以生命表方法解析高齡死亡率與壽命上限的爭論

Mortality Deceleration at Extreme Old Age and Life Expectancy at Birth: a Life Table Analysis

董宜禎(Yi-Jhen Dong)

指導教授 ： 王德睦 ； 陳寬政

國立中正大學/社會科學院/社會福利學系/博士(2014年)

摘要

人類壽命自1750年死亡率下降以來不斷攀升，然而當人口轉型晚期，長期慢性疾病取代傳染性疾病，出生時平均餘命持續增漲的空間落在中高齡時，高齡死亡率的變化乃成為壽命成長與否的重要關鍵。晚近高齡死亡率隨年齡上昇的幅度變緩（高齡死亡率趨平），老年人口的存活數量愈來愈多，使得壽命上限的相關議題在學術上蔚為風潮。事實上，壽命上限的爭論圍繞著兩個不同觀點，一則高齡死亡率之趨平發展將伴隨出生時平均餘命持續成長，二則高齡死亡率之趨平發展並存於出生時平均餘命向定點收斂。另一方面，高齡死亡率趨平使學者、精算師與政府最常使用的Gompertz死亡函數之適用性備受質疑，並衍生死亡率交錯問題，長久以來卻無對當的理論考察與完整分析。1998年台灣人口的死亡統計最高齡組延伸至100歲以上，基於台灣戶籍資料相對準確與完整，提供我們深入考察這現象，進一步予以解析的機會。

我們檢視Gompertz死亡函數與存活函數之適用高齡死亡率，雖然Gompertz死亡函數於高齡端已無法抓住死亡率之趨平，但使用Gompertz存活函數推算的死亡機率卻可解決高齡死亡率「趨平發展」的問題。本文進一步指出以邏輯函數取代Gompertz函數的主張不恰當，邏輯函數侷限在左右對稱的條件下，於高齡端明顯低估生命表數值，表現不如Gompertz函數。然而，即使高齡死亡率確實趨平，不必然蘊含壽命水準的大幅增漲，由於高齡死亡率之下降不但較為困難，對平均餘命的影響幅度也相對縮小。死亡率每年下滑2%的樂觀情況下，平均餘命也僅在一百歲左右，過程中在在顯示死亡率愈迫近於零變化速度愈緩慢，區間餘命愈趨近於組距變化愈顯遲滯，呼應死亡率趨平發展伴隨平均餘命之上升趨緩的數據熵觀點。而生命表最高齡組的區間餘命為該年齡組開端年齡的平均餘命，對出生時平均餘命之估算難免產生干擾。本文顯示生命表最高齡組愈往高齡方向移動，出生時平均餘命的數值愈低，長期以來可能壓抑出生時平均餘命上揚的幅度，顯得較為保守的生命統計未適時調整最高齡組時，會傾向於高估出生時平均餘命。另一方面，死亡率趨平衍生年齡與性別的死亡率交錯，但檢視政府統計資料後，我們懷疑可能與資料誤差有關。經由內政部老人清查資料驗證，指出此一現象不同於物競天擇的演化熵概念，事實上是資料誤差的結果。我們認為高齡人口的登錄問題頗值得政府正視，尤其目前人口老化的影響下，未來獨居家戶將持續增加，登錄問題也隨之加劇，有必要針對高齡登錄問題規劃一套因應措施。

關鍵字

區間餘命 ； Gompertz函數 ； 高齡死亡率趨平 ； 生命表 ； 死亡率交錯

並列摘要

The decline of human mortality since 1750 has resulted in a dramatic increase in the life expectancy at birth. Recently the mortality deceleration at extreme old age is becoming a focus in the debates of the path the growing life expectancy has displayed. The issue is critical to the demographic literature because it also relates to replacing Gompertz function(Gompertz 1825, 1971) by something else in fitting, explainning, and predicting human mortality along the age axis, and to the newly discovered phenomenon of mortality crossover by age, sex, and race.
The analysis in this thesis indicates that the Gompertz function, as often exploited in the literature, is actually composed of two mathematical formulations, one for the life table probability of dying q(x) and the other for the life table survivors l(x). The former being the rate of decrease of the later. While it was obviosly noted by Gompertz himself that the parameters of the q(x) function might be different for 3 or 4 broad age groups, the thesis shows that without controversy the Gompertz’ survival function fits well to the life table survivors. Consequently the derived probability of dying q(x)=-Δl(x)/l(x) must follow a deceleration pattern at extreme old age. Compared to the proposed Beard(1963) and Perks formulations, the derived probability of dying stands out in its robustness. The results imply that the mortality deceleration could be more numerical than substantial.
The thesis further indicates that mortality deceleration at extreme old age does coexist with a flattening in the growth of life expectancy at birth, as Coale(1996) has suggested. Adopting the notion of temporary life expectancy, the thesis explains that while the temporary life expectancies approached the age interval as a natural upper limit, the momentum of mortality decline began to fade away. The life expectancy at birth is nothing but a sum product of the temporary life expectancies and life table survival ratios. Extending the terminal age to higher end of age spectrum tends to bring down, instead of boosting up, the life expectancy at birth. It is argued that with the temporary life expectancies serially reaching the upper limit, flattening of life expectancy at birth appears unavoidable.
It is further noted that the mortality statistics for the extreme old can be deceiving, simply because the errors in mortality compilation tend to escalate as the denominator of the rate dwindles. Given a modification in the declaration of statue death in Taiwan, the obvious deceleration of mortality at extreme old age has become less pronounced, even eliminated, along with the mortality crossover by age and sex in recent years. A government survey commenced in 2003 attempting to locate the very olds in Taiwan found 18.3% dead, missing, or unreachable among the registered population of 90+ in age; and the older the age, the higher the proportion. Obviously the under- and over-registration of death and population should be taken into account in the discussion of mortality at the extreme old age.

並列關鍵字

temporary life expectancy ； Gompertz function ； mortality deceleration at extreme old age ； life table ； mortality crossover

參考文獻

陳寬政與董宜禎（2012）高齡死亡率的減速發展，台灣人口學會年會論文，台北政治大學。

董宜禎與王德睦（2011）台灣的死亡率步調效果：步調調整後的平均餘命，人口學刊 第42期：43-79。

陳寬政、林子瑜、邱毅潔與紀筱涵（2009）人口老化、疾病擴張、與健保醫療費用，人口學刊 第39期：59-83。

王德睦與李大正（2009）台灣的存活曲線矩型化與壽命延長，人口學刊 第38期：1-31。

楊靜利、黃奕綺、蔡宏政與王香蘋（2012）台灣外籍配偶與本及配偶的生育率數量與品質，人文及社會科學集刊 第42卷第1期：83-120。.