Calibrating SVD-Comp to the Relationship Between 5q0 and Mortality at Other Ages in the HMD

All computation is carried out using the R statistical programming environment (R Foundation for Statistical Computing 2016).

The life tables of the HMD are arranged into two A × L matrices (Q_z) of single-year, age-specific life table probabilities of dying (₁q_x), one for each sex. A = number of age groups = 110 L = number of life tables = 4,610; and z ∈ {female, male}. The SVD⁵ of each Q_z yields ρ LSVs, u_zi; RSVs, v_zi; and SVs, s_z. To ensure that all age groups have approximately the same influence when calculating the SVDs, each mortality schedule is offset from the origin⁶ by −10, and the offset is added back to predicted mortality schedules. Four of the new dimensions identified by each SVD are retained—that is, c = 4 in Eq. (11). For females, those account for 0.998328, 0.000936, 0.000071, and 0.000058 of the total sum of squares, respectively, or together 0.999392. Corresponding figures for males are 0.998595, 0.000824, 0.000103, and 0.000052, and together 0.999575. Section C of the online appendix contains additional information on the total sum of squares explained by each component of the SVD.

Based on Eqs. (9) and (10), regression models are defined that relate the RSVs v_zi to ₅q_0z and ₄₅q₁₅z. Scatterplots of the elements of the RSVs versus logit(₅q₀) in Figs. E1 and E2 in the online appendix make it clear that the relationships are not linear or simple. With no theory to guide the choice of predictors, I tried all combinations of simple transformations of logit(₅q₀) and logit(₄₅q₁₅) and their interactions. The resulting models explain almost all the variance in the elements of v₁ R² ≈ 97% for both sexes for both sexes), the vast majority of the variance in the elements of v₂ (R² ≈ 87 % for both sexes), and one-third to one-half the variance in the elements of v₃ and v₄. Additionally, I tried to avoid overfitting or creating odd boundary effects in the predicted values that would have made out-of-sample predictions immediately implausible. These models behave sensibly up to the edges of the sample. The final models are

where i ∈ {1 : 4} indexes the SVD dimensions, and ℓ indexes mortality schedules and elements of v_zi. OLS regression is used to estimate coefficients for the eight regression models defined in Eq. (12), and the estimated values are contained in online appendix D, Tables D1 and D2. With new values for both ₅q₀ and ₄₅q₁₅ as inputs, these models are used to predict values for the weights in Eq. (11)—that is, for prediction, v_zℓi on the left-hand side is replaced with W^zi.

To accommodate a one-parameter model that uses only ₅q₀ as an input, I define a regression model that relates adult mortality logit(₄₅q₁₅)z to child mortality ₅q_0Z. The scatterplot of logit(₄₅q₁₅) versus logit(₅q₀) in Fig. E3 in the online appendix reveals a slightly complicated relationship that is neither linear nor systematically curvilinear. Again, without theory as a guide, I tried a variety of models, including various simple transformations of ₅q₀. The resulting models explain most of the variance in logit(₄₅q₁₅) (R² = 93 % for females, and 79 % for males). The final models are

OLS regression is used to estimate coefficients for the two regression models defined by Eq. (13), and the estimated coefficients are contained in Table D3 in the online appendix. This model is used to predict values for ₄₅q₁₅ when only ₅q₀ is supplied as an input. Then both the input value for ₅q₀ and the predicted value for ₄₅q₁₅ are used in Eq. (12) to predict the weights in Eq. (11).

Figure E4 in the online appendix displays the relationship between logit(₁q₀) and logit(₅q₀). Mortality falls very rapidly in the first few years of life. Using the child mortality rate (₅q₀), a five-year summary of mortality between ages 0 and 5, as a predictor of single-year mortality within that same five-year age group is relatively uninformative. Experimentation reveals that ₅q₀ predicts ₁q₁ through ₁q₄ well and ₁q₀ slightly less well. The prediction of ₁q₀ can be improved by modeling the relationship between logit(₁q₀) and logit(₅q₀) separately as

OLS regression is used to estimate the coefficients of this model, displayed in Table D4 of the online appendix. The model explains essentially all the variance in logit(₁q₀) (R² > 99 % for both sexes) and is used to predict values for ₁q₀ directly from the input value of ₅q₀.