Structural and Cognitive Aspects of Social Capital and All-Cause Mortality: A Meta-Analysis of Cohort Studies

Abstract

Social capital covers different characteristics such as social networks, social participation, social support and trust. The aim of this study was to explore which aspects of social capital were predictive of mortality. Criteria for inclusion in the meta-analysis were: population based observational cohort studies (follow-up ≥5 years); study sample included the adult population; parts of social capital as the primary exposure variable of interest; reported a mortality outcome; and sample size >1,000 individuals. Twenty studies provided eligible data for the meta-analyses. A random effect model was used to estimate the combined overall hazard rate ratio effects of structural social capital such as social participation and social networks, and cognitive social capital including social support and trust in relation to mortality. The results showed that social participation and social networks were negatively associated with mortality. The impact of social networks attenuated somewhat when controlling for gender and age. While trust also appeared to be negatively associated with mortality, we remain cautious with this conclusion, since only two studies provided eligible data. Perceived social support failed to show a significant impact upon mortality. The findings suggest that people who engage socially and report frequent contacts with friends and family live longer.

Appendices

Appendix 1: Transforming Logistic Regression Betas (log OR’s) into Cox Regression Betas (log HR’s)

An excellent exposition of the relationship between odds ratios (OR) derived from logistic regressions and hazard rate ratios (HR) derived from Cox proportional hazard regressions may be found in Symons and Moore (2002). While the first is a ratio of odds at the study endpoint, the latter is a ratio of rates, that is conditional probability per unit time, assumed to remain constant under the whole study period.

Both risk measures may be calculated by exponentiating the beta coefficient of the respective regression type. They approximate each other closely as long as baseline risk may be considered negligible but diverge increasingly with increasing event probability in the reference class. Symons and Moore provide the following formula for calculating odds ratios from hazard rate ratios in their equation (8)²:

$$ OR = \frac{{(1 - P_{0} )^{ - HR} - 1}}{{(1 - P_{0} )^{ - 1} - 1}}, $$

where P ₀ is the event probability in the reference class (baseline probability). Solving this equation for the hazard rate ratio yields

$$ HR = \frac{{ - \log (OR((1 - P_{0} )^{ - 1} - 1) + 1)}}{{\log (1 - P_{0} )}}, $$

where log denotes the natural logarithm with base e. Taking the natural logarithm of the hazard rate ratio will yield the regression beta, which one would have obtained using Cox proportional hazard model. Note that the transformation formula does not only depend upon the odds ratio obtained from the logistic regression, but also the reference probability P ₀, which is unambiguously defined only in univariate regressions with a single dummy variable. When multiple covariates are included in the regression, then the reference probability will be contingent upon the values of the other covariates and the tables in Cornman et al. (2003) and Orth-Gomér and Johnson (1987) do not permit their recalculation. We helped ourselves by approximating baseline risk with overall mortality in both studies: 12 % in Cornman et al. (2003) and 4.8 % in Orth-Gomér and Johnson (1987). While this is only an approximation, it is clearly more appropriate than postulating equality of odds ratios and hazard rate ratios, which would implicitly assume a baseline mortality of 0 %.

In order to perform the meta-analysis, the standard errors of the logistic regression betas need to be transformed as well, which is done using the so called delta method. It rests on a theorem (see e.g. Theorem A in Chapter 3 of Serfling (1980)) asserting that transforming asymptotically normally distributed random variables by a differentiable function yields again an asymptotically normally distributed random variable however with new expectation and standard deviation. Applying a Taylor approximation to the transformation of the standard deviation then boils down to multiplying the original standard error with the derivative of the new measure (here log HR) with respect to the original measure (here log OR), which in the case of transforming logistic regression betas into Cox regression betas is given by

$$ \frac{\partial \log HR}{\partial \log OR} = \frac{{P_{0} OR}}{{(1 - P_{0} )\left( {1 + \frac{{P_{0} OR}}{{1 - P_{0} }}} \right)\log \left( {1 + \frac{{P_{0} OR}}{{1 - P_{0} }}} \right)}}. $$

Since the transformation rules for the standard errors and for the regression betas themselves do not coincide, it follows that strictly speaking the level of significance is not preserved when transforming log OR’s into log HR’s, but that it becomes a function of both the value of the logistic regression beta and the reference probability P ₀. This is a consequence of keeping only the linear term in the Taylor approximation to the transformation of standard deviations.

The numerical impact of that is however small. The largest absolute change in p value we experienced was when transforming the logistic regression beta of −0.153 with a standard error of 0.080 for social networks in Cornman et al. (2003) into a Cox regression beta of −0.144 with a standard error of 0.076. This increased the original p value of 5.58 % by 0.1 percentage points to a p value 5.68 % with no practical influence upon the meta-analysis whatsoever. It is thus practically legitimate to regard the p value of an original log OR and the transformed log HR as identical, especially when keeping in mind the normal approximation used in calculating confidence intervals in both regression models is strictly valid first in the limit of infinitely large sample size.

Appendix 2: DerSimonian and Laird Random Effects Model

The random effects model by DerSimonian and Laird (1986) assumes that the study-specific treatment effects θ _i (log HR’s of individual studies) are normally distributed around an unknown mean effect θ (the sought overall log HR) with unknown variance τ ². This stands in contrast to the fixed effects model, where the overall effect is assumed to be a fixed albeit unknown quantity and all variation in measured effects is attributed to sampling error only. In the random effects model, the estimate of the additional variance in measured effects due to their random nature is given by

$$ \hat{\tau }^{2} = \frac{Q - (k - 1)}{{\sum {w_{i} } - {{\left( {\sum {w_{i}^{2} } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum {w_{i}^{2} } } \right)} {\sum {w_{i} } }}} \right. \kern-0pt} {\sum {w_{i} } }}}}, $$

where

$$ w_{i} = {1 \mathord{\left/ {\vphantom {1 {SE(\hat{\theta }_{i} }}} \right. \kern-0pt} {SE(\hat{\theta }_{i} }})^{2} $$

are the reciprocals of the squared standard errors of the observed logarithmic hazard ratios $\hat{\theta }_{i}$ obtained from the individual studies, all sums extend to the number of studies k, and Q is given by

$$ Q = \sum\limits_{i = 1}^{k} {w_{i} } \left( {\hat{\theta }_{i} - \frac{{\sum {w_{i} \hat{\theta }_{i} } }}{{\sum {w_{i} } }}} \right)^{2} . $$

The estimate of the sought mean effect (log HR) θ is given by the weighted sum

$$ \hat{\theta } = \frac{{\sum {w_{i} '\hat{\theta }_{i} } }}{{\sum {w_{i} '} }},\quad {\text{where}}\quad w_{i} ' = \frac{1}{{SE(\hat{\theta }_{i} )^{2} + \hat{\tau }^{2} }}. $$

Its standard error is given by

$$ SE\left( {\hat{\theta }} \right) = {1 \mathord{\left/ {\vphantom {1 {\sqrt {\sum {w_{i} '} } }}} \right. \kern-0pt} {\sqrt {\sum {w_{i} '} } }}, $$

which may be used to obtain 100(1−α) % confidence intervals as

$$ \hat{\theta } \pm SE\left( {\hat{\theta }} \right) \cdot \Upphi \left( {1 - \alpha /2} \right), $$

where Ф denotes the cumulative distribution function of the standard normal.

Note that the standard error of the mean effect decreases with the number of studies and may therefore be expected to fall below the average individual standard error, once sufficiently many studies have been collected.

Appendix 3: Publication Bias

Publication bias is the overestimation of an aggregate effect size due to using a biased sample of studies, which results when the publication process favors studies with significant effects at the cost of equally valid but unpublished studies, which do not find significant effects.

We assessed possible presence of publication bias by means of funnel plots according to standard errors and performing cumulative meta-analyses according to study size. The idea of both methods is to compare the effect sizes of small studies with those of large studies, where also small effects are deemed significant due to large sample size and are therefore more likely to be published. If effect size decreases with study size, then this is an indication that smaller studies with small effect sizes might have been lost under the publication process. Funnel plots employ standard errors as reciprocal measures of study size, which are plotted against the effect size. Publication bias appears then as a shift towards larger effect sizes for larger standard errors, in contrast to symmetric effect sizes independent of standard errors when publication bias is not present.

The funnel plot for social participation (Figure IX of ESM) is slightly asymmetric with increasing effect size for larger standard errors, which we take as evidence that the overall effect in the meta-analysis is probably overstated due to publication bias.

On the other hand, in a cumulative meta-analysis starting with the largest study and adding smaller studies one by one, inclusion of the three largest studies only (Seeman et al. 1987; Iwasaki et al. 2002; Hyyppä 2007) generates already a significant summary HR. So while the size of the impact of social participation upon mortality might be overstated, we cannot find any reason to doubt the overall significance of the result.

The funnel plot for social networks (Figure X of ESM) is symmetric and the cumulative meta-analysis, if anything, indicates a tendency for smaller studies to provide less significant results. Hence publication bias does not appear to be a problem here.

The presence of publication bias for social support (Figure XI of ESM) is difficult to judge due to the small numbers of eligible studies, however it would be of no consequences since all eight studies provided insignificant results anyway.