Table 1 Summary of methods identified for replacing missing variance/SD/SE
Method | Category | Description | Statistics required | Assumptions | Software implementation |
|---|---|---|---|---|---|
Abrams et al. (2005) | 3 | Bayesian meta-analysis estimates within-patient correlation between baseline and follow-up; enables imputation of mean change from baseline and its SD when only baseline and follow-up means and SDs reported | • baseline mean/SD • follow-up mean/SD • change from baseline mean/SD in some included studies | SD at baseline same as SD at follow-up; within subject correlation comes from same distribution for all studies and treatment arms; careful choice of prior distribution for variance parameters | Example WinBUGS [24] code provided in paper |
Hozo et al. (2005) | 1 | Missing variance estimated for example as: | • minimum • median • maximum • sample size | Data normally distributed | Excel spreadsheet provided by Wan et al. (2014) |
\( Var\approx \frac{\left(n+1\right)}{48n{\left(n-1\right)}^2}+\left(\left({n}^2+3\right){\left(a-2m+b\right)}^2+4{n}^2{\left(b-a\right)}^2\right) \) | |||||
Sung et al. (2006) | 3 | Imputation of missing variances within Bayesian meta-analysis assuming distributed as True variance * χ2 (n-1)/(n-1) where true variance distributed as log-normal | • variances reported for other included studies | Assume missing variances come from same lognormal distribution as reported variances | Implemented in WinBUGS; code supplied in online supplement to article |
Walter and Yao (2007) | 1 | Improved version of “range” method which calculates SD = (b-a)/4 | • sample size • range or min/max | Approximate normality | Lookup table in paper could readily be implemented in standard software; RevMan [8] could accommodate in update |
Ma et al. (2008) | 2 | Impute weighted average of variances observed in other studies; or calculate a range of pooled estimates for efficacy based on the smallest and largest variances observed | • sample size • variances of other studies in meta-analysis | Unobserved and observed variances come from the same underlying distribution | Could readily be implemented in any statistical software |
Nixon et al. (2009) | 3 | Impute missing change from baseline SD values in Bayesian random effects meta-regression | • baseline SD • follow-up SD | Log transform of baseline SD, follow-up SD and change from baseline SD follow trivariate normal distribution. Where follow-up SD is based on complete cases, imputation assumes non-informative drop-out | Applied in WinBUGS |
Dakin et al. (2010) | 3 | Bayesian hierarchical modelling estimating SD values in context of network meta-analysis. SD assumed to follow gamma distribution; parameters estimated from studies reporting SDs | • observed SDs | Observed and missing SD values come from the same gamma distribution | WinBUGS code provided in publication |
MacNeil et al. (2010) | 3 | Impute missing SDs in hierarchical Bayesian meta-analysis based on posterior predictive distribution | • observed SDs | Observed, missing SDs arise from same gamma distribution | Implemented in PyMC Markov chain Monte Carlo (MCMC) toolkit [31] of Python [32]; code given in online supplement |
Stevens (2011), Stevens et al. (2012) | 3 | Bayesian network meta-analysis that enables imputation of missing SDs via posterior predictive distribution (variances assumed to follow gamma distribution) | • observed variances | Variances follow gamma distribution; log(SD) given weak uniform prior distribution | WinBUGS code provided |
Boucher (2012) | 3 | Emax model of SDs; implemented using either maximum likelihood or hierarchical Bayesian model | • observed SDs over time in longitudinal study | longitudinal modelling of SDs using Emax mixed effects model; differences by treatment group permitted in SDs; weak uniform prior for SD used in Bayesian approach | SAS (SAS Institute Inc., Cary, NC) PROC NLMIXED and WinBUGS code provided for maximum likelihood and Bayesian approaches respectively |
Wan et al. (2014)* | 1 | \( SD\approx \frac{q3-q1}{2{\Phi}^{-1}\left(\frac{0.75n-0.125}{n+0.25}\right)} \) | • lower quartile • upper quartile • sample size | Data normally distributed | Excel spreadsheet provided by Wan et al. (2014) |
Bland (2015) | 1 | Missing variance estimated as: | • minimum • lower quartile • median • upper quartile • maximum • sample mean | Data normally distributed | Excel spreadsheet provided by Wan et al. (2014) |
\( Var\approx \frac{1}{n-1}\left(\frac{\left[2\left(\mathrm{n}+3\right)\left(\mathrm{q}12+\mathrm{m}2+\mathrm{q}32\right)+2\left(\mathrm{n}-5\right)\left(\mathrm{a}.\mathrm{q}1+\mathrm{m}.\mathrm{q}1+\mathrm{m}.\mathrm{q}3+\mathrm{q}3.\mathrm{b}\right)+\left(\mathrm{n}+11\right)\left(\mathrm{a}2+\mathrm{b}2\right)\right]}{16}-\mathrm{n}{\overline{x}}^2\right) \) | |||||
Kwon and Reis (2015) | 1 | Approximate Bayesian computation to estimate SD | • available summary statistics | Underlying distribution of data | R code provided |
Chowdhry et al. (2016) | 2 | Meta-regression assuming sample variances follow gamma distribution | • observed variances from other studies in meta-analysis • study covariates | Variances missing at random (MAR) and follow a gamma distribution | Can be fitted in SAS PROC NLMIXED |