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Appendix A
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Procedure for random effects meta-analysis
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Random-effects meta-analysis is a two step process. Step one involves conducting a fixed
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effects meta-analysis to determine the between study variance. Step two produces the random
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effects estimates, which involves using the estimate in step one to partition the total variance into
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among-study and within-study components.
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In the first step, the combined slope, mc is expressed as a weighted average of slopes:
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k
k
i 1
i 1
mc   wi mi /  wi
where k is number of studies, mi is the estimated slope of the calcification-arag relationship
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from the ith study, and wi is the study-specific weight of study i (Borenstein et al 2009). Weights
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are simply the reciprocal of the variance (i.e. the square of the slope standard error) of the slope
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estimate, wi=1/vi. The variance of the combined slope, mc, is thus given by:
k
S m2c  1/  wi
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i 1
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(note that this represents measurement uncertainty around the mean slope, not the total amount
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of among-study variation).
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The Q statistic, which is used to test for heterogeneity of variance, was then calculated
by:
𝑘
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𝑄=
∑ 𝑤𝑖 𝑚𝑖2
𝑖=1
(∑𝑘𝑖=1 𝑤𝑖 𝑚𝑖 )
−
∑𝑘𝑖=1 𝑤𝑖
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And the between-study variance, τ2, was calculated as:
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𝑄 − 𝑑𝑓
𝑖𝑓 𝑄 > 𝑑𝑓
𝜏 ={ 𝐶
0
𝑖𝑓 𝑄 ≤ 𝑑𝑓
2
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where C is a scaling factor to ensure that τ2 has the same units as the within-study variance, and
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calculated by
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∑ 𝑤𝑖2
𝐶 = ∑ 𝑤𝑖 −
∑𝑤
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and df (degrees of freedom) is number of studies minus 1.
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In the second step, the combined (within and between study) variance for each study is
𝑣𝑖∗ = 𝑣𝑖 + 𝜏 2
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Combined slope, variance and 95% confidence intervals are then re-calculated as done in
step one but with the study weights wi* set equal to 1/𝑣𝑖∗ , instead of 1/𝑣𝑖 , as in step one.
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Quantifying variance between studies
The I2 statistic, the ratio of excess dispersion to total dispersion, was calculated by
𝑸−𝒅𝒇
𝑰𝟐 = (
𝑸
) × 𝟏𝟎𝟎%
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I2 values of 25%, 50% and 75% have been suggested as indicative of low, moderate and high
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variance respectively (Higgins et al 2003).
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Testing for between-group differences
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The estimated difference between two effects is
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𝜇𝐷𝑖𝑓𝑓 = 𝑀𝐵 − 𝑀𝐴
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where MA & MB are the mean estimated slopes for groups A & B, calculated according to eq. 1
The standard error of this estimated difference is
𝜎𝐷𝑖𝑓𝑓 = √𝑉𝑀𝐴 + 𝑉𝑀𝐵
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And thus the Z-statistic is
𝑍𝐷𝑖𝑓𝑓 =
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𝜇𝐷𝑖𝑓𝑓
𝜎𝐷𝑖𝑓𝑓
Under the null hypothesis that the true mean effect size μ is the same for both groups
𝐻0 : 𝜇𝐴 = 𝜇𝐵
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ZDiff would follow the normal distribution. For a two-tailed test the p-value is given by
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𝑝 = 2 [1 − (𝛷(|𝑍𝐷𝑖𝑓𝑓 |))]
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Meta-regression procedure
As with the original meta-analysis, meta-regression involves, first, a fixed effects meta-
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regression to determine the between study variance, followed by a random effects step that
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partitions the total variance into among-study and within-study components.
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In step one, the fixed effects weighted regression uses the model:
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𝑚𝑖 ~N(𝛼̂ + 𝛽̂ 𝑥𝑖 , 𝑣𝑖 )
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Where mi is the estimated slope of the calcification-arag relationship from study i, xi is the value
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of the explanatory variable (log(study duration) or log(irradiance level), depending on the meta-
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regression) in study i, vi is the variance of the estimated slope within study i, 𝛽̂ represents the
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change in the calcification-arag slope per unit change in the explanatory variable, and 𝛼̂ is the
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intercept of the relationship between the calcification-arag slope, and the explanatory variable
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(Thompson & Sharp 1999). N(𝛼̂ + 𝛽̂ 𝑥𝑖 , 𝑣𝑖 ) denotes a normal distribution with mean 𝛼̂ + 𝛽̂ 𝑥𝑖 and
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variance 𝑣𝑖 . Maximum likelihood estimates of 𝛼̂ and 𝛽̂ were obtained by weighted least squares
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regression (glm() in R) of mi on xi, with weights wi = 1/vi
The moment estimator of between-study variance, 𝜏 2 , was calculated as
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𝜏2 =
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𝑄−(𝑘−2)
𝐹(𝑤,𝑥)
if Q > k – 2, or 0 otherwise
where
𝑄 = ∑ 𝑤𝑖 (𝑦𝑖 − 𝛼̂ − 𝛽̂ 𝑥𝑖 )2
k is the number of studies, and
𝐹(𝑤, 𝑥) = ∑ 𝑤𝑖 −
∑ 𝑤𝑖2 ∑ 𝑤𝑖 𝑥𝑖2 −2 ∑ 𝑤𝑖2 𝑥𝑖 ∑ 𝑤𝑖 𝑥𝑖 +∑ 𝑤𝑖 ∑ 𝑤𝑖2 𝑥𝑖2
∑ 𝑤𝑖 ∑ 𝑤𝑖 𝑥𝑖2 −(∑ 𝑤𝑖 𝑥𝑖 )2
In step two, the random effects weighted regression uses the model
̂∗ 𝑥𝑖 , 𝑣𝑖∗ )
̂∗ + 𝛽
𝑚𝑖 ~𝑁(𝛼
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̂∗ were obtained in the same way as 𝛼̂ and 𝛽̂ , except
̂∗ and 𝛽
Where vi*=vi+2, and estimates of 𝛼
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with weights 𝑤𝑖∗ = 1/(𝑣𝑖 + 𝜏 2 ) used in place of 𝑤𝑖 = 1/𝑣𝑖 .
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Publication Bias
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The fail-safe number (X) is:
2
(∑ 𝑍𝑗 )
𝑋=
−𝐾
2.706
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where Zj = Zrj√(Nj – 3), Zrj is Fisher’s z-transformed correlation coefficient for the relationship
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between calcification and ΩAragonite for sample j, Nj is sample size for sample j and K is the
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number of studies (Moller and Jennions 2001). The z-test value 2.706 (=1.6452) is based on a
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one-tailed p value of 0.05 (Moller and Jennions 2001). Fisher’s z-transformed correlation
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coefficient is:
1 1+𝑟
𝑍𝑟𝑗 = 𝑙𝑛
2 1−𝑟
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where r is the correlation coefficient between calcification and ΩArag.
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References
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Borenstein M, Hedges LV, Higgins JPT, Rothstein HR (2009) Introduction to Meta-Analysis. John Wiley &
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Sons, Hoboken, NJ, USA
Higgins J, Thompson SG, Deeks JJ, Altman DG (2003) Measuring inconsistency in meta-analysis. BMJ 327:
557-560
Moller AP, Jennions MD (2001) Testing and adjusting for publication bias. TRENDS in Ecology & Evolution
16: 580-586
Thompson SG, Sharp SJ (1999) Explaining heterogeneity in meta-analysis: A comparison of methods.
Statist. Med. 18: 2693-2708
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