# Packages
library(pacman); p_load(meta, metapower, metafor, sjmisc, DT)
#Recoded Race Variables
data$RaceRec <- rec(data$Race, rec = "White = White; Black = Black; Hispanic = Hispanic; Asian/Other, Multi-racial, Non-White = Other")
#The maximum SE, as a robustness check
data$HighSE <- rep(max(data$SESA), 42)
# Data
datatable(data, extensions = c("Buttons", "FixedColumns"), options = list(dom = 'Bfrtip', buttons = c('copy', 'csv', 'print'), scrollX = T, fixedColumns = list(leftColumns = 3))) # Function
MinimumMetaCorrelation <- function(E1 = 0.5, step = 0.01, SS, K, I2, power = 0.8, p = 0.05){
library(pacman); p_load(metapower, tidyverse)
DEF = seq(-1, 1, step)
result_list = lapply(DEF, function(x) {
tryCatch(
expr = {
res = subgroup_power(
n_groups = 2,
effect_sizes = c(E1, E1 + x),
study_size = SS,
k = K,
i2 = I2,
es_type = "r",
p = p)
res_ = data.frame(
DEF = x,
FP = res$subgroup_power$fixed_power_b,
RP = res$subgroup_power$random_power_b)
return(res_)},
error = function(cond) {
return(data.frame(
DEF = x,
FP = NA,
RP = NA))})})
FPP <- result_list %>%
bind_rows() %>%
filter(FP > power) %>%
arrange((DEF < 0)) %>%
head(n = 1); FPP
FPN <- result_list %>%
bind_rows() %>%
filter(FP > power) %>%
arrange((DEF < 0)) %>%
tail(n = 1); FPN
RPP <- result_list %>%
bind_rows() %>%
filter(RP > power) %>%
arrange((DEF < 0)) %>%
head(n = 1); RPP
RPN <- result_list %>%
bind_rows() %>%
filter(RP > power) %>%
arrange((DEF < 0)) %>%
tail(n = 1); RPN
PR <- data.frame("FPP" = FPP[1:2], "FPN" = FPN[1:2], "RPP" = RPP[c(1,3)], "RPN" = RPN[c(1,3)])
return(PR)}
# Subsets
dataWB <- subset(data, Race == "White" | Race == "Black")
dataWH <- subset(data, Race == "White" | Race == "Hispanic")
dataWO <- subset(data, Race == "White" | Race == "Asian/Other" | Race == "Multi-racial" | Race == "Non-White"); dataWO$Race <- rec(dataWO$Race, rec = "White = White; Asian/Other, Multi-racial, Non-White = Other")AMeta <- metagen(data = data,
TE = SA,
seTE = SESA,
studlab = Study,
sm = "COR",
fixed = F,
random = T,
method.tau = "HS",
hakn = T,
backtransf = F,
title = "The Heritability of Cognitive Ability in the U.S."); summary(AMeta)## Review: The Heritability of Cognitive Ability in the U.S.
##
## COR 95%-CI %W(random)
## Scarr-Salapatek (1971) 0.5300 [-0.5872; 1.6472] 1.1
## Beaver et al. (2013) 0.6900 [ 0.0432; 1.3368] 3.1
## Hodges (1976) 0.5800 [-0.2236; 1.3836] 2.0
## Osborne (1980) 0.7300 [-0.0540; 1.5140] 2.1
## Scarr (1981) 0.7300 [-0.2108; 1.6708] 1.5
## Scarr et al. (1993) 0.6400 [-0.5948; 1.8748] 0.9
## Beaver et al. (2013) 0.7200 [ 0.1712; 1.2688] 4.4
## Rhemtulla & Tucker-Drob (2012) 0.6100 [-0.0368; 1.2568] 3.1
## Hart et al. (2013) 0.8900 [ 0.2628; 1.5172] 3.3
## Woodley of Menie et al. (2015) 0.3700 [-0.2768; 1.0168] 3.1
## Figlio et al. (2017) 0.7500 [ 0.3188; 1.1812] 7.1
## Figlio et al. (2017) 0.7700 [ 0.2604; 1.2796] 5.1
## Mollon et al. (2018) 0.8500 [ 0.3404; 1.3596] 5.1
## Engelhardt et al. (2019) 0.6800 [ 0.0528; 1.3072] 3.3
## Pesta et al. (2019) 0.8200 [ 0.1732; 1.4668] 3.1
## Pesta et al. (2019) 0.7800 [-0.4352; 1.9952] 0.9
## Hodges (1976) 0.6100 [-0.1936; 1.4136] 2.0
## Hart et al. (2013) 0.7100 [ 0.0240; 1.3960] 2.8
## Figlio et al. (2017) 0.8500 [ 0.2228; 1.4772] 3.3
## Figlio et al. (2017) 0.8500 [ 0.1640; 1.5360] 2.8
## Engelhardt et al. (2019) 0.8100 [ 0.0064; 1.6136] 2.0
## Pesta et al. (2019) 0.8900 [ 0.2628; 1.5172] 3.3
## Pesta et al. (2019) 0.4900 [-0.9408; 1.9208] 0.6
## Scarr-Salapatek (1971) 0.5600 [-0.4984; 1.6184] 1.2
## Beaver et al. (2013) 0.7100 [ 0.0828; 1.3372] 3.3
## Hodges (1976) 0.4500 [-0.5104; 1.4104] 1.4
## Osborne (1980) 0.7700 [-0.2884; 1.8284] 1.2
## Scarr (1981) 0.6900 [-0.2704; 1.6504] 1.4
## Scarr et al. (1993) 0.7200 [-0.5736; 2.0136] 0.8
## Beaver et al. (2013) 0.6700 [-0.2120; 1.5520] 1.7
## Hart et al. (2013) 0.9400 [ 0.1560; 1.7240] 2.1
## Woodley of Menie et al. (2015) 0.9700 [-0.8332; 2.7732] 0.4
## Figlio et al. (2017) 0.7500 [ 0.1620; 1.3380] 3.8
## Figlio et al. (2017) 0.6900 [ 0.0432; 1.3368] 3.1
## Mollon et al. (2018) 0.7800 [ 0.0548; 1.5052] 2.5
## Engelhardt et al. (2019) 0.3600 [-0.7180; 1.4380] 1.1
## Pesta et al. (2019) 0.9000 [ 0.3120; 1.4880] 3.8
## Pesta et al. (2019) 0.7900 [-0.2684; 1.8484] 1.2
## Hart et al. (2013) 0.4900 [-0.7448; 1.7248] 0.9
## Engelhardt et al. (2019) 0.6200 [-0.6540; 1.8940] 0.8
## Engelhardt et al. (2019) 0.6600 [-0.6924; 2.0124] 0.7
## Rhemtulla & Tucker-Drob (2012) 0.5900 [-0.1744; 1.3544] 2.2
##
## Number of studies combined: k = 42
##
## COR 95%-CI t p-value
## Random effects model 0.7283 [0.6875; 0.7691] 36.02 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0; tau = 0; I^2 = 0.0% [0.0%; 35.5%]; H = 1.00 [1.00; 1.24]
##
## Test of heterogeneity:
## Q d.f. p-value
## 4.90 41 1.0000
##
## Details on meta-analytical method:
## - Inverse variance method
## - Hunter-Schmidt estimator for tau^2
## - Hartung-Knapp adjustment for random effects model
## - Untransformed correlationsGAMeta <- update.meta(AMeta,
subgroup = RaceRec,
tau.common = F); GAMeta## Review: The Heritability of Cognitive Ability in the U.S.
##
## Number of studies combined: k = 42
##
## COR 95%-CI t p-value
## Random effects model 0.7283 [0.6875; 0.7691] 36.02 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0; tau = 0; I^2 = 0.0% [0.0%; 35.5%]; H = 1.00 [1.00; 1.24]
##
## Test of heterogeneity:
## Q d.f. p-value
## 4.90 41 1.0000
##
## Results for subgroups (random effects model):
## k COR 95%-CI tau^2 tau Q I^2
## RaceRec = White 16 0.7173 [0.6494; 0.7852] 0 0 2.19 0.0%
## RaceRec = Hispanic 7 0.7877 [0.6799; 0.8955] 0 0 0.58 0.0%
## RaceRec = Black 15 0.7346 [0.6554; 0.8139] 0 0 1.62 0.0%
## RaceRec = Other 4 0.5875 [0.4908; 0.6842] 0 0 0.04 0.0%
##
## Test for subgroup differences (random effects model):
## Q d.f. p-value
## Between groups 18.50 3 0.0003
##
## Details on meta-analytical method:
## - Inverse variance method
## - Hunter-Schmidt estimator for tau^2
## - Hartung-Knapp adjustment for random effects model
## - Untransformed correlationsThe White H is 0.7173, which translates to
GAMeta$TE.random.w[1]^2 - GAMeta$TE.random.w[3]^2 #B - W Difference## [1] -0.02514989GAMeta$TE.random.w[1]^2 - GAMeta$TE.random.w[2]^2 #H - W Difference## [1] -0.105884GAMeta$TE.random.w[1]^2 - GAMeta$TE.random.w[4]^2 #O - W Difference## [1] 0.1693885K = 15; median Nh = 448; \(I^2\) = 0.402
MinimumMetaCorrelation(E1 = 0.7173, step = 0.001, SS = 448, K = 15, I2 = 0.402)GAMeta$TE.random.w[1]^2 - (GAMeta$TE.random.w[1] - 0.047)^2## [1] 0.06521878#Post Hoc
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], GAMeta$TE.random.w[3]),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.93873
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.3199935
## Random-Effects Model (i2 = 40.2%): 0.2108719#-5%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.05)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8320481
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.8137628
## Random-Effects Model (i2 = 40.2%): 0.596924#-10%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.10)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.764723
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9998598
## Random-Effects Model (i2 = 40.2%): 0.9909839#-20%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.20)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.6340603
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 1
## Random-Effects Model (i2 = 40.2%): 1#-30%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.30)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.5013629
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 1
## Random-Effects Model (i2 = 40.2%): 1#-40%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.40)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.3523327
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 1
## Random-Effects Model (i2 = 40.2%): 1K = 7; median Nh = 245; \(I^2\) = 0.454. Odd sample sizes are rounded down.
MinimumMetaCorrelation(E1 = 0.7173, step = 0.001, SS = 244, K = 7, I2 = 0.454)GAMeta$TE.random.w[1]^2 - (GAMeta$TE.random.w[1] - 0.103)^2## [1] 0.1371583#Post Hoc
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], GAMeta$TE.random.w[2]),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 1.065269
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9146766
## Random-Effects Model (i2 = 45.4%): 0.6917142#-5%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.05)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8320481
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.2982646
## Random-Effects Model (i2 = 45.4%): 0.1843728#-10%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.10)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.764723
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.8005507
## Random-Effects Model (i2 = 45.4%): 0.5444694#-20%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.20)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.6340603
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9997761
## Random-Effects Model (i2 = 45.4%): 0.9813303#-30%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.30)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.5013629
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 1
## Random-Effects Model (i2 = 45.4%): 0.9999778#-40%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.40)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.3523327
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 1
## Random-Effects Model (i2 = 45.4%): 1K = 4; median Nh = 65; \(I^2\) = 0.480.
MinimumMetaCorrelation(E1 = 0.7173, step = 0.001, SS = 64, K = 4, I2 = 0.480)GAMeta$TE.random.w[1]^2 - (GAMeta$TE.random.w[1] - 0.345)^2## [1] 0.3759236#Post Hoc
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], GAMeta$TE.random.w[4]),
study_size = 64,
k = 4,
i2 = 0.480,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.673838
## Expected Study Size: 64
## Expected Number of Studies: 4
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.4124193
## Random-Effects Model (i2 = 48%): 0.2406206#-5%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.05)),
study_size = 64,
k = 4,
i2 = 0.480,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8320481
## Expected Study Size: 64
## Expected Number of Studies: 4
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.08318825
## Random-Effects Model (i2 = 48%): 0.06711886#-10%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.10)),
study_size = 64,
k = 4,
i2 = 0.480,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.764723
## Expected Study Size: 64
## Expected Number of Studies: 4
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.1817434
## Random-Effects Model (i2 = 48%): 0.1173177#-20%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.20)),
study_size = 64,
k = 4,
i2 = 0.480,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.6340603
## Expected Study Size: 64
## Expected Number of Studies: 4
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.5324432
## Random-Effects Model (i2 = 48%): 0.3130875#-30%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.30)),
study_size = 64,
k = 4,
i2 = 0.480,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.5013629
## Expected Study Size: 64
## Expected Number of Studies: 4
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.8625677
## Random-Effects Model (i2 = 48%): 0.5951553#-40%
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - 0.40)),
study_size = 64,
k = 4,
i2 = 0.480,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.3523327
## Expected Study Size: 64
## Expected Number of Studies: 4
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9870233
## Random-Effects Model (i2 = 48%): 0.8552532To estimate Scarr-Rowe effects based on the weak/false Scarr-Rowe hypothesis, we need estimates for levels of racial/ethnic socioeconomic status. Because SES data was not available for all studies in the meta-analysis, we must outsource our estimates. This will yield strictly wrong predictions for adoption studies and studies that have been selective with respect to SES by race, and it will also be wrong for different years to the extent differences in socioeconomic status have changed over time. So, the following estimates are approximations with varying degrees of accuracy.
Warne (2021) provided estimates of Black-White SES differences from the National Educational Longitudinal Study of 1988, Early Childhood Longitudinal Study, Kindergarten Class of 1998-1999, Education Longitudinal Study of 2002, and High School Longitudinal Study of 2009. His weighted mean was 0.658. The values for the Hispanic sample take a range, because sources vary so wildly about the SES gaps between them and Whites in the U.S. and I did not have access to all of those datasets. For Hispanics, I assessed the power to detect the weak Scarr-Rowe effect with SES gaps of 0.4, 0.5, 0.6, 0.7, and 1 d, and for both groups, I assumed equal variances with Whites.
The Scarr-Rowe values were taken from Tucker-Drob & Bates (2016). All values were assumed invariant by race. First, the main U.S. value was used. Next, each of the moderation model estimates for the U.S. was used. Since the samples used in this study generally matched on age and measures, those effects are more likely to be realistic. Because “Other” is not a real group, analyses were not conducted with respect to it.
#Black
BSES <- 0.658
B1 <- 0.074 * BSES
B2 <- 0.076 * BSES
B3 <- 0.078 * BSES
B4 <- 0.123 * BSES
B5 <- 0.092 * BSES
B6 <- 0.085 * BSES
B7 <- 0.060 * BSES
#Hispanic
H41 <- 0.074 * 0.400
H42 <- 0.076 * 0.400
H43 <- 0.078 * 0.400
H44 <- 0.123 * 0.400
H45 <- 0.092 * 0.400
H46 <- 0.085 * 0.400
H47 <- 0.060 * 0.400
H51 <- 0.074 * 0.500
H52 <- 0.076 * 0.500
H53 <- 0.078 * 0.500
H54 <- 0.123 * 0.500
H55 <- 0.092 * 0.500
H56 <- 0.085 * 0.500
H57 <- 0.060 * 0.500
H61 <- 0.074 * 0.600
H62 <- 0.076 * 0.600
H63 <- 0.078 * 0.600
H64 <- 0.123 * 0.600
H65 <- 0.092 * 0.600
H66 <- 0.085 * 0.600
H67 <- 0.060 * 0.600
H71 <- 0.074 * 0.700
H72 <- 0.076 * 0.700
H73 <- 0.078 * 0.700
H74 <- 0.123 * 0.700
H75 <- 0.092 * 0.700
H76 <- 0.085 * 0.700
H77 <- 0.060 * 0.700
HF1 <- 0.074 * 1
HF2 <- 0.076 * 1
HF3 <- 0.078 * 1
HF4 <- 0.123 * 1
HF5 <- 0.092 * 1
HF6 <- 0.085 * 1
HF7 <- 0.060 * 1#Black
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B1)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8338411
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.7935651
## Random-Effects Model (i2 = 40.2%): 0.5749366subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B2)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8320372
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.8138824
## Random-Effects Model (i2 = 40.2%): 0.5970576subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B3)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8302351
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.8328924
## Random-Effects Model (i2 = 40.2%): 0.6188505subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B4)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7901434
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.99531
## Random-Effects Model (i2 = 40.2%): 0.9411668subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B5)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8176719
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9301806
## Random-Effects Model (i2 = 40.2%): 0.7574015subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B6)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8239426
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.8891165
## Random-Effects Model (i2 = 40.2%): 0.691658subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B7)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8465226
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.6188969
## Random-Effects Model (i2 = 40.2%): 0.4168014The range of power to detect the weak Scarr-Rowe effect was 0.42 to 0.94.
#Hispanic
subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF1)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7994594
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.55359
## Random-Effects Model (i2 = 45.4%): 0.3403229subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF2)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7967681
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.5752076
## Random-Effects Model (i2 = 45.4%): 0.355284subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF3)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7940803
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.5965751
## Random-Effects Model (i2 = 45.4%): 0.3704576subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF4)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7343745
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9282593
## Random-Effects Model (i2 = 45.4%): 0.7154233subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF5)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7753565
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.7345331
## Random-Effects Model (i2 = 45.4%): 0.4806233subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF6)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.7846991
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.6685542
## Random-Effects Model (i2 = 45.4%): 0.4249021subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - HF7)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8184005
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.4006882
## Random-Effects Model (i2 = 45.4%): 0.2432776All Hispanic group results were not presented, but can easily be run using the above code and the provided values. Estimates ran from 0.08 to 0.72.
#Black
BV <- subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - B7)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r"); BV##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8465226
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.6188969
## Random-Effects Model (i2 = 40.2%): 0.4168014tanh(BV$effect_sizes[1])^2-tanh(BV$effect_sizes[2])^2## [1] 0.03948subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[3], sqrt(GAMeta$TE.random.w[1]^2 - B7)),
study_size = 448,
k = 15,
i2 = 0.402,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.93873 0.8465226
## Expected Study Size: 448
## Expected Number of Studies: 15
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.963595
## Random-Effects Model (i2 = 40.2%): 0.8271625#Hispanic
HV <- subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[1], sqrt(GAMeta$TE.random.w[1]^2 - H47)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r"); HV##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 0.9020958 0.8680618
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.1068371
## Random-Effects Model (i2 = 45.4%): 0.08068195tanh(HV$effect_sizes[1])^2-tanh(HV$effect_sizes[2])^2## [1] 0.024subgroup_power(n_groups = 2,
effect_sizes = c(GAMeta$TE.random.w[2], sqrt(GAMeta$TE.random.w[1]^2 - H47)),
study_size = 244,
k = 7,
i2 = 0.454,
es_type = "r")##
## Power Analysis for Subgroup analysis:
##
## Effect Size Metric: r
## Number of Subgroups: 2
## Groups:
## Expected Effect Sizes: 1.065269 0.8680618
## Expected Study Size: 244
## Expected Number of Studies: 7
##
## Esimated Power to detect subgroup differences
##
## Fixed-Effects Model: 0.9805245
## Random-Effects Model (i2 = 45.4%): 0.8446939With observed heritabilities, there were post hoc power levels of 0.83 and 0.84 for the Black and Hispanic comparisons, respectively.
This first robustness check involves setting the SEs to the maximum SE and rerunning the analysis.
AMeta <- metagen(data = data,
TE = SA,
seTE = HighSE,
studlab = Study,
sm = "COR",
fixed = F,
random = T,
method.tau = "HS",
hakn = T,
backtransf = F,
title = "The Heritability of Cognitive Ability in the U.S."); summary(AMeta)## Review: The Heritability of Cognitive Ability in the U.S.
##
## COR 95%-CI %W(random)
## Scarr-Salapatek (1971) 0.5300 [-1.2732; 2.3332] 2.4
## Beaver et al. (2013) 0.6900 [-1.1132; 2.4932] 2.4
## Hodges (1976) 0.5800 [-1.2232; 2.3832] 2.4
## Osborne (1980) 0.7300 [-1.0732; 2.5332] 2.4
## Scarr (1981) 0.7300 [-1.0732; 2.5332] 2.4
## Scarr et al. (1993) 0.6400 [-1.1632; 2.4432] 2.4
## Beaver et al. (2013) 0.7200 [-1.0832; 2.5232] 2.4
## Rhemtulla & Tucker-Drob (2012) 0.6100 [-1.1932; 2.4132] 2.4
## Hart et al. (2013) 0.8900 [-0.9132; 2.6932] 2.4
## Woodley of Menie et al. (2015) 0.3700 [-1.4332; 2.1732] 2.4
## Figlio et al. (2017) 0.7500 [-1.0532; 2.5532] 2.4
## Figlio et al. (2017) 0.7700 [-1.0332; 2.5732] 2.4
## Mollon et al. (2018) 0.8500 [-0.9532; 2.6532] 2.4
## Engelhardt et al. (2019) 0.6800 [-1.1232; 2.4832] 2.4
## Pesta et al. (2019) 0.8200 [-0.9832; 2.6232] 2.4
## Pesta et al. (2019) 0.7800 [-1.0232; 2.5832] 2.4
## Hodges (1976) 0.6100 [-1.1932; 2.4132] 2.4
## Hart et al. (2013) 0.7100 [-1.0932; 2.5132] 2.4
## Figlio et al. (2017) 0.8500 [-0.9532; 2.6532] 2.4
## Figlio et al. (2017) 0.8500 [-0.9532; 2.6532] 2.4
## Engelhardt et al. (2019) 0.8100 [-0.9932; 2.6132] 2.4
## Pesta et al. (2019) 0.8900 [-0.9132; 2.6932] 2.4
## Pesta et al. (2019) 0.4900 [-1.3132; 2.2932] 2.4
## Scarr-Salapatek (1971) 0.5600 [-1.2432; 2.3632] 2.4
## Beaver et al. (2013) 0.7100 [-1.0932; 2.5132] 2.4
## Hodges (1976) 0.4500 [-1.3532; 2.2532] 2.4
## Osborne (1980) 0.7700 [-1.0332; 2.5732] 2.4
## Scarr (1981) 0.6900 [-1.1132; 2.4932] 2.4
## Scarr et al. (1993) 0.7200 [-1.0832; 2.5232] 2.4
## Beaver et al. (2013) 0.6700 [-1.1332; 2.4732] 2.4
## Hart et al. (2013) 0.9400 [-0.8632; 2.7432] 2.4
## Woodley of Menie et al. (2015) 0.9700 [-0.8332; 2.7732] 2.4
## Figlio et al. (2017) 0.7500 [-1.0532; 2.5532] 2.4
## Figlio et al. (2017) 0.6900 [-1.1132; 2.4932] 2.4
## Mollon et al. (2018) 0.7800 [-1.0232; 2.5832] 2.4
## Engelhardt et al. (2019) 0.3600 [-1.4432; 2.1632] 2.4
## Pesta et al. (2019) 0.9000 [-0.9032; 2.7032] 2.4
## Pesta et al. (2019) 0.7900 [-1.0132; 2.5932] 2.4
## Hart et al. (2013) 0.4900 [-1.3132; 2.2932] 2.4
## Engelhardt et al. (2019) 0.6200 [-1.1832; 2.4232] 2.4
## Engelhardt et al. (2019) 0.6600 [-1.1432; 2.4632] 2.4
## Rhemtulla & Tucker-Drob (2012) 0.5900 [-1.2132; 2.3932] 2.4
##
## Number of studies combined: k = 42
##
## COR 95%-CI t p-value
## Random effects model 0.7014 [0.6564; 0.7464] 31.48 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0; tau = 0; I^2 = 0.0% [0.0%; 35.5%]; H = 1.00 [1.00; 1.24]
##
## Test of heterogeneity:
## Q d.f. p-value
## 1.01 41 1.0000
##
## Details on meta-analytical method:
## - Inverse variance method
## - Hunter-Schmidt estimator for tau^2
## - Hartung-Knapp adjustment for random effects model
## - Untransformed correlationsGAMeta <- update.meta(AMeta,
subgroup = RaceRec,
tau.common = F); GAMeta## Review: The Heritability of Cognitive Ability in the U.S.
##
## Number of studies combined: k = 42
##
## COR 95%-CI t p-value
## Random effects model 0.7014 [0.6564; 0.7464] 31.48 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0; tau = 0; I^2 = 0.0% [0.0%; 35.5%]; H = 1.00 [1.00; 1.24]
##
## Test of heterogeneity:
## Q d.f. p-value
## 1.01 41 1.0000
##
## Results for subgroups (random effects model):
## k COR 95%-CI tau^2 tau Q I^2
## RaceRec = White 16 0.6962 [0.6270; 0.7655] 0 0 0.30 0.0%
## RaceRec = Hispanic 7 0.7443 [0.6072; 0.8813] 0 0 0.16 0.0%
## RaceRec = Black 15 0.7167 [0.6245; 0.8088] 0 0 0.46 0.0%
## RaceRec = Other 4 0.5900 [0.4745; 0.7055] 0 0 0.02 0.0%
##
## Test for subgroup differences (random effects model):
## Q d.f. p-value
## Between groups 8.39 3 0.0387
##
## Details on meta-analytical method:
## - Inverse variance method
## - Hunter-Schmidt estimator for tau^2
## - Hartung-Knapp adjustment for random effects model
## - Untransformed correlationsGAMeta$TE.random.w[1]^2 - GAMeta$TE.random.w[3]^2 #B - W Difference## [1] -0.02884705GAMeta$TE.random.w[1]^2 - GAMeta$TE.random.w[2]^2 #H - W Difference## [1] -0.06919716GAMeta$TE.random.w[1]^2 - GAMeta$TE.random.w[4]^2 #O - W Difference## [1] 0.1366641This second robustness check involves assessing whether Giangrande & Turkheimer’s implication that a given study was dubious actually affected the final result. Dubious labels were given to studies whose faults they specifically mentioned, excluding Mollon et al. (2021), because the issue they raised was spurious, as their issue was with the content of the tests they used, which nonetheless tap g. All “Other” group studies were considered problematic because of issues with the definition of the goruping variable, so obviously a meta-regression for them alone could not run.
summary(rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
subset = (RaceRec == "White"),
mods =~ Problematic))##
## Mixed-Effects Model (k = 16; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## 0.4413 2.1890 5.1174 7.4352 7.1174
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0367)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 14) = 2.1890, p-val = 0.9999
##
## Test of Moderators (coefficient 2):
## F(df1 = 1, df2 = 14) = 0.0239, p-val = 0.8793
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.7118 0.0487 14.6201 14 <.0001 0.6074 0.8162 ***
## Problematic 0.0102 0.0661 0.1546 14 0.8793 -0.1316 0.1521
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
subset = (RaceRec == "Black"),
mods =~ Problematic))##
## Mixed-Effects Model (k = 15; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## -2.6614 1.2288 11.3228 13.4470 13.5046
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0599)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 13) = 1.2288, p-val = 1.0000
##
## Test of Moderators (coefficient 2):
## F(df1 = 1, df2 = 13) = 4.1907, p-val = 0.0614
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.6503 0.0530 12.2680 13 <.0001 0.5358 0.7648 ***
## Problematic 0.1396 0.0682 2.0471 13 0.0614 -0.0077 0.2868 .
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
subset = (RaceRec == "Hispanic"),
mods =~ Problematic))##
## Mixed-Effects Model (k = 7; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## -0.1980 0.4838 6.3961 6.2338 14.3961
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0637)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 5) = 0.4838, p-val = 0.9927
##
## Test of Moderators (coefficient 2):
## F(df1 = 1, df2 = 5) = 0.9758, p-val = 0.3686
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.7100 0.0902 7.8732 5 0.0005 0.4782 0.9418 ***
## Problematic 0.1022 0.1034 0.9878 5 0.3686 -0.1637 0.3680
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Unproblematic <- rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
mods =~ RaceRec)
AProblem <- rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
mods =~ RaceRec + Problematic); summary(AProblem)##
## Mixed-Effects Model (k = 42; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## -4.0235 4.1718 20.0470 30.4730 22.4470
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0294)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 37) = 4.1718, p-val = 1.0000
##
## Test of Moderators (coefficients 2:5):
## F(df1 = 4, df2 = 37) = 1.6103, p-val = 0.1922
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.6963 0.0442 15.7362 37 <.0001 0.6067 0.7860 ***
## RaceRecHispanic 0.0432 0.0603 0.7154 37 0.4788 -0.0791 0.1654
## RaceRecOther -0.1722 0.0996 -1.7298 37 0.0920 -0.3739 0.0295 .
## RaceRecWhite -0.0134 0.0460 -0.2904 37 0.7732 -0.1065 0.0798
## Problematic 0.0634 0.0416 1.5233 37 0.1362 -0.0209 0.1477
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(Unproblematic, AProblem)##
## df AIC BIC AICc logLik LRT pval QE tau^2 R^2
## Full 6 20.0470 30.4730 22.4470 -4.0235 4.1718 0.0000
## Reduced 5 18.3087 26.9970 19.9753 -4.1543 0.2616 0.6090 4.4334 0.0000 0.0000%As a third robustness check, we can multiply the “problematic” variable by the number of complaints Giangrande & Turkheimer had about specific studies and then perform the meta-regression again with something a bit more continuous.
summary(rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
subset = (RaceRec == "White"),
mods =~ ProblematicCon))##
## Mixed-Effects Model (k = 16; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## 0.4403 2.1909 5.1193 7.4371 7.1193
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0367)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 14) = 2.1909, p-val = 0.9999
##
## Test of Moderators (coefficient 2):
## F(df1 = 1, df2 = 14) = 0.0119, p-val = 0.9149
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.7209 0.0463 15.5556 14 <.0001 0.6215 0.8203 ***
## ProblematicCon -0.0057 0.0522 -0.1089 14 0.9149 -0.1176 0.1062
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
subset = (RaceRec == "Black"),
mods =~ ProblematicCon))##
## Mixed-Effects Model (k = 15; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## -2.7837 1.4735 11.5675 13.6916 13.7493
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0599)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 13) = 1.4735, p-val = 1.0000
##
## Test of Moderators (coefficient 2):
## F(df1 = 1, df2 = 13) = 1.3365, p-val = 0.2684
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.6883 0.0542 12.6933 13 <.0001 0.5711 0.8054 ***
## ProblematicCon 0.0651 0.0563 1.1561 13 0.2684 -0.0565 0.1867
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
subset = (RaceRec == "Hispanic"),
mods =~ ProblematicCon))##
## Mixed-Effects Model (k = 7; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## -0.1980 0.4838 6.3961 6.2338 14.3961
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0637)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 5) = 0.4838, p-val = 0.9927
##
## Test of Moderators (coefficient 2):
## F(df1 = 1, df2 = 5) = 0.9758, p-val = 0.3686
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.7100 0.0902 7.8732 5 0.0005 0.4782 0.9418 ***
## ProblematicCon 0.1022 0.1034 0.9878 5 0.3686 -0.1637 0.3680
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1AProblem <- rma(data = data, yi = SA, sei = SESA, method = "HS", test = "knha",
mods =~ RaceRec + ProblematicCon); summary(AProblem)##
## Mixed-Effects Model (k = 42; tau^2 estimator: HS)
##
## logLik deviance AIC BIC AICc
## -4.1199 4.3646 20.2399 30.6659 22.6399
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0294)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 37) = 4.3646, p-val = 1.0000
##
## Test of Moderators (coefficients 2:5):
## F(df1 = 4, df2 = 37) = 1.1304, p-val = 0.3572
##
## Model Results:
##
## estimate se tval df pval ci.lb ci.ub
## intrcpt 0.7164 0.0443 16.1801 37 <.0001 0.6266 0.8061 ***
## RaceRecHispanic 0.0518 0.0614 0.8440 37 0.4041 -0.0725 0.1761
## RaceRecOther -0.1593 0.1017 -1.5667 37 0.1257 -0.3653 0.0467
## RaceRecWhite -0.0151 0.0471 -0.3201 37 0.7507 -0.1104 0.0803
## ProblematicCon 0.0257 0.0336 0.7637 37 0.4499 -0.0425 0.0938
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(Unproblematic, AProblem)##
## df AIC BIC AICc logLik LRT pval QE tau^2 R^2
## Full 6 20.2399 30.6659 22.6399 -4.1199 4.3646 0.0000
## Reduced 5 18.3087 26.9970 19.9753 -4.1543 0.0688 0.7931 4.4334 0.0000 0.0000%There are (at least) two notes here for Giangrande & Turkheimer. Moving forward, they need to always remember at least these two things. First, complaints about power must be specific. There is no objective sense in which a study has low or high power. Power is always with respect to certain cutoffs and effect sizes. If I say a study has low power, I have an obligation to say what I mean - I need to say if they have <80% power to detect d = 0.5 with \(\alpha\) = 0.01, or if they have <99% power to detect r = 0.2 with \(\alpha\) = 0.10. Power complaints can never be left in a vacuum because power does not exist in one. Secondly, complaints about quality, grouping, constraints, approximations, etc., are only useful in so far as they actually have an effect. Anyone can cast doubt on any study for a million arbitrary reasons. The question is not whether doubt can be sewn, it is whether those doubts can manifest into meaningful effects on the results. If they cannot, they are no better than screaming into space.
In Giangrande & Turkheimer’s paper, they alleged there was low power and a number of other methodological issues. It did not appear that their complaints about power were meaningful or that their complaints about methodological issues mattered. The qualitative style of nagging complaint that they embraced in their critique is not unique to them. Bouchard dubbed it “pseudo-analysis”; described in his own words (1984):
A principal feature of the many critiques of hereditarian research is an excessive concern for purity, both in terms of meeting every last assumption of the models being tested an in terms of eliminating all possible errors. The various assumptions and potential errors that may, or may not, be of concern are enumerated and discussed at great length. The longer the discussion of potential biasing factors, the more likely the critic is to conclude that they are actual sources of bias. By the time a chapter summary or conclusion section is reached, the critic asserts that it is impossible to learn anything using the design under discussion. There is often, however, a considerable amount known about the possible effect of the violation of assumptions. As my colleague Paul Meehl has observed, ‘Why these constraints are regularly treated as “assumptions” instead of refutable conjectures is itself a deep and fascinating question…’ (Meehl, 1978, p. 810). In addition, potential systematic errors sometimes have testable consequences that can be estimated. They are, unfortunately, seldom evaluated. In other instances the data themselves are simply abused. As I have pointed out elsewhere:
The data are subgrouped using a variety of criteria that, although plausible on their face, yield the smallest genetic estimates that can be squeezed out. Statistical significance tests are liberally applied and those favorable to the investigator’s prior position are emphasized. Lack of statistical significance is overlooked when it is convenient to do so, and multiple measurements of the same construct (constructive replication within a study) are ignored. There is repeated use of significance tests on data chosen post hoc. The sample sizes are often very small and t he problem of sampling error is entirely ignored. (Bouchard, 1982, p. 190)
This fallacious line of reasoning is so endemic that I have given it a name, ‘pseudo-analysis’ (Bouchard, 1982a, 1982b).
And later
Critiques of the MZA data often have a pseudo-scientific flavor. They sometimes assert that the correlations could be explained by factors about which we have little knowledge. Of course anything in the world can be explained by factors which we do not understand. These criticisms are also used in a contradictory manner. […] The argument that ignorance prevents us from gaining knowledge because we cannot conduct perfect experiments is fallacious. The reason that critics introduce such arguments is a problem belonging to the sociology of knowledge and has very little to do with the search for causes of individual differences in behavior. Imperfect evidence is the most common variety of evidence in science. It is used in conjunction with other imperfect evidence in an attempt to generate theoretical structures that make the world comprehensible on a theoretical rather than on an ad hoc or a priori basis.
And, when discussing the value of meta-analysis, he said
Models allow us to test the fit of data to theory. They do not, as some people appear to assume, allow us to evaluate the quality of the data. Peculiar results due to the application of a model to data will sometimes alert the investigator to problems in the data; this, however, is only an indirect and very fallible test of the quality of data. Hereditarians have regularly proceeded to analyze extensive bodies of data with little concern for the quality of the information they are analyzing. Consequently, critics of the hereditarian literature have had a field day. As my analyses of their treatment of the MZA data show, they have often carried out their critiques in an ad hoc and biased manner. One of the consequences of their critiques has been to discredit model-building per se as part of the scientific enterprise. In discussions with colleagues and students I have often found that the pseudo-analysis of studies is regarded as evidence that both the hereditarian position and model-building have been discredited.
In addition to carrying out a variety of pseudo-analyses of the evidence, critics of the hereditarian program have correctly pointed out the tremendous heterogeneity of the data that are often used by models. The models are generally fitted to medians or means without regard for the underlying variance. Nevertheless, rather than attempt to understand the sources of this variance, the critics simply assert that it invalidates model-fitting. Bouchard and McGue highlighted this problem of heterogeneity in their review of the world literature on familial resemblance in measured intelligence, and attempted to explore some of the sources of heterogeneity. They found that neither sex of familial pairing nor type of IQ test used moderated the heterogeneity, and warned that until the heterogeneity was better understood, models fitted to such data should be interpreted cautiously.
And finally, Bouchard outlined the “basic philosophy” that researchers needed to embrace
What Giangrande & Turkheimer did was to throw caution to the wind. They alleged a number of issues, but provided no reason to think those were in any way critical or even had any effect on the results they criticized. Moreover, they made claims about power without actually specifying what they meant, in effect leaving it open to the reader to determine just how underpowered the study could have been. Because their criticisms were necessarily qualitative rather than quantitative, it might be of no surprise that they amounted to nothing. The quality they needed to substantiate the errors they alleged was quantity, but they failed to provide it, and when it was assessed, it did not bear out what they inferred were issues.
In summation, reasonable degrees of difference in heritability were, in fact, detectable, even when we purposely reduced our expected power. The effect of supposedly biased studies was nonsignificant, but because of limited sample sizes, could not be investigated with more granularity. Careless criticism like Giangrande & Turkheimer’s ought to be bundled with the label “pseudo-analysis”, and this label ought to also be applied to their work in general, as they have shown themselves to be perfectly willing to publish incautious works of motivated reasoning and negative merit. Contrarily, there may be one good thing to have come from their careless and at times clearly (extremely) libelous critique: hopefully, there will be more studies of this topic. The most major issue with the Pesta et al. meta-analysis was the lack of data. If more data can be mustered, the questions that study asked can be better addressed. Specific forms of heterogeneity can be investigated, precision can be greatly increased, and a blossoming of investigation can be had. However, more likely, their criticisms will be unduly well-received and there will be a chilling effect on research on this topic. Fewer people will publish their summary statistics grouped by variables like sex and race, and people will avoid doing well-reported multigroup modeling with demographic variables. By poisoning the well and overstating their claims, they have harmed the state of knowledge in behavior genetics.
Warne, Russell T. 2021. “Between-Group Mean Differences in Intelligence in the United States Are >0% Genetically Caused: Five Converging Lines of Evidence.” The American Journal of Psychology 134(4):480–501.
Tucker-Drob, Elliot M., and Timothy C. Bates. 2016. “Large Cross-National Differences in Gene × Socioeconomic Status Interaction on Intelligence.” Psychological Science 27(2):138–49. doi: 10.1177/0956797615612727.
Bouchard, T. (1984). The Hereditarian Research Program: Triumphs and Tribulations In Modgil & C. Modgil (Eds.), Arthur Jensen: Consensus and Controversy. Lewes, Sussex, Falmer Press.
Meehl, Paul E. 1978. “Theoretical Risks and Tabular Asterisks: Sir Karl, Sir Ronald, and the Slow Progress of Soft Psychology.” Journal of Consulting and Clinical Psychology 46(4):806–34. doi: 10.1037/0022-006X.46.4.806.
Bouchard Thomas J. (1982). Identical Twins Reared Apart: Reanalysis or Pseudo-analysis? 27(3), 190-191. https://doi.org/10.1037/021001
Bouchard, Thomas J. 1982. Review of “The Intelligence Controversy.” The American Journal of Psychology 95(2):346. doi: 10.2307/1422481.
sessionInfo()## R version 4.1.2 (2021-11-01)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
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## [1] stats graphics grDevices utils datasets methods base
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## other attached packages:
## [1] forcats_0.5.1 stringr_1.4.0 dplyr_1.0.7 purrr_0.3.4
## [5] readr_2.0.2 tidyr_1.1.4 tibble_3.1.5 ggplot2_3.3.5
## [9] tidyverse_1.3.1 DT_0.20 sjmisc_2.8.7 metafor_3.0-2
## [13] Matrix_1.3-4 metapower_0.2.2 meta_5.1-1 pacman_0.5.1
## [17] haven_2.4.3
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## loaded via a namespace (and not attached):
## [1] httr_1.4.2 sass_0.4.0 jsonlite_1.7.2 splines_4.1.2
## [5] modelr_0.1.8 bslib_0.3.1 assertthat_0.2.1 cellranger_1.1.0
## [9] yaml_2.2.1 pillar_1.6.4 backports_1.3.0 lattice_0.20-45
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## [25] generics_0.1.1 sjlabelled_1.1.8 ellipsis_0.3.2 withr_2.4.3
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## [33] evaluate_0.14 fs_1.5.2 fansi_0.5.0 nlme_3.1-153
## [37] MASS_7.3-54 xml2_1.3.3 tools_4.1.2 hms_1.1.1
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