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Calculating Prediction Intervals

pima.Rd

This function can estimate prediction intervals (PIs) as follows: A parametric bootstrap PI based on confidence distribution (Nagashima et al., 2018). A parametric bootstrap confidence interval is also calculated based on the same sampling method for bootstrap PI. The Higgins–Thompson–Spiegelhalter (2009) prediction interval. The Partlett–Riley (2017) prediction intervals.

Usage

pima(
  y,
  se,
  v = NULL,
  alpha = 0.05,
  method = c("boot", "HTS", "HK", "SJ", "KR", "CL", "APX", "WL"),
  theta0 = 0,
  side = c("lt", "gt"),
  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 error estimates 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 prediction interval (Nagashima et al., 2018).

  • HTS: the Higgins–Thompson–Spiegelhalter (2009) prediction interval / (the DerSimonian & Laird estimator for \(\tau^2\) with an approximate variance estimator for the average effect, \((1/\sum{\hat{w}_i})^{-1}\), \(df=K-2\)).

  • HK: Partlett–Riley (2017) prediction interval (the REML estimator for \(\tau^2\) with the Hartung (1999)'s variance estimator [the Hartung and Knapp (2001)'s estimator] for the average effect, \(df=K-2\)).

  • SJ: Partlett–Riley (2017) prediction interval / (the REML estimator for \(\tau^2\) with the Sidik and Jonkman (2006)'s bias coreccted variance estimator for the average effect, \(df=K-2\)).

  • KR: Partlett–Riley (2017) prediction interval / (the REML estimator for \(\tau^2\) with the Kenward and Roger (1997)'s approach for the average effect, \(df=\nu-1\)).

  • APX: Partlett–Riley (2017) prediction interval / (the REML estimator for \(\tau^2\) with an approximate variance estimator for the average effect, \(df=K-2\)). for the average effect, \(df=\nu-1\)).

  • WL: Wang–Lee (2019) prediction interval / (a method of sample quantiles of ensemble estimates).

theta0

threshold \(\theta_0\), for the cumulative probability of effect \(\theta_{new}\) less or greater than \(\theta_0\); \(\Pr(\theta_{new} < \theta_0)\) or \(\Pr(\theta_{new} > \theta_0)\).

side

either the cumulative probability of effect less (default = "lt") or greater ("gt") then \(\theta_0\)

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 upper limit 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 \(\hat{\mu}_u\).

  • lpi, upi: the lower and upper prediction limits \(\hat{c}_l\) and \(\hat{c}_u\).

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

  • i2h: the estimate for \(I^2\).

  • nup: degrees of freedom for the prediction interval.

  • nuc: degrees of freedom for the confidence interval.

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

Details

The functions bootPI, pima_boot, pima_hts, htsdl, pima_htsreml, htsreml are deprecated, and integrated to the pima function.

References

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

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.

Wang, C-C and Lee, W-C. (2019). A simple method to estimate prediction intervals and predictive distributions. Res Syn Meth. 30(28): 3304-3312. https://doi.org/10.1002/jrsm.1345.

Hartung, J. (1999). An alternative method for meta-analysis. Biom J. 41(8): 901-916. https://doi.org/10.1002/(SICI)1521-4036(199912)41:8<901::AID-BIMJ901>3.0.CO;2-W.

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.

Kenward, M. G., and Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics. 53(3): 983-997. https://www.ncbi.nlm.nih.gov/pubmed/9333350.

DerSimonian, R., and Laird, N. (1986). Meta-analysis in clinical trials. Control Clin Trials. 7(3): 177-188.

See also

print.pima, plot.pima, cima.

Examples

data(sbp, package = "pimeta")

# Nagashima-Noma-Furukawa prediction interval
# is sufficiently accurate when I^2 >= 10% and K >= 3
pimeta::pima(sbp$y, sbp$sigmak, seed = 3141592, parallel = 4)
#> 
#> Prediction & Confidence Intervals for Random-Effects Meta-Analysis
#> 
#> A parametric bootstrap prediction and confidence intervals
#>  Heterogeneity variance: DerSimonian-Laird
#>  Variance for average treatment effect: Hartung (Hartung-Knapp)
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% prediction interval]:
#>  -0.3341 [-0.8789, 0.2165]
#>  d.f.: 9
#> 
#> 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%
#> 
#> Estimated cumulative probability of effect `theta_new`
#>  Pr(theta_new < 0): 0.9157
#> 

# Higgins-Thompson-Spiegelhalter prediction interval and
# Partlett-Riley prediction intervals
# are accurate when I^2 > 30% and K > 25
pimeta::pima(sbp$y, sbp$sigmak, method = "HTS")
#> 
#> Prediction & Confidence Intervals for Random-Effects Meta-Analysis
#> 
#> Higgins-Thompson-Spiegelhalter prediction and confidence intervals
#>  Heterogeneity variance: DerSimonian-Laird
#>  Variance for average treatment effect: approximate
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% prediction interval]:
#>  -0.3341 [-0.7598, 0.0917]
#>  d.f.: 8
#> 
#> 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%
#> 
#> Estimated cumulative probability of effect `theta_new`
#>  Pr(theta_new < 0): 0.9460
#> 
pimeta::pima(sbp$y, sbp$sigmak, method = "HK")
#> 
#> Prediction & Confidence Intervals for Random-Effects Meta-Analysis
#> 
#> Partlett-Riley prediction and confidence intervals
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: Hartung (Hartung-Knapp)
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% prediction interval]:
#>  -0.3287 [-0.9887, 0.3312]
#>  d.f.: 8
#> 
#> 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%
#> 
#> Estimated cumulative probability of effect `theta_new`
#>  Pr(theta_new < 0): 0.8581
#> 
pimeta::pima(sbp$y, sbp$sigmak, method = "SJ")
#> 
#> Prediction & Confidence Intervals for Random-Effects Meta-Analysis
#> 
#> Partlett-Riley prediction and confidence intervals
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: bias corrected Sidik-Jonkman
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% prediction interval]:
#>  -0.3287 [-0.9835, 0.3261]
#>  d.f.: 8
#> 
#> 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%
#> 
#> Estimated cumulative probability of effect `theta_new`
#>  Pr(theta_new < 0): 0.8598
#> 
pimeta::pima(sbp$y, sbp$sigmak, method = "KR")
#> 
#> Prediction & Confidence Intervals for Random-Effects Meta-Analysis
#> 
#> Partlett-Riley prediction and confidence intervals
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: Kenward-Roger
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% prediction interval]:
#>  -0.3287 [-1.0280, 0.3706]
#>  d.f.: 5.9508
#> 
#> 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%
#> 
#> Estimated cumulative probability of effect `theta_new`
#>  Pr(theta_new < 0): 0.8534
#> 
pimeta::pima(sbp$y, sbp$sigmak, method = "APX")
#> 
#> Prediction & Confidence Intervals for Random-Effects Meta-Analysis
#> 
#> Partlett-Riley prediction and confidence intervals
#>  Heterogeneity variance: REML
#>  Variance for average treatment effect: approximate
#> 
#> No. of studies: 10
#> 
#> Average treatment effect [95% prediction interval]:
#>  -0.3287 [-0.9843, 0.3268]
#>  d.f.: 8
#> 
#> 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%
#> 
#> Estimated cumulative probability of effect `theta_new`
#>  Pr(theta_new < 0): 0.8596
#>