Package 'metaConvert'

metaConvert: An R Package Dedicated to Automated Effect Size Calculations

Description

The metaConvert package automatically estimates 11 effect size measures from a well-formatted dataframe. Various other functions can help, for example, removing dependency between several effect sizes, or identifying differences between two dataframes. This package is mainly designed to assist in conducting a systematic review with a meta-analysis, but it can be useful to any researcher interested in estimating an effect size.

Overview of the package

To visualize all the types of input data that can be used to estimate the 11 effect size measures available in metaConvert, you can use the see_input_data() function.

Estimate effect sizes

To automatically estimate effect sizes directly from a dataset, you can use the convert_df() function.

Aggregate dependent effect sizes

To automatically aggregate dependent effect sizes using Borenstein's formulas, you can use the aggregate_df() function. This function can handle dependent effect sizes from multiple subgroups, or dependent effect sizes from the same participants.

Flag differences between two datasets

If pairs of data extractors have generated similar datasets that should be compared, you can use the compare_df() function.

Prepare a dataset extraction sheet

If you have not started data extraction yet, you can use the data_extraction_sheet() function to obtain a perfectly formatted data extraction sheet.

Well-formatted dataset

One of the specificities of the metaConvert package is that its core function (convert_df) does not have arguments to specify the names of the variables contained in the dataset. While this allow using a convenient automatic process in the calculations, this requires that the datasets passed to this function respect a very precise formatting (which we will refer to as well-formatted dataset).

Rather than a long description of all column names, we built several tools that help you find required information.

1. You can use the data_extraction_sheet() function that generates an excel/csv/txt file containing all the column names available, as well as a description of the information it should contain.

2. You can use the see_input_data() function that generates a list of all available types of input data as well as their estimated/converted effect size measures. This function also points out to the corresponding helper tables available in https://metaconvert.org

Effect size measures available

Eleven effect size measures are accepted:

• "d": standardized mean difference (i.e., Cohen's d)

• "g": Hedges' g

• "md": mean difference

• "r": Correlation coefficient

• "z": Fisher's r-to-z correlation

• "or" or "logor": odds ratio or its logarithm

• "rr" or "logrr": risk ratio or its logarithm

• "irr" or "logirr": incidence rate ratio or its logarithm

• "nnt": number needed to treat

• "logcvr": log coefficient of variation

• "logvr": log variability ratio

Output

All the functions of the metaConvert package that are dedicated to effect size calculations (i.e., all the functions named es_from_*) return a dataframe that contain, depending on the function - some of the following columns:

info_used	input data used to generate the effect size.
md	value of the mean difference.
md_se	standard error of the mean difference.
md_ci_lo	lower bound of the 95% CI of the mean difference.
md_ci_up	upper bound of the 95% CI of the mean difference.
d	value of the Cohen's d.
d_se	standard error of the Cohen's d.
d_ci_lo	lower bound of the 95% CI of the Cohen's d.
d_ci_up	upper bound of the 95% CI of the Cohen's d.
g	value of the Hedges' g.
g_se	standard error of the Hedges' g.
g_ci_lo	lower bound of the 95% CI of the Hedges' g.
g_ci_up	upper bound of the 95% CI of the Hedges' g.
r	value of the correlation coefficient.
r_se	standard error of the correlation coefficient.
r_ci_lo	lower bound of the 95% CI of the correlation coefficient.
r_ci_up	upper bound of the 95% CI of the correlation coefficient.
z	value of the r-to-z transformed correlation coefficient.
z_se	standard error of the r-to-z transformed correlation coefficient.
z_ci_lo	lower bound of the 95% CI of the r-to-z transformed correlation coefficient.
z_ci_up	upper bound of the 95% CI of the r-to-z transformed correlation coefficient.
logor	value of the log odds ratio.
logor_se	standard error of the log odds ratio.
logor_ci_lo	lower bound of the 95% CI of the log odds ratio.
logor_ci_up	upper bound of the 95% CI of the log odds ratio.
logrr	value of the log risk ratio.
logrr_se	standard error of the log risk ratio.
logrr_ci_lo	lower bound of the 95% CI of the log risk ratio.
logrr_ci_up	upper bound of the 95% CI of the log risk ratio.
logirr	value of the log incidence rate ratio.
logirr_se	standard error of the log incidence rate ratio.
logirr_ci_lo	lower bound of the 95% CI of the log incidence rate ratio.
logirr_ci_up	upper bound of the 95% CI of the log incidence rate ratio.
logvr	value of the log variability ratio.
logvr_se	standard error of the log variability ratio.
logvr_ci_lo	lower bound of the 95% CI of the log variability ratio.
logvr_ci_up	upper bound of the 95% CI of the log variability ratio.
logcvr	value of the log coefficient of variation.
logcvr_se	standard error of the log coefficient of variation.
logcvr_ci_lo	lower bound of the 95% CI of the log coefficient of variation.
logcvr_ci_up	upper bound of the 95% CI of the log coefficient of variation.
nnt	number needed to treat.

---

Aggregate a dataframe containing dependent effect sizes

Description

Aggregate a dataframe containing dependent effect sizes

Usage

aggregate_df(
  x,
  dependence = "outcomes",
  cor_unit = 0.8,
  agg_fact,
  es = "es",
  se = "se",
  col_mean = NA,
  col_weighted_mean = NA,
  weights = NA,
  col_sum = NA,
  col_min = NA,
  col_max = NA,
  col_fact = NA,
  na.rm = TRUE
)

Arguments

x	a dataframe that should be aggregated (must contain effect size values and standard errors).
dependence	The type of dependence in your dataframe (can be either "outcomes" or "subgroups"). See details.
cor_unit	The correlation between effect sizes coming from the same clustering unit (only used when dependence = "times" or dependence = "outcomes").
agg_fact	A character string identifying the column name that contains the clustering units (all rows with the same agg_fact value will be aggregated together).
es	A character string identifying the column name containing the effect size values. Default is "es".
se	A character string identifying the column name containing the standard errors of the effect size. Default is "se".
col_mean	a vector of character strings identifying the column names for which the dependent values are summarized by taking their mean.
col_weighted_mean	a vector of character strings identifying the column names for which the dependent values are summarized by taking their weighted mean.
weights	The weights that will be used to estimated the weighted means.
col_sum	a vector of character strings identifying the column names for which the dependent values are summarized by taking their sum.
col_min	a vector of character strings identifying the column names for which the dependent values are summarized by taking their minimum.
col_max	a vector of character strings identifying the column names for which the dependent values are summarized by taking their maximum.
col_fact	a vector of character strings identifying the column names that are factors (different values will be separated by a "/" character).
na.rm	a logical vector indicating whether missing values should be ignored in the calculations for the col_mean, col_weighted_mean, col_sum, col_min and col_max arguments.

Details

1. In the dependence argument, you should indicate "outcomes" if the dependence within the same clustering unit (e.g., study) is due to the presence of multiple effect sizes produced from the same participants at the same time-point (e.g., multiple outcome measures)

2. In the dependence argument, you should indicate "times" if the dependence within the same clustering unit (e.g., study) is due to the presence of multiple effect sizes produced from the same participants at the different time-points (e.g., an RCT with several follow-up waves).

3. In the dependence argument, you should indicate "subgroups" if the dependence within the same clustering unit (e.g., study) is due to the presence of multiple effect sizes produced by independent subgroups (e.g., one effect size for boys, and one for girls).

If you are working with ratio measures, make sure that the information on the effect size estimates (i.e., the column passed to the es argument of the function) is presented on the log scale.

Value

The object returned by the aggregate_df contains, is a dataframe containing at the very least, the aggregating factor, and the aggregated effect size values and standard errors. All columns indicated in the col_* arguments will also be included in this dataframe.

row_id	the row number in the original dataset.
es	the aggregated effect size value.
se	the standard error of the aggregated effect size.
...	any columns indicated in the col_* arguments.

Examples

res <- summary(convert_df(df.haza, measure = "d"))
aggregate_df(res, dependence = "outcomes", cor_unit = 0.8,
             agg_fact = "study_id", es = "es_crude", se = "se_crude",
             col_fact = c("outcome", "type_publication"))

---

Flag the differences between two dataframes.

Description

Flag the differences between two dataframes.

Usage

compare_df(
  df_extractor_1,
  df_extractor_2,
  ordering_columns = NULL,
  tolerance = 0,
  tolerance_type = "ratio",
  output = "html",
  file_name = "comparison.xlsx"
)

Arguments

df_extractor_1	a first dataset. Differences with the second dataset will be flagged in green.
df_extractor_2	a second dataset. Differences with the first dataset will be flagged in red.
ordering_columns	column names that should be used to re-order the two datasets before running the comparisons
tolerance	the cut-off value used to flag differences between two numeric values
tolerance_type	must be either 'difference' or 'ratio'
output	type of object returned by the function (see 'Value' section). Must be either 'wide', 'long', 'html', 'html2' or 'xlsx'.
file_name	the name of the generated file (only used when output="xlsx")

Details

This function aims to facilitate the comparison of two datasets created by blind data extractors during a systematic review. It is a wrapper of several functions from the 'compareDF' package.

Value

This function returns a dataframe composed of the rows that include a difference (all identical rows are removed). Several outputs can be requested :

1. setting output="xlsx" returns an excel file. A message indicates the location of the generated file on your computer.

2. setting output="html" returns an html file

3. setting output="html2" returns an html file (only useful when the "html" command did not make the html pane appear in R studio).

4. setting output="wide" a wide dataframe

5. setting output="long" a long dataframe

References

Alex Joseph (2022). compareDF: Do a Git Style Diff of the Rows Between Two Dataframes with Similar Structure. R package version 2.3.3. https://CRAN.R-project.org/package=compareDF

Examples

df.compare1 = df.compare1[order(df.compare1$author), ]
df.compare2 = df.compare2[order(df.compare2$year), ]

compare_df(
  df_extractor_1 = df.compare1,
  df_extractor_2 = df.compare2,
  ordering_columns = c("author", "year")
)

---

Automatically compute effect sizes from a well formatted dataset

Description

Automatically compute effect sizes from a well formatted dataset

Usage

convert_df(
  x,
  measure = c("d", "g", "md", "logor", "logrr", "logirr", "nnt", "r", "z", "logvr",
    "logcvr"),
  main_es = TRUE,
  es_selected = c("auto", "hierarchy", "minimum", "maximum"),
  selection_auto = c("crude", "paired", "adjusted"),
  split_adjusted = TRUE,
  format_adjusted = c("wide", "long"),
  verbose = TRUE,
  max_asymmetry = 10,
  hierarchy = "means_sd > means_se > means_ci",
  table_2x2_to_cor = "tetrachoric",
  rr_to_or = "metaumbrella",
  or_to_rr = "metaumbrella_cases",
  or_to_cor = "bonett",
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "bonett",
  r_pre_post = 0.5,
  cor_to_smd = "viechtbauer",
  unit_type = "raw_scale",
  yates_chisq = FALSE
)

Arguments

x	a well formatted dataset
measure	the effect size measure that will be estimated from the information stored in the dataset. See details.
main_es	a logical variable indicating whether a main effect size should be selected when overlapping data are present. See details.
es_selected	the method used to select the main effect size when several information allows to estimate an effect size for the same association/comparison. Must be either "minimum" (the smallest effect size will be selected), "maximum" (the largest effect size will be selected) or "hierarchy" (the effect size computed from the information specified highest in the hierarchy will be selected). See details.
selection_auto	a character string giving details on the best "auto" hierarchy to use (only useful when hierarchy="auto" and measure= "d", "g" or "md"). See details.
split_adjusted	a logical value indicating whether crude and adjusted effect sizes should be presented separately. See details.
format_adjusted	presentation format of the adjusted effect sizes. See details.
verbose	a logical variable indicating whether text outputs and messages should be generated. We recommend turning this option to FALSE only after having carefully read all the generated messages.
max_asymmetry	A percentage indicating the tolerance before detecting asymmetry in the 95% CI bounds.
hierarchy	a character string indicating the hierarchy in the information to be prioritized for the effect size calculations. See details.
table_2x2_to_cor	formula used to obtain a correlation coefficient from the contingency table. For now only 'tetrachoric' is available.
rr_to_or	formula used to convert the rr value into an odds ratio.
or_to_rr	formula used to convert the or value into a risk ratio.
or_to_cor	formula used to convert the or value into a correlation coefficient.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation.
pre_post_to_smd	formula used to obtain a SMD from pre/post means and SD of two independent groups.
r_pre_post	pre-post correlation across the two groups (use this argument only if the precise correlation in each group is unknown)
cor_to_smd	formula used to convert a correlation coefficient value into a SMD.
unit_type	the type of unit for the unit_increase_iv argument. Must be either "sd" or "value" (see es_from_pearson_r).
yates_chisq	a logical value indicating whether the Chi square has been performed using Yate's correction for continuity.

Details

This function automatically computes or converts between 11 effect sizes measures from any relevant type of input data stored in the dataset you pass to this function.

Effect size measures

Possible effect size measures are:

1. Cohen's d ("d")

2. Hedges' g ("g")

3. mean difference ("md")

4. (log) odds ratio ("or" and "logor")

5. (log) risk ratio ("rr" and "logrr")

6. (log) incidence rate ratio ("irr" and "logirr")

7. correlation coefficient ("r")

8. transformed r-to-z correlation coefficient ("z")

9. log variability ratio ("logvr")

10. log coefficient of variation ("logcvr")

11. number needed to treat ("nnt")

Computation of a main effect size

If you enter multiple types of input data (e.g., means/sd of two groups and a student t-test value) for the same comparison i.e., for the same row of the dataset, the convert_df() function can have two behaviours. If you set:

• main_es = FALSE the function will estimate all possible effect sizes from all types of input data (which implies that if a comparison has several types of input data, it will result in multiple rows in the dataframe returned by the function)

• main_es = TRUE the function will select one effect size per comparison (which implies that if a comparison has several types of input data, it will result in a unique row in the dataframe returned by the function)

Selection of input data for the computation of the main effect size

If you choose to estimate one main effect size (i.e., by setting main_es = TRUE), you have several options to select this main effect size. If you set:

• es_selected = "auto": the main effect size will be automatically selected, by prioritizing specific types of input data over other (see next section "Hierarchy").

• es_selected = "hierarchy": the main effect size will be selected, by prioritizing specific types of input data over other (see next section "Hierarchy").

• es_selected = "minimum": the main effect size will be selected, by selecting the lowest effect size available.

• es_selected = "maximum": the main effect size will be selected, by selecting the highest effect size available.

Hierarchy

More than 70 different combinations of input data can be used to estimate an effect size. You can retrieve the effect size measures estimated by each combination of input data in the see_input_data() function and online https://metaconvert.org/input.html.

You have two options to use a hierarchy in the types of input data.

• an automatic way (es_selected = "auto")

• an manual way (es_selected = "hierarchy")

Automatic

If you select an automatic hierarchy, here are the types of input data that will be prioritized.

Crude SMD or MD (measure=c("d", "g", "md") and selection_auto="crude")

1. User's input effect size value

2. SMD value

3. Means at post-test

4. ANOVA/Student's t-test/point biserial correlation statistics

5. Linear regression estimates

6. Mean difference values

7. Quartiles/median/maximum values

8. Post-test means extracted from a plot

9. Pre-test+post-test means or mean change

10. Paired ANOVA/t-test statistics

11. Odds ratio value

12. Contingency table

13. Correlation coefficients

14. Phi/chi-square value

Paired SMD or MD (measure=c("d", "g", "md") and selection_auto="paired")

1. User's input effect size value

2. Paired SMD value

3. Pre-test+post-test means or mean change

4. Paired ANOVA/t-test statistics

5. Means at post-test

6. ANOVA/Student's t-test/point biserial correlation

7. Linear regression estimates

8. Mean difference values

9. Quartiles/median/maximum values

10. Odds ratio value

11. Contingency table

12. Correlation coefficients

13. Phi/chi-square value

Adjusted SMD or MD (measure=c("d", "g", "md") and selection_auto="adjusted")

1. User's input adjusted effect size value

2. Adjusted SMD value

3. Estimated marginal means from ANCOVA

4. F- or t-test value from ANCOVA

5. Adjusted mean difference from ANCOVA

6. Estimated marginal means from ANCOVA extracted from a plot

Odds Ratio (measure=c("or"))

1. User's input effect size value

2. Odds ratio value

3. Contingency table

4. Risk ratio values

5. Phi/chi-square value

6. Correlation coefficients

7. (Then hierarchy as for "d" or "g" option crude)

Risk Ratio (measure=c("rr"))

1. User's input effect size value

2. Risk ratio values

3. Contingency table

4. Odds ratio values

5. Phi/chi-square value

Incidence rate ratio (measure=c("irr"))

1. User's input effect size value

2. Number of cases and time of disease free observation time

Correlation (measure=c("r", "z"))

1. User's input effect size value

2. Correlation coefficients

3. Contingency table

4. Odds ratio value

5. Phi/chi-square value

6. SMD value

7. Means at post-test

8. ANOVA/Student's t-test/point biserial correlation

9. Linear regression estimates

10. Mean difference values 11 Quartiles/median/maximum values

11. Post-test means extracted from a plot

12. Pre-test+post-test means or mean change

13. Paired ANOVA/t-test

Variability ratios (measure=c("vr", "cvr"))

1. User's input effect size value

2. means/variability indices at post-test

3. means/variability indices at post-test extracted from a plot

Number needed to treat (measure=c("nnt"))

1. User's input effect size value

2. Contingency table

3. Odds ratio values

4. Risk ratio values

5. Phi/chi-square value

Manual

If you select a manual hierarchy, you can specify the order in which you want to use each type of input data. You can prioritize some types of input data by placing them at the begining of the hierarchy argument, and you must separate all input data with a ">" separator. For example, if you set:

• hierarchy = "means_sd > means_se > student_t", the convert_df function will prioritize the means + SD, then the means + SE, then the Student's t-test to estimate the main effect size.

• hierarchy = "2x2 > or_se > phi", the convert_df function will prioritize the contigency table, then the odds ratio value + SE, then the phi coefficient to estimate the main effect size.

Importantly, if none of the types of input data indicated in the hierarchy argument can be used to estimate the target effect size measure, the convert_df() function will automatically try to use other types of input data to estimate an effect size.

Adjusted effect sizes

Some datasets will be composed of crude (i.e., non-adjusted) types of input data (such as standard means + SD, Student's t-test, etc.) and adjusted types of input data (such as means + SE from an ANCOVA model, a t-test from an ANCOVA, etc.).

In these situations, you can decide to:

• treat crude and adjusted input data the same way split_adjusted = FALSE

• split calculations for crude and adjusted types of input data split_adjusted = TRUE

If you want to split the calculations, you can decide to present the final dataset:

• in a long format (i.e., crude and adjusted effect sizes presented in separate rows format_adjusted = "long")

• in a wide format (i.e., crude and adjusted effect sizes presented in separate columns format_adjusted = "wide")

Value

The convert_df() function returns a list of more than 70 dataframes (one for each function automatically applied to the dataset). These dataframes systematically contain the columns described in metaConvert-package. The list of dataframes can be easily converted to a single, calculations-ready dataframe using the summary function (see summary.metaConvert).

Examples

res <- convert_df(df.haza,
  measure = "g",
  split_adjusted = TRUE,
  es_selected = "minimum",
  format_adjusted = "long"
)
summary(res)

---

Data extraction sheet generator

Description

Data extraction sheet generator

Usage

data_extraction_sheet(
  measure = c("d", "g", "md", "or", "rr", "nnt", "r", "z", "logvr", "logcvr", "irr"),
  type_of_measure = c("natural", "natural+converted"),
  name = "mcv_data_extraction",
  extension = c("data.frame", ".txt", ".csv", ".xlsx"),
  verbose = TRUE
)

Arguments

measure	Target effect size measure (one of the 11 available in metaConvert). Default is "all".
type_of_measure	One of "natural+converted" or "natural" (see details).
name	Name of the file created
extension	Extension of the file created. Most common are ".xlsx", ".csv" or ".txt". It is also possible to generate an R dataframe object by using the "data.frame" extension.
verbose	logical variable indicating whether some information should be printed (e.g., the location where the sheet is created when using ".xlsx", ".csv" or ".txt" extensions)

Details

This function generates, on your computer, a data extraction sheet that contains the name of columns that can be used by our tools to estimate various effect size measures.

If you select a specific measure (e.g., measure = "g"), you will be presented only with most common information allowing to estimate this measure (e.g., you will not be provided with columns for contigency tables if you request a data extraction sheet for measure = "g").

Measure

You can specify a specific effect size measures (among those available in the metaConvert-package). Doing this, the data extraction sheet will contain only the columns of the input data allowing a natural estimation of the effect size measure. For example, if you request measure="d" the data extraction sheet will not contain the columns for the contingency table since, although the convert_df function allows you to convert a contingency table into a "d", this requires to convert the "OR" that is naturally estimated from the contingency table into a "d".

This table is designed to be used in combination with tables showing the combination of input data leading to estimate each of the effect size measures (https://metaconvert.org/html/input.html)

Extension

You can export a file in various formats outside R (by indicating, for example, ".txt", ".xlsx", or ".csv") in the extension argument. You can also visualise this dataset directly in R by setting extension = "data.frame".

Value

This function returns a data extraction sheet that contains all the information necessary to estimate any effect size using the metaConvert tools.

Examples

data_extraction_sheet(measure = "md", extension = "data.frame")

---

Fictitious dataset 1

Description

First fictitious dataset aiming to understand how the compare_df function works. Slightly different from df.compare1

Usage

df.compare1

Format

An object of class data.frame with 5 rows and 7 columns.

---

Fictitious dataset 2

Description

First fictitious dataset aiming to understand how the compare_df function works. Slightly different from df.compare2

Usage

df.compare2

Format

An object of class data.frame with 6 rows and 7 columns.

---

Meta-analytic dataset inspired from Haza and colleagues (2024)

Description

Dataset of a meta-analysis exploring the specificity of social functioning of children with ADHD (compared to healthy controls) in case-control studies. This dataset contains: 1. several information coming from the same participants (due to the completion of multiple outcomes). 1. several information coming from the same study (due to the presence of multiple subgroups). 1. overlapping information for the same comparison 1. several information types from which a standardized mean difference can be estimated/converted

Usage

df.haza

Format

An object of class data.frame with 170 rows and 106 columns.

Source

Haza B, Gosling CJ, Conty L & Pinabiaux C (2024). Social Functioning in Children and Adolescents with ADHD: A Meta-analysis. Journal of Child Psychology and Psychiatry and Allied Disciplines.

---

Short version of the df.haza dataset

Description

This dataset is a shoter version of the df.haza dataset.

Usage

df.short

Format

An object of class grouped_df (inherits from tbl_df, tbl, data.frame) with 37 rows and 109 columns.

Source

Haza B, Gosling CJ, Conty L & Pinabiaux C (2024). Social Functioning in Children and Adolescents with ADHD: A Meta-analysis. Journal of Child Psychology and Psychiatry and Allied Disciplines.

---

Convert a 2x2 table into several effect size measures

Description

Convert a 2x2 table into several effect size measures

Usage

es_from_2x2(
  n_cases_exp,
  n_cases_nexp,
  n_controls_exp,
  n_controls_nexp,
  table_2x2_to_cor = "tetrachoric",
  reverse_2x2
)

Arguments

n_cases_exp	number of cases/events in the exposed group
n_cases_nexp	number of cases/events in the non exposed group
n_controls_exp	number of controls/no-event in the exposed group
n_controls_nexp	number of controls/no-event in the non exposed group
table_2x2_to_cor	formula used to obtain a correlation coefficient from the contingency table (see details).
reverse_2x2	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function first computes (log) odds ratio (OR), (log) risk ratio (RR) and number needed to treat (NNT) from the 2x2 table. Note that if a cell is equal to 0, we applied the typical adjustment (add 0.5) to all cells. Cohen's d (D), Hedges' g (G) and correlation coefficients (R/Z) are then estimated from the OR.

To estimate an OR, the formulas used (Box 6.4.a in the Cochrane Handbook) are:

$logor = log(\frac{n\_cases\_exp / n\_cases\_nexp}{n\_controls\_exp / n\_controls\_nexp})$

$logor\_se = \sqrt{\frac{1}{n\_cases\_exp} + \frac{1}{n\_cases\_nexp} + \frac{1}{n\_controls\_exp} + \frac{1}{n\_controls\_nexp}}$

To estimate an RR, the formulas used (Box 6.4.a in the Cochrane Handbook) are:

$logrr = log(\frac{n\_cases\_exp / n\_exp}{n\_cases\_nexp / n\_nexp})$

$logrr\_se = \sqrt{\frac{1}{n\_cases\_exp} - \frac{1}{n\_exp} + \frac{1}{n\_cases\_nexp} - \frac{1}{n\_nexp}}$

To estimate a NNT, the formulas used are (Sedwick, 2013) :

$pt = \frac{n\_cases\_exp}{n\_cases\_exp + n\_controls\_exp}$

$pc = \frac{n\_cases\_nexp}{n\_cases\_nexp + n\_controls\_nexp}$

$AAR = pc - pt$

$nnt = \frac{1}{AAR}$

To convert the 2x2 table into a SMD, the function estimates an OR value from the 2x2 table (formula above) that is then converted to a SMD (see formula in es_from_or_se()).

To convert the 2x2 table into a correlation coefficient, For now, only the tetrachoric correlation is currently proposed

• table_2x2_to_cor = "tetrachoric". Given the heavy calculations required for this effect size measure, we relied on the implementation of the formulas of the 'metafor' package. More information can be retrieved here (https://wviechtb.github.io/metafor/reference/escalc.html#-b-measures-for-two-dichotomous-variables).

Value

This function estimates and converts between several effect size measures.

natural effect size measure	OR + RR + NNT
converted effect size measure	D + G + R + Z
required input data	See 'Section 7. Contingency (2x2) table or proportions'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Available from www.training.cochrane.org/handbook.

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Sedgwick, P. (2013). What is number needed to treat (NNT)? Bmj, 347.

Examples

es_from_2x2(n_cases_exp = 467, n_cases_nexp = 22087, n_controls_exp = 261, n_controls_nexp = 8761)

---

Convert the proportion of occurrence of a binary event in two independent groups into several effect size measures

Description

Convert the proportion of occurrence of a binary event in two independent groups into several effect size measures

Usage

es_from_2x2_prop(
  prop_cases_exp,
  prop_cases_nexp,
  n_exp,
  n_nexp,
  table_2x2_to_cor = "tetrachoric",
  reverse_prop
)

Arguments

prop_cases_exp	proportion of cases/events in the exposed group (ranging from 0 to 1)
prop_cases_nexp	proportion of cases/events in the non-exposed group (ranging from 0 to 1)
n_exp	total number of participants in the exposed group
n_nexp	total number of participants in the non exposed group
table_2x2_to_cor	formula used to obtain a correlation coefficient from the contigency table (see details).
reverse_prop	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function uses the proportions and sample size to recreate the 2x2 table, and then relies on the calculations of the es_from_2x2_sum() function.

The formulas used is to obtain the 2x2 table are

$n\_cases\_exp = prop\_cases\_exp * n\_exp$

$n\_cases\_nexp = prop\_cases\_nexp * n\_nexp$

$n\_controls\_exp = (1 - prop\_cases\_exp) * n\_exp$

$n\_controls\_nexp = (1 - prop\_cases\_nexp) * n\_nexp$

Value

This function estimates and converts between several effect size measures.

natural effect size measure	OR + RR + NNT
converted effect size measure	D + G + R + Z
required input data	See 'Section 7. Contingency (2x2) table or proportions'
	https://metaconvert.org/input.html

Examples

es_from_2x2_prop(prop_cases_exp = 0.80, prop_cases_nexp = 0.60, n_exp = 10, n_nexp = 20)

---

Convert a table with the number of cases and row marginal sums into several effect size measures

Description

Convert a table with the number of cases and row marginal sums into several effect size measures

Usage

es_from_2x2_sum(
  n_cases_exp,
  n_exp,
  n_cases_nexp,
  n_nexp,
  table_2x2_to_cor = "tetrachoric",
  reverse_2x2
)

Arguments

n_cases_exp	number of cases/events in the exposed group
n_exp	total number of participants in the exposed group
n_cases_nexp	number of cases/events in the non exposed group
n_nexp	total number of participants in the non exposed group
table_2x2_to_cor	formula used to obtain a correlation coefficient from the contigency table (see details).
reverse_2x2	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function uses the number of cases in both the exposed and non-exposed groups and the total number of participants exposed and non-exposed to recreate a 2x2 table. Then relies on the calculations of the es_from_2x2 function.

$n\_controls\_exp = n\_exp - n\_cases\_exp$

$n\_controls\_nexp = n\_nexp - n\_cases\_nexp$

Value

This function estimates and converts between several effect size measures.

natural effect size measure	OR + RR + NNT
converted effect size measure	D + G + R + Z
required input data	See 'Section 7. Contingency (2x2) table or proportions'
	https://metaconvert.org/input.html

Examples

es_from_2x2_sum(n_cases_exp = 10, n_exp = 40, n_cases_nexp = 25, n_nexp = 47)

---

Convert a F-statistic obtained from an ANCOVA model into several effect size measures.

Description

Convert a F-statistic obtained from an ANCOVA model into several effect size measures.

Usage

es_from_ancova_f(
  ancova_f,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_f
)

Arguments

ancova_f	a F-statistic from an ANCOVA (binary predictor)
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_f	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function first computes an "adjusted" Cohen's d (D), and Hedges' g (G) from the F-value of an ANCOVA (binary predictor). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a Cohen's d the formula used is (table 12.3 in Cooper):

$cohen\_d = \sqrt{ancova\_f * \frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp}} * \sqrt{1 - cov\_out\_cor^2}$

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 18. Adjusted: ANCOVA statistics, eta-squared'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L. V., & Valentine, J. C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_f(ancova_f = 4, cov_outcome_r = 0.2, n_cov_ancova = 3, n_exp = 20, n_nexp = 20)

---

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Description

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Usage

es_from_ancova_f_pval(
  ancova_f_pval,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_f_pval
)

Arguments

ancova_f_pval	a two-tailed p-value of an F-test in an ANCOVA (binary predictor)
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_f_pval	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function converts the p-value of an ANCOVA (binary predictor) into a t value, and then relies on the calculations of the es_from_ancova_t() function.

To convert the p-value into a t-value, the following formula is used (table 12.3 in Cooper):

$df = n\_exp + n\_nexp + n\_exp - 2 - n\_cov\_ancova$

$t = | pt(ancova\_f\_pval/2, df = df) |$

Then, calculations of the es_from_ancova_t() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 18. Adjusted: ANCOVA statistics, eta-squared'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_f_pval(
  ancova_f_pval = 0.05, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

---

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_ci(
  ancova_md,
  ancova_md_ci_lo,
  ancova_md_ci_up,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  max_asymmetry = 10,
  smd_to_cor = "viechtbauer",
  reverse_ancova_md
)

Arguments

ancova_md	adjusted mean difference between two independent groups
ancova_md_ci_lo	lower bound of the covariate-adjusted 95% CI of the mean difference
ancova_md_ci_up	upper bound of the covariate-adjusted 95% CI of the mean difference
cov_outcome_r	correlation between the outcome and covariate (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
max_asymmetry	A percentage indicating the tolerance before detecting asymmetry in the 95% CI bounds.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_ancova_md	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts the mean difference (MD) 95% CI into a standard error, and then relies on the calculations of the es_from_ancova_md_se function.

To convert the 95% CI into a standard error, the following formula is used (table 12.3 in Cooper):

$md\_se = \frac{ancova\_md\_ci\_up - ancova\_md\_ci\_lo}{(2 * qt(0.975, n\_exp + n\_nexp - 2 - n\_cov\_ancova))}$

Calculations of the es_from_ancova_md_se() are then applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 20. Adjusted: Mean difference and dispersion'
	https://metaconvert.org/input.html

Examples

es_from_ancova_md_ci(
  ancova_md = 4, ancova_md_ci_lo = 2,
  ancova_md_ci_up = 6,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

---

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_pval(
  ancova_md,
  ancova_md_pval,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_md
)

Arguments

ancova_md	adjusted mean difference between two independent groups
ancova_md_pval	p-value (two-tailed) of the adjusted mean difference
cov_outcome_r	correlation between the outcome and covariate (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_ancova_md	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts the mean difference (MD) p-value into a standard error, and then relies on the calculations of the es_from_ancova_md_se() function.

To convert the p-value into a standard error, the following formula is used (table 12.3 in Cooper):

$t = qt(p = \frac{ancova\_md\_pval}{2}, df = n\_exp + n\_nexp - 2 - n\_cov\_ancova)$

$ancova\_md\_se = | \frac{ancova\_md}{t} |$

Calculations of the es_from_ancova_md_se() are then applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 20. Adjusted: Mean difference and dispersion'
	https://metaconvert.org/input.html

Examples

es_from_ancova_md_pval(
  ancova_md = 4, ancova_md_pval = 0.05,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

---

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_sd(
  ancova_md,
  ancova_md_sd,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_md
)

Arguments

ancova_md	adjusted mean difference between two independent groups
ancova_md_sd	covariate-adjusted standard deviation of the mean difference
cov_outcome_r	correlation between the outcome and covariate (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_ancova_md	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function first computes an "adjusted" Cohen's d (D), Hedges' g (G) from the adjusted mean difference (MD). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate the unadjusted variance of MD (table 12.3 in Cooper):

$md\_sd = \frac{ancova\_md\_sd}{\sqrt{1 - cor\_outcome\_r^2}}$

$md\_se = md\_sd * \sqrt{\frac{1}{n\_exp} + \frac{1}{n\_nexp}}$

$md\_lo = md - md\_se * qt(.975, n\_exp + n\_nexp-2-n\_cov\_ancova)$

$md\_up = md + md\_se * qt(.975, n\_exp + n\_nexp-2-n\_cov\_ancova)$

To estimate the Cohen's d (table 12.3 in Cooper):

$d = \frac{ancova\_md}{md\_sd}$

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 20. Adjusted: Mean difference and dispersion'
	https://metaconvert.org/input.html

Examples

es_from_ancova_md_sd(
  ancova_md = 4, ancova_md_sd = 2,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

---

Convert an adjusted mean difference and standard error between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and standard error between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_se(
  ancova_md,
  ancova_md_se,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_md
)

Arguments

ancova_md	adjusted mean difference between two independent groups
ancova_md_se	covariate-adjusted standard error of the mean difference
cov_outcome_r	correlation between the outcome and covariate (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_ancova_md	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts the mean difference (MD) standard error into a standard deviation, and then relies on the calculations of the es_from_ancova_md_sd function.

To convert the standard error into a standard deviation, the following formula is used.

$ancova\_md\_sd = \frac{ancova\_md\_se}{\sqrt{1 / n_exp + 1 / n_nexp}}$

Calculations of the es_from_ancova_md_sd() are then applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 20. Adjusted: Mean difference and dispersion'
	https://metaconvert.org/input.html

Examples

es_from_ancova_md_se(
  ancova_md = 4, ancova_md_se = 2,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

---

Convert means and 95% CIs of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and 95% CIs of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_ci(
  n_exp,
  n_nexp,
  ancova_mean_exp,
  ancova_mean_ci_lo_exp,
  ancova_mean_ci_up_exp,
  ancova_mean_nexp,
  ancova_mean_ci_lo_nexp,
  ancova_mean_ci_up_nexp,
  cov_outcome_r,
  n_cov_ancova,
  max_asymmetry = 10,
  smd_to_cor = "viechtbauer",
  reverse_ancova_means
)

Arguments

n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
ancova_mean_exp	adjusted mean of participants in the experimental/exposed group.
ancova_mean_ci_lo_exp	lower bound of the adjusted 95% CI of the mean of the experimental/exposed group
ancova_mean_ci_up_exp	upper bound of the adjusted 95% CI of the mean of the experimental/exposed group
ancova_mean_nexp	adjusted mean of participants in the non-experimental/non-exposed group.
ancova_mean_ci_lo_nexp	lower bound of the adjusted 95% CI of the mean of the non-experimental/non-exposed group.
ancova_mean_ci_up_nexp	upper bound of the adjusted 95% CI of the mean of the non-experimental/non-exposed group.
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
max_asymmetry	A percentage indicating the tolerance before detecting asymmetry in the 95% CI bounds.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_means	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function converts the adjusted means 95% CI of two independent groups into a standard error, and then relies on the calculations of the es_from_ancova_means_se() function.

To convert the 95% CIs into standard errors, the following formula is used (table 12.3 in Cooper):

$ancova\_mean\_se\_exp = \frac{ancova\_mean\_ci\_up\_exp - ancova\_mean\_ci\_lo\_exp}{2 * qt(0.975, df = n\_exp - 1)}$

$ancova\_mean\_se\_nexp = \frac{ancova\_mean\_ci\_up\_nexp - ancova\_mean\_ci\_lo\_nexp}{2 * qt(0.975, df = n\_nexp - 1)}$

Calculations of the es_from_ancova_means_se() are then applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 19. Adjusted: Means and dispersion'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_ci(
  n_exp = 55, n_nexp = 55, cov_outcome_r = 0.5, n_cov_ancova = 4,
  ancova_mean_exp = 25, ancova_mean_ci_lo_exp = 15, ancova_mean_ci_up_exp = 35,
  ancova_mean_nexp = 18, ancova_mean_ci_lo_nexp = 12, ancova_mean_ci_up_nexp = 24
)

---

Convert means and standard deviations of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and standard deviations of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_sd(
  n_exp,
  n_nexp,
  ancova_mean_exp,
  ancova_mean_nexp,
  ancova_mean_sd_exp,
  ancova_mean_sd_nexp,
  cov_outcome_r,
  n_cov_ancova,
  smd_to_cor = "viechtbauer",
  reverse_ancova_means
)

Arguments

n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
ancova_mean_exp	adjusted mean of participants in the experimental/exposed group.
ancova_mean_nexp	adjusted mean of participants in the non-experimental/non-exposed group.
ancova_mean_sd_exp	adjusted standard deviation of participants in the experimental/exposed group.
ancova_mean_sd_nexp	adjusted standard deviation of participants in the non-experimental/non-exposed group.
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_means	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function first computes an "adjusted" mean difference (MD), Cohen's d (D) and Hedges' g (G) from the adjusted means and standard deviations. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

This function start by estimating the non-adjusted standard deviation of the two groups (formula 12.24 in Cooper);

$mean\_sd\_exp = \frac{ancova\_mean\_sd\_exp}{\sqrt{1 - cov\_outcome\_r^2}}$

$mean\_sd\_nexp = \frac{ancova\_mean\_sd\_nexp}{\sqrt{1 - cov\_outcome\_r^2}}$

To obtain the mean difference, the following formulas are used (authors calculations):

$md = ancova\_mean\_exp - ancova\_mean\_nexp$

$md\_se = \sqrt{\frac{mean\_sd\_exp^2}{n\_exp} + \frac{mean\_sd\_nexp^2}{n\_nexp}}$

$md\_ci\_lo = md - md\_se * qt(.975, n\_exp+n\_nexp-2-n\_cov\_ancova)$

$md\_ci\_up = md + md\_se * qt(.975, n\_exp+n\_nexp-2-n\_cov\_ancova)$

To obtain the Cohen's d, the following formulas are used (table 12.3 in Cooper):

$mean\_sd\_pooled = \sqrt{\frac{(n\_exp - 1) * ancova\_mean\_exp^2 + (n\_nexp - 1) * ancova\_mean\_nexp^2}{n\_exp+n\_nexp-2}}$

$cohen\_d = \frac{ancova\_mean\_exp - ancova\_mean\_nexp}{mean\_sd\_pooled}$

$cohen\_d\_se = \frac{(n\_exp+n\_nexp)*(1-cov\_outcome\_r^2)}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2(n\_exp+n\_nexp)}$

$cohen\_d\_ci\_lo = cohen\_d - cohen\_d\_se * qt(.975, n\_exp + n\_nexp - 2 - n\_cov\_ancova)$

$cohen\_d\_ci\_up = cohen\_d + cohen\_d\_se * qt(.975, n\_exp + n\_nexp - 2 - n\_cov\_ancova)$

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 19. Adjusted: Means and dispersion'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_sd(
  n_exp = 55, n_nexp = 55,
  ancova_mean_exp = 2.3, ancova_mean_sd_exp = 1.2,
  ancova_mean_nexp = 1.9, ancova_mean_sd_nexp = 0.9,
  cov_outcome_r = 0.2, n_cov_ancova = 3
)

---

Convert means and adjusted pooled standard deviation of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and adjusted pooled standard deviation of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_sd_pooled_adj(
  ancova_mean_exp,
  ancova_mean_nexp,
  ancova_mean_sd_pooled,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_means
)

Arguments

ancova_mean_exp	adjusted mean of participants in the experimental/exposed group.
ancova_mean_nexp	adjusted mean of participants in the non-experimental/non-exposed group.
ancova_mean_sd_pooled	adjusted pooled standard deviation.
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_means	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function converts the adjusted pooled standard deviations of two independent groups into a crude pooled standard deviation. and then relies on the calculations of the es_from_ancova_means_sd_pooled_crude() function.

To convert the adjusted pooled SD into a crude pooled SD (table 12.3 in Cooper):

$mean\_sd\_pooled = \frac{ancova\_mean\_sd\_pooled}{\sqrt{1 - cov\_outcome\_r^2}}$

Calculations of the es_from_ancova_means_sd_pooled_crude() are then applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 19. Adjusted: Means and dispersion'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_sd_pooled_adj(
  ancova_mean_exp = 98, ancova_mean_nexp = 87,
  ancova_mean_sd_pooled = 17, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

---

Convert adjusted means obtained from an ANCOVA model and crude pooled standard deviation of two independent groups into several effect size measures

Description

Convert adjusted means obtained from an ANCOVA model and crude pooled standard deviation of two independent groups into several effect size measures

Usage

es_from_ancova_means_sd_pooled_crude(
  ancova_mean_exp,
  ancova_mean_nexp,
  mean_sd_pooled,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_means
)

Arguments

ancova_mean_exp	adjusted mean of participants in the experimental/exposed group.
ancova_mean_nexp	adjusted mean of participants in the non-experimental/non-exposed group.
mean_sd_pooled	crude pooled standard deviation.
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_means	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function first computes an "adjusted" mean difference (MD) and Cohen's d (D) from the adjusted means and crude pooled standard deviation of two independent groups. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate the Cohen's d:

$d = \frac{ancova\_mean\_exp - ancova\_mean\_nexp\_adj}{mean\_sd\_pooled}$

To estimate the mean difference:

$md = ancova\_mean\_exp - ancova\_mean\_nexp\_adj$

$md\_se = \sqrt{\frac{n\_exp + n\_nexp}{n\_exp * n\_nexp} * (1 - cov\_outcome\_r^2) * mean\_sd\_pooled^2}$

Then, calculations of the es_from_ancova_means_sd() and es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 19. Adjusted: Means and dispersion'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_sd_pooled_crude(
  ancova_mean_exp = 29, ancova_mean_nexp = 34,
  mean_sd_pooled = 7, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

---

Convert means and standard errors of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and standard errors of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_se(
  n_exp,
  n_nexp,
  ancova_mean_exp,
  ancova_mean_nexp,
  ancova_mean_se_exp,
  ancova_mean_se_nexp,
  cov_outcome_r,
  n_cov_ancova,
  smd_to_cor = "viechtbauer",
  reverse_ancova_means
)

Arguments

n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
ancova_mean_exp	adjusted mean of participants in the experimental/exposed group.
ancova_mean_nexp	adjusted mean of participants in the non-experimental/non-exposed group.
ancova_mean_se_exp	adjusted standard error of participants in the experimental/exposed group.
ancova_mean_se_nexp	adjusted standard error of participants in the non-experimental/non-exposed group.
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_means	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function converts the adjusted means standard errors of two independent groups into standard deviations, and then relies on the calculations of the es_from_ancova_means_sd function.

To convert the standard errors into standard deviations, the following formula is used.

$ancova\_mean\_sd\_exp = ancova\_mean\_se\_exp * \sqrt{n\_exp}$

$ancova\_mean\_sd\_nexp = ancova\_mean\_se\_nexp * \sqrt{n\_nexp}$

Calculations of the es_from_ancova_means_sd() are then applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	MD + D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 19. Adjusted: Means and dispersion'
	https://metaconvert.org/input.html

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_ancova_means_se(
  n_exp = 55, n_nexp = 55,
  ancova_mean_exp = 2.3, ancova_mean_se_exp = 1.2,
  ancova_mean_nexp = 1.9, ancova_mean_se_nexp = 0.9,
  cov_outcome_r = 0.2, n_cov_ancova = 3
)

---

Convert a t-statistic obtained from an ANCOVA model into several effect size measures.

Description

Convert a t-statistic obtained from an ANCOVA model into several effect size measures.

Usage

es_from_ancova_t(
  ancova_t,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_t
)

Arguments

ancova_t	a t-statistic from an ANCOVA (binary predictor)
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_t	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function first computes an "adjusted" Cohen's d (D), and Hedges' g (G) from the t-value of an ANCOVA (binary predictor). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a Cohen's d the formula used is (table 12.3 in Cooper):

$cohen\_d = ancova\_t* \sqrt{\frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp}}\sqrt{1 - cov\_out\_cor^2}$

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 18. Adjusted: ANCOVA statistics, eta-squared'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L. V., & Valentine, J. C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_t(ancova_t = 2, cov_outcome_r = 0.2, n_cov_ancova = 3, n_exp = 20, n_nexp = 20)

---

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Description

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Usage

es_from_ancova_t_pval(
  ancova_t_pval,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_ancova_t_pval
)

Arguments

ancova_t_pval	a two-tailed p-value of a t-test in an ANCOVA (binary predictor)
cov_outcome_r	correlation between the outcome and covariate(s) (multiple correlation when multiple covariates are included in the ANCOVA model).
n_cov_ancova	number of covariates in the ANCOVA model.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the adjusted cohen_d value into a coefficient correlation (see details).
reverse_ancova_t_pval	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function converts the p-value of an ANCOVA (binary predictor) into a t value, and then relies on the calculations of the es_from_ancova_t() function.

To convert the p-value into a t-value, the following formula is used (table 12.3 in Cooper):

$df = n\_exp + n\_nexp + n\_exp - 2 - n\_cov\_ancova$

$t = | pt(ancova\_f\_pval/2, df = df) |$

Then, calculations of the es_from_ancova_t() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 18. Adjusted: ANCOVA statistics, eta-squared'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_t_pval(
  ancova_t_pval = 0.05, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

---

Convert a one-way independent ANOVA F-value to several effect size measures

Description

Convert a one-way independent ANOVA F-value to several effect size measures

Usage

es_from_anova_f(
  anova_f,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_anova_f
)

Arguments

anova_f	ANOVA F-value (one-way, binary predictor).
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the anova_f value into a coefficient correlation (see details).
reverse_anova_f	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts the F-value (one-way, binary predictor) into a t-value, and then relies on the calculations of the es_from_student_t() function.

To convert the F-value into a t-value, the following formula is used (table 12.1 in Cooper):

$student\_t = \sqrt{anova\_f}$

Then, calculations of the es_from_student_t() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 11. ANOVA statistics, Student's t-test, or point-bis correlation'
	https://metaconvert.org/html/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_anova_f(anova_f = 2.01, n_exp = 20, n_nexp = 22)

---

Convert a p-value from a one-way independent ANOVA to several effect size measures

Description

Convert a p-value from a one-way independent ANOVA to several effect size measures

Usage

es_from_anova_pval(
  anova_f_pval,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_anova_f_pval
)

Arguments

anova_f_pval	p-value (two-tailed) from an ANOVA (binary predictor). If your p-value is one-tailed, simply multiply it by two.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the anova_f_pval value into a coefficient correlation (see details).
reverse_anova_f_pval	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts the p-value from the F-value of an ANOVA (one-way, binary predictor) into a t-value, and then relies on the calculations of the es_from_student_t() function.

To convert the p-value into a t-value, the following formula is used (table 12.1 in Cooper):

$student\_t = qt(\frac{anova\_f\_pval}{2}, df = n\_exp + n\_nexp - 2)$

Then, calculations of the es_from_student_t() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 11. ANOVA statistics, Student's t-test, or point-bis correlation'
	https://metaconvert.org/html/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_anova_pval(anova_f_pval = 0.0012, n_exp = 20, n_nexp = 22)

---

Convert a standardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Description

Convert a standardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Usage

es_from_beta_std(
  beta_std,
  sd_dv,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_beta_std
)

Arguments

beta_std	a standardized regression coefficient value (binary predictor, no other covariables in the model)
sd_dv	standard deviation of the dependent variable
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_beta_std	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function converts a standardized linear regression coefficient (coming from a model with only one binary predictor), into an unstandardized linear regression coefficient.

$sd\_dummy = \sqrt{\frac{n_exp - (n_exp^2 / (n_exp + n_nexp))}{(n_exp + n_nexp - 1)}}$

$unstd\_beta = beta\_std * \frac{sd\_dv}{sd\_dummy}$

Calculations of the es_from_beta_unstd functions are then used.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 13. (Un-)Standardized regression coefficient'
	https://metaconvert.org/input.html

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_beta_std(beta_std = 2.1, sd_dv = 0.98, n_exp = 20, n_nexp = 22)

---

Convert an unstandardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Description

Convert an unstandardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Usage

es_from_beta_unstd(
  beta_unstd,
  sd_dv,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  reverse_beta_unstd
)

Arguments

beta_unstd	an unstandardized regression coefficient value (binary predictor, no other covariables in the model)
sd_dv	standard deviation of the dependent variable
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_beta_unstd	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function estimates a Cohen's d (D) and Hedges' g (G) from an unstandardized linear regression coefficient (coming from a model with only one binary predictor), and the standard deviation of the dependent variable. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

The formula used to obtain the Cohen's d is:

$N = n\_exp + n\_nexp$

$sd\_pooled = \sqrt{\frac{sd\_dv^2 * (N - 1) - unstd\_beta^2 * \frac{n\_exp * n\_nexp}{N}}{N - 2}}$

$cohen\_d = \frac{unstd\_beta}{sd\_pooled}$

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 13. (Un-)Standardized regression coefficient'
	https://metaconvert.org/input.html

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_beta_unstd(beta_unstd = 2.1, sd_dv = 0.98, n_exp = 20, n_nexp = 22)

---

Convert the number of cases and the person-time of disease-free observation in two independent groups into an incidence rate ratio (IRR)

Description

Convert the number of cases and the person-time of disease-free observation in two independent groups into an incidence rate ratio (IRR)

Usage

es_from_cases_time(n_cases_exp, n_cases_nexp, time_exp, time_nexp, reverse_irr)

Arguments

n_cases_exp	number of cases in the exposed group
n_cases_nexp	number of cases in the non-exposed group
time_exp	person-time of disease-free observation in the exposed group
time_nexp	person-time of disease-free observation in the non-exposed group
reverse_irr	a logical value indicating whether the direction of the generated effect sizes should be flipped.

Details

This function estimates the incidence rate ratio from the number of cases and the person-time of disease-free observation in two independent groups.

The formula used to obtain the IRR and its standard error are (Cochrane Handbook (section 6.7.1):

$logirr = log(\frac{n\_cases\_exp / time\_exp}{n\_cases\_nexp / time\_nexp)}$

$logirr\_se = \sqrt{\frac{1}{n\_cases\_exp} + \frac{1}{n\_cases\_nexp}}$

Value

This function estimates IRR.

natural effect size measure	IRR
converted effect size measure	N/A
required input data	See 'Section 5. Incidence Ratio Ratio'
	https://metaconvert.org/input.html

References

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_cases_time(
  n_cases_exp = 241, n_cases_nexp = 554,
  time_exp = 12.764, time_nexp = 19.743
)

---

Convert a chi-square value to several effect size measures

Description

Convert a chi-square value to several effect size measures

Usage

es_from_chisq(
  chisq,
  n_sample,
  n_cases,
  n_exp,
  yates_chisq = FALSE,
  reverse_chisq
)

Arguments

chisq	value of the chi-squared
n_sample	total number of participants in the sample
n_cases	total number of cases/events
n_exp	total number of participants in the exposed group
yates_chisq	a logical value indicating whether the Chi square has been performed using Yate's correction for continuity.
reverse_chisq	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts a chi-square value (with one degree of freedom) into a phi coefficient (Lipsey et al. 2001):

$phi = \sqrt{\frac{chisq^2}{n\_sample}}$

.

Note that if yates_chisq = "TRUE", a small correction is added.

Then, the phi coefficient is converted to other effect size measures (see es_from_phi).

Value

This function estimates and converts between several effect size measures.

natural effect size measure	OR + RR + NNT
converted effect size measure	D + G + R + Z
required input data	See 'Section 8. Phi or chi-square'
	https://metaconvert.org/input.html

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_chisq(chisq = 4.21, n_sample = 78, n_cases = 51, n_exp = 50)

---

Convert a p-value of a chi-square to several effect size measures

Description

Convert a p-value of a chi-square to several effect size measures

Usage

es_from_chisq_pval(
  chisq_pval,
  n_sample,
  n_cases,
  n_exp,
  yates_chisq = FALSE,
  reverse_chisq_pval
)

Arguments

chisq_pval	p-value of a chi-square coefficient
n_sample	total number of participants in the sample
n_cases	total number of cases/events
n_exp	total number of participants in the exposed group
yates_chisq	a logical value indicating whether the Chi square has been performed using Yate's correction for continuity.
reverse_chisq_pval	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function converts a chi-square value (with one degree of freedom) into a chi-square coefficient (Section 3.12 in Lipsey et al., 2001):

$chisq = qchisq(chisq\_pval, df = 1, lower.tail = FALSE)$

Note that if yates_chisq = "TRUE", a small correction is added.

Then, the chisq coefficient is converted to other effect size measures (see es_from_chisq).

Value

This function estimates and converts between several effect size measures.

natural effect size measure	OR + RR + NNT
converted effect size measure	D + G + R + Z
required input data	See 'Section 8. Phi or chi-square'
	https://metaconvert.org/input.html

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_chisq_pval(chisq_pval = 0.2, n_sample = 42, n_exp = 25, n_cases = 13)

---

Convert a Cohen's d value to several effect size measures

Description

Convert a Cohen's d value to several effect size measures

Usage

es_from_cohen_d(cohen_d, n_exp, n_nexp, smd_to_cor = "viechtbauer", reverse_d)

Arguments

cohen_d	Cohen's d (i.e., standardized mean difference) value.
n_exp	number of participants in the experimental/exposed group.
n_nexp	number of participants in the non-experimental/non-exposed group.
smd_to_cor	formula used to convert the cohen_d value into a coefficient correlation (see details).
reverse_d	a logical value indicating whether the direction of generated effect sizes should be flipped.

Details

This function estimates the standard error of a Cohen's d value and computes a Hedges' g (G). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate the standard error of Cohen's d, the following formula is used (formula 12.13 in Cooper):

$cohen\_d\_se = \sqrt{\frac{n\_exp+n\_nexp}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2*(n\_exp+n\_nexp)}}$

$cohen\_d\_ci\_lo = cohen\_d - cohen\_d\_se * qt(.975, df = n\_exp+n\_nexp-2)$

$cohen\_d\_ci\_up = cohen\_d + cohen\_d\_se * qt(.975, df = n\_exp+n\_nexp-2)$

To estimate the Hedges' g and its standard error, the following formulas are used (Hedges, 1981):

$df = n\_exp + n\_nexp - 2$

$J = exp(\log_{gamma}(\frac{df}{2}) - 0.5 * \log(\frac{df}{2}) - \log_{gamma}(\frac{df - 1}{2}))$

$hedges\_g = cohen\_d * J$

$hedges\_g\_se = \sqrt{cohen\_d\_se^2 * J^2}$

$hedges\_g\_ci\_lo = hedges\_g - hedges\_g\_se * qt(.975, df = n\_exp+n\_nexp-2)$

$hedges\_g\_ci\_up = hedges\_g + hedges\_g\_se * qt(.975, df = n\_exp+n\_nexp-2)$

To estimate the log odds ratio and its standard error, the following formulas are used (formulas 12.34-12.35 in Cooper):

$logor = \frac{cohen\_d * \pi}{\sqrt{3}}$

$logor\_se = \sqrt{\frac{cohen\_d\_se^2 * \pi^2}{3}}$

$logor\_lo = logor - logor\_se * qnorm(.975)$

$logor\_up = logor + logor\_se * qnorm(.975)$

Note that this conversion assumes that responses within the two groups follow logistic distributions.

To estimate the correlation coefficient and its standard error, various formulas can be used.

A. To estimate the 'biserial' correlation (smd_to_cor="viechtbauer"), the following formulas are used (formulas 5, 8, 13, 17, 18, 19 in Viechtbauer):

$h = \frac{n\_exp + n\_nexp}{n\_exp} + \frac{n\_exp + n\_nexp}{n\_nexp}$

$r.pb = \frac{cohen\_d}{\sqrt{cohen\_d^2 + h}}$

$p = \frac{n\_exp}{n\_exp + n\_nexp}$

$q = 1 - p$

$R = \frac{\sqrt{p*q}}{dnorm(qnorm(1-p)) * r.pb}$

$R\_var = \frac{1}{n\_exp + n\_nexp - 1} * (\frac{\sqrt{p*q}}{dnorm(qnorm(1-p))} - R^2)^2$

$R\_se = \sqrt{R\_var}$

$a = \frac{\sqrt{dnorm(qnorm(1-p))}}{(p*q)^\frac{1}{4}}$

$Z = \frac{a}{2} * \log(\frac{1+a*R}{1-a*R})$

$Z\_var = \frac{1}{n - 1}$

$Z\_se = \sqrt{Z\_var}$

$Z\_ci\_lo = Z - qnorm(.975) * Z\_se$

$Z\_ci\_up = Z + qnorm(.975) * Z\_se$

$R\_ci\_lo = tanh(Z\_lo)$

$R\_ci\_up = tanh(Z\_up)$

B. To estimate the correlation coefficient according to Cooper et al. (2019) (formulas 12.40-42) and Borenstein et al. (2009) (formulas 54-56), the following formulas are used (smd_to_cor="lipsey_cooper"):

$p = \frac{n\_exp}{n\_exp + n\_nexp}$

$R = \frac{cohen\_d}{\sqrt{cohen\_d^2 + 1 / (p * (1 - p))}}$

$a = \frac{(n\_exp + n\_nexp)^2}{(n\_exp*n\_nexp)}$

$var\_R = \frac{a^2 * cohen\_d\_se^2}{(cohen\_d^2 + a)^3}$

$R\_se = \sqrt{R\_var}$

$R\_ci\_lo = R - qt(.975, n\_exp+n\_nexp- 2) * R\_se$

$R\_ci\_up = R + qt(.975, n\_exp+n\_nexp- 2) * R\_se$

$Z = atanh(R)$

$Z\_var = \frac{cohen\_d\_se^2}{cohen\_d\_se^2 + (1 / p*(1-p))}$

$Z\_se = \sqrt{Z\_var}$

$Z\_ci\_lo = Z - qnorm(.975) * Z\_se$

$Z\_ci\_up = Z + qnorm(.975) * Z\_se$

Value

This function estimates and converts between several effect size measures.

natural effect size measure	D + G
converted effect size measure	OR + R + Z
required input data	See 'Section 1. Cohen's d or Hedges' g'
	https://metaconvert.org/input.html

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Borenstein, M., Hedges, L. V., Higgins, J. P., & Rothstein, H. R. (2021). Introduction to meta-analysis. John Wiley & Sons.

Hedges LV (1981): Distribution theory for Glass’s estimator of effect size and related estimators. Journal of Educational and Behavioral Statistics, 6, 107–28

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research synthesis methods, 8(2), 161–180.

Examples

es_from_cohen_d(cohen_d = 1, n_exp = 20, n_nexp = 20)

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