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Calculating Confidence Intervals

cima.Rd

This function calculates confidence intervals.

Usage

cima(
  y,
  se,
  v = NULL,
  alpha = 0.05,
  method = c("boot", "DL", "HK", "SJ", "KR", "APX", "PL", "BC"),
  B = 25000,
  parallel = FALSE,
  seed = NULL,
  maxit1 = 1e+05,
  eps = 10^(-10),
  lower = 0,
  upper = 1000,
  maxit2 = 1000,
  tol = .Machine$double.eps^0.25,
  rnd = NULL,
  maxiter = 100
)

Arguments

y

the effect size estimates vector

se

the within studies standard errors vector

v

the within studies variance estimates vector

alpha

the alpha level of the prediction interval

method

the calculation method for the pretiction interval (default = "boot").

  • boot: A parametric bootstrap confidence interval (Nagashima et al., 2018).

  • DL: A Wald-type t-distribution confidence interval (the DerSimonian & Laird estimator for \(\tau^2\) with an approximate variance estimator for the average effect, \((1/\sum{\hat{w}_i})^{-1}\), \(df=K-1\)).

  • HK: A Wald-type t-distribution confidence interval (the REML estimator for \(\tau^2\) with the Hartung (1999)'s varance estimator [the Hartung and Knapp (2001)'s estimator] for the average effect, \(df=K-1\)).

  • SJ: A Wald-type t-distribution confidence interval (the REML estimator for \(\tau^2\) with the Sidik and Jonkman (2006)'s bias coreccted SE estimator for the average effect, \(df=K-1\)).

  • KR: Partlett–Riley (2017) confidence interval / (the REML estimator for \(\tau^2\) with the Kenward and Roger (1997)'s approach for the average effect, \(df=\nu\)).

  • APX: A Wald-type t-distribution confidence interval / (the REML estimator for \(\tau^2\) with an approximate variance estimator for the average effect, \(df=K-1\)).

  • PL: Profile likelihood confidence interval (Hardy & Thompson, 1996).

  • BC: Profile likelihood confidence interval with Bartlett-type correction (Noma, 2011).

B

the number of bootstrap samples

parallel

the number of threads used in parallel computing, or FALSE that means single threading

seed

set the value of random seed

maxit1

the maximum number of iteration for the exact distribution function of \(Q\)

eps

the desired level of accuracy for the exact distribution function of \(Q\)

lower

the lower limit of random numbers of \(\tau^2\)

upper

the lower upper of random numbers of \(\tau^2\)

maxit2

the maximum number of iteration for numerical inversions

tol

the desired level of accuracy for numerical inversions

rnd

a vector of random numbers from the exact distribution of \(\tau^2\)

maxiter

the maximum number of iteration for REML estimation

Value

  • K: the number of studies.

  • muhat: the average treatment effect estimate \(\hat{\mu}\).

  • lci, uci: the lower and upper confidence limits \(\hat{\mu}_l\) and \(\hat{\mu}_u\).

  • tau2h: the estimate for \(\tau^2\).

  • i2h: the estimate for \(I^2\).

  • nuc: degrees of freedom for the confidence interval.

  • vmuhat: the variance estimate for \(\hat{\mu}\).

Details

Excellent reviews of heterogeneity variance estimation have been published (e.g., Veroniki, et al., 2018).

References

Veroniki, A. A., Jackson, D., Bender, R., Kuss, O., Langan, D., Higgins, J. P. T., Knapp, G., and Salanti, J. (2019). Methods to calculate uncertainty in the estimated overall effect size from a random-effects meta-analysis Res Syn Meth. 10(1): 23-43. https://doi.org/10.1002/jrsm.1319.

Nagashima, K., Noma, H., and Furukawa, T. A. (2019). Prediction intervals for random-effects meta-analysis: a confidence distribution approach. Stat Methods Med Res. 28(6): 1689-1702. https://doi.org/10.1177/0962280218773520.

Higgins, J. P. T, Thompson, S. G., Spiegelhalter, D. J. (2009). A re-evaluation of random-effects meta-analysis. J R Stat Soc Ser A Stat Soc. 172(1): 137-159. https://doi.org/10.1111/j.1467-985X.2008.00552.x

Partlett, C, and Riley, R. D. (2017). Random effects meta-analysis: Coverage performance of 95 confidence and prediction intervals following REML estimation. Stat Med. 36(2): 301-317. https://doi.org/10.1002/sim.7140

Hartung, J., and Knapp, G. (2001). On tests of the overall treatment effect in meta-analysis with normally distributed responses. Stat Med. 20(12): 1771-1782. https://doi.org/10.1002/sim.791

Sidik, K., and Jonkman, J. N. (2006). Robust variance estimation for random effects meta-analysis. Comput Stat Data Anal. 50(12): 3681-3701. https://doi.org/10.1016/j.csda.2005.07.019

Noma H. (2011) Confidence intervals for a random-effects meta-analysis based on Bartlett-type corrections. Stat Med. 30(28): 3304-3312. https://doi.org/10.1002/sim.4350

See also

pima

Examples

data(sbp, package = "pimeta")
set.seed(20161102)

# Nagashima-Noma-Furukawa confidence interval
pimeta::cima(sbp$y, sbp$sigmak, seed = 3141592)
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> A parametric bootstrap confidence interval
#>  Heterogeneity variance: DerSimonian-Laird
#>  Variance for average treatment effect: Hartung
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3341 [-0.5673, -0.0985]
#>  d.f.: 9
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0282
#>  I-squared: 70.5%
#> 

# A Wald-type t-distribution confidence interval
# An approximate variance estimator & DerSimonian-Laird estimator for tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "DL")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3341 [-0.5068, -0.1613]
#>  d.f.: 9
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0282
#>  I-squared: 70.5%
#> 

# A Wald-type t-distribution confidence interval
# The Hartung variance estimator & REML estimator for tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "HK")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> A Wald-type t-distribution confidence interval
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: Hartung-Knapp
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3287 [-0.5761, -0.0814]
#>  d.f.: 9
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0700
#>  I-squared: 85.5%
#> 

# A Wald-type t-distribution confidence interval
# The Sidik-Jonkman variance estimator & REML estimator for tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "SJ")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> A Wald-type t-distribution confidence interval
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: bias corrected Sidik-Jonkman
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3287 [-0.5625, -0.0950]
#>  d.f.: 9
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0700
#>  I-squared: 85.5%
#> 

# A Wald-type t-distribution confidence interval
# The Kenward-Roger approach & REML estimator for tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "KR")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> A Wald-type t-distribution confidence interval
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: Kenward-Roger
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3287 [-0.5815, -0.0759]
#>  d.f.: 6.9508
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0700
#>  I-squared: 85.5%
#> 

# A Wald-type t-distribution confidence interval
# An approximate variance estimator & REML estimator for tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "APX")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3287 [-0.5646, -0.0929]
#>  d.f.: 9
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0700
#>  I-squared: 85.5%
#> 

# Profile likelihood confidence interval
# Maximum likelihood estimators of variance for the average effect & tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "PL")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> A profile likelihood confidence interval
#>  Heterogeneity variance: ML
#>  Variance for average treatment effect: ML
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3301 [-0.5435, -0.1089]
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0547
#>  I-squared: 82.2%
#> 

# Profile likelihood confidence interval with a Bartlett-type correction
# Maximum likelihood estimators of variance for the average effect & tau^2
pimeta::cima(sbp$y, sbp$sigmak, method = "BC")
#> 
#> Confidence Interval for Random-Effects Meta-Analysis
#> 
#> A profile likelihood confidence interval with a Bartlett-type correction
#>  Heterogeneity variance: ML
#>  Variance for average treatment effect: ML
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% confidence interval]:
#>  -0.3301 [-0.5733, -0.0781]
#> 
#> Heterogeneity measure
#>  tau-squared: 0.0547
#>  I-squared: 82.2%
#>