Package 'altmeta'

A Meta-Analysis on the Effects of Fluvastatin 20 mg/day on High-Density Lipoprotein

Description

This meta-analysis serves as an example to illustrate the usage of the function meta.pen.

Usage

data("dat.ha")

Format

A data frame containing 32 studies.

y

the observed effect size (mean difference) for each study in the meta-analysis.

s2

the within-study variance for each study.

Source

Adams SP, Sekhon SS, Tsang M, Wright JM (2018). "Fluvastatin for lowering lipids." Cochrane Database of Systematic Reviews, 7. Art. No.: CD012282. <doi:10.1002/14651858.CD012282.pub2>

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A Meta-Analysis for Evaluating the Effect of Aerobic Exercise on Visceral Adipose Tissue Content/Volume

Description

This meta-analysis serves as an example to illustrate function usage in the package altmeta.

Usage

data("dat.aex")

Format

A data frame containing 29 studies with the observed effect sizes and their within-study variances.

y

the observed effect size for each collected study in the meta-analysis.

s2

the within-study variance for each study.

Source

Ismail I, Keating SE, Baker MK, Johnson NA (2012). "A systematic review and meta-analysis of the effect of aerobic vs. resistance exercise training on visceral fat." Obesity Reviews, 13(1), 68–91. <doi:10.1111/j.1467-789X.2011.00931.x>

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A Meta-Analysis for Comparing the Effect of Steroids vs. Control in the Length of Intensive Care Unit (ICU) Stay

Description

This dataset serves as an example of meta-analysis of mean differences.

Usage

data("dat.annane")

Format

A data frame with 12 studies with the following 5 variables within each study.

y

point esimates of mean differences.

s2

sample variances of mean differences.

n1

sample sizes in treatment group 1 (steroids).

n2

sample sizes in treatment group 2 (control).

n

total sample sizes.

Source

Annane D, Bellissant E, Bollaert PE, Briegel J, Keh D, Kupfer Y (2015). "Corticosteroids for treating sepsis." Cochrane Database of Systematic Reviews, 12, Art. No.: CD002243. <doi:10.1002/14651858.CD002243.pub3>

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A Network Meta-Analysis on Effects of Pharmacologic Treatments for Chronic Obstructive Pulmonary Disease

Description

This dataset serves as an example of network meta-analysis with binary outcomes.

Usage

data("dat.baker")

Format

A dataset of network meta-analysis with binary outcomes, containing 38 studies and 5 treatments.

sid

study IDs.

tid

treatment IDs.

r

event counts.

n

sample sizes.

Details

Treatment IDs represent: 1) placebo; 2) inhaled corticosteroid; 3) inhaled corticosteroid + long-acting $\beta_2$-agonist; 4) long-acting $\beta_2$-agonist; and 5) tiotropium.

Source

Baker WL, Baker EL, Coleman CI (2009). "Pharmacologic treatments for chronic obstructive pulmonary disease: a mixed-treatment comparison meta-analysis." Pharmacotherapy, 29(8), 891–905. <doi:10.1592/phco.29.8.891>

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A Meta-Analysis on the Effect of Parent Training Programs vs. Control for Improving Parental Psychosocial Health Within 4 Weeks After Intervention

Description

This dataset serves as an example of meta-analysis of standardized mean differences.

Usage

data("dat.barlow")

Format

A data frame with 26 studies with the following 5 variables within each study.

y

point esimates of standardized mean differences.

s2

sample variances of standardized mean differences.

n1

sample sizes in treatment group 1 (parent training programs).

n2

sample sizes in treatment group 2 (control).

n

total sample sizes.

Source

Barlow J, Smailagic N, Huband N, Roloff V, Bennett C (2014). "Group-based parent training programmes for improving parental psychosocial health." Cochrane Database of Systematic Reviews, 5, Art. No.: CD002020. <doi:10.1002/14651858.CD002020.pub4>

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A Meta-Analysis of Prevalence of Depression or Depressive Symptoms Among Medical Students

Description

This dataset serves as an example of meta-analysis of proportions.

Usage

data("dat.beck17")

Format

A data frame with 6 studies with the following 2 variables within each study.

e

event counts of samples with depression or depressive symptoms.

n

sample sizes.

Details

The original article by Rotenstein et al. (2016) stratified all extracted studies based on various screening instruments and cutoff scores. This dataset focuses on the meta-analysis of 6 studies with Beck Depression Inventory Score $\geq$17.

Source

Rotenstein LS, Ramos MA, Torre M, Segal JB, Peluso MJ, Guille C, Sen S, Mata DA (2016). "Prevalence of depression, depressive symptoms, and suicidal ideation among medical students: a systematic review and meta-analysis." JAMA, 316(21), 2214–2236. <doi:10.1001/jama.2016.17324>

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A Meta-Analysis on Type 2 Diabetes Mellitus After Gestational Diabetes

Description

This meta-analysis serves as an example of meta-analysis with binary outcomes.

Usage

data("dat.bellamy")

Format

A data frame containing 20 cohort studies with the following 4 variables.

sid

study IDs.

tid

treatment/exposure IDs (0: non-exposure; 1: exposure).

e

event counts.

n

sample sizes.

Source

Bellamy L, Casas JP, Hingorani AD, Williams D (2009). "Type 2 diabetes mellitus after gestational diabetes: a systematic review and meta-analysis." Lancet, 373(9677), 1773–1779. <doi:10.1016/S0140-6736(09)60731-5>

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A Meta-Analysis on the Beneficial and Harmful Effects of Vitamin D Supplementation for the Prevention of Mortality in Healthy Adults and Adults in a Stable Phase of Disease

Description

This meta-analysis serves as an example to illustrate the usage of the function meta.pen.

Usage

data("dat.ha")

Format

A data frame containing 53 studies.

n1

the sample size in the treatment group in each study.

n2

the sample size in the control group in each study.

r1

the event count in the treatment group in each study.

r2

the event count in the control group in each study.

y

the observed effect size (log odds ratio) for each study in the meta-analysis.

s2

the within-study variance for each study.

Source

Bjelakovic G, Gluud LL, Nikolova D, Whitfield K, Wetterslev J, Simonetti RG, Bjelakovic M, Gluud C (2014). "Vitamin D supplementation for prevention of mortality in adults." Cochrane Database of Systematic Reviews, 7. Art. No.: CD007470. <doi:10.1002/14651858.CD007470.pub3>

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A Meta-Analysis on the Effects of Continuous, One-To-One Intrapartum Support on Spontaneous Vaginal Births, Compared With Usual Care

Description

This meta-analysis serves as an example to illustrate the usage of the function meta.pen.

Usage

data("dat.ha")

Format

A data frame containing 21 studies.

n1

the sample size in the treatment group in each study.

n2

the sample size in the control group in each study.

r1

the event count in the treatment group in each study.

r2

the event count in the control group in each study.

y

the observed effect size (log odds ratio) for each study in the meta-analysis.

s2

the within-study variance for each study.

Source

Bohren MA, Hofmeyr GJ, Sakala C, Fukuzawa RK, Cuthbert A (2017). "Continuous support for women during childbirth." Cochrane Database of Systematic Reviews, 7. Art. No.: CD003766. <doi:10.1002/14651858.CD003766.pub6>

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A Meta-Analysis on the Overall Response of the Addition of Drugs to a Chemotherapy Regimen for Metastatic Breast Cancer

Description

This dataset serves as an example of meta-analysis of (log) odds ratios.

Usage

data("dat.butters")

Format

A data frame with 16 studies with the following 7 variables within each study.

y

point esimates of log odds ratios.

s2

sample variances of log odds ratios.

n1

sample sizes in treatment group 1 (addition of drug).

n2

sample sizes in treatment group 2 (control).

r1

event counts in treatment group 1.

r2

event counts in treatment group 2.

n

total sample sizes.

Source

Butters DJ, Ghersi D, Wilcken N, Kirk SJ, Mallon PT (2010). "Addition of drug/s to a chemotherapy regimen for metastatic breast cancer." Cochrane Database of Systematic Reviews, 11, Art. No.: CD003368. <doi:10.1002/14651858.CD003368.pub3>

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A Meta-Analysis on the Efficacy of Platelet-Rich-Plasmapheresis in Reducing Peri-Operative Allogeneic Red Blood Cell Transfusion in Cardiac Surgery

Description

This meta-analysis serves as an example to illustrate the usage of the function meta.pen.

Usage

data("dat.ha")

Format

A data frame containing 20 studies.

n1

the sample size in the treatment group in each study.

n2

the sample size in the control group in each study.

r1

the event count in the treatment group in each study.

r2

the event count in the control group in each study.

y

the observed effect size (log odds ratio) for each study in the meta-analysis.

s2

the within-study variance for each study.

Source

Carless PA, Rubens FD, Anthony DM, O'Connell D, Henry DA (2011). "Platelet-rich-plasmapheresis for minimising peri-operative allogeneic blood transfusion." Cochrane Database of Systematic Reviews, 3. Art. No.: CD004172. <doi:10.1002/14651858.CD004172.pub2>

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A Meta-Analysis of Proportions on Chorioamnionitis

Description

This dataset serves as an example of meta-analysis of proportions.

Usage

data("dat.chor")

Format

A data frame with 21 studies with the following 2 variables within each study.

e

event counts of horioamnionitis.

n

sample sizes.

Source

Woodd SL, Montoya A, Barreix M, Pi L, Calvert C, Rehman AM, Chou D, Campbell OMR (2019). "Incidence of maternal peripartum infection: a systematic review and meta-analysis." PLOS Medicine, 16(12), e1002984. <doi:10.1371/journal.pmed.1002984>

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A Meta-Analysis of Binary and Continuous Outcomes on Depression

Description

This dataset serves as an example of meta-analysis of combining standardized mean differences and odds ratios.

Usage

data("dat.dep")

Format

A data frame with 6 studies with the following 15 variables within each study.

author

The first author of each study.

year

The publication year of each study.

treatment

The treatment group.

control

The control group.

y1

The sample mean in the treatment group for the continuous outcome.

sd1

The sample standard deviation in the treatment group for the continuous outcome.

n1

The sample size in the treatment group for the continuous outcome.

y0

The sample mean in the control group for the continuous outcome.

sd0

The sample standard deviation in the control group for the continuous outcome.

n0

The sample size in the control group for the continuous outcome.

r1

The event count in the treatment group for the binary outcome.

m1

The sample size in the treatment group for the binary outcome.

r0

The event count in the control group for the binary outcome.

m0

The sample size in the control group for the binary outcome.

id.bin

An indicator of whether the outcome is binary (1) or continuous (0).

Details

This dataset is from Cipriani et al. (2016), comparing the efficacy and tolerability of antidepressants for major depressive disorders in children and adolescents. Our case study focuses on efficacy. The authors originally performed a network meta-analysis; however, here we restrict the comparison to fluoxetine and placebo. The continuous outcomes are measured by the mean overall changes in depressive symptoms from baseline to endpoint. For the binary outcomes, events are defined as whether patients' depression rating scores were reduced by at least a specified cutoff value.

Source

Cipriani A, Zhou X, Del Giovane C, Hetrick SE, Qin B, Whittington C, Coghill D, Zhang Y, Hazell P, Leucht S, Cuijpers P, Pu J, Cohen D, Ravindran AV, Liu Y, Michael KD, Yang L, Liu L, Xie P (2016). "Comparative efficacy and tolerability of antidepressants for major depressive disorder in children and adolescents: a network meta-analysis." The Lancet, 388(10047), 881–890. <doi:10.1016/S0140-6736(16)30385-3>

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A Meta-Analysis on the Effect of Long-Acting Inhaled Beta2-Agonists vs. Control for Chronic Asthma

Description

This meta-analysis serves as an example of meta-analysis with binary outcomes.

Usage

data("dat.ducharme")

Format

A data frame containing 33 studies with the following 4 variables within each study.

n00

counts of non-events in treatment group 0 (placebo).

n01

counts of events in treatment group 0 (placebo).

n10

counts of non-events in treatment group 1 (beta2-agonists).

n11

counts of events in treatment group 1 (beta2-agonists).

Note

The original review collected 35 studies; two double-zero-counts studies are excluded from this dataset because their odds ratios are not estimable.

Source

Ducharme FM, Ni Chroinin M, Greenstone I, Lasserson TJ (2010). "Addition of long-acting beta2-agonists to inhaled corticosteroids versus same dose inhaled corticosteroids for chronic asthma in adults and children." Cochrane Database of Systematic Reviews, 5, Art. No.: CD005535. <doi:10.1002/14651858.CD005535.pub2>

---

A Multivariate Meta-Analysis by the Fibrinogen Studies Collaboration

Description

This multivariate meta-analysis serves as an example to illustrate function usage in the package altmeta. It consists of 31 studies with 4 outcomes.

Usage

data("dat.fib")

Format

A list containing three elements, y, S, and sd.

y

a 31 x 4 numeric matrix containing the observed effect sizes; the rows represent studies and the columns represent outcomes.

S

a list containing 31 elements; each element is within-study covariance matrix of the corresponding study.

sd

a 31 x 4 numeric matrix containing the within-study standard deviations; the rows represent studies and the columns represent outcomes.

Source

Fibrinogen Studies Collaboration (2004). "Collaborative meta-analysis of prospective studies of plasma fibrinogen and cardiovascular disease." European Journal of Cardiovascular Prevention and Rehabilitation, 11(1), 9–17. <doi:10.1097/01.hjr.0000114968.39211.01>

Fibrinogen Studies Collaboration (2005). "Plasma fibrinogen level and the risk of major cardiovascular diseases and nonvascular mortality: an individual participant meta-analysis." JAMA, 294(14), 1799–1809. <doi:10.1001/jama.294.14.1799>

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A Meta-Analysis on the Effect of Placebo Interventions for All Clinical Conditions Regarding Patient-Reported Outcomes

Description

This meta-analysis serves as an example to illustrate function usage in the package altmeta.

Usage

data("dat.ha")

Format

A data frame containing 109 studies with the observed effect sizes and their within-study variances.

y

the observed effect size for each collected study in the meta-analysis.

s2

the within-study variance for each study.

Source

Hrobjartsson A, Gotzsche PC (2010). "Placebo interventions for all clinical conditions." Cochrane Database of Systematic Reviews, 1. Art. No.: CD003974. <doi:10.1002/14651858.CD003974.pub3>

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A Meta-Analysis for Evaluating the Effect of Tranexamic Acid on Perioperative Allogeneic Blood Transfusion

Description

This meta-analysis serves as an example of meta-analysis with binary outcomes.

Usage

data("dat.henry")

Format

A data frame containing 26 studies with the following 4 variables within each study.

n00

counts of non-events in treatment group 0 (placebo).

n01

counts of events in treatment group 0 (placebo).

n10

counts of non-events in treatment group 1 (tranexamic acid).

n11

counts of events in treatment group 1 (tranexamic acid).

Note

The original review collected 27 studies; one double-zero-counts study is excluded from this dataset because its odds ratio is not estimable.

Source

Henry DA, Carless PA, Moxey AJ, O'Connell, Stokes BJ, Fergusson DA, Ker K (2011). "Anti-fibrinolytic use for minimising perioperative allogeneic blood transfusion." Cochrane Database of Systematic Reviews, 1, Art. No.: CD001886. <doi:10.1002/14651858.CD001886.pub3>

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A Meta-Analysis on the Magnitude and Duration of Excess Mortality After Hip Fracture Among Older Men

Description

This meta-analysis serves as an example to illustrate function usage in the package altmeta.

Usage

data("dat.hipfrac")

Format

A data frame containing 17 studies with the observed effect sizes and their within-study variances.

y

the observed effect size for each collected study in the meta-analysis.

s2

the within-study variance for each study.

Source

Haentjens P, Magaziner J, Colon-Emeric CS, Vanderschueren D, Milisen K, Velkeniers B, Boonen S (2010). "Meta-analysis: excess mortality after hip fracture among older women and men". Annals of Internal Medicine, 152(6), 380–390. <doi:10.7326/0003-4819-152-6-201003160-00008>

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A Meta-Analysis on the Effect of Brief Alcohol Interventions vs. Control in Primary Care Populations

Description

This dataset serves as an example of meta-analysis of risk differences.

Usage

data("dat.kaner")

Format

A data frame with 13 studies with the following 7 variables within each study.

y

point esimates of risk differences.

s2

sample variances of risk differences.

n1

sample sizes in treatment group 1 (brief alcohol interventions).

n2

sample sizes in treatment group 2 (control).

r1

event counts in treatment group 1.

r2

event counts in treatment group 2.

n

total sample sizes.

Source

Kaner EF, Dickinson HO, Beyer FR, Campbell F, Schlesinger C, Heather N, Saunders JB, Burnand B, Pienaar ED (2007). "Effectiveness of brief alcohol interventions in primary care populations." Cochrane Database of Systematic Reviews, 2, Art. No.: CD004148. <doi:10.1002/14651858.CD004148.pub3>

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A Meta-Analysis on the Effect of Progressive Resistance Strength Training Exercise vs. Control

Description

This meta-analysis serves as an example to illustrate function usage in the package altmeta.

Usage

data("dat.lcj")

Format

A data frame containing 33 studies with the observed effect sizes and their within-study variances.

y

the observed effect size for each collected study in the meta-analysis.

s2

the within-study variance for each study.

Source

Liu CJ, Latham NK (2009). "Progressive resistance strength training for improving physical function in older adults." Cochrane Database of Systematic Reviews, 3. Art. No.: CD002759. <doi:10.1002/14651858.CD002759.pub2>

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A Meta-Analysis on the Effectiveness of Spinal Manipulative Therapies (Other Than Sham)

Description

This dataset serves as an example of meta-analysis of standardized mean differences.

Usage

data("dat.paige")

Format

A data frame with 6 studies with the following 5 variables within each study.

y

point esimates of standardized mean differences.

s2

sample variances of standardized mean differences.

n1

sample sizes in treatment group 1 (spinal manipulation).

n2

sample sizes in treatment group 2 (comparator).

n

total sample sizes.

Source

Paige NM, Miake-Lye IM, Booth MS, Beroes JM, Mardian AS, Dougherty P, Branson R, Tang B, Morton SC, Shekelle PG (2017). "Association of spinal manipulative therapy with clinical benefit and harm for acute low back pain: systematic review and meta-analysis." JAMA, 317(14), 1451–1460. <doi:10.1001/jama.2017.3086>

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A Meta-Analysis for Comparing the Fluoroscopy Time in Percutaneous Coronary Intervention Between Radial and Femoral Accesses

Description

This dataset serves as an example of meta-analysis of mean differences.

Usage

data("dat.plourde")

Format

A data frame with 19 studies with the following 5 variables within each study.

y

point esimates of mean differences.

s2

sample variances of mean differences.

n1

sample sizes in treatment group 1 (radial).

n2

sample sizes in treatment group 2 (femoral).

n

total sample sizes.

Source

Plourde G, Pancholy SB, Nolan J, Jolly S, Rao SV, Amhed I, Bangalore S, Patel T, Dahm JB, Bertrand OF (2015). "Radiation exposure in relation to the arterial access site used for diagnostic coronary angiography and percutaneous coronary intervention: a systematic review and meta-analysis." Lancet, 386(10009), 2192–2203. <doi:10.1016/S0140-6736(15)00305-0>

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A Meta-Analysis for Evaluating the Effect of Mucolytic on Bronchitis/Chronic Obstructive Pulmonary Disease

Description

This meta-analysis serves as an example of meta-analysis with binary outcomes.

Usage

data("dat.poole")

Format

A data frame containing 24 studies with the following 4 variables within each study.

n00

counts of non-events in treatment group 0 (placebo).

n01

counts of events in treatment group 0 (placebo).

n10

counts of non-events in treatment group 1 (mucolytic).

n11

counts of events in treatment group 1 (mucolytic).

Source

Poole P, Chong J, Cates CJ (2015). "Mucolytic agents versus placebo for chronic bronchitis or chronic obstructive pulmonary disease." Cochrane Database of Systematic Reviews, 7, Art. No.: CD001287. <doi:10.1002/14651858.CD001287.pub5>

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Meta-Analysis of Multiple Risk Factors for Pterygium

Description

This dataset serves as an example to illustrate network meta-analysis of multiple factors. It consists of 29 studies on a total of 8 risk factors: area of residence (rural vs. urban); education attainment (low vs. high); latitude of residence (low vs. high); occupation type (outdoor vs. indoor); smoking status (yes vs. no); use of hat (yes vs. no); use of spectacles (yes vs. no); and use of sunglasses (yes vs. no). Each study only investigates a subset of the 8 risk factors, so the dataset contains many missing values.

Usage

data("dat.pte")

Format

A list containing two elements, y and se.

y

a 29 x 8 numeric matrix containing the observed effect sizes; the rows represent studies and the columns represent outcomes.

se

a 29 x 8 numeric matrix containing the within-study standard errors; the rows represent studies and the columns represent outcomes.

Source

Serghiou S, Patel CJ, Tan YY, Koay P, Ioannidis JPA (2016). "Field-wide meta-analyses of observational associations can map selective availability of risk factors and the impact of model specifications." Journal of Clinical Epidemiology, 71, 58–67. <doi:10.1016/j.jclinepi.2015.09.004>

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A Network Meta-Analysis on Smoking Cessation

Description

The dataset is extracted from Lu and Ades (2006); it was initially reported by Hasselblad (1998) (without performing a formal network meta-analysis).

Usage

data("dat.sc")

Format

A dataset of network meta-analysis with binary outcomes, containing 24 studies and 4 treatments.

sid

study IDs.

tid

treatment IDs.

r

event counts.

n

sample sizes.

Details

Treatment IDs represent: 1)no contact; 2) selfhelp; 3) individual counseling; and 4) group counseling.

Source

Hasselblad V (1998). "Meta-analysis of multitreatment studies." Medical Decision Making, 18(1), 37–43. <doi:10.1177/0272989X9801800110>

Lu G, Ades AE (2006). "Assessing evidence inconsistency in mixed treatment comparisons." Journal of the American Statistical Association, 101(474), 447–459. <doi:10.1198/016214505000001302>

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Meta-Analysis on the Utility of Lymphangiography, Computed Tomography, and Magnetic Resonance Imaging for the Diagnosis of Lymph Node Metastasis

Description

This meta-analysis serves as an example of meta-analyses of diagnostic tests.

Usage

data("dat.scheidler")

Format

A data frame with 44 studies with the following 5 variables; each row represents a study.

dt

types of diagnostic tests; CT: computed tomography; LAG: lymphangiography; and MRI: magnetic resonance imaging.

tp

counts of true positives.

fp

counts of false positives.

fn

counts of false negatives.

tn

counts of true negatives.

Source

Scheidler J, Hricak H, Yu KK, Subak L, Segal MR (1997). "Radiological evaluation of lymph node metastases in patients with cervical cancer: a meta-analysis." JAMA, 278(13), 1096–1101. <doi:10.1001/jama.1997.03550130070040>

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A Meta-Analysis on the Effect of Nicotine Gum for Smoking Cessation

Description

This meta-analysis serves as an example to illustrate function usage in the package altmeta.

Usage

data("dat.slf")

Format

A data frame containing 56 studies with the observed effect sizes and their within-study variances.

y

the observed effect size for each collected study in the meta-analysis.

s2

the within-study variance for each study.

Source

Stead LF, Perera R, Bullen C, Mant D, Hartmann-Boyce J, Cahill K, Lancaster T (2012). "Nicotine replacement therapy for smoking cessation." Cochrane Database of Systematic Reviews, 11. Art. No.: CD000146. <doi:10.1002/14651858.CD000146.pub4>

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Meta-Analysis on the Diagnostic Accuracy of Ultrasound for Detecting Partial Thickness Rotator Cuff Tears

Description

This meta-analysis serves as an example of meta-analyses of diagnostic tests.

Usage

data("dat.smith")

Format

A data frame with 30 studies with the following 4 variables; each row represents a study.

tp

counts of true positives.

fp

counts of false positives.

fn

counts of false negatives.

tn

counts of true negatives.

Source

Smith TO, Back T, Toms AP, Hing CB (2011). "Diagnostic accuracy of ultrasound for rotator cuff tears in adults: a systematic review and meta-analysis." Clinical Radiology, 66(11), 1036–1048. <doi:10.1016/j.crad.2011.05.007>

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A Meta-Analysis on Adverse Events for the Comparison Cannabinoid vs. Placebo

Description

This dataset serves as an example of meta-analysis of (log) odds ratios.

Usage

data("dat.whiting")

Format

A data frame with 29 studies with the following 9 variables within each study.

y

point esimates of log odds ratios.

s2

sample variances of log odds ratios.

n00

counts of non-events in treatment group 0 (placebo).

n01

counts of events in treatment group 0.

n10

counts of non-events in treatment group 1 (cannabinoid).

n11

counts of events in treatment group 1 (cannabinoid).

n0

sample sizes in treatment group 0.

n1

sample sizes in treatment group 1.

n

total sample sizes.

Source

Whiting PF, Wolff RF, Deshpande S, Di Nisio M, Duffy S, Hernandez AV, Keurentjes JC, Lang S, Misso K, Ryder S, Schmidlkofer S, Westwood M, Kleijnen J (2015). "Cannabinoids for medical use: a systematic review and meta-analysis." JAMA, 313(24), 2456–2473. <doi:10.1001/jama.2015.6358>

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A Meta-Analysis on the Effect of Pharmacotherapy for Social Anxiety Disorder

Description

This dataset serves as an example of meta-analysis of (log) relative risks.

Usage

data("dat.williams")

Format

A data frame with 20 studies with the following 7 variables within each study.

y

point esimates of log relative risks.

s2

sample variances of log relative risks.

n1

sample sizes in treatment group 1 (medication).

n2

sample sizes in treatment group 2 (placebo).

r1

event counts in treatment group 1.

r2

event counts in treatment group 2.

n

total sample sizes.

Source

Williams T, Hattingh CJ, Kariuki CM, Tromp SA, van Balkom AJ, Ipser JC, Stein DJ (2017). "Pharmacotherapy for social anxiety disorder (SAnD)." Cochrane Database of Systematic Reviews, 10, Art. No.: CD001206. <doi:10.1002/14651858.CD001206.pub3>

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A Network Meta-Analysis on Immune Checkpoint Inhibitor Drugs

Description

This network meta-analysis investigates the effects of seven immune checkpoint inhibitor (ICI) drugs on all-grade treatment-related adverse events (TrAEs). It aimed to provide a safety ranking of the ICI drugs for the treatment of cancer.

Usage

data("dat.xu")

Format

A dataset of network meta-analysis with binary outcomes, containing 23 studies and 7 treatments.

sid

study IDs.

tid

treatment IDs.

r

event counts.

n

sample sizes.

Details

Treatment IDs represent: 1) conventional therapy; 2) nivolumab; 3) pembrolizumab; 4) two ICIs; 5) ICI and conventional therapy; 6) atezolizumab; and 7) ipilimumab.

Source

Xu C, Chen YP, Du XJ, Liu JQ, Huang CL, Chen L, Zhou GQ, Li WF, Mao YP, Hsu C, Liu Q, Lin AH, Tang LL, Sun Y, Ma J (2018). "Comparative safety of immune checkpoint inhibitors in cancer: systematic review and network meta-analysis." BMJ, 363, k4226. <doi:10.1136/bmj.k4226>

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Meta-Analysis of Proportions Using Generalized Linear Mixed Models

Description

Performs a meta-analysis of proportions using generalized linear mixed models (GLMMs) with various link functions.

Usage

maprop.glmm(e, n, data, link = "logit", alpha = 0.05,
            pop.avg = TRUE, int.approx = 10000, b.iter = 1000,
            seed = 1234, ...)

Arguments

e	a numeric vector specifying the event counts in the collected studies.
n	a numeric vector specifying the sample sizes in the collected studies.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, e and n, should be specified as their corresponding column names in data.
link	a character string specifying the link function used in the GLMM, which can be one of "log" (log link), "logit" (logit link, the default), "probit" (probit link), "cauchit" (cauchit link), and "cloglog" (complementary log-log link).
alpha	a numeric value specifying the statistical significance level.
pop.avg	a logical value indicating whether the population-averaged proportion and its confidence interval are to be produced. This quantity is the marginal mean of study-specific proportions, while the commonly-reported overall proportion usually represents the median (or interpreted as a conditional measure); see more details about this quantity in Section 13.2.3 in Agresti (2013), Chu et al. (2012), Lin and Chu (2020), and Zeger et al. (1988). If pop.avg = TRUE (the default), the bootstrap resampling is used to produce the confidence interval of the population-averaged proportion; the confidence interval of the commonly-reported median proportion will be also produced, in addition to its conventional confidence interval (by back-transforming the Wald-type confidence interval derived on the scale specified by link).
int.approx	an integer specifying the number of independent standard normal samples for numerically approximating the integration involved in the calculation of the population-averaged proportion; see details in Lin and Chu (2020). It is only used when pop.avg = TRUE and link is not "probit". The probit link leads to a closed form of the population-averaged proportion, so it does not need the numerical approximation; for other links, the population-averaged proportion does not have a closed form.
b.iter	an integer specifying the number of bootstrap iterations; it is only used when pop.avg = TRUE.
seed	an integer for specifying the seed of the random number generation for reproducibility during the bootstrap resampling (and numerical approximation for the population-averaged proportion); it is only used when pop.avg = TRUE.
...	other arguments that can be passed to the function glmer in the package lme4.

Value

This function returns a list containing the point and interval estimates of the overall proportion. Specifically, prop.c.est is the commonly-reported median (or conditional) proportion, and prop.c.ci is its confidence interval. It also returns information about AIC, BIC, log likelihood, deviance, and residual degrees-of-freedom. If pop.avg = TRUE, the following additional elements will be also in the produced list: prop.c.ci.b is the bootstrap confidence interval of the commonly-reported median (conditional) proportion, prop.m.est is the point estimate of the population-averaged (marginal) proportion, prop.m.ci.b is the bootstrap confidence interval of the population-averaged (marginal) proportion, and b.w.e is a vector of two numeric values, indicating the counts of warnings and errors occurred during the bootstrap iterations.

Note

This function implements the GLMM for the meta-analysis of proportions via the function glmer in the package lme4. It is possible that the algorithm of the GLMM estimation may not converge for some bootstrapped meta-analyses when pop.avg = TRUE, and the function glmer may report warnings or errors about the convergence issue. The bootstrap iterations are continued until b.iter replicates without any warnings or errors are obtained; those replicates with any warnings or errors are discarded.

References

Agresti A (2013). Categorical Data Analysis. Third edition. John Wiley & Sons, Hoboken, NJ.

Bakbergenuly I, Kulinskaya E (2018). "Meta-analysis of binary outcomes via generalized linear mixed models: a simulation study." BMC Medical Research Methodology, 18, 70. <doi:10.1186/s12874-018-0531-9>

Chu H, Nie L, Chen Y, Huang Y, Sun W (2012). "Bivariate random effects models for meta-analysis of comparative studies with binary outcomes: methods for the absolute risk difference and relative risk." Statistical Methods in Medical Research, 21(6), 621–633. <doi:10.1177/0962280210393712>

Hamza TH, van Houwelingen HC, Stijnen T (2008). "The binomial distribution of meta-analysis was preferred to model within-study variability." Journal of Clinical Epidemiology, 61(1), 41–51. <doi:10.1016/j.jclinepi.2007.03.016>

Lin L, Chu H (2020). "Meta-analysis of proportions using generalized linear mixed models." Epidemiology, 31(5), 713–717. <doi:10.1097/ede.0000000000001232>

Stijnen T, Hamza TH, Ozdemir P (2010). "Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data." Statistics in Medicine, 29(29), 3046–3067. <doi:10.1002/sim.4040>

Zeger SL, Liang K-Y, Albert PS (1988). "Models for longitudinal data: a generalized estimating equation approach." Biometrics, 44(4), 1049–1060. <doi:10.2307/2531734>

See Also

maprop.twostep

Examples

# chorioamnionitis data
data("dat.chor")
# GLMM with the logit link with only 10 bootstrap iterations
out.chor.glmm.logit <- maprop.glmm(e, n, data = dat.chor,
  link = "logit", b.iter = 10, seed = 1234)
out.chor.glmm.logit
# not calculating the population-averaged (marginal) proportion,
#  without bootstrap resampling
out.chor.glmm.logit <- maprop.glmm(e, n, data = dat.chor,
  link = "logit", pop.avg = FALSE)
out.chor.glmm.logit

# increases the number of bootstrap iterations to 1000,
#  taking longer time
out.chor.glmm.logit <- maprop.glmm(e, n, data = dat.chor,
  link = "logit", b.iter = 1000, seed = 1234)
out.chor.glmm.logit

# GLMM with the log link
out.chor.glmm.log <- maprop.glmm(e, n, data = dat.chor,
  link = "log", b.iter = 10, seed = 1234)
out.chor.glmm.log
# GLMM with the probit link
out.chor.glmm.probit <- maprop.glmm(e, n, data = dat.chor,
  link = "probit", b.iter = 10, seed = 1234)
out.chor.glmm.probit
# GLMM with the cauchit link
out.chor.glmm.cauchit <- maprop.glmm(e, n, data = dat.chor,
  link = "cauchit", b.iter = 10, seed = 1234)
out.chor.glmm.cauchit
# GLMM with the cloglog link
out.chor.glmm.cloglog <- maprop.glmm(e, n, data = dat.chor,
  link = "cloglog", b.iter = 10, seed = 1234)
out.chor.glmm.cloglog


# depression data
data("dat.beck17")
out.beck17.glmm.log <- maprop.glmm(e, n, data = dat.beck17,
  link = "log", b.iter = 10, seed = 1234)
out.beck17.glmm.log
out.beck17.glmm.logit <- maprop.glmm(e, n, data = dat.beck17,
  link = "logit", b.iter = 10, seed = 1234)
out.beck17.glmm.logit
out.beck17.glmm.probit <- maprop.glmm(e, n, data = dat.beck17,
  link = "probit", b.iter = 10, seed = 1234)
out.beck17.glmm.probit
out.beck17.glmm.cauchit <- maprop.glmm(e, n, data = dat.beck17,
  link = "cauchit", b.iter = 10, seed = 1234)
out.beck17.glmm.cauchit
out.beck17.glmm.cloglog<- maprop.glmm(e, n, data = dat.beck17,
  link = "cloglog", b.iter = 10, seed = 1234)
out.beck17.glmm.cloglog

---

Meta-Analysis of Proportions Using Two-Step Methods

Description

Performs a meta-analysis of proportions using conventional two-step methods with various data transformations.

Usage

maprop.twostep(e, n, data, link = "logit", method = "ML", alpha = 0.05,
               pop.avg = TRUE, int.approx = 10000, b.iter = 1000,
               seed = 1234)

Arguments

e	a numeric vector specifying the event counts in the collected studies.
n	a numeric vector specifying the sample sizes in the collected studies.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, e and n, should be specified as their corresponding column names in data.
link	a character string specifying the data transformation for each study's proportion used in the two-step method, which can be one of "log" (log transformation), "logit" (logit transformation, the default), "arcsine" (arcsine transformation), and "double.arcsine" (Freeman–Tukey double-arcsine transformation).
method	a character string specifying the method to perform the meta-analysis, which is passed to the argument method in the function rma.uni in the package metafor. It can be one of "ML" (maximum likelihood, the default), "REML" (restricted maximum likelihood), and many other options; see more details in the manual of metafor. The default is set to "ML" for consistency with the function maprop.glmm, where generalized linear mixed models are often estimated via the maximum likelihood approach. For the two-step method, users might also use "REML" because the restricted maximum likelihood estimation may have superior performance in many cases.
alpha	a numeric value specifying the statistical significance level.
pop.avg	a logical value indicating whether the population-averaged proportion and its confidence interval are to be produced. This quantity is the marginal mean of study-specific proportions, while the commonly-reported overall proportion usually represents the median (or interpreted as a conditional measure); see more details about this quantity in Section 13.2.3 in Agresti (2013), Chu et al. (2012), Lin and Chu (2020), and Zeger et al. (1988). If pop.avg = TRUE (the default), the bootstrap resampling is used to produce the confidence interval of the population-averaged proportion; the confidence interval of the commonly-reported median proportion will be also produced, in addition to its conventional confidence interval (by back-transforming the Wald-type confidence interval derived on the scale specified by link).
int.approx	an integer specifying the number of independent standard normal samples for numerically approximating the integration involved in the calculation of the population-averaged proportion; see details in Lin and Chu (2020). It is only used when pop.avg = TRUE. For the commonly-used data transformations available for link, the population-averaged proportion does not have a closed form.
b.iter	an integer specifying the number of bootstrap iterations; it is only used when pop.avg = TRUE.
seed	an integer for specifying the seed of the random number generation for reproducibility during the bootstrap resampling (and numerical approximation for the population-averaged proportion); it is only used when pop.avg = TRUE.

Value

This function returns a list containing the point and interval estimates of the overall proportion. Specifically, prop.c.est is the commonly-reported median (or conditional) proportion, and prop.c.ci is its confidence interval. If pop.avg = TRUE, the following additional elements will be also in the produced list: prop.c.ci.b is the bootstrap confidence interval of the commonly-reported median (conditional) proportion, prop.m.est is the point estimate of the population-averaged (marginal) proportion, prop.m.ci.b is the bootstrap confidence interval of the population-averaged (marginal) proportion, and b.w.e is a vector of two numeric values, indicating the counts of warnings and errors occurred during the bootstrap iterations. Moreover, if the Freeman–Tukey double-arcsine transformation (link = "double.arcsine") is used, the back-transformation will be implemented at four values as the overall sample size: the harmonic, geometric, and arithmetic means of the study-specific sample sizes, and the inverse of the synthesized result's variance. See details in Barendregt et al. (2013) and Schwarzer et al. (2019).

Note

This function implements the two-step method for the meta-analysis of proportions via the rma.uni function in the package metafor. It is possible that the algorithm of the maximum likelihood or restricted maximum likelihood estimation may not converge for some bootstrapped meta-analyses when pop.avg = TRUE, and the rma.uni function may report warnings or errors about the convergence issue. The bootstrap iterations are continued until b.iter replicates without any warnings or errors are obtained; those replicates with any warnings or errors are discarded.

References

Agresti A (2013). Categorical Data Analysis. Third edition. John Wiley & Sons, Hoboken, NJ.

Barendregt JJ, Doi SA, Lee YY, Norman RE, Vos T (2013). "Meta-analysis of prevalence." Journal of Epidemiology and Community Health, 67(11), 974–978. <doi:10.1136/jech-2013-203104>

Freeman MF, Tukey JW (1950). "Transformations related to the angular and the square root." The Annals of Mathematical Statistics, 21(4), 607–611. <doi:10.1214/aoms/1177729756>

Lin L, Chu H (2020). "Meta-analysis of proportions using generalized linear mixed models." Epidemiology, 31(5), 713–717. <doi:10.1097/ede.0000000000001232>

Miller JJ (1978). "The inverse of the Freeman–Tukey double arcsine transformation." The American Statistician, 32(4), 138. <doi:10.1080/00031305.1978.10479283>

Schwarzer G, Chemaitelly H, Abu-Raddad LJ, Rucker G (2019). "Seriously misleading results using inverse of Freeman-Tukey double arcsine transformation in meta-analysis of single proportions." Research Synthesis Methods, 10(3), 476–483. <doi:10.1002/jrsm.1348>

Viechtbauer W (2010). "Conducting meta-analyses in R with the metafor package." Journal of Statistical Software, 36, 3. <doi:10.18637/jss.v036.i03>

Zeger SL, Liang K-Y, Albert PS (1988). "Models for longitudinal data: a generalized estimating equation approach." Biometrics, 44(4), 1049–1060. <doi:10.2307/2531734>

See Also

maprop.glmm

Examples

# chorioamnionitis data
data("dat.chor")
# two-step method with the logit transformation
out.chor.twostep.logit <- maprop.twostep(e, n, data = dat.chor,
  link = "logit", b.iter = 10, seed = 1234)
out.chor.twostep.logit
# not calculating the population-averaged (marginal) proportion,
#  without bootstrap resampling
out.chor.twostep.logit <- maprop.twostep(e, n, data = dat.chor,
  link = "logit", pop.avg = FALSE)
out.chor.twostep.logit

# increases the number of bootstrap iterations to 1000,
#  taking longer time
out.chor.twostep.logit <- maprop.twostep(e, n, data = dat.chor,
  link = "logit", b.iter = 1000, seed = 1234)
out.chor.twostep.logit

# two-step method with the log transformation
out.chor.twostep.log <- maprop.twostep(e, n, data = dat.chor,
  link = "log", b.iter = 10, seed = 1234)
out.chor.twostep.log
# two-step method with the arcsine transformation
out.chor.twostep.arcsine <- maprop.twostep(e, n, data = dat.chor,
  link = "arcsine", b.iter = 10, seed = 1234)
out.chor.twostep.arcsine
# two-step method with the Freeman--Tukey double-arcsine transformation
out.chor.twostep.double.arcsine <- maprop.twostep(e, n, data = dat.chor,
  link = "double.arcsine", b.iter = 10, seed = 1234)
out.chor.twostep.double.arcsine


# depression data
data("dat.beck17")
out.beck17.twostep.log <- maprop.twostep(e, n, data = dat.beck17,
  link = "log", b.iter = 10, seed = 1234)
out.beck17.twostep.log
out.beck17.twostep.logit <- maprop.twostep(e, n, data = dat.beck17,
  link = "logit", b.iter = 10, seed = 1234)
out.beck17.twostep.logit
out.beck17.twostep.arcsine <- maprop.twostep(e, n, data = dat.beck17,
  link = "arcsine", b.iter = 10, seed = 1234)
out.beck17.twostep.arcsine
out.beck17.twostep.double.arcsine <- maprop.twostep(e, n, data = dat.beck17,
  link = "double.arcsine", b.iter = 10, seed = 1234)
out.beck17.twostep.double.arcsine

---

Bivariate Method for Meta-Analysis.

Description

Performs a meta-analysis with a binary outcome using a bivariate generalized linear mixed model (GLMM) described in Chu et al. (2012).

Usage

meta.biv(sid, tid, e, n, data, link = "logit", alpha = 0.05,
         b.iter = 1000, seed = 1234, ...)

Arguments

sid	a vector specifying the study IDs.
tid	a vector of 0/1 specifying the treatment/exposure IDs (0: control/non-exposure; 1: treatment/exposure).
e	a numeric vector specifying the event counts.
n	a numeric vector specifying the sample sizes.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, sid, tid, e, and n, should be specified as their corresponding column names in data.
link	a character string specifying the link function used in the GLMM, which can be either "logit" (the default) or "probit".
alpha	a numeric value specifying the statistical significance level.
b.iter	an integer specifying the number of bootstrap iterations, which are used to produce confidence intervals of marginal results.
seed	an integer for specifying the seed of the random number generation for reproducibility during the bootstrap resampling.
...	other arguments that can be passed to the function glmer in the package lme4.

Details

Suppose a meta-analysis with a binary outcome contains $N$ studies. Let $n_{i0}$ and $n_{i1}$ be the sample sizes in the control/non-exposure and treatment/exposure groups in study $i$, respectively, and let $e_{i0}$ and $e_{i1}$ be the event counts ($i = 1, \ldots, N$). The event counts are assumed to independently follow binomial distributions:

$e_{i0} \sim Bin(n_{i0}, p_{i0});$

$e_{i1} \sim Bin(n_{i1}, p_{i1}),$

where $p_{i0}$ and $p_{i1}$ represent the true event probabilities. They are modeled jointly as follows:

$g(p_{i0}) = \mu_0 + \nu_{i0};$

$g(p_{i1}) = \mu_1 + \nu_{i1};$

$(\nu_{i0}, \nu_{i1})^\prime \sim N ((0, 0)^\prime, \mathbf{\Sigma}).$

Here, $g(\cdot)$ denotes the link function that transforms the event probabilities to linear forms. The fixed effects $\mu_0$ and $\mu_1$ represent the overall event probabilities on the transformed scale. The study-specific parameters $\nu_{i0}$ and $\nu_{i1}$ are random effects, which are assumed to follow the bivariate normal distribution with zero means and variance-covariance matrix $\mathbf{\Sigma}$. The diagonal elements of $\mathbf{\Sigma}$ are $\sigma_0^2$ and $\sigma_1^2$ (between-study variances due to heterogeneity), and the off-diagonal elements are $\rho \sigma_0 \sigma_1$, where $\rho$ is the correlation coefficient.

When using the logit link, $\mu_1 - \mu_0$ represents the log odds ratio (Van Houwelingen et al., 1993; Stijnen et al., 2010; Jackson et al., 2018); $\exp(\mu_1 - \mu_0)$ may be referred to as the conditional odds ratio (Agresti, 2013). Alternatively, we can obtain the marginal event probabilities (Chu et al., 2012):

$p_k = E[p_{ik}] \approx \left[1 + \exp\left(-\mu_k/\sqrt{1 + C^2 \sigma_k^2}\right)\right]^{-1}$

for $k$ = 0 and 1, where $C = 16 \sqrt{3} / (15 \pi)$. The marginal odds ratio, relative risk, and risk difference are subsequently obtained as $[p_1/(1 - p_1)]/[p_0/(1 - p_0)]$, $p_1/p_0$, and $p_1 - p_0$, respectively.

When using the probit link, the model does not yield the conditional odds ratio. The marginal probabilities have closed-form solutions:

$p_k = E[p_{ik}] = \Phi\left(\mu_k/\sqrt{1 + \sigma_k^2}\right)$

for $k$ = 0 and 1, where $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution. They further lead to the marginal odds ratio, relative risk, and risk difference.

Value

This function returns a list containing the point and interval estimates of the marginal event rates (p0.m, p0.m.ci, p1.m, and p1.m.ci), odds ratio (OR.m and OR.m.ci), relative risk (RR.m and RR.m.ci), risk difference (RD.m and RD.m.ci), and correlation coefficient between the two treatment/exposure groups (rho and rho.ci). These interval estimates are obtained using the bootstrap resampling. During the bootstrap resampling, computational warnings or errors may occur for implementing the bivariate GLMM in some resampled meta-analyses. This function returns the counts of warnings and errors (b.w.e). The resampled meta-analyses that lead to warnings and errors are not used for producing the bootstrap confidence intervals; the bootstrap iterations stop after obtaining b.iter resampled meta-analyses without warnings and errors. If the logit link is used (link = "logit"), it also returns the point and interval estimates of the conditional odds ratio (OR.c and OR.c.ci), which are more frequently reported in the current literature than the marginal odds ratios. Unlike the marginal results that use the bootstrap resampling to produce their confidence intervals, the Wald-type confidence interval is calculated for the log conditional odds ratio; it is then transformed to the odds ratio scale.

References

Agresti A (2013). Categorical Data Analysis. Third edition. John Wiley & Sons, Hoboken, NJ.

Chu H, Nie L, Chen Y, Huang Y, Sun W (2012). "Bivariate random effects models for meta-analysis of comparative studies with binary outcomes: methods for the absolute risk difference and relative risk." Statistical Methods in Medical Research, 21(6), 621–633. <doi:10.1177/0962280210393712>

Jackson D, Law M, Stijnen T, Viechtbauer W, White IR (2018). "A comparison of seven random-effects models for meta-analyses that estimate the summary odds ratio." Statistics in Medicine, 37(7), 1059–1085. <doi:10.1002/sim.7588>

Stijnen T, Hamza TH, Ozdemir P (2010). "Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data." Statistics in Medicine, 29(29), 3046–3067. <doi:10.1002/sim.4040>

Van Houwelingen HC, Zwinderman KH, Stijnen T (1993). "A bivariate approach to meta-analysis." Statistics in Medicine, 12(24), 2273–2284. <doi:10.1002/sim.4780122405>

See Also

maprop.glmm, meta.dt

Examples

data("dat.bellamy")
out.bellamy.logit <- meta.biv(sid, tid, e, n, data = dat.bellamy,
  link = "logit", b.iter = 1000)
out.bellamy.logit
out.bellamy.probit <- meta.biv(sid, tid, e, n, data = dat.bellamy,
  link = "probit", b.iter = 1000)
out.bellamy.probit

---

Meta-Analysis of Diagnostic Tests

Description

Performs a meta-analysis of diagnostic tests using approaches described in Reitsma et al. (2005) and Chu and Cole (2006).

Usage

meta.dt(tp, fp, fn, tn, data, method = "biv.glmm", alpha = 0.05, ...)

Arguments

tp	counts of true positives.
fp	counts of false positives.
fn	counts of false negatives.
tn	counts of true negatives.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, tp, fp, fn, and tn, should be specified as their corresponding column names in data.
method	a character string specifying the method used to implement the meta-analysis of diagnostic tests. It should be one of "s.roc" (summary ROC approach), "biv.lmm" (bivariate linear mixed model), and "biv.glmm" (bivariate generalized linear mixed model, the default). See details.
alpha	a numeric value specifying the statistical significance level.
...	other arguments that can be passed to the function lm (when method = "s.roc"), the function rma.mv in the package metafor (when method = "biv.lmm"), or the function glmer in the package lme4 (when method = "biv.glmm").

Details

Suppose a meta-analysis of diagnostic tests contains $N$ studies. Each study reports the counts of true positives, false positives, false negatives, and true negatives, denoted by $TP_i$, $FP_i$, $FN_i$, and $TN_i$, respectively. The study-specific estimates of sensitivity and specificity are calculated as $Se_i = TP_i/(TP_i + FN_i)$ and $Sp_i = TN_i/(FP_i + TN_i)$ for $i = 1, \cdots, N$. They are analyzed on the logarithmic scale in the meta-analysis. When using the summary ROC (receiver operating characteristic) approach or the bivariate linear mixed model, 0.5 needs to be added to all four counts in a study when at least one count is zero.

The summary ROC approach first calculates

$D_i = \log \left( \frac{Se_i}{1 - Se_i} \right) + \log \left( \frac{Sp_i}{1 - Sp_i} \right);$

$S_i = \log \left( \frac{Se_i}{1 - Se_i} \right) - \log \left( \frac{Sp_i}{1 - Sp_i} \right),$

where $D_i$ represents the log diagnostic odds ratio (DOR) in study $i$. A linear regression is then fitted:

$D_i = \alpha + \beta \cdot S_i.$

The regression could be either unweighted or weighted; this function performs both versions. If weighted, the study-specific weights are the inverse of the variances of $D_i$, i.e., $1/TP_i + 1/FP_i + 1/FN_i + 1/TN_i$. Based on the estimated regression intercept $\hat{\alpha}$ and slope $\hat{\beta}$, one may obtain the DOR at mean of $S_i$, Q point, summary ROC curve, and area under the curve (AUC). The Q point is the point on the summary ROC curve where sensitivity and specificity are equal. The ROC curve is given by

$Se = \left\{ 1 + e^{-\hat{\alpha}/(1 - \hat{\beta})} \cdot [Sp/(1 - Sp)]^{(1 + \hat{\beta})/(1 - \hat{\beta})} \right\}^{-1}.$

See more details of the summary ROC approach in Moses et al. (1993) and Irwig et al. (1995).

The bivariate linear mixed model described in Reitsma et al. (2005) assumes that the logit sensitivity and logit specificity independently follow normal distributions within each study: $g(Se_i) \sim N(\theta_{i, Se}, s_{i, Se}^2)$ and $g(Sp_i) \sim N(\theta_{i, Sp}, s_{i, Sp}^2)$, where $g(\cdot)$ denotes the logit function. The within-study variances are calculated as $s_{i, Se}^2 = 1/TP_i + 1/FN_i$ and $s_{i, Sp}^2 = 1/FP_i + 1/TN_i$. The parameters $\theta_{i, Se}$ and $\theta_{i, Sp}$ are the underlying true sensitivity and specificity (on the logit scale) in study $i$. They are assumed to be random effects, jointly following a bivariate normal distribution:

$\left(\theta_{i, Se}, \theta_{i, Sp}\right)^\prime \sim N\left((\mu_{Se}, \mu_{Sp})^\prime, \mathbf{\Sigma} \right),$

where $\mathbf{\Sigma}$ is the between-study variance-covariance matrix. The diagonal elements of $\mathbf{\Sigma}$ are $\sigma_{Se}^2$ and $\sigma_{Sp}^2$, representing the heterogeneity variances of sensitivities and specificities (on the logit scale), respectively. The correlation coefficient is $\rho$.

The bivariate generalized linear mixed model described in Chu and Cole (2006) refines the bivariate linear mixed model by directly modeling the counts of true positives and true negatives. This approach does not require the assumption that the logit sensitivity and logit specificity approximately follow normal distributions within studies, which could be seriously violated in the presence of small data counts. It also avoids corrections for zero counts. Specificially, the counts of true positives and true negatives are modeled using binomial likelihoods:

$TP_i \sim Bin(TP_i + FN_i, Se_i);$

$TN_i \sim Bin(FP_i + TN_i, Sp_i);$

$\left( g(Se_i), g(Sp_i) \right)^\prime \sim N\left((\mu_{Se}, \mu_{Sp})^\prime, \mathbf{\Sigma} \right).$

See more details in Chu and Cole (2006) and Ma et al. (2016).

For both the bivariate linear mixed model and bivariate generalized linear mixed model, $\mu_{Se}$ and $\mu_{Sp}$ represent the overall sensitivity and specificity (on the logit scale) across studies, respectively, and $\mu_{Se} + \mu_{Sp}$ represents the log DOR. The summary ROC curve may be constructed as

$Se = \left\{ 1 + e^{-\hat{\mu}_{Se} + \hat{\mu}_{Sp} \cdot \hat{\rho} \hat{\sigma}_{Se}/\hat{\sigma}_{Sp}} \cdot [Sp/(1 - Sp)]^{-\rho \hat{\sigma}_{Se}/\hat{\sigma}_{Sp}} \right\}^{-1}.$

Value

This function returns a list of the meta-analysis results. When method = "s.roc", the list consists of the regression intercept (inter.unwtd), slope (slope.unwtd), their variance-covariance matrix (vcov.unwtd), DOR at mean of $S_i$ (DOR.meanS.unwtd) with its confidence interval (DOR.meanS.unwtd.ci), Q point (Q.unwtd) with its confidence interval (Q.unwtd.ci), and AUC (AUC.unwtd) for the unweighted regression; it also consists of the counterparts for the weighted regression. When method = "biv.lmm" or "biv.glmm", the list consists of the overall sensitivity (sens.overall) with its confidence interval (sens.overall.ci), overall specificity (spec.overall) with its confidence interval (spec.overall.ci), overall DOR (DOR.overall) with its confidence interval (DOR.overall.ci), AUC (AUC), estimated $\mu_{Se}$ (mu.sens), $\mu_{Sp}$ (mu.spec), their variance-covariance matrix (mu.vcov), estimated $\sigma_{Se}$ (sig.sens), $\sigma_{Sp}$ (sig.spec), and $\rho$ (rho). In addition, the list includes the method used to perform the meta-analysis of diagnostic tests (method), significance level (alpha), and original data (data).

Note

The original articles by Reitsma et al. (2005) and Chu and Cole (2006) used SAS to implement (generalized) linear mixed models (specifically, PROC MIXED and PROC NLMIXED); this function imports rma.mv from the package metafor and glmer from the package lme4 for implementing these models. The estimation approaches adopted in SAS and the R packages metafor and lme4 may differ, which may impact the results. See, for example, Zhang et al. (2011).

Author(s)

Lifeng Lin, Kristine J. Rosenberger

References

Chu H, Cole SR (2006). "Bivariate meta-analysis of sensitivity and specificity with sparse data: a generalized linear mixed model approach." Journal of Clinical Epidemiology, 59(12), 1331–1332. <doi:10.1016/j.jclinepi.2006.06.011>

Irwig L, Macaskill P, Glasziou P, Fahey M (1995). "Meta-analytic methods for diagnostic test accuracy." Journal of Clinical Epidemiology, 48(1), 119–130. <doi:10.1016/0895-4356(94)00099-C>

Ma X, Nie L, Cole SR, Chu H (2016). "Statistical methods for multivariate meta-analysis of diagnostic tests: an overview and tutorial." Statistical Methods in Medical Research, 25(4), 1596–1619. <doi:10.1177/0962280213492588>

Moses LE, Shapiro D, Littenberg B (1993). "Combining independent studies of a diagnostic test into a summary ROC curve: data-analytic approaches and some additional considerations." Statistics in Medicine, 12(14), 1293–1316. <doi:10.1002/sim.4780121403>

Reitsma JB, Glas AS, Rutjes AWS, Scholten RJPM, Bossuyt PM, Zwinderman AH (2005). "Bivariate analysis of sensitivity and specificity produces informative summary measures in diagnostic reviews." Journal of Clinical Epidemiology, 58(10), 982–990. <doi:10.1016/j.jclinepi.2005.02.022>

Zhang H, Lu N, Feng C, Thurston SW, Xia Y, Zhu L, Tu XM (2011). "On fitting generalized linear mixed-effects models for binary responses using different statistical packages." Statistics in Medicine, 30(20), 2562–2572. <doi:10.1002/sim.4265>

See Also

maprop.twostep, meta.biv, plot.meta.dt, print.meta.dt

Examples

data("dat.scheidler")
out1 <- meta.dt(tp, fp, fn, tn, data = dat.scheidler[dat.scheidler$dt == "MRI",],
  method = "s.roc")
out1
plot(out1)
out2 <- meta.dt(tp, fp, fn, tn, data = dat.scheidler[dat.scheidler$dt == "MRI",],
  method = "biv.lmm")
out2
plot(out2, predict = TRUE)
out3 <- meta.dt(tp, fp, fn, tn, data = dat.scheidler[dat.scheidler$dt == "MRI",],
  method = "biv.glmm")
out3
plot(out3, add = TRUE, studies = FALSE,
  col.roc = "blue", col.overall = "blue", col.confid = "blue",
  predict = TRUE,col.predict = "blue")

data("dat.smith")
out4 <- meta.dt(tp, fp, fn, tn, data = dat.smith, method = "biv.glmm")
out4
plot(out4, predict = TRUE)

---

Meta-Analysis of Combining Standardized Mean Differences and Odds Ratios

Description

Performs a Bayesian meta-analysis to synthesize standardized mean differences (SMDs) for a continuous outcome and odds ratios (ORs) for a binary outcome.

Usage

meta.or.smd(y1, sd1, n1, y0, sd0, n0, r1, m1, r0, m0, id.bin, data,
            n.adapt = 1000, n.chains = 3, n.burnin = 5000, n.iter = 20000, n.thin = 2,
            seed = 1234)

Arguments

y1	a vector specifying the sample means in the treatment group for the continuous outcome. NA is allowed and indicates that data are not available for this outcome; the same applies to the other arguments, including sd1, n1, y0, sd0, n0, r1, m1, r0, and m0.
sd1	a vector specifying the sample standard deviations in the treatment group for the continuous outcome.
n1	a vector specifying the sample sizes in the treatment group for the continuous outcome.
y0	a vector specifying the sample means in the control group for the continuous outcome.
sd0	a vector specifying the sample standard deviations in the control group for the continuous outcome.
n0	a vector specifying the sample sizes in the control group for the continuous outcome.
r1	a vector specifying the event counts in the treatment group for the binary outcome.
m1	a vector specifying the sample sizes in the treatment group for the binary outcome.
r0	a vector specifying the event counts in the control group for the binary outcome.
m0	a vector specifying the sample sizes in the control group for the binary outcome.
id.bin	a vector indicating whether the outcome is binary (1) or continuous (0).
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, y1, sd1, n1, y0, sd0, n0, r1, m1, r0, m0, and id.bin should be specified as their corresponding column names in data.
n.adapt	the number of iterations for adaptation in the Markov chain Monte Carlo (MCMC) algorithm. The default is 1,000. This argument and the following n.chains, n.burnin, n.iter, and n.thin are passed to the functions in the package rjags.
n.chains	the number of MCMC chains. The default is 3.
n.burnin	the number of iterations for burn-in period. The default is 5,000.
n.iter	the total number of iterations in each MCMC chain after the burn-in period. The default is 20,000.
n.thin	a positive integer specifying thinning rate. The default is 2.
seed	an integer for specifying the seed of the random number generation for reproducibility during the MCMC algorithm for performing the Bayesian meta-analysis model.

Details

The Bayesian meta-analysis model implemented by this function is detailed in Section 2.5 of Jing et al. (2023).

Value

"This function returns a list of Bayesian estimates, including posterior medians and 95% credible intervals (comprising the 2.5% and 97.5% posterior quantiles) for the overall SMD (d), the between-study standard deviation (tau), and the individual studies' SMDs (theta).

Author(s)

Yaqi Jing, Lifeng Lin

References

Jing Y, Murad MH, Lin L (2023). "A Bayesian model for combining standardized mean differences and odds ratios in the same meta-analysis." Journal of Biopharmaceutical Statistics, 33(2), 167–190. <doi:10.1080/10543406.2022.2105345>

Examples

data("dat.dep")
out <- meta.or.smd(y1, sd1, n1, y0, sd0, n0, r1, m1, r0, m0, id.bin, data = dat.dep)
out

---

A penalization approach to random-effects meta-analysis

Description

Performs the penalization methods introduced in Wang et al. (2022) to achieve a compromise between the common-effect and random-effects model.

Usage

meta.pen(y, s2, data, tuning.para = "tau", upp = 1, n.cand = 100, tol = 1e-10)

Arguments

y	a numeric vector or the corresponding column name in the argument data, specifying the observed effect sizes in the collected studies.
s2	a numeric vector or the corresponding column name in the argument data, specifying the within-study variances.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, y and s2, should be specified as their corresponding column names in data.
tuning.para	a character string specifying the type of tuning parameter used in the penalization method. It should be one of "lambda" (use lambda as a tuning parameter) or "tau" (use the standard deviation as a tuning parameter). The default is "tau".
upp	a positive scalar used to control the upper bound of the range for the tuning parameter. Specifically, [0, T*upp] is used as the range of a tuning parameter. T is the upper threshold value of a tuning parameter. The default value of upp is 1.
n.cand	the total number of candidate values of the tuning parameter within the specified range. The default is 100.
tol	the desired accuracy (convergence tolerance). The default is 1e-10.

Details

Suppose a meta-analysis collects $n$ independent studies. Let $\mu_{i}$ be the true effect size in study $i$ ($i$ = 1, ..., $n$). Each study reports an estimate of the effect size and its sample variance, denoted by $y_{i}$ and $s_{i}^{2}$, respectively. These data are commonly modeled as $y_{i}$ from $N(\mu_{i},s_{i}^{2})$. If study-specific true effect sizes $\mu_{i}$ are assumed i.i.d. from $N(\mu, \tau^{2})$, this is the random-effects (RE) model, where $\mu$ is the overall effect size and $\tau^{2}$ is the between-study variance. If $\tau^{2}=0$ and thus $\mu_{i}=\mu$ for all studies, this implies that studies are homogeneous and the RE model is reduced to the common-effect (CE) model.

Marginally, the RE model yields $y_{i} \sim N(\mu,s_{i}^2+\tau^{2})$, and its log-likelihood is

$l(\mu, \tau^{2}) = -\frac{1}{2} \sum_{i=1}^{n} \left[\log(s_{i}^{2} + \tau^{2}) + \frac{(y_{i} - \mu)^2}{s_{i}^2 + \tau^2}\right] + C,$

where $C$ is a constant. In the past two decades, penalization methods have been rapidly developed for variable selection in high-dimensional data analysis to control model complexity and reduce the variance of parameter estimates. Borrowing the idea from the penalization methods, we employ a penalty term on the between-study variance $\tau^2$ when the heterogeneity is overestimated. The penalty term increases with $\tau^2$. Specifically, we consider the following optimization problem:

$(\hat{\mu}(\lambda), \hat{\tau}^2(\lambda)) = \min_{\mu, \tau^2 \geq 0} \left\{\sum_{i=1}^{n} \left[ \log(s_{i}^{2} + \tau^{2}) + \frac{(y_{i} - \mu)^2}{s_{i}^2 + \tau^2}\right] + \lambda \tau^2\right\},$

where $\lambda \geq 0$ is a tuning parameter that controls the penalty strength. Using the technique of profile likelihood by taking the target function's derivative in the above equation with respect to $\mu$ for a given $\tau^2$. The bivariate optimization problem is reduced to a univariate minimization problem. When $\lambda=0$, the minimization problem is equivalent to minimizing the log-likelihood without penalty, so the penalized-likelihood method is identical to the conventional RE model. By contrast, it can be shown that a sufficiently large $\lambda$ produces the estimated between-study variance as 0, leading to the conventional CE model.

As different tuning parameters lead to different estimates of $\hat{\mu}(\lambda)$ and $\hat{\tau}^2(\lambda)$, it is important to select the optimal $\lambda$ among a set of candidate values. We perform the cross-validation process and construct a loss function of $\lambda$ to measure the performance of specific $\lambda$ values. The $\lambda$ corresponding to the smallest loss is considered optimal. The threshold, denoted by $\lambda_{\max}$, based on the penalty function $p(\tau^2)=\tau^2$ can be calculated. For all $\lambda > \lambda_{\max}$, the estimated between-study variance is 0. Consequently, we select a certain number of candidate values (e.g., 100) from the range $[0,\lambda_{\max}]$ for the tuning parameter. For a set of tuning parameters, the leave-one-study-out (i.e., $n$-fold) cross-validation is used to construct the loss function. Specifically, we use the following loss function for the penalization method by tuning $\lambda$:

$\hat{L}(\lambda) = \left[\frac{1}{n} \sum_{i=1}^{n} \frac{(y_{i} - \hat{\mu}_{(-i)}(\lambda))^2}{s_{i}^2 + \hat{\tau}_{RE(-i)}^2 + \frac{\sum_{j \ne i}(s_{j}^2 + \hat{\tau}_{RE(-i)}^2) / (s_{j}^2 + \hat{\tau}^2_{(-i)}(\lambda))^2}{(\sum_{j \ne i} 1 / (s_{j}^2 + \hat{\tau}^2_{(-i)}(\lambda)))^2}}\right]^{1/2},$

where the subscript $(-i)$ indicates that study $i$ is removed, $\hat{\tau}^2_{(-i)}(\lambda)$ is the estimated between-study variance for a given $\lambda$, $\hat{\tau}_{RE(-i)}^2$ is the corresponding between-study variance estimate of the RE model, and

$\hat{\mu}_{(-i)}(\lambda) = \frac{\sum_{j \ne i} y_{j} / (s_{j}^2 + \hat{\tau}^2_{(-i)}(\lambda))}{\sum_{j \ne i} 1 / (s_{j}^2 + \hat{\tau}^2_{(-i)}(\lambda))}.$

The above procedure focuses on tuning the parameter, $\lambda$, to control the penalty strength for the between-study variance. Alternatively, for the purpose of shrinking the potentially overestimated heterogeneity, we can directly treat the between-study standard deviation (SD), $\tau$, as the tuning parameter. A set of candidate values of $\tau$ are considered, and the value that produces the minimum loss function is selected. Compared with tuning $\lambda$ from the perspective of penalized likelihood, tuning $\tau$ is more straightforward and intuitive from the practical perspective. The candidate values of $\tau$ can be naturally chosen from $[0,\hat{\tau}_{RE}]$, with the lower and upper bounds corresponding to the CE and RE models, respectively. Denoted the candidate SDs as $\tau_{t}$, the loss function with respect to $\tau_{t}$ is similarly defined as

$\hat{L}(\tau_{t}) = \left[\frac{1}{n} \sum_{i=1}^{n} \frac{(y_i - \hat{\mu}_{(-i)}(\tau_{t}))^2}{s_{i}^2 + \hat{\tau}_{RE(-i)} + \frac{\sum_{j \ne i}(s_{j}^2 + \hat{\tau}^2_{RE(-i)}) / (s_{j}^2 + \tau_{t}^2)^2}{[\sum_{j \ne i} 1 / (s_{j}^2 + \tau_{t}^2)]^2}}\right]^{1/2},$

where the overall effect size estimate (excluding study $i$) is

$\hat{\mu}_{(-i)}(\tau_{t}) = \frac{\sum_{j \ne i} y_j / (s_{j}^2 + \tau_{t}^2)}{\sum_{j \ne i} 1 / (s_{j}^2 + \tau_{t}^2)}.$

Value

This function returns a list containing estimates of the overall effect size and their 95% confidence intervals. Specifically, the components include:

n	the number of studies in the meta-analysis.
I2	the $I^2$ statistic for quantifying heterogeneity.
tau2.re	the maximum likelihood estimate of the between-study variance.
mu.fe	the estimated overall effect size of the CE model.
se.fe	the standard deviation of the overall effect size estimate of the CE model.
mu.re	the estimated overall effect size of the RE model.
se.re	the standard deviation of the overall effect size estimate of the RE model.
loss	the values of the loss function for candidate tuning parameters.
tau.cand	the candidate values of the tuning parameter $\tau$.
lambda.cand	the candidate values of the tuning parameter $\lambda$.
tau.opt	the estimated between-study standard deviation of the penalization method.
mu.opt	the estimated overall effect size of the penalization method.
se.opt	the standard error estimate of the estimated overall effect size of the penalization method.

Author(s)

Yipeng Wang [email protected]

References

Wang Y, Lin L, Thompson CG, Chu H (2022). "A penalization approach to random-effects meta-analysis." Statistics in Medicine, 41(3), 500–516. <doi:10.1002/sim.9261>

Examples

data("dat.bohren") ## log odds ratio
## perform the penaliztion method by tuning tau
out11 <- meta.pen(y, s2, dat.bohren)
## plot the loss function and candidate taus
plot(out11$tau.cand, out11$loss, xlab = NA, ylab = NA, 
  lwd = 1.5, type = "l", cex.lab = 0.8, cex.axis = 0.8)
title(xlab = expression(paste(tau[t])),
  ylab = expression(paste("Loss function ", ~hat(L)(tau[t]))),
  cex.lab = 0.8, line = 2)
idx <- which(out11$loss == min(out11$loss))
abline(v = out11$tau.cand[idx], col = "gray", lwd = 1.5, lty = 2)
     
## perform the penaliztion method by tuning lambda     
out12 <- meta.pen(y, s2, dat.bohren, tuning.para = "lambda")
## plot the loss function and candidate lambdas
plot(log(out12$lambda.cand + 1), out12$loss, xlab = NA, ylab = NA, 
  lwd=1.5, type = "l", cex.lab = 0.8, cex.axis = 0.8)
title(xlab = expression(log(lambda + 1)),
  ylab = expression(paste("Loss function ", ~hat(L)(lambda))),
  cex.lab = 0.8, line = 2)
idx <- which(out12$loss == min(out12$loss))
abline(v = log(out12$lambda.cand[idx] + 1), col = "gray", lwd = 1.5, lty = 2)


data("dat.bjelakovic") ## log odds ratio
## perform the penaliztion method by tuning tau
out21 <- meta.pen(y, s2, dat.bjelakovic)
## plot the loss function and candidate taus
plot(out21$tau.cand, out21$loss, xlab = NA, ylab = NA, 
  lwd=1.5, type = "l", cex.lab = 0.8, cex.axis = 0.8)
title(xlab = expression(paste(tau[t])),
  ylab = expression(paste("Loss function ", ~hat(L)(tau[t]))),
  cex.lab = 0.8, line = 2)
idx <- which(out21$loss == min(out21$loss))
abline(v = out21$tau.cand[idx], col = "gray", lwd = 1.5, lty = 2)

out22 <- meta.pen(y, s2, dat.bjelakovic, tuning.para = "lambda")

data("dat.carless") ## log odds ratio
## perform the penaliztion method by tuning tau
out31 <- meta.pen(y, s2, dat.carless)
## plot the loss function and candidate taus
plot(out31$tau.cand, out31$loss, xlab = NA, ylab = NA, 
  lwd=1.5, type = "l", cex.lab = 0.8, cex.axis = 0.8)
title(xlab = expression(paste(tau[t])),
  ylab = expression(paste("Loss function ", ~hat(L)(tau[t]))),
  cex.lab = 0.8, line = 2)
idx <- which(out31$loss == min(out31$loss))
abline(v = out31$tau.cand[idx], col = "gray", lwd = 1.5, lty = 2)

out32 <- meta.pen(y, s2, dat.carless, tuning.para = "lambda")

data("dat.adams") ## mean difference
out41 <- meta.pen(y, s2, dat.adams)
## plot the loss function and candidate taus
plot(out41$tau.cand, out41$loss, xlab = NA, ylab = NA, 
  lwd=1.5, type = "l", cex.lab = 0.8, cex.axis = 0.8)
title(xlab = expression(paste(tau[t])),
  ylab = expression(paste("Loss function ", ~hat(L)(tau[t]))),
  cex.lab = 0.8, line = 2)
idx <- which(out41$loss == min(out41$loss))
abline(v = out41$tau.cand[idx], col = "gray", lwd = 1.5, lty = 2)

out42 <- meta.pen(y, s2, dat.adams, tuning.para = "lambda")

---

Meta-Analysis Heterogeneity Measures

Description

Calculates various between-study heterogeneity measures in meta-analysis, including the conventional measures (e.g., $I^2$) and the alternative measures (e.g., $I_r^2$) which are robust to outlying studies; p-values of various tests are also calculated.

Usage

metahet(y, s2, data, n.resam = 1000)

Arguments

y	a numeric vector specifying the observed effect sizes in the collected studies; they are assumed to be normally distributed.
s2	a numeric vector specifying the within-study variances.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, y and s2, should be specified as their corresponding column names in data.
n.resam	a positive integer specifying the number of resampling iterations for calculating p-values of test statistics and 95% confidence interval of heterogeneity measures.

Details

Suppose that a meta-analysis collects $n$ studies. The observed effect size in study $i$ is $y_i$ and its within-study variance is $s^{2}_{i}$. Also, the inverse-variance weight is $w_i = 1 / s^{2}_{i}$. The fixed-effect estimate of overall effect size is $\bar{\mu} = \sum_{i = 1}^{n} w_i y_i / \sum_{i = 1}^{n} w_i$. The conventional test statistic for heterogeneity is

$Q = \sum_{i = 1}^{n} w_i (y_{i} - \bar{\mu})^2.$

Based on the $Q$ statistic, the method-of-moments estimate of the between-study variance $\hat{\tau}_{DL}^2$ is (DerSimonian and Laird, 1986)

$\hat{\tau}^2_{DL} = \max \left\{ 0, \frac{Q - (n - 1)}{\sum_{i = 1}^{n} w_{i} - \sum_{i = 1}^{n} w_{i}^{2} / \sum_{i = 1}^{n} w_{i}} \right\}.$

Also, the $H$ and $I^2$ statistics (Higgins and Thompson, 2002; Higgins et al., 2003) are widely used in practice because they do not depend on the number of collected studies $n$ and the effect size scale; these two statistics are defined as

$H = \sqrt{Q/(n - 1)};$

$I^{2} = \frac{Q - (n - 1)}{Q}.$

Specifically, the $H$ statistic reflects the ratio of the standard deviation of the underlying mean from a random-effects meta-analysis compared to the standard deviation from a fixed-effect meta-analysis; the $I^2$ statistic describes the proportion of total variance across studies that is due to heterogeneity rather than sampling error.

Outliers are frequently present in meta-analyses, and they may have great impact on the above heterogeneity measures. Alternatively, to be more robust to outliers, the test statistic may be modified as (Lin et al., 2017):

$Q_{r} = \sum_{i = 1}^{n} \sqrt{w_i} |y_{i} - \bar{\mu}|.$

Based on the $Q_r$ statistic, the method-of-moments estimate of the between-study variance $\hat{\tau}_r^2$ is defined as the solution to

$Q_r \sqrt{\frac{\pi}{2}} = \sum_{i = 1}^{n} \left\{1 - \frac{w_{i}}{\sum_{j = 1}^{n} w_{j}} + \tau^{2} \left[ w_{i} - \frac{2 w_{i}^{2}}{\sum_{j = 1}^{n} w_{j}} + \frac{w_{i} \sum_{j = 1}^{n} w_{j}^{2}}{(\sum_{j = 1}^{n} w_{j})^2} \right]\right\}.$

If no positive solution exists to the equation above, set $\hat{\tau}_{r}^{2} = 0$. The counterparts of the $H$ and $I^2$ statistics are defined as

$H_{r} = Q_r \sqrt{\pi/[2 n (n - 1)]};$

$I_{r}^{2} = \frac{Q_{r}^{2} - 2 n (n - 1) / \pi}{Q_{r}^{2}}.$

To further improve the robustness of heterogeneity assessment, the weighted mean in the $Q_r$ statistic may be replaced by the weighted median $\hat{\mu}_m$, which is the solution to $\sum_{i = 1}^{n} w_i [I (\theta \geq y_i) - 0.5] = 0$ with respect to $\theta$. The new test statistic is

$Q_m = \sum_{i = 1}^{n} \sqrt{w_i} |y_{i} - \hat{\mu}_m|.$

Based on $Q_m$, the new estimator of the between-study variance $\hat{\tau}_m^2$ is the solution to

$Q_m \sqrt{\pi/2} = \sum_{i = 1}^{n} \sqrt{(s_i^2 + \tau^2)/s_i^2}.$

The counterparts of the $H$ and $I^2$ statistics are

$H_m = \frac{Q_m}{n} \sqrt{\pi/2};$

$I_m^2 = \frac{Q_m^2 - 2 n^2/\pi}{Q_m^2}.$

Value

This function returns a list containing p-values of various heterogeneity tests and various heterogeneity measures with 95% confidence intervals. Specifically, the components include:

p.Q	p-value of the $Q$ statistic (using the resampling method).
p.Q.theo	p-value of the $Q$ statistic using the $Q$'s theoretical chi-squared distribution.
p.Qr	p-value of the $Q_r$ statistic (using the resampling method).
p.Qm	p-value of the $Q_m$ statistic (using the resampling method).
Q	the $Q$ statistic.
ci.Q	95% CI of the $Q$ statistic.
tau2.DL	DerSimonian–Laird estimate of the between-study variance.
ci.tau2.DL	95% CI of the between-study variance based on the DerSimonian–Laird method.
H	the $H$ statistic.
ci.H	95% CI of the $H$ statistic.
I2	the $I^2$ statistic.
ci.I2	95% CI of the $I^2$ statistic.
Qr	the $Q_r$ statistic.
ci.Qr	95% CI of the $Q_r$ statistic.
tau2.r	the between-study variance estimate based on the $Q_r$ statistic.
ci.tau2.r	95% CI of the between-study variance based on the $Q_r$ statistic.
Hr	the $H_r$ statistic.
ci.Hr	95% CI of the $H_r$ statistic.
Ir2	the $I_r^2$ statistic.
ci.Ir2	95% CI of the $I_r^2$ statistic.
Qm	the $Q_m$ statistic.
ci.Qm	95% CI of the $Q_m$ statistic.
tau2.m	the between-study variance estimate based on the $Q_m$ statistic.
ci.tau2.m	95% CI of the between-study variance based on the $Q_m$ statistic
Hm	the $H_m$ statistic.
ci.Hm	95% CI of the $H_m$ statistic.
Im2	the $I_m^2$ statistic.
ci.Im2	95% CI of the $I_m^2$ statistic.

References

DerSimonian R, Laird N (1986). "Meta-analysis in clinical trials." Controlled Clinical Trials, 7(3), 177–188. <doi:10.1016/0197-2456(86)90046-2>

Higgins JPT, Thompson SG (2002). "Quantifying heterogeneity in a meta-analysis." Statistics in Medicine, 21(11), 1539–1558. <doi:10.1002/sim.1186>

Higgins JPT, Thompson SG, Deeks JJ, Altman DG (2003). "Measuring inconsistency in meta-analyses." BMJ, 327(7414), 557–560. <doi:10.1136/bmj.327.7414.557>

Lin L, Chu H, Hodges JS (2017). "Alternative measures of between-study heterogeneity in meta-analysis: reducing the impact of outlying studies." Biometrics, 73(1), 156–166. <doi:10.1111/biom.12543>

Examples

data("dat.aex")
set.seed(1234)
metahet(y, s2, dat.aex, 100)
metahet(y, s2, dat.aex, 1000)

data("dat.hipfrac")
set.seed(1234)
metahet(y, s2, dat.hipfrac, 100)
metahet(y, s2, dat.hipfrac, 1000)

---

Outlier Detection in Meta-Analysis

Description

Calculates the standardized residual for each study in meta-analysis using the methods desribed in Chapter 12 in Hedges and Olkin (1985) and Viechtbauer and Cheung (2010). A study is considered as an outlier if its standardized residual is greater than 3 in absolute magnitude.

Usage

metaoutliers(y, s2, data, model)

Arguments

y	a numeric vector specifying the observed effect sizes in the collected studies; they are assumed to be normally distributed.
s2	a numeric vector specifying the within-study variances.
data	an optional data frame containing the meta-analysis dataset. If data is specified, the previous arguments, y and s2, should be specified as their corresponding column names in data.
model	a character string specified as either "FE" or "RE". If model = "FE", this function uses the outlier detection procedure for the fixed-effect meta-analysis desribed in Chapter 12 in Hedges and Olkin (1985); If model = "RE", the procedure for the random-effects meta-analysis desribed in Viechtbauer and Cheung (2010) is used. See Details for the two approaches. If the argument model is not specified, this function sets model = "FE" if $I_r^2 < 30\%$ and sets model = "RE" if $I_r^2 \geq 30\%$.

Details

Suppose that a meta-analysis collects $n$ studies. The observed effect size in study $i$ is $y_i$ and its within-study variance is $s^{2}_{i}$. Also, the inverse-variance weight is $w_i = 1 / s^{2}_{i}$.

Chapter 12 in Hedges and Olkin (1985) describes the outlier detection procedure for the fixed-effect meta-analysis (model = "FE"). Using the studies except study $i$, the pooled estimate of the overall effect size is $\bar{\mu}_{(-i)} = \sum_{j \neq i} w_j y_j / \sum_{j \neq i} w_j$. The residual of study $i$ is $e_{i} = y_i - \bar{\mu}_{(-i)}$. The variance of $e_{i}$ is $v_{i} = s_{i}^{2} + (\sum_{j \neq i} w_{j})^{-1}$, so the standardized residual of study $i$ is $\epsilon_{i} = e_{i} / \sqrt{v_{i}}$.

Viechtbauer and Cheung (2010) describes the outlier detection procedure for the random-effects meta-analysis (model = "RE"). Using the studies except study $i$, let the method-of-moments estimate of the between-study variance be $\hat{\tau}_{(-i)}^{2}$. The pooled estimate of the overall effect size is $\bar{\mu}_{(-i)} = \sum_{j \neq i} \tilde{w}_{(-i)j} y_j / \sum_{j \neq i} \tilde{w}_{(-i)j}$, where $\tilde{w}_{(-i)j} = 1/(s_{j}^{2} + \hat{\tau}_{(-i)}^{2})$. The residual of study $i$ is $e_{i} = y_i - \bar{\mu}_{(-i)}$, and its variance is $v_{i} = s_{i}^2 + \hat{\tau}_{(-i)}^{2} + (\sum_{j \neq i} \tilde{w}_{(-i)j})^{-1}$. Then, the standardized residual of study $i$ is $\epsilon_{i} = e_{i} / \sqrt{v_{i}}$.

Value

This functions returns a list which contains standardized residuals and identified outliers. A study is considered as an outlier if its standardized residual is greater than 3 in absolute magnitude.

References

Hedges LV, Olkin I (1985). Statistical Method for Meta-Analysis. Academic Press, Orlando, FL.

Viechtbauer W, Cheung MWL (2010). "Outlier and influence diagnostics for meta-analysis." Research Synthesis Methods, 1(2), 112–125. <doi:10.1002/jrsm.11>

Examples

data("dat.aex")
metaoutliers(y, s2, dat.aex, model = "FE")
metaoutliers(y, s2, dat.aex, model = "RE")

data("dat.hipfrac")
metaoutliers(y, s2, dat.hipfrac)

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