Polygenic Score Attenuation by Assortative Mating
Setup
library(pacman); p_load(ggplot2, DT, meta, metafor, dmetar, DescTools, tools, metasens)Rationale
The predictive validity of polygenic scores within the general population is reduced within families for a multitude of reasons. One of these reasons is assortative mating. Assortative mating strikes against within-family prediction because it results in positive correlations between genetic variants such that the effect of one also contains signal from others. Similarly, assortative mating leads to overrated shared environmental effects in classical behavior genetic designs because it increases trait-relevant genetic similarity in a regime of additive effects. Lee et al. (2018) described this phenomenon with respect to PGS betas in their supplement:
The assumption \(\beta_j = \beta_{WF,j}\) could be violated for a number of reasons, one of which is assortative mating with respect to genotypes at causal loci. Such assortative mating has been documented in analyses of genome-wide SNP data on spousal pairs and is substantial at both the genetic and phenotypic level. One recent paper reports a spousal phenotypic correlation of 0.41 for educational attainment in the UK Biobank. In section 2.10 below, we show that in a stylized model of phenotypic assortment, it is possible to calculate analytically the attenuation of within-family coefficients relative to GWAS coefficients expected due to assortative mating. We show that for a trait with a large number of causal loci, within-family coefficients are deflated by a factor of \([1 - rh^2_0]\), where r is the spousal phenotypic correlation and \(h^2_0\) is the SNP heritability in a hypothetical base population without assortative mating. Accordingly, in several of our analyses below, we compare GWAS effect estimates to assortative-mating-adjusted within-family effect estimates. Specifically, we calculate the theoretical benchmark assuming that \(\beta_{WF,j}/\beta_j = R\) and Var(\(s_j\)) = 0, where R is an assortative-mating adjustment parameter. In those analyses, we set R = 0.83, a ballpark estimate based on published estimates of the relevant parameters for EduYears.
Thus,
\[AM Attenuation = [1 - rh^2_0]\]
To get at realistic values for the part of the reduction in within-family coefficients for intelligence, lets just run meta-analyses for phenotypic spousal correlations for IQ. This is not necessary to do with \(h^2_0\) because we can just impute various values to check the effects. Moreover, it is difficult because those values must be adjusted for assortative mating (the typical way for behavior genetic models is appropriate), but it is unnecessary, as it is better to edify about a variety of scenarios, from low to high heritability. Because SNP heritability is lower than the true heritability ascertained with twins for reasons to do with sample size, measurement issues that are common when they are used, array size, and much else, I will utilize the values 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6.
Meta-analysis of Spousal IQ Correlations
This meta-analysis makes no claims to comprehensiveness, although an effort was made to be as comprehensive as possible. All data used is available for download below. In each case, the best available measure of cognitive performance was used. In this way, I weighted g factor scores above FSIQs, and FSIQs above individual scale scores, but when only scale scores were available, I used the most reliable/g-loaded of those, although there is still little clarity about group factor impacts on assortative mating. Unadjusted correlations are used when available. the quality rating denotes the quality of the data provided: if only one terrible test was involved, the quality was poor; if the value presented was from a well-recognized test but the sample was modest in size, it was given a moderate quality rating; and if the test was very good and the sample size good enough, it got a good rating. A large sample size would not make up for a poor-quality test, unfortunately. The quality ratings were assigned based on personal judgments that I may formalize more later if I ever decide to take this further.
Here’s the data. It is the most comprehensive meta-analysis of assortative mating for IQ to date. Half of the excluded datapoints (Jones, 1928; Watkins & Meredith, 1981; Wadsworth et al., 2002) were not included because they repeated results from the same datasets, which were fuller or had better measures with another datapoint. Bouchard & McGue (1981) was excluded because it was a review datapoint and the underlying studies were uncertain. Guttman (1970) was excluded because the tests were odd and the data was difficult to find; specifically, the reason I excluded it was the same as why I excluded Burt & Howard (1956): I could not find the number of couple pairs.
Data here.
datatable(data, extensions = c("Buttons", "FixedColumns"), options = list(dom = 'Bfrtip', buttons = c('copy', 'csv', 'print'), scrollX = T, fixedColumns = list(leftColumns = 3)))Aggregate meta-analysis here.
The first method of dealing with each study’s variance involves weighting by \(\sqrt{N}\), the second with assuming equal variances for men and women, and the third with going based off of reviews of large datasets (i.e., \(VR_0\) = 1.16, a stylized fact about the variance ratio for IQ by sex; see https://rpubs.com/JLLJ/SDVR). This sort of approximation had to be daone because of the paucity of requisite summary stats in much of the literature. A fourth strategy of mixing and imputing is certainly possible and might marginally increase accuracy. Ignoring that fourth option for ease, for this meta-analysis, the second is almost certainly the most defensible, for a few reasons. The first is that greater male variability is modest, at 16% greater variance when there are no score gaps on tests; combined with the observation that these samples are small, and certainly much smaller than the N > 1000 requirement needed to reliably detect a difference of that size, we arrive at the conclusion that samples will more often be statistically nonsignificantly different. It is true that nonsignificant differences can produce significant subsequent differences, but that is also less likely than if they significantly differed regardless. Moreover, because couples are correlated, the variance of their distributions ought to be shifted towards one another accordingly. To confirm this, compare the derived SEs based on equal variances and stylized variances to the provided SEs from the UK Biobank and Generation Scotland in Okbay et al.’s (2022) supplementary table 14. They found SEs of 0.067 and 0.024, respectively. The assumed equal variance SEs were 0.067 and 0.025, while the stylized SEs were 0.073 and 0.027. These are almost certainly inconsequential differences, so the equal variance SEs will be used for conclusions. All results are presented to make the lack of consequence evident and to showcase robustness.
data$SE_Equal <- (1)/(sqrt(data$NPairs - 2)/sqrt(1 - data$R^2))
data$SE_Stylized <- (sqrt(1.16)/1)/(sqrt(data$NPairs - 2)/sqrt(1 - data$R^2))MetaSqrtN <- metacor(R,
sqrt(NPairs),
data = data,
studlab = StudYear,
sm = "ZCOR",
method.tau = "SJ",
hakn = T,
backtrans = T,
title = "Spousal IQ Correlations with Square-root N Weighting"); summary(MetaSqrtN)## Review: Spousal IQ Correlations with Square-root N Weighting
##
## COR 95%-CI %W(common)
## Pan & Wang 2011 0.2809 [-0.4627; 0.7925] 1.0
## Dufouil et al. 2000 0.3600 [-0.1313; 0.7093] 2.3
## Schooley 1936 0.5560 [-0.1750; 0.8919] 0.9
## Conrad & Jones 1940 0.5200 [-0.0910; 0.8466] 1.4
## Outhit 1933 0.7400 [-0.0126; 0.9574] 0.7
## Willoughby 1928 0.4400 [-0.2888; 0.8460] 1.0
## Carter 1932 0.2100 [-0.4682; 0.7325] 1.2
## Burt 1972 0.6780 [ 0.0707; 0.9186] 1.1
## Burks 1928 A 0.4200 [-0.1648; 0.7863] 1.6
## Burks 1928 B 0.5500 [-0.1218; 0.8762] 1.1
## Leahy 1935 0.5700 [ 0.0335; 0.8515] 1.6
## Swagerman et al. 2015 0.3800 [-0.2777; 0.7952] 1.3
## Leahy 1935 0.4100 [-0.1880; 0.7862] 1.6
## Freeman et al. 1928 0.4900 [-0.1080; 0.8276] 1.5
## Burt 1972 0.3790 [-0.3414; 0.8189] 1.1
## Reed & Reed 1965 0.3260 [ 0.0292; 0.5700] 6.4
## Halprin 1946 0.7600 [ 0.4543; 0.9056] 2.4
## Spuhler 1962 0.3990 [-0.1827; 0.7738] 1.6
## Guttman 1974 0.2600 [-0.4420; 0.7645] 1.1
## Spuhler 1962 0.5185 [-0.0687; 0.8389] 1.5
## Williams 1973 0.6900 [-0.0708; 0.9433] 0.7
## Stanley 1972 0.1500 [-0.8755; 0.9300] 0.3
## Smith 1941 0.1900 [-0.2656; 0.5762] 2.8
## Higgins et al. 1962 0.3300 [-0.0219; 0.6092] 4.6
## Penrose 1933 0.4400 [-0.2623; 0.8376] 1.1
## Cattell & Nesselroade 1967 0.3100 [-0.3927; 0.7842] 1.1
## Terman & Oden 1959 0.4810 [ 0.1174; 0.7309] 3.7
## Escorial & Martin-Buro 2012 0.4050 [-0.1215; 0.7536] 2.0
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 0.4920 [ 0.0407; 0.7766] 2.5
## Gruber-Baldini, Schaie & Willis 1995 0.2500 [-0.3706; 0.7163] 1.5
## Phillips et al. 1988 0.2000 [-0.2902; 0.6070] 2.4
## Vinkhuyzen et al. 2012 0.3700 [-0.1858; 0.7464] 1.8
## van Bergen et al. 2015 0.3600 [-0.2072; 0.7461] 1.8
## DeFries et al. 1979 0.2300 [-0.1455; 0.5476] 4.2
## Benbow, Zonderman & Stanley 1983 0.4400 [-0.5886; 0.9246] 0.5
## Defries et al. 1979 0.1500 [-0.3463; 0.5807] 2.3
## Bouchard & McGee 1977 0.0600 [-0.5322; 0.6128] 1.4
## Bock & Kolakowski 1973 0.2600 [-0.3418; 0.7106] 1.6
## Stafford 1961 0.0300 [-0.6128; 0.6489] 1.1
## Corah 1965 0.1500 [-0.6343; 0.7821] 0.8
## Loehlin, Sharan & Jacoby 1978 0.3200 [-0.2573; 0.7290] 1.7
## Scarr & Weinberg 1978 A 0.3000 [-0.3230; 0.7416] 1.5
## Scarr & Weinberg 1978 B 0.2000 [-0.3399; 0.6407] 2.0
## Lewak, Wakefield & Briggs 1985 0.3500 [-0.4092; 0.8229] 0.9
## Mascie-Taylor 1989 A 0.4030 [-0.1651; 0.7703] 1.7
## Mascie-Taylor 1989 B 0.3720 [-0.2485; 0.7760] 1.5
## Price & Vandenberg 1980 A 0.3400 [-0.1770; 0.7099] 2.1
## Price & Vandenberg 1980 B 0.3600 [-0.2843; 0.7803] 1.4
## Smith 1941 0.1930 [-0.2627; 0.5783] 2.8
## Woods 1906 0.0800 [-0.4483; 0.5668] 1.9
## Moorees 1924 0.1000 [-0.7247; 0.8069] 0.6
## Okbay et al. 2022 A 0.1390 [-0.4066; 0.6115] 1.9
## Okbay et al. 2022 B 0.2420 [-0.0788; 0.5174] 5.7
## Keller et al. 2013 0.3500 [-0.0237; 0.6379] 4.0
## %W(random)
## Pan & Wang 2011 1.1
## Dufouil et al. 2000 2.4
## Schooley 1936 1.0
## Conrad & Jones 1940 1.5
## Outhit 1933 0.7
## Willoughby 1928 1.1
## Carter 1932 1.3
## Burt 1972 1.2
## Burks 1928 A 1.7
## Burks 1928 B 1.2
## Leahy 1935 1.7
## Swagerman et al. 2015 1.4
## Leahy 1935 1.7
## Freeman et al. 1928 1.6
## Burt 1972 1.2
## Reed & Reed 1965 5.1
## Halprin 1946 2.4
## Spuhler 1962 1.7
## Guttman 1974 1.2
## Spuhler 1962 1.6
## Williams 1973 0.8
## Stanley 1972 0.3
## Smith 1941 2.8
## Higgins et al. 1962 4.0
## Penrose 1933 1.2
## Cattell & Nesselroade 1967 1.2
## Terman & Oden 1959 3.4
## Escorial & Martin-Buro 2012 2.1
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 2.5
## Gruber-Baldini, Schaie & Willis 1995 1.6
## Phillips et al. 1988 2.4
## Vinkhuyzen et al. 2012 1.9
## van Bergen et al. 2015 1.9
## DeFries et al. 1979 3.8
## Benbow, Zonderman & Stanley 1983 0.5
## Defries et al. 1979 2.3
## Bouchard & McGee 1977 1.5
## Bock & Kolakowski 1973 1.7
## Stafford 1961 1.2
## Corah 1965 0.8
## Loehlin, Sharan & Jacoby 1978 1.8
## Scarr & Weinberg 1978 A 1.6
## Scarr & Weinberg 1978 B 2.0
## Lewak, Wakefield & Briggs 1985 1.1
## Mascie-Taylor 1989 A 1.8
## Mascie-Taylor 1989 B 1.6
## Price & Vandenberg 1980 A 2.2
## Price & Vandenberg 1980 B 1.5
## Smith 1941 2.8
## Woods 1906 2.0
## Moorees 1924 0.7
## Okbay et al. 2022 A 1.9
## Okbay et al. 2022 B 4.7
## Keller et al. 2013 3.7
##
## Number of studies combined: k = 54
## Number of observations: o = 793.5918
##
## COR 95%-CI z|t p-value
## Common effect model 0.3492 [0.2789; 0.4157] 9.16 < 0.0001
## Random effects model 0.3520 [0.3053; 0.3970] 14.09 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0.0121; tau = 0.1101; I^2 = 0.0% [0.0%; 31.9%]; H = 1.00 [1.00; 1.21]
##
## Test of heterogeneity:
## Q d.f. p-value
## 21.88 53 1.0000
##
## Details on meta-analytical method:
## - Inverse variance method
## - Sidik-Jonkman estimator for tau^2
## - Hartung-Knapp adjustment for random effects model
## - Fisher's z transformation of correlationsMetaEqual <- metagen(data = data,
TE = R,
seTE = SE_Equal,
studlab = StudYear,
sm = "ZCOR",
fixed = T,
random = T,
method.tau = "SJ",
hakn = T,
backtransf = T,
title = "Spousal IQ Correlations with Equal Variances"); summary(MetaEqual)## Review: Spousal IQ Correlations with Equal Variances
##
## COR 95%-CI %W(common)
## Pan & Wang 2011 0.2737 [ 0.0730; 0.4531] 0.5
## Dufouil et al. 2000 0.3452 [ 0.2516; 0.4324] 2.1
## Schooley 1936 0.5050 [ 0.3553; 0.6294] 0.7
## Conrad & Jones 1940 0.4777 [ 0.3582; 0.5818] 1.1
## Outhit 1933 0.6291 [ 0.5018; 0.7298] 0.6
## Willoughby 1928 0.4136 [ 0.2472; 0.5564] 0.6
## Carter 1932 0.2070 [ 0.0239; 0.3766] 0.6
## Burt 1972 0.5902 [ 0.4843; 0.6791] 1.0
## Burks 1928 A 0.3969 [ 0.2769; 0.5047] 1.2
## Burks 1928 B 0.5005 [ 0.3667; 0.6140] 0.8
## Leahy 1935 0.5154 [ 0.4196; 0.5998] 1.5
## Swagerman et al. 2015 0.3627 [ 0.2132; 0.4956] 0.8
## Leahy 1935 0.3885 [ 0.2632; 0.5009] 1.1
## Freeman et al. 1928 0.4542 [ 0.3360; 0.5584] 1.1
## Burt 1972 0.3618 [ 0.1886; 0.5132] 0.6
## Reed & Reed 1965 0.3149 [ 0.2758; 0.3530] 12.1
## Halprin 1946 0.6411 [ 0.5973; 0.6810] 4.4
## Spuhler 1962 0.3791 [ 0.2583; 0.4882] 1.2
## Guttman 1974 0.2543 [ 0.0687; 0.4229] 0.6
## Spuhler 1962 0.4765 [ 0.3638; 0.5756] 1.2
## Williams 1973 0.5980 [ 0.4611; 0.7071] 0.6
## Stanley 1972 0.1489 [-0.2760; 0.5251] 0.1
## Smith 1941 0.1877 [ 0.0970; 0.2754] 2.6
## Higgins et al. 1962 0.3185 [ 0.2654; 0.3697] 6.6
## Penrose 1933 0.4136 [ 0.2564; 0.5496] 0.7
## Cattell & Nesselroade 1967 0.3004 [ 0.1230; 0.4592] 0.6
## Terman & Oden 1959 0.4470 [ 0.3931; 0.4979] 5.2
## Escorial & Martin-Buro 2012 0.3842 [ 0.2820; 0.4779] 1.7
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 0.4558 [ 0.3795; 0.5260] 2.6
## Gruber-Baldini, Schaie & Willis 1995 0.2449 [ 0.0937; 0.3851] 0.9
## Phillips et al. 1988 0.1974 [ 0.0943; 0.2962] 2.0
## Vinkhuyzen et al. 2012 0.3540 [ 0.2396; 0.4587] 1.4
## van Bergen et al. 2015 0.3452 [ 0.2261; 0.4542] 1.3
## DeFries et al. 1979 0.2260 [ 0.1638; 0.2865] 5.3
## Benbow, Zonderman & Stanley 1983 0.4136 [ 0.1328; 0.6330] 0.2
## Defries et al. 1979 0.1489 [ 0.0397; 0.2545] 1.8
## Bouchard & McGee 1977 0.0599 [-0.1038; 0.2205] 0.8
## Bock & Kolakowski 1973 0.2543 [ 0.1122; 0.3862] 1.0
## Stafford 1961 0.0300 [-0.1673; 0.2250] 0.6
## Corah 1965 0.1489 [-0.1041; 0.3837] 0.3
## Loehlin, Sharan & Jacoby 1978 0.3095 [ 0.1832; 0.4258] 1.2
## Scarr & Weinberg 1978 A 0.2913 [ 0.1453; 0.4249] 0.9
## Scarr & Weinberg 1978 B 0.1974 [ 0.0746; 0.3143] 1.4
## Lewak, Wakefield & Briggs 1985 0.3364 [ 0.1425; 0.5054] 0.5
## Mascie-Taylor 1989 A 0.3825 [ 0.2666; 0.4875] 1.3
## Mascie-Taylor 1989 B 0.3557 [ 0.2189; 0.4789] 1.0
## Price & Vandenberg 1980 A 0.3275 [ 0.2242; 0.4235] 1.8
## Price & Vandenberg 1980 B 0.3452 [ 0.1982; 0.4770] 0.9
## Smith 1941 0.1906 [ 0.1000; 0.2781] 2.6
## Woods 1906 0.0798 [-0.0496; 0.2067] 1.3
## Moorees 1924 0.0997 [-0.1949; 0.3777] 0.3
## Okbay et al. 2022 A 0.1381 [ 0.0069; 0.2646] 1.3
## Okbay et al. 2022 B 0.2374 [ 0.1911; 0.2827] 9.5
## Keller et al. 2013 0.3364 [ 0.2777; 0.3925] 5.3
## %W(random)
## Pan & Wang 2011 1.6
## Dufouil et al. 2000 2.1
## Schooley 1936 1.7
## Conrad & Jones 1940 1.9
## Outhit 1933 1.7
## Willoughby 1928 1.7
## Carter 1932 1.7
## Burt 1972 1.9
## Burks 1928 A 1.9
## Burks 1928 B 1.8
## Leahy 1935 2.0
## Swagerman et al. 2015 1.8
## Leahy 1935 1.9
## Freeman et al. 1928 1.9
## Burt 1972 1.7
## Reed & Reed 1965 2.3
## Halprin 1946 2.2
## Spuhler 1962 1.9
## Guttman 1974 1.6
## Spuhler 1962 1.9
## Williams 1973 1.6
## Stanley 1972 0.7
## Smith 1941 2.1
## Higgins et al. 1962 2.2
## Penrose 1933 1.7
## Cattell & Nesselroade 1967 1.7
## Terman & Oden 1959 2.2
## Escorial & Martin-Buro 2012 2.0
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 2.1
## Gruber-Baldini, Schaie & Willis 1995 1.8
## Phillips et al. 1988 2.1
## Vinkhuyzen et al. 2012 2.0
## van Bergen et al. 2015 2.0
## DeFries et al. 1979 2.2
## Benbow, Zonderman & Stanley 1983 1.1
## Defries et al. 1979 2.0
## Bouchard & McGee 1977 1.8
## Bock & Kolakowski 1973 1.9
## Stafford 1961 1.6
## Corah 1965 1.3
## Loehlin, Sharan & Jacoby 1978 1.9
## Scarr & Weinberg 1978 A 1.8
## Scarr & Weinberg 1978 B 2.0
## Lewak, Wakefield & Briggs 1985 1.6
## Mascie-Taylor 1989 A 2.0
## Mascie-Taylor 1989 B 1.9
## Price & Vandenberg 1980 A 2.0
## Price & Vandenberg 1980 B 1.8
## Smith 1941 2.1
## Woods 1906 2.0
## Moorees 1924 1.2
## Okbay et al. 2022 A 1.9
## Okbay et al. 2022 B 2.3
## Keller et al. 2013 2.2
##
## Number of studies combined: k = 54
##
## COR 95%-CI z|t p-value
## Common effect model 0.3362 [0.3229; 0.3494] 45.89 0
## Random effects model 0.3409 [0.3004; 0.3802] 15.77 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0.0226 [0.0135; 0.0359]; tau = 0.1503 [0.1163; 0.1895]
## I^2 = 86.3% [82.9%; 89.0%]; H = 2.70 [2.42; 3.02]
##
## Test of heterogeneity:
## Q d.f. p-value
## 387.54 53 < 0.0001
##
## Details on meta-analytical method:
## - Inverse variance method
## - Sidik-Jonkman estimator for tau^2
## - Q-profile method for confidence interval of tau^2 and tau
## - Hartung-Knapp adjustment for random effects model
## - Fisher's z transformation of correlationsMetaStylized <- metagen(data = data,
TE = R,
seTE = SE_Stylized,
studlab = StudYear,
sm = "ZCOR",
fixed = T,
random = T,
method.tau = "SJ",
hakn = T,
backtransf = T,
title = "Spousal IQ Correlations with Stylized Variances"); summary(MetaStylized)## Review: Spousal IQ Correlations with Stylized Variances
##
## COR 95%-CI %W(common)
## Pan & Wang 2011 0.2737 [ 0.0571; 0.4657] 0.5
## Dufouil et al. 2000 0.3452 [ 0.2442; 0.4388] 2.1
## Schooley 1936 0.5050 [ 0.3429; 0.6379] 0.7
## Conrad & Jones 1940 0.4777 [ 0.3484; 0.5891] 1.1
## Outhit 1933 0.6291 [ 0.4908; 0.7365] 0.6
## Willoughby 1928 0.4136 [ 0.2335; 0.5663] 0.6
## Carter 1932 0.2070 [ 0.0095; 0.3889] 0.6
## Burt 1972 0.5902 [ 0.4755; 0.6852] 1.0
## Burks 1928 A 0.3969 [ 0.2673; 0.5125] 1.2
## Burks 1928 B 0.5005 [ 0.3557; 0.6219] 0.8
## Leahy 1935 0.5154 [ 0.4118; 0.6058] 1.5
## Swagerman et al. 2015 0.3627 [ 0.2012; 0.5050] 0.8
## Leahy 1935 0.3885 [ 0.2531; 0.5089] 1.1
## Freeman et al. 1928 0.4542 [ 0.3264; 0.5658] 1.1
## Burt 1972 0.3618 [ 0.1746; 0.5238] 0.6
## Reed & Reed 1965 0.3149 [ 0.2727; 0.3559] 12.1
## Halprin 1946 0.6411 [ 0.5938; 0.6839] 4.4
## Spuhler 1962 0.3791 [ 0.2486; 0.4961] 1.2
## Guttman 1974 0.2543 [ 0.0540; 0.4349] 0.6
## Spuhler 1962 0.4765 [ 0.3545; 0.5826] 1.2
## Williams 1973 0.5980 [ 0.4494; 0.7144] 0.6
## Stanley 1972 0.1489 [-0.3065; 0.5488] 0.1
## Smith 1941 0.1877 [ 0.0899; 0.2820] 2.6
## Higgins et al. 1962 0.3185 [ 0.2612; 0.3736] 6.6
## Penrose 1933 0.4136 [ 0.2435; 0.5591] 0.7
## Cattell & Nesselroade 1967 0.3004 [ 0.1089; 0.4705] 0.6
## Terman & Oden 1959 0.4470 [ 0.3888; 0.5017] 5.2
## Escorial & Martin-Buro 2012 0.3842 [ 0.2738; 0.4847] 1.7
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 0.4558 [ 0.3734; 0.5311] 2.6
## Gruber-Baldini, Schaie & Willis 1995 0.2449 [ 0.0818; 0.3953] 0.9
## Phillips et al. 1988 0.1974 [ 0.0863; 0.3036] 2.0
## Vinkhuyzen et al. 2012 0.3540 [ 0.2305; 0.4663] 1.4
## van Bergen et al. 2015 0.3452 [ 0.2166; 0.4621] 1.3
## DeFries et al. 1979 0.2260 [ 0.1589; 0.2911] 5.3
## Benbow, Zonderman & Stanley 1983 0.4136 [ 0.1096; 0.6469] 0.2
## Defries et al. 1979 0.1489 [ 0.0313; 0.2624] 1.8
## Bouchard & McGee 1977 0.0599 [-0.1163; 0.2325] 0.8
## Bock & Kolakowski 1973 0.2543 [ 0.1010; 0.3958] 1.0
## Stafford 1961 0.0300 [-0.1822; 0.2395] 0.6
## Corah 1965 0.1489 [-0.1234; 0.4003] 0.3
## Loehlin, Sharan & Jacoby 1978 0.3095 [ 0.1731; 0.4342] 1.2
## Scarr & Weinberg 1978 A 0.2913 [ 0.1337; 0.4346] 0.9
## Scarr & Weinberg 1978 B 0.1974 [ 0.0650; 0.3229] 1.4
## Lewak, Wakefield & Briggs 1985 0.3364 [ 0.1268; 0.5172] 0.5
## Mascie-Taylor 1989 A 0.3825 [ 0.2573; 0.4951] 1.3
## Mascie-Taylor 1989 B 0.3557 [ 0.2079; 0.4877] 1.0
## Price & Vandenberg 1980 A 0.3275 [ 0.2160; 0.4306] 1.8
## Price & Vandenberg 1980 B 0.3452 [ 0.1864; 0.4865] 0.9
## Smith 1941 0.1906 [ 0.0930; 0.2847] 2.6
## Woods 1906 0.0798 [-0.0596; 0.2162] 1.3
## Moorees 1924 0.0997 [-0.2168; 0.3972] 0.3
## Okbay et al. 2022 A 0.1381 [-0.0032; 0.2740] 1.3
## Okbay et al. 2022 B 0.2374 [ 0.1874; 0.2861] 9.5
## Keller et al. 2013 0.3364 [ 0.2731; 0.3967] 5.3
## %W(random)
## Pan & Wang 2011 1.5
## Dufouil et al. 2000 2.1
## Schooley 1936 1.6
## Conrad & Jones 1940 1.9
## Outhit 1933 1.6
## Willoughby 1928 1.6
## Carter 1932 1.6
## Burt 1972 1.9
## Burks 1928 A 1.9
## Burks 1928 B 1.8
## Leahy 1935 2.0
## Swagerman et al. 2015 1.8
## Leahy 1935 1.9
## Freeman et al. 1928 1.9
## Burt 1972 1.6
## Reed & Reed 1965 2.4
## Halprin 1946 2.3
## Spuhler 1962 1.9
## Guttman 1974 1.6
## Spuhler 1962 1.9
## Williams 1973 1.6
## Stanley 1972 0.7
## Smith 1941 2.2
## Higgins et al. 1962 2.3
## Penrose 1933 1.7
## Cattell & Nesselroade 1967 1.6
## Terman & Oden 1959 2.3
## Escorial & Martin-Buro 2012 2.0
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 2.2
## Gruber-Baldini, Schaie & Willis 1995 1.8
## Phillips et al. 1988 2.1
## Vinkhuyzen et al. 2012 2.0
## van Bergen et al. 2015 2.0
## DeFries et al. 1979 2.3
## Benbow, Zonderman & Stanley 1983 1.1
## Defries et al. 1979 2.1
## Bouchard & McGee 1977 1.8
## Bock & Kolakowski 1973 1.9
## Stafford 1961 1.6
## Corah 1965 1.3
## Loehlin, Sharan & Jacoby 1978 1.9
## Scarr & Weinberg 1978 A 1.8
## Scarr & Weinberg 1978 B 2.0
## Lewak, Wakefield & Briggs 1985 1.5
## Mascie-Taylor 1989 A 2.0
## Mascie-Taylor 1989 B 1.9
## Price & Vandenberg 1980 A 2.1
## Price & Vandenberg 1980 B 1.8
## Smith 1941 2.2
## Woods 1906 2.0
## Moorees 1924 1.1
## Okbay et al. 2022 A 2.0
## Okbay et al. 2022 B 2.3
## Keller et al. 2013 2.3
##
## Number of studies combined: k = 54
##
## COR 95%-CI z|t p-value
## Common effect model 0.3362 [0.3218; 0.3504] 42.61 0
## Random effects model 0.3411 [0.3006; 0.3803] 15.81 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0.0219 [0.0127; 0.0349]; tau = 0.1481 [0.1126; 0.1868]
## I^2 = 84.1% [80.0%; 87.4%]; H = 2.51 [2.24; 2.82]
##
## Test of heterogeneity:
## Q d.f. p-value
## 334.08 53 < 0.0001
##
## Details on meta-analytical method:
## - Inverse variance method
## - Sidik-Jonkman estimator for tau^2
## - Q-profile method for confidence interval of tau^2 and tau
## - Hartung-Knapp adjustment for random effects model
## - Fisher's z transformation of correlationsforest.meta(MetaEqual, layout = "JAMA",
col.diamond = "gold",
sortvar = TE,
predict = T,
print.tau2 = T,
leftlabs = c("Author", "Correlation", "95% CI"))
Subgrouped by quality of the study here.
MetaQual <- subgroup.analysis.mixed.effects(MetaEqual, data$Quality); MetaQual## Warning: Use argument 'fixed' instead of 'comb.fixed' (deprecated).## Warning: Use argument 'random' instead of 'comb.random' (deprecated).## Warning: Use argument 'subgroup' instead of 'byvar' (deprecated).## Warning: Use argument 'fixed' instead of 'comb.fixed' (deprecated).## Warning: Use argument 'subgroup' instead of 'byvar' (deprecated).## Subgroup Results:
## --------------
## k ZCOR SE LLCI ULCI p Q I2
## Good 23 0.3417661 0.02718542 0.288 0.395 3.022606e-36 91.13497 0.76
## Moderate 15 0.4623770 0.04088256 0.382 0.543 1.172449e-29 143.44342 0.90
## Poor 16 0.2575627 0.04090791 0.177 0.338 3.051100e-10 85.93032 0.83
## I2.lower I2.upper
## Good 0.64 0.84
## Moderate 0.86 0.93
## Poor 0.73 0.89
##
## Test for subgroup differences (mixed/fixed-effects (plural) model):
## --------------
## Q df p
## Between groups 12.7532 2 0.001700897
##
## - Total number of studies included in subgroup analysis: 54
## - Tau estimator used for within-group pooling: SJforest(MetaQual)
A meta-regression by year to check for trends over time.
MetaYear <- rma.uni(yi = R, vi = SE_Equal, data = data, method = "ML", mods = ~ RawYear); MetaYear##
## Mixed-Effects Model (k = 54; tau^2 estimator: ML)
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 0.0108)
## 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 = 52) = 20.8399, p-val = 1.0000
##
## Test of Moderators (coefficient 2):
## QM(df = 1) = 0.9874, p-val = 0.3204
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## intrcpt 2.6322 2.2934 1.1477 0.2511 -1.8628 7.1273
## RawYear -0.0012 0.0012 -0.9937 0.3204 -0.0034 0.0011
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In terms of mean absolute differences, this amounts to
2 * 15 * sqrt(1 - 0.34) / sqrt(pi)## [1] 13.7504913.75 points.
Plots
With an aggregate value for r of 0.34 in hand, here is a function to say how much within-family PGS validity is expected to be reduced by given a specific value of r and AM-free \(h^2\), or \(h^2_0\).
AMPart <- function(r, h20, Y, ro = 3){
Attenuation = 1 - r * h20
Part = 1 - Attenuation
Reduction = Part * 100
Out <- capture.output(cat(paste0("Your within-family PGS validity is expected to be reduced by ", round(Reduction, ro), "% due to assortative mating.\n")))
relist <- list("Attenuation" = Attenuation, "Part" = Part, "Reduction" = Reduction, "Mess" = Out)
return(relist)}
AMPart(0.34, c(0.1, 0.2, 0.3, 0.4, 0.5, 0.6))$Mess## [1] "Your within-family PGS validity is expected to be reduced by 3.4% due to assortative mating."
## [2] " Your within-family PGS validity is expected to be reduced by 6.8% due to assortative mating."
## [3] " Your within-family PGS validity is expected to be reduced by 10.2% due to assortative mating."
## [4] " Your within-family PGS validity is expected to be reduced by 13.6% due to assortative mating."
## [5] " Your within-family PGS validity is expected to be reduced by 17% due to assortative mating."
## [6] " Your within-family PGS validity is expected to be reduced by 20.4% due to assortative mating."AMPart(0.23, c(0.1, 0.2, 0.3, 0.4, 0.5, 0.6))$Mess #Pooled EA4 samples' values## [1] "Your within-family PGS validity is expected to be reduced by 2.3% due to assortative mating."
## [2] " Your within-family PGS validity is expected to be reduced by 4.6% due to assortative mating."
## [3] " Your within-family PGS validity is expected to be reduced by 6.9% due to assortative mating."
## [4] " Your within-family PGS validity is expected to be reduced by 9.2% due to assortative mating."
## [5] " Your within-family PGS validity is expected to be reduced by 11.5% due to assortative mating."
## [6] " Your within-family PGS validity is expected to be reduced by 13.8% due to assortative mating."Or, plotted:
PlotData = data.frame("Source" = c(rep("Simulated", 6), "Empirical"), "Position" = c("0.1", "0.2", "0.3", "0.4", "0.5", "0.6", "EA4"),
"Values" = c(AMPart(0.34, 0.1)$Part,
AMPart(0.34, 0.2)$Part,
AMPart(0.34, 0.3)$Part,
AMPart(0.34, 0.4)$Part,
AMPart(0.34, 0.5)$Part,
AMPart(0.34, 0.6)$Part, 0.1758)) #Empirical from EA4ggplot(data = PlotData, aes(x = Position, y = Values, fill = Source)) +
geom_bar(stat = "identity", color = "black") +
scale_fill_manual(values = c("red", "steelblue")) +
xlab("Within-Family Prediction Attenuation due to AM by SNP Heritability sans AM versus EA4 Attenuation") + ylab("Within-Family Coefficient Attenuation") +
theme_bw() +
theme(legend.position = c(0.06, 0.92), legend.background = element_blank())
Conclusion
Because the predictive validity of cognitive performance is barely affected when EA4 is used within families to predict IQ, a considerable proportion of its degraded validity is likely to stem from assortative mating. For example, if the SNP heritability is 10%, 20%, 30%, 40%, or 50%, and the phenotypic coefficients for assortment in Okbay et al.’s (2022) supplement are correct (r = 0.230), then 13.08%, 26.17%, 39.25%, 52.33%, 65.42%, or 78.50% of the reduction can be attributed to assortative mating. Because it is more likely that the poor cognitive measurements used in the UKBB and Generation Scotland have attenuated phenotypic similarity, moving that r value a bit higher may be in order. If it is 0.34, then assortative mating explains 19.34%, 38.68%, 58.02%, 77.36%, 96.70%, or >100% of the reduction in validity within families. The latter option seems to suggest a SNP heritability of 60% is not in the cards, that the reduction has high variance, or that some other countervailing force may be at play, among other options. If, again, the coefficient for r rises to 0.4 - which seems to be a likely value in studies which use latent g, given apparent Jensen effects for assortative mating and associations with heritability - assortative mating takes care of 22.75%, 45.51%, 68.26%, 91.01%, and >100% of the reduction in validity within families.
References
Lee, James J., Robbee Wedow, Aysu Okbay, Edward Kong, Omeed Maghzian, Meghan Zacher, Tuan Anh Nguyen-Viet, Peter Bowers, Julia Sidorenko, Richard Karlsson Linnér, Mark Alan Fontana, Tushar Kundu, Chanwook Lee, Hui Li, Ruoxi Li, Rebecca Royer, Pascal N. Timshel, Raymond K. Walters, Emily A. Willoughby, Loïc Yengo, Maris Alver, Yanchun Bao, David W. Clark, Felix R. Day, Nicholas A. Furlotte, Peter K. Joshi, Kathryn E. Kemper, Aaron Kleinman, Claudia Langenberg, Reedik Mägi, Joey W. Trampush, Shefali Setia Verma, Yang Wu, Max Lam, Jing Hua Zhao, Zhili Zheng, Jason D. Boardman, Harry Campbell, Jeremy Freese, Kathleen Mullan Harris, Caroline Hayward, Pamela Herd, Meena Kumari, Todd Lencz, Jian’an Luan, Anil K. Malhotra, Andres Metspalu, Lili Milani, Ken K. Ong, John R. B. Perry, David J. Porteous, Marylyn D. Ritchie, Melissa C. Smart, Blair H. Smith, Joyce Y. Tung, Nicholas J. Wareham, James F. Wilson, Jonathan P. Beauchamp, Dalton C. Conley, Tõnu Esko, Steven F. Lehrer, Patrik K. E. Magnusson, Sven Oskarsson, Tune H. Pers, Matthew R. Robinson, Kevin Thom, Chelsea Watson, Christopher F. Chabris, Michelle N. Meyer, David I. Laibson, Jian Yang, Magnus Johannesson, Philipp D. Koellinger, Patrick Turley, Peter M. Visscher, Daniel J. Benjamin, and David Cesarini. 2018. “Gene Discovery and Polygenic Prediction from a Genome-Wide Association Study of Educational Attainment in 1.1 Million Individuals.” Nature Genetics 50(8):1112–21. doi: 10.1038/s41588-018-0147-3.
Okbay, Aysu, Yeda Wu, Nancy Wang, Hariharan Jayashankar, Michael Bennett, Seyed Moeen Nehzati, Julia Sidorenko, Hyeokmoon Kweon, Grant Goldman, Tamara Gjorgjieva, Yunxuan Jiang, Barry Hicks, Chao Tian, David A. Hinds, Rafael Ahlskog, Patrik K. E. Magnusson, Sven Oskarsson, Caroline Hayward, Archie Campbell, David J. Porteous, Jeremy Freese, Pamela Herd, Chelsea Watson, Jonathan Jala, Dalton Conley, Philipp D. Koellinger, Magnus Johannesson, David Laibson, Michelle N. Meyer, James J. Lee, Augustine Kong, Loic Yengo, David Cesarini, Patrick Turley, Peter M. Visscher, Jonathan P. Beauchamp, Daniel J. Benjamin, and Alexander I. Young. 2022. “Polygenic Prediction of Educational Attainment within and between Families from Genome-Wide Association Analyses in 3 Million Individuals.” Nature Genetics 1–13. doi: 10.1038/s41588-022-01016-z.
sessionInfo()## R version 4.1.2 (2021-11-01)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19042)
##
## Matrix products: default
##
## locale:
## [1] LC_COLLATE=English_United States.1252
## [2] LC_CTYPE=English_United States.1252
## [3] LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C
## [5] LC_TIME=English_United States.1252
##
## attached base packages:
## [1] tools stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] metasens_1.0-1 DescTools_0.99.44 dmetar_0.0.9000 metafor_3.0-2
## [5] Matrix_1.3-4 meta_5.1-1 DT_0.20 ggplot2_3.3.5
## [9] pacman_0.5.1
##
## loaded via a namespace (and not attached):
## [1] magic_1.5-9 sass_0.4.0 jsonlite_1.7.2 splines_4.1.2
## [5] bslib_0.3.1 assertthat_0.2.1 expm_0.999-6 highr_0.9
## [9] lmom_2.8 gld_2.6.4 stats4_4.1.2 yaml_2.2.1
## [13] robustbase_0.93-9 ggrepel_0.9.1 pillar_1.6.4 lattice_0.20-45
## [17] glue_1.4.2 digest_0.6.28 minqa_1.2.4 colorspace_2.0-2
## [21] MuMIn_1.43.17 htmltools_0.5.2 netmeta_2.0-1 pkgconfig_2.0.3
## [25] mvtnorm_1.1-3 purrr_0.3.4 scales_1.1.1 rootSolve_1.8.2.3
## [29] lme4_1.1-27.1 proxy_0.4-26 tibble_3.1.5 farver_2.1.0
## [33] generics_0.1.1 ellipsis_0.3.2 withr_2.4.3 nnet_7.3-16
## [37] magrittr_2.0.1 crayon_1.4.2 mclust_5.4.9 evaluate_0.14
## [41] fansi_0.5.0 nlme_3.1-153 MASS_7.3-54 xml2_1.3.3
## [45] class_7.3-19 data.table_1.14.2 lifecycle_1.0.1 Exact_3.1
## [49] stringr_1.4.0 kernlab_0.9-29 munsell_0.5.0 cluster_2.1.2
## [53] fpc_2.2-9 e1071_1.7-9 compiler_4.1.2 jquerylib_0.1.4
## [57] rlang_0.4.12 grid_4.1.2 nloptr_1.2.2.2 rstudioapi_0.13
## [61] CompQuadForm_1.4.3 htmlwidgets_1.5.4 crosstalk_1.2.0 labeling_0.4.2
## [65] rmarkdown_2.11 boot_1.3-28 gtable_0.3.0 abind_1.4-5
## [69] flexmix_2.3-17 DBI_1.1.1 R6_2.5.1 gridExtra_2.3
## [73] knitr_1.36 prabclus_2.3-2 dplyr_1.0.7 fastmap_1.1.0
## [77] utf8_1.2.2 mathjaxr_1.4-0 poibin_1.5 modeltools_0.2-23
## [81] stringi_1.7.5 parallel_4.1.2 Rcpp_1.0.7 vctrs_0.3.8
## [85] DEoptimR_1.0-9 tidyselect_1.1.1 xfun_0.27 diptest_0.76-0Postscript: Publication Bias
A variety of methods fail to evidence publication bias.
dfZ <- data.frame(MetaEqual$seTE, MetaEqual$TE); dfZ$se <- FisherZInv(dfZ$MetaEqual.seTE); dfZ$r <- FisherZInv(dfZ$MetaEqual.TE)
estimateT = FisherZInv(MetaEqual$TE.random); set = FisherZInv(MetaEqual$seTE.random); estimateT## [1] 0.3409045se.seq = seq(0, max(dfZ$se), 0.001)
ll95 = estimateT - (1.96*se.seq)
ul95 = estimateT + (1.96*se.seq)
ll95a = FisherZInv(MetaEqual$lower.random)
ul95a = FisherZInv(MetaEqual$upper.random)
ll99 = estimateT - (3.29*se.seq)
ul99 = estimateT + (3.29*se.seq)
ll99a = 1.67857*FisherZInv(MetaEqual$lower.random)
ul99a = 1.67857*FisherZInv(MetaEqual$upper.random)
meanll95 = estimateT - (1.96*set)
meanul95 = estimateT + (1.96*set)
dfZCI <- data.frame(ll95, ul95, ll99, ul99, se.seq, estimateT, meanll95, meanul95, ll95a, ul95a, ll99a, ul99a)
STAND <- ggplot(aes(x = se, y = r), data = dfZ) +
geom_point(shape = 16, size = 3, colour = "#800020") +
xlab('Standard Error') + ylab('z-to-r Transformed Correlations') +
geom_line(aes(x = se.seq, y = ll95), linetype = 'dotted', colour = "#666666", size = 1, data = dfZCI) +
geom_line(aes(x = se.seq, y = ul95), linetype = 'dotted', colour = "#666666", size = 1, data = dfZCI) +
geom_line(aes(x = se.seq, y = ll99), linetype = 'dashed', colour = "#666666", size = 1, data = dfZCI) +
geom_line(aes(x = se.seq, y = ul99), linetype = 'dashed', colour = "#666666", size = 1, data = dfZCI) +
geom_segment(aes(x = min(se.seq), y = estimateT, xend = max(se.seq), yend = estimateT), linetype='dotted', colour = "#E9C535", size = 1, data=dfZCI) +
geom_segment(aes(x = min(se.seq), y = ll95a, xend = max(se.seq), yend = ll95a), linetype='dotted' , colour = "gold", size = 1, data=dfZCI) +
geom_segment(aes(x = min(se.seq), y = ul95a, xend = max(se.seq), yend = ul95a), linetype='dotted' , colour = "gold", size = 1, data=dfZCI) +
scale_x_reverse() +
coord_flip() +
theme_bw() +
theme(text = element_text(family = "serif", size = 12))
metabias(MetaEqual, method.bias = "linreg")## Review: Spousal IQ Correlations with Equal Variances
##
## Linear regression test of funnel plot asymmetry
##
## Test result: t = 0.36, df = 52, p-value = 0.7183
##
## Sample estimates:
## bias se.bias intercept se.intercept
## 0.2967 0.8181 0.3350 0.0458
##
## Details:
## - multiplicative residual heterogeneity variance (tau^2 = 7.4339)
## - predictor: standard error
## - weight: inverse variance
## - reference: Egger et al. (1997), BMJSTAND
pcurve(MetaEqual)
## P-curve analysis
## -----------------------
## - Total number of provided studies: k = 54
## - Total number of p<0.05 studies included into the analysis: k = 48 (88.89%)
## - Total number of studies with p<0.025: k = 46 (85.19%)
##
## Results
## -----------------------
## pBinomial zFull pFull zHalf pHalf
## Right-skewness test 0 -31.878 0 -31.439 0
## Flatness test 1 26.498 1 28.870 1
## Note: p-values of 0 or 1 correspond to p<0.001 and p>0.999, respectively.
## Power Estimate: 99% (99%-99%)
##
## Evidential value
## -----------------------
## - Evidential value present: yes
## - Evidential value absent/inadequate: noTF <- trimfill(MetaEqual, side = "right"); summary(TF); funnel(TF, legend = T)## Review: Spousal IQ Correlations with Equal Variances
##
## COR 95%-CI %W(random)
## Pan & Wang 2011 0.2737 [ 0.0730; 0.4531] 1.6
## Dufouil et al. 2000 0.3452 [ 0.2516; 0.4324] 2.1
## Schooley 1936 0.5050 [ 0.3553; 0.6294] 1.7
## Conrad & Jones 1940 0.4777 [ 0.3582; 0.5818] 1.9
## Outhit 1933 0.6291 [ 0.5018; 0.7298] 1.7
## Willoughby 1928 0.4136 [ 0.2472; 0.5564] 1.7
## Carter 1932 0.2070 [ 0.0239; 0.3766] 1.7
## Burt 1972 0.5902 [ 0.4843; 0.6791] 1.9
## Burks 1928 A 0.3969 [ 0.2769; 0.5047] 1.9
## Burks 1928 B 0.5005 [ 0.3667; 0.6140] 1.8
## Leahy 1935 0.5154 [ 0.4196; 0.5998] 2.0
## Swagerman et al. 2015 0.3627 [ 0.2132; 0.4956] 1.8
## Leahy 1935 0.3885 [ 0.2632; 0.5009] 1.9
## Freeman et al. 1928 0.4542 [ 0.3360; 0.5584] 1.9
## Burt 1972 0.3618 [ 0.1886; 0.5132] 1.7
## Reed & Reed 1965 0.3149 [ 0.2758; 0.3530] 2.3
## Halprin 1946 0.6411 [ 0.5973; 0.6810] 2.2
## Spuhler 1962 0.3791 [ 0.2583; 0.4882] 1.9
## Guttman 1974 0.2543 [ 0.0687; 0.4229] 1.6
## Spuhler 1962 0.4765 [ 0.3638; 0.5756] 1.9
## Williams 1973 0.5980 [ 0.4611; 0.7071] 1.6
## Stanley 1972 0.1489 [-0.2760; 0.5251] 0.7
## Smith 1941 0.1877 [ 0.0970; 0.2754] 2.1
## Higgins et al. 1962 0.3185 [ 0.2654; 0.3697] 2.2
## Penrose 1933 0.4136 [ 0.2564; 0.5496] 1.7
## Cattell & Nesselroade 1967 0.3004 [ 0.1230; 0.4592] 1.7
## Terman & Oden 1959 0.4470 [ 0.3931; 0.4979] 2.2
## Escorial & Martin-Buro 2012 0.3842 [ 0.2820; 0.4779] 2.0
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 0.4558 [ 0.3795; 0.5260] 2.1
## Gruber-Baldini, Schaie & Willis 1995 0.2449 [ 0.0937; 0.3851] 1.8
## Phillips et al. 1988 0.1974 [ 0.0943; 0.2962] 2.1
## Vinkhuyzen et al. 2012 0.3540 [ 0.2396; 0.4587] 2.0
## van Bergen et al. 2015 0.3452 [ 0.2261; 0.4542] 2.0
## DeFries et al. 1979 0.2260 [ 0.1638; 0.2865] 2.2
## Benbow, Zonderman & Stanley 1983 0.4136 [ 0.1328; 0.6330] 1.1
## Defries et al. 1979 0.1489 [ 0.0397; 0.2545] 2.0
## Bouchard & McGee 1977 0.0599 [-0.1038; 0.2205] 1.8
## Bock & Kolakowski 1973 0.2543 [ 0.1122; 0.3862] 1.9
## Stafford 1961 0.0300 [-0.1673; 0.2250] 1.6
## Corah 1965 0.1489 [-0.1041; 0.3837] 1.3
## Loehlin, Sharan & Jacoby 1978 0.3095 [ 0.1832; 0.4258] 1.9
## Scarr & Weinberg 1978 A 0.2913 [ 0.1453; 0.4249] 1.8
## Scarr & Weinberg 1978 B 0.1974 [ 0.0746; 0.3143] 2.0
## Lewak, Wakefield & Briggs 1985 0.3364 [ 0.1425; 0.5054] 1.6
## Mascie-Taylor 1989 A 0.3825 [ 0.2666; 0.4875] 2.0
## Mascie-Taylor 1989 B 0.3557 [ 0.2189; 0.4789] 1.9
## Price & Vandenberg 1980 A 0.3275 [ 0.2242; 0.4235] 2.0
## Price & Vandenberg 1980 B 0.3452 [ 0.1982; 0.4770] 1.8
## Smith 1941 0.1906 [ 0.1000; 0.2781] 2.1
## Woods 1906 0.0798 [-0.0496; 0.2067] 2.0
## Moorees 1924 0.0997 [-0.1949; 0.3777] 1.2
## Okbay et al. 2022 A 0.1381 [ 0.0069; 0.2646] 1.9
## Okbay et al. 2022 B 0.2374 [ 0.1911; 0.2827] 2.3
## Keller et al. 2013 0.3364 [ 0.2777; 0.3925] 2.2
##
## Number of studies combined: k = 54 (with 0 added studies)
##
## COR 95%-CI t p-value
## Random effects model 0.3409 [0.3004; 0.3802] 15.77 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0.0226 [0.0135; 0.0359]; tau = 0.1503 [0.1163; 0.1895]
## I^2 = 86.3% [82.9%; 89.0%]; H = 2.70 [2.42; 3.02]
##
## Test of heterogeneity:
## Q d.f. p-value
## 387.54 53 < 0.0001
##
## Details on meta-analytical method:
## - Inverse variance method
## - Sidik-Jonkman estimator for tau^2
## - Q-profile method for confidence interval of tau^2 and tau
## - Hartung-Knapp adjustment for random effects model
## - Trim-and-fill method to adjust for funnel plot asymmetry
## - Fisher's z transformation of correlations
limit <- limitmeta(MetaEqual); summary(limit)## Results for individual studies
## (left: original data; right: shrunken estimates)
##
## COR 95%-CI COR
## Pan & Wang 2011 0.2737 [ 0.0730; 0.4531] 0.3174
## Dufouil et al. 2000 0.3452 [ 0.2516; 0.4324] 0.3542
## Schooley 1936 0.5050 [ 0.3553; 0.6294] 0.5035
## Conrad & Jones 1940 0.4777 [ 0.3582; 0.5818] 0.4795
## Outhit 1933 0.6291 [ 0.5018; 0.7298] 0.6100
## Willoughby 1928 0.4136 [ 0.2472; 0.5564] 0.4269
## Carter 1932 0.2070 [ 0.0239; 0.3766] 0.2557
## Burt 1972 0.5902 [ 0.4843; 0.6791] 0.5806
## Burks 1928 A 0.3969 [ 0.2769; 0.5047] 0.4064
## Burks 1928 B 0.5005 [ 0.3667; 0.6140] 0.4998
## Leahy 1935 0.5154 [ 0.4196; 0.5998] 0.5138
## Swagerman et al. 2015 0.3627 [ 0.2132; 0.4956] 0.3803
## Leahy 1935 0.3885 [ 0.2632; 0.5009] 0.3994
## Freeman et al. 1928 0.4542 [ 0.3360; 0.5584] 0.4583
## Burt 1972 0.3618 [ 0.1886; 0.5132] 0.3840
## Reed & Reed 1965 0.3149 [ 0.2758; 0.3530] 0.3170
## Halprin 1946 0.6411 [ 0.5973; 0.6810] 0.6375
## Spuhler 1962 0.3791 [ 0.2583; 0.4882] 0.3902
## Guttman 1974 0.2543 [ 0.0687; 0.4229] 0.2964
## Spuhler 1962 0.4765 [ 0.3638; 0.5756] 0.4783
## Williams 1973 0.5980 [ 0.4611; 0.7071] 0.5825
## Stanley 1972 0.1489 [-0.2760; 0.5251] 0.3113
## Smith 1941 0.1877 [ 0.0970; 0.2754] 0.2034
## Higgins et al. 1962 0.3185 [ 0.2654; 0.3697] 0.3221
## Penrose 1933 0.4136 [ 0.2564; 0.5496] 0.4258
## Cattell & Nesselroade 1967 0.3004 [ 0.1230; 0.4592] 0.3329
## Terman & Oden 1959 0.4470 [ 0.3931; 0.4979] 0.4482
## Escorial & Martin-Buro 2012 0.3842 [ 0.2820; 0.4779] 0.3923
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 0.4558 [ 0.3795; 0.5260] 0.4576
## Gruber-Baldini, Schaie & Willis 1995 0.2449 [ 0.0937; 0.3851] 0.2766
## Phillips et al. 1988 0.1974 [ 0.0943; 0.2962] 0.2166
## Vinkhuyzen et al. 2012 0.3540 [ 0.2396; 0.4587] 0.3660
## van Bergen et al. 2015 0.3452 [ 0.2261; 0.4542] 0.3588
## DeFries et al. 1979 0.2260 [ 0.1638; 0.2865] 0.2329
## Benbow, Zonderman & Stanley 1983 0.4136 [ 0.1328; 0.6330] 0.4394
## Defries et al. 1979 0.1489 [ 0.0397; 0.2545] 0.1732
## Bouchard & McGee 1977 0.0599 [-0.1038; 0.2205] 0.1205
## Bock & Kolakowski 1973 0.2543 [ 0.1122; 0.3862] 0.2820
## Stafford 1961 0.0300 [-0.1673; 0.2250] 0.1173
## Corah 1965 0.1489 [-0.1041; 0.3837] 0.2407
## Loehlin, Sharan & Jacoby 1978 0.3095 [ 0.1832; 0.4258] 0.3276
## Scarr & Weinberg 1978 A 0.2913 [ 0.1453; 0.4249] 0.3163
## Scarr & Weinberg 1978 B 0.1974 [ 0.0746; 0.3143] 0.2235
## Lewak, Wakefield & Briggs 1985 0.3364 [ 0.1425; 0.5054] 0.3670
## Mascie-Taylor 1989 A 0.3825 [ 0.2666; 0.4875] 0.3926
## Mascie-Taylor 1989 B 0.3557 [ 0.2189; 0.4789] 0.3718
## Price & Vandenberg 1980 A 0.3275 [ 0.2242; 0.4235] 0.3392
## Price & Vandenberg 1980 B 0.3452 [ 0.1982; 0.4770] 0.3644
## Smith 1941 0.1906 [ 0.1000; 0.2781] 0.2061
## Woods 1906 0.0798 [-0.0496; 0.2067] 0.1187
## Moorees 1924 0.0997 [-0.1949; 0.3777] 0.2272
## Okbay et al. 2022 A 0.1381 [ 0.0069; 0.2646] 0.1726
## Okbay et al. 2022 B 0.2374 [ 0.1911; 0.2827] 0.2412
## Keller et al. 2013 0.3364 [ 0.2777; 0.3925] 0.3404
## 95%-CI
## Pan & Wang 2011 [ 0.1204; 0.4903]
## Dufouil et al. 2000 [ 0.2612; 0.4407]
## Schooley 1936 [ 0.3536; 0.6282]
## Conrad & Jones 1940 [ 0.3602; 0.5833]
## Outhit 1933 [ 0.4781; 0.7149]
## Willoughby 1928 [ 0.2622; 0.5674]
## Carter 1932 [ 0.0752; 0.4199]
## Burt 1972 [ 0.4730; 0.6711]
## Burks 1928 A [ 0.2873; 0.5131]
## Burks 1928 B [ 0.3659; 0.6134]
## Leahy 1935 [ 0.4179; 0.5984]
## Swagerman et al. 2015 [ 0.2326; 0.5109]
## Leahy 1935 [ 0.2752; 0.5105]
## Freeman et al. 1928 [ 0.3405; 0.5619]
## Burt 1972 [ 0.2133; 0.5319]
## Reed & Reed 1965 [ 0.2778; 0.3550]
## Halprin 1946 [ 0.5934; 0.6777]
## Spuhler 1962 [ 0.2704; 0.4981]
## Guttman 1974 [ 0.1139; 0.4596]
## Spuhler 1962 [ 0.3657; 0.5771]
## Williams 1973 [ 0.4422; 0.6950]
## Stanley 1972 [-0.1109; 0.6383]
## Smith 1941 [ 0.1131; 0.2904]
## Higgins et al. 1962 [ 0.2691; 0.3732]
## Penrose 1933 [ 0.2701; 0.5598]
## Cattell & Nesselroade 1967 [ 0.1584; 0.4872]
## Terman & Oden 1959 [ 0.3944; 0.4990]
## Escorial & Martin-Buro 2012 [ 0.2906; 0.4851]
## Colom, Aluja-Fabregat & Garcia-Lopez 2002 [ 0.3815; 0.5276]
## Gruber-Baldini, Schaie & Willis 1995 [ 0.1274; 0.4137]
## Phillips et al. 1988 [ 0.1141; 0.3144]
## Vinkhuyzen et al. 2012 [ 0.2526; 0.4696]
## van Bergen et al. 2015 [ 0.2408; 0.4664]
## DeFries et al. 1979 [ 0.1709; 0.2932]
## Benbow, Zonderman & Stanley 1983 [ 0.1636; 0.6515]
## Defries et al. 1979 [ 0.0646; 0.2777]
## Bouchard & McGee 1977 [-0.0431; 0.2778]
## Bock & Kolakowski 1973 [ 0.1416; 0.4113]
## Stafford 1961 [-0.0809; 0.3066]
## Corah 1965 [-0.0089; 0.4621]
## Loehlin, Sharan & Jacoby 1978 [ 0.2026; 0.4421]
## Scarr & Weinberg 1978 A [ 0.1721; 0.4472]
## Scarr & Weinberg 1978 B [ 0.1017; 0.3387]
## Lewak, Wakefield & Briggs 1985 [ 0.1765; 0.5310]
## Mascie-Taylor 1989 A [ 0.2776; 0.4965]
## Mascie-Taylor 1989 B [ 0.2364; 0.4930]
## Price & Vandenberg 1980 A [ 0.2367; 0.4343]
## Price & Vandenberg 1980 B [ 0.2192; 0.4938]
## Smith 1941 [ 0.1160; 0.2929]
## Woods 1906 [-0.0104; 0.2439]
## Moorees 1924 [-0.0661; 0.4843]
## Okbay et al. 2022 A [ 0.0423; 0.2972]
## Okbay et al. 2022 B [ 0.1949; 0.2863]
## Keller et al. 2013 [ 0.2819; 0.3963]
##
## Result of limit meta-analysis:
##
## Random effects model COR 95%-CI z pval
## Adjusted estimate 0.3584 [0.2948; 0.4189] 10.32 < 0.0001
## Unadjusted estimate 0.3409 [0.3004; 0.3802] 15.77 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0.0226; I^2 = 86.3% [82.9%; 89.0%]; G^2 = 47.2%
##
## Test of heterogeneity:
## Q d.f. p-value
## 387.54 53 0
##
## Test of small-study effects:
## Q-Q' d.f. p-value
## 0.98 1 0.3228
##
## Test of residual heterogeneity beyond small-study effects:
## Q' d.f. p-value
## 386.56 52 0
##
## Details on adjustment method:
## - expectation (beta0)funnel.limitmeta(limit, xlim = c(-0.2, 0.85), shrunken = T)