kermack, McKendrick, and McKinlay (1934) have been recognized as a pioneering example of research demonstrating that long-term mortality change tends to follow the birth cohort dimension (Finch and Crimmins 2004; Hobcraft, Menken, and Preston 1982; Preston and Wang 2006).”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“Given this limitation, the use of graphical analyses is often regarded as a more transparent approach to identify APC effects than the use of statistical modeling (Murphy 2010; Preston and Wang 2006; Willets 2004).”

“2018), MortalitySmooth (Camarda 2012), and HMDHFDplus (Riffe 2015).”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“Indicative Lexis diagrams (Willets 2004) and colored Lexis surfaces (Rau et al. 2008, 2018; Richards, Kirkby, and Currie 2006) of these derivatives have been proposed as visualization techniques that could be applied to identify the presence of such patterns in Lexis plots.”

“’ This term should not be confused with ‘local curvature,’ also referred to as ‘contrast’ and ‘second-order difference’ (Acosta et al. 2019b; Clayton and Schifflers 1987; Holford 1991; Pullum 1980; Tango and Kurashina 1987; Tarone and Chu 1996), which are used to measure the change in the slope of the effects.”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“Consistent with this idea, the hand-drawn contours indicating cohort mortality improvement patterns presented in the work of Kermack, McKendrick, and McKinlay (1934) have been recognized as a pioneering example of research demonstrating that long-term mortality change tends to follow the birth…”

“Indicative Lexis diagrams (Willets 2004) and colored Lexis surfaces (Rau et al. 2008, 2018; Richards, Kirkby, and Currie 2006) of these derivatives have been proposed as visualization techniques that could be applied to identify the presence of such patterns in Lexis plots.”

“Goldstein (2011) presented a plot of the hump peaks over time for several countries, and Remund and colleagues (2018) decomposed and plotted the contributions by cause of death in order to analyze several features of the hump in the United States.”

“While there are infinitely many ways to partition the linear effects into APC components, the curvature components have the same shape and magnitude regardless of the parameterization used to fit the model (Clayton and Schifflers 1987; Holford 1983, 1991).”

“Several arithmetic and statistical models have been proposed to identify deviation from trends, such as the Median Polish technique (Keyes and Li 2010; Selvin 2001; Tukey 1977) and the Detrended Age-Period-Cohort models (dAPC) (Carstensen 2007; Clayton and Schifflers 1987; Holford 1983).”

“’ This term should not be confused with ‘local curvature,’ also referred to as ‘contrast’ and ‘second-order difference’ (Acosta et al. 2019b; Clayton and Schifflers 1987; Holford 1991; Pullum 1980; Tango and Kurashina 1987; Tarone and Chu 1996), which are used to measure the change in the slope of the effects.”

“These approaches include applying interpolation techniques (Camarda 2012), extracting the irregularity using decomposition techniques (Remund, Camarda, and Riffe 2018), using detrended APC models (Chauvel, Leist, and Smith 2017), or simply detrending the smoothed vital rates over the selected…”

“Given this limitation, the use of graphical analyses is often regarded as a more transparent approach to identify APC effects than the use of statistical modeling (Murphy 2010; Preston and Wang 2006; Willets 2004).”

“Several arithmetic and statistical models have been proposed to identify deviation from trends, such as the Median Polish technique (Keyes and Li 2010; Selvin 2001; Tukey 1977) and the Detrended Age-Period-Cohort models (dAPC) (Carstensen 2007; Clayton and Schifflers 1987; Holford 1983).”

“Demographic Research: Volume 41, Article 42 http://www.demographic-research.org 1219 their life course (Acosta et al. 2019a; Miech, Koester, and Dorsey-Holliman 2011; Zang et al. 2019).”

“Indicative Lexis diagrams (Willets 2004) and colored Lexis surfaces (Rau et al. 2008, 2018; Richards, Kirkby, and Currie 2006) of these derivatives have been proposed as visualization techniques that could be applied to identify the presence of such patterns in Lexis plots.”

“Demographic Research: Volume 41, Article 42 http://www.demographic-research.org 1219 their life course (Acosta et al. 2019a; Miech, Koester, and Dorsey-Holliman 2011; Zang et al. 2019).”

“In the case of Russia, the cohort pattern could be indicative of a longterm mortality disadvantage for those cohorts who entered adulthood in the early 1990s, when the country was undergoing a profound sociohistorical transformation, as well as a severe alcohol abuse epidemic (Keenan et al. 2015).”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“It has long been recognized that populations change along the three dimensions of age, period (typically calendar year), and cohort (typically year of birth) (Caselli and Vallin 2005; Keiding 2011).”

“Demographic Research: Volume 41, Article 42 http://www.demographic-research.org 1219 their life course (Acosta et al. 2019a; Miech, Koester, and Dorsey-Holliman 2011; Zang et al. 2019).”

“Although identifiable, the curvature estimates are average effects that are fixed over the whole length of time (Chauvel 2013).”

“Cohort age-specific fertility rates (ASFR) were obtained from the Human Fertility Database (HFD 2019) and extracted using the R package HMDHFDplus (Riffe 2015).”

“Recently, discussions of this topic became even more heated in reaction to the set of methods proposed by Yang and colleagues (Bell and Jones 2013; Fosse and Winship 2018, 2019a; Luo 2013; Masters et al. 2016; Reither et al. 2015; Yang and Land 2013).”

“There is also scope to combine the effects found in different countries on to single Lexis ‘curvature’ plots, allowing for interesting cross-national comparison (Acosta and van Raalte 2019).”

“…produced a Lexis Surface of the smoothed age-specific death rates (using the smooth functions that the Epi package provides in the apc.fit function), to visually inspect the presence of period or cohort features based on the curvature of the surfaces (Acosta & van Raalte, 2019; Rau et al., 2018).”