pima: Calculating Prediction Intervals
In pimeta: Prediction Intervals for Random-Effects Meta-Analysis
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
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
1
2
3
4
5
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 τ^2 with
an approximate variance estimator for the average effect,
(1/∑{\hat{w}_i})^{-1}, df=K-2).
-
HK: Partlett–Riley (2017) prediction interval
(the REML estimator for τ^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 τ^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 τ^2 with
the Kenward and Roger (1997)'s approach
for the average effect, df=ν-1).
-
APX: Partlett–Riley (2017) prediction interval /
(the REML estimator for τ^2 with
an approximate variance estimator for the average
effect, df=K-2).
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 τ^2
upper
the upper limit of random numbers of τ^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 τ^2
maxiter
the maximum number of iteration for REML estimation
Details
The functions bootPI, pima_boot,
pima_hts, htsdl, pima_htsreml, htsreml
are deprecated, and integrated to the pima function.
Value
-
K: the number of studies.
-
muhat: the average treatment effect estimate \hat{μ}.
-
lci, uci: the lower and upper confidence limits \hat{μ}_l and \hat{μ}_u.
-
lpi, upi: the lower and upper prediction limits \hat{c}_l and \hat{c}_u.
-
tau2h: the estimate for τ^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{μ}.
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. (2018).
Prediction intervals for random-effects meta-analysis:
a confidence distribution approach.
Stat Methods Med Res.
In press.
https://doi.org/10.1177/0962280218773520.
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
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
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)
# 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")
pimeta::pima(sbp$y, sbp$sigmak, method = "HK")
pimeta::pima(sbp$y, sbp$sigmak, method = "SJ")
pimeta::pima(sbp$y, sbp$sigmak, method = "KR")
pimeta::pima(sbp$y, sbp$sigmak, method = "APX")
Example output
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%
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%
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%
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%
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%
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%
pimeta documentation built on Sept. 17, 2019, 5:03 p.m.
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Description Usage Arguments Details Value References See Also Examples
Description
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
1 2 3 4 5 |
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").
|
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 τ^2 |
upper |
the upper limit of random numbers of τ^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 τ^2 |
maxiter |
the maximum number of iteration for REML estimation |
Details
The functions bootPI, pima_boot,
pima_hts, htsdl, pima_htsreml, htsreml
are deprecated, and integrated to the pima function.
Value
-
K: the number of studies. -
muhat: the average treatment effect estimate \hat{μ}. -
lci,uci: the lower and upper confidence limits \hat{μ}_l and \hat{μ}_u. -
lpi,upi: the lower and upper prediction limits \hat{c}_l and \hat{c}_u. -
tau2h: the estimate for τ^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{μ}.
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. (2018). Prediction intervals for random-effects meta-analysis: a confidence distribution approach. Stat Methods Med Res. In press. https://doi.org/10.1177/0962280218773520.
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
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | 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)
# 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")
pimeta::pima(sbp$y, sbp$sigmak, method = "HK")
pimeta::pima(sbp$y, sbp$sigmak, method = "SJ")
pimeta::pima(sbp$y, sbp$sigmak, method = "KR")
pimeta::pima(sbp$y, sbp$sigmak, method = "APX")
|
Example output
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%
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%
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%
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%
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%
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%
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