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J-Curve? A Meta-Analysis and Meta-Regression of Parity and Parental Mortality


Research suggests that parity and parental health and mortality are associated significantly, although the pattern of association varies across studies. Studies ascribe long-term poor health (and mortality) to either low or high parity, and some studies show that both low and high parity increase the risk of adverse health for parents (i.e., forming a “J-shaped curve”). While a recent meta-analysis (Zeng et al., Sci Rep 6:19351, 2016) has partially addressed this gap in the literature, the present study further extends the literature by using a methodology that allows for more robust control of study heterogeneity and potential confounders. Using data on 223 measures of relative mortality risk from 37 studies, from samples gathered after 1945 from developed nations, meta-analysis and meta-regression (weighted linear regression) results show a nonlinear association (J-shaped curve) between parity and all-cause parental mortality, though the strength of the association varies by both sex and cohort. The results also suggest that the mortality hazard is partially explained by health selection effects.

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Section 1: Full Search Algorithms for Medline (Comparable Search Algorithms Used for the Other Database Searches)

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  83. 83.


  84. 84.

    82 not 83

  85. 85.


  86. 86.

    8 or 85

  87. 87.

    exp Cohort Studies/

  88. 88.

    Controlled Clinical Trials/

  89. 89.

    controlled clinical

  90. 90.

    ((incidence or concurrent) adj (study or studies)).tw.

  91. 91.


  92. 92.


  93. 93.


  94. 94.


  95. 95.


  96. 96.


  97. 97.


  98. 98.


  99. 99.

    86 and 98

  100. 100.

    limit 99 to humans

Section 2: Calculated Margins of Error for Figs. 2 and 3

As noted in the text accompanying Figs. 2 and 3, confidence limits for the predicted hazard ratios, though useful, cannot be easily added to the graphs without rendering the overall figure incomprehensible. With this in mind, we first present the formulas needed to calculate these confidence limits and the calculations.

First, note that meta-regression is a form of weighted OLS regression. For weighted OLS, where the dependent variable is the log hazard ratio, the expected (predicted) values for the log hazard are given by the following matrix formula:

$$\hat{Y} = X\left( {X^{T} WX} \right)^{ - 1} X^{T} WY,$$

where \(\hat{Y}\) denotes an n by 1 vector of predicted values, X denotes the matrix of independent variables (an n by p matrix where the 1st column consists of only 1s and the remaining \(p - 1 = k\) columns contain the observed values for the k independent variables in the model), W denotes an n by n diagonal matrix of regression weights (i.e., a square matrix with the inverse variance weights for all n cases in the data located on the upper-left to lower-right diagonal and with 0 s everywhere else in the matrix), and Y denotes an n by 1 vector of observed log hazard ratios. Note also that any matrix with a superscript T indicates the transpose of that matrix and that any matrix with a superscript −1 indicates the inverse of that matrix.

The mean square error is therefore given by

$$MSE = \sqrt {\frac{1}{n - k - 1}\left( {Y - \hat{Y}} \right)^{T} \left( {Y - \hat{Y}} \right)}$$

and the standard error for the predicted mean response [calculated using an p by 1 vector (denoted as \(X_{h}\) in the formula below), where the 1st value is a 1, the values for the independent variables used for the x-axis and (if applicable) z-axis of the graph set equal to the desired prediction value, and all other values (for the remaining IVs) set equal to the means for the corresponding variables] equals

$$\hat{\sigma }_{{\hat{y}}} = \sqrt {MSE*X_{h}^{T} \left( {X^{T} WX} \right)^{ - 1} X_{h} } .$$

From these equations we can derive the 95% confidence intervals for Figs. 2 and 3 as shown in Tables 5 and 6 below.

Table 5 Predicted mean hazard ratio with 95% confidence interval corresponding to Fig. 2 (a graph of the predicted mean HR only)
Table 6 Predicted mean hazard ratio with 95% confidence interval corresponding to Fig. 3 (a surface graph of the predicted mean HR only)

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Högnäs, R.S., Roelfs, D.J., Shor, E. et al. J-Curve? A Meta-Analysis and Meta-Regression of Parity and Parental Mortality. Popul Res Policy Rev 36, 273–308 (2017).

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  • Parity
  • Mortality
  • Meta-analysis
  • Meta-regression