---
title: "Converting t-Values to Partial Correlations: Ballesteros et al. (2013)"
author: "Danilo Assis Pereira, Ph.D."
date: "2025-01-28"
output: html_document
---Part I: Converting t-Values to Partial Correlations
Gizem wrote: “We have 1 study having correlation data between GSH and Total symptom and functioning, but the study did not indicate their r values, which we are using r values to do meta-analysis. They gave t and F values with p value. I am wondering if you can convert these F and t values into r values to be used into our meta-analysis.”
Below we use the data reported in the Ballesteros et al. (2013) paper, specifically the t‐values for GSH and GSSG each predicting UPSA‐2 (a measure of everyday functioning) in the same sample. Note that using two effect sizes from a single paper is not a typical “meta‐analysis” in the standard sense (since a meta‐analysis usually combines results from multiple distinct samples). However, the following example demonstrates how to:
- Convert the reported t‐values into partial correlations
(given known residual degrees of freedom and sample size).
- Convert those partial r values to Fisher’s z.
- Compute a simple random‐effects meta‐analysis (with
k=2 effect sizes).
- Interpret the results.
1. Load Package
# Uncomment to install if needed:
# install.packages("metafor")
library(metafor)Loading required package: MatrixLoading required package: metadatLoading required package: numDeriv
Loading the 'metafor' package (version 4.6-0). For an
introduction to the package please type: help(metafor)2. Define Helper Functions
# Convert (t_value, df_resid) to partial correlation
convert_t_to_partial_r <- function(t_value, df_resid) {
r_mag <- sqrt(t_value^2 / (t_value^2 + df_resid))
r_signed <- sign(t_value) * r_mag
return(r_signed)
}
# Convert correlation (r) to Fisher’s z
r_to_z <- function(r) {
0.5 * log((1 + r)/(1 - r))
}
# Approximate standard error of Fisher’s z for correlation
# (Strictly for zero-order correlations, used as an approximation for partial r.)
z_se <- function(n) {
1 / sqrt(n - 3)
}3. The Ballesteros Data
From the article (Ballesteros et al., 2013), we know:
- Residual df in that regression = 49
- Sample size (N) for that analysis ≈ 52
- t-values for each partial correlation:
| Measure | t-value | df_resid | n_total |
|---|---|---|---|
| GSH vs. UPSA-2 | +2.08 | 49 | 52 |
| GSSG vs. UPSA-2 | -2.22 | 49 | 52 |
df_ballesteros <- data.frame(
measure = c("GSH", "GSSG"),
t_value = c( 2.08, -2.22),
df_resid = c(49, 49),
n_total = c(52, 52)
)
df_ballesteros measure t_value df_resid n_total
1 GSH 2.08 49 52
2 GSSG -2.22 49 524. Compute Partial r, Fisher’s z, and SE
# 1) Partial r
df_ballesteros$partial_r <- mapply(
convert_t_to_partial_r,
df_ballesteros$t_value,
df_ballesteros$df_resid
)
# 2) Convert partial r to Fisher’s z
df_ballesteros$z <- sapply(df_ballesteros$partial_r, r_to_z)
# 3) Approximate SE for z
df_ballesteros$z_se <- sapply(df_ballesteros$n_total, z_se)
df_ballesteros measure t_value df_resid n_total partial_r z z_se
1 GSH 2.08 49 52 0.2848342 0.2929353 0.1428571
2 GSSG -2.22 49 52 -0.3023042 -0.3120536 0.1428571Interpretation: - partial_r: the
partial correlations.
- z: Fisher’s z.
- z_se: approximate standard error of z.
5. Random-Effects Meta-Analysis (k=2)
We will illustrate the metafor approach:
res <- rma(yi = df_ballesteros$z,
sei = df_ballesteros$z_se,
method = "REML")
summary(res)
Random-Effects Model (k = 2; tau^2 estimator: REML)
logLik deviance AIC BIC AICc
-0.5698 1.1396 5.1396 1.1396 17.1396
tau^2 (estimated amount of total heterogeneity): 0.1626 (SE = 0.2588)
tau (square root of estimated tau^2 value): 0.4032
I^2 (total heterogeneity / total variability): 88.85%
H^2 (total variability / sampling variability): 8.97
Test for Heterogeneity:
Q(df = 1) = 8.9673, p-val = 0.0027
Model Results:
estimate se zval pval ci.lb ci.ub
-0.0096 0.3025 -0.0316 0.9748 -0.6024 0.5833
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Interpretation
- Estimate: The pooled z‐value for these two
partial correlations.
- tau^2, I^2: With k=2, these heterogeneity statistics must be interpreted very carefully.
6. Back-Transform to Correlation
predict(res, transf = transf.ztor)
pred ci.lb ci.ub pi.lb pi.ub
-0.0096 -0.5388 0.5251 -0.7606 0.7524 This yields: - pred: the mean partial correlation.
- ci.lb, ci.ub: the 95% CI in the correlation
metric.
7. Forest Plot
forest(
x = df_ballesteros$z,
sei = df_ballesteros$z_se,
slab = df_ballesteros$measure,
transf = transf.ztor,
xlab = "Partial correlation (r)",
main = "Forest Plot of Partial r (Ballesteros et al. 2013)"
)
- This shows the two partial correlations (GSH vs. UPSA‐2, GSSG vs. UPSA‐2) and a pooled diamond.
Discussion (Part I)
- Partial Correlations
- Based on t = 2.08 with df=49, partial r ~ 0.28
(positive).
- Based on t = −2.22 with df=49, partial r ~ −0.30 (negative).
- Based on t = 2.08 with df=49, partial r ~ 0.28
(positive).
- Significance
- The paper reported each predictor was significant in the model (p < 0.05).
Conclusions (Part I)
- We converted t‐values from Ballesteros et al. (2013) into
partial correlations for two relationships (GSH → functioning, GSSG →
functioning).
- GSH and GSSG showed partial r ≈ +0.28 and −0.30 with
UPSA‐2, respectively, in that single study’s regression.
- We used metafor to combine these into an overall effect size purely for demonstration.
References
- Ballesteros, A., et al. (2013). No Evidence of Exogenous Origin
for the Abnormal Glutathione Redox State in Schizophrenia.
Schizophrenia Research, 146, 184–189.
- Viechtbauer, W. (2010). Conducting meta-analyses in R with the
metafor package. Journal of Statistical
Software, 36(3), 1–48.
Part II: Converting Overall F to a Multiple R
Below is a step-by-step explanation of how to convert the overall model F(2,26)=0.30 into an \(r\) value (i.e., the multiple correlation \(R\) of the full two‐predictor model). However, this approach does not provide a unique correlation (partial or zero‐order) for GSH alone, because the paper only gives us the overall F for both GSH and GSSG together. If you specifically want the partial correlation between GSH and total symptoms (controlling for GSSG) or the zero‐order correlation for GSH alone, the paper would have needed to report an individual t (or Pearson \(r\)) for GSH vs. total symptoms. Only the overall F for the 2‐predictor model is given.
1. Converting the Overall F to a Multiple R
For a multiple regression with \(p\) predictors and \(\mathrm{df}_{\text{resid}}\) residual degrees of freedom, the relationship between F (the overall model test), \(R^2\) (the multiple coefficient of determination), and sample size is:
\[ F = \frac{(R^2 / p)}{\bigl[(1 - R^2) / (n - p - 1)\bigr]} \quad\Longrightarrow\quad R^2 = \frac{p \cdot F}{\,p \cdot F \;+\;\mathrm{df}_{\text{resid}}\,}. \]
Here:
- \(p = 2\) (two predictors: GSH and
GSSG)
- \(\mathrm{df}_{\text{resid}} =
26\)
- Reported: \(F(2,26) = 0.30, p=0.74\)
Thus:
\[ R^2 = \frac{2 \times 0.30}{(2 \times 0.30) + 26} = \frac{0.60}{26.60} \approx 0.0226 \]
Hence the overall multiple correlation \(R\) is:
\[ R = \sqrt{0.0226} \approx 0.15 \]
This indicates the entire 2‐predictor model (GSH + GSSG) has \(R \approx 0.15\) with BPRS total severity—consistent with the non-significant p=0.74.
2. Converting \(R\) to Fisher’s z
Often in meta‐analysis we prefer the Fisher’s z form of \(R\):
\[ z = \frac{1}{2} \ln\!\Bigl(\frac{1+R}{1-R}\Bigr) \]
With \(R = 0.15\):
\[ z = 0.5 \,\times\, \ln\bigl(1.15/0.85\bigr) \approx 0.5 \,\times\, 0.302 = 0.151 \]
Thus, the overall model correlation in Fisher’s z is ~0.15.
3. Critical Caveat: No GSH‐Specific r
- The only statistic we have from the paper for BPRS total is
the overall F for GSH + GSSG together.
- This F → overall multiple correlation \(R
\approx 0.15\), not the correlation for GSH
alone (nor for GSSG alone).
- If you need “GSH vs. total symptom” correlation, you would need the t (partial or zero‐order) for GSH alone or the raw Pearson \(r\). The paper only reports the overall F.
Bottom Line:
- We can convert F(2,26)=0.30 to \(R\approx
0.15\).
- That lumps both predictors into the model’s overall correlation.
4. Example R Snippet
Below is optional R code implementing the overall \(R^2\) formula, converting it to \(R\) and then to Fisher’s z.
#############################
## A. Fill in the known values
#############################
p <- 2 # number of predictors
f_value <- 0.30
df_resid <- 26
#############################
## B. Compute R^2 from F
#############################
r_sq <- (p * f_value) / ((p * f_value) + df_resid)
cat("Overall R^2 =", r_sq, "\n")Overall R^2 = 0.02255639 # Overall R:
R <- sqrt(r_sq)
cat("Overall R =", R, "\n")Overall R = 0.1501879 #############################
## C. Convert R to Fisher z
#############################
z <- 0.5 * log((1 + R)/(1 - R))
z[1] 0.1513326When you run this, you get:
- R^2 = 0.0226
- R = 0.15
- z = 0.151
Part III: DL vs. REML: Explaining Small Discrepancies
Below we discuss why Gamze sees a small discrepancy between her Stata result (r = –0.175, p = 0.104) and an R result (r = –0.166, p = 0.0655). Essentially, Stata’s default random‐effects meta‐analysis uses the DerSimonian–Laird (DL) method to estimate between‐study variance, whereas metafor in R defaults to REML. Different estimation methods for \(\tau^2\) produce slightly different pooled estimates and p‐values.
We can replicate Gamze’s Stata result by setting
method = "DL" in metafor. That yields
r ≈ –0.175, p ≈ 0.104. If we stick with the default
(method = "REML"), we get r ≈ –0.166, p ≈ 0.066.
Both are legitimate; they differ only in how \(\tau^2\) is computed. Choose
one and report it consistently.
Introduction
Gamze’s Stata result: - Pooled correlation = –0.175
(p = 0.104).
- This uses DerSimonian–Laird (DL), Stata’s default for random‐effects
meta‐analysis.
R’s metafor result (default): - Pooled correlation =
–0.166 (p = 0.066).
- Uses REML (Restricted Maximum Likelihood).
We illustrate how to replicate both results in R:
1) Create the Data Frame
Imagine five studies reporting GSH–symptom correlations:
# Example data frame with 5 studies
df <- data.frame(
author = c("Matsuzawa et al. 2008",
"Reyes-Madrigal et al. 2019",
"Iwata et al. 2021",
"Lesh et al. 2021",
"Ravanfar et al. 2022"),
r = c(-0.41, 0.42, -0.08, -0.293, -0.286),
n = c(20, 10, 67, 33, 12)
)
df author r n
1 Matsuzawa et al. 2008 -0.410 20
2 Reyes-Madrigal et al. 2019 0.420 10
3 Iwata et al. 2021 -0.080 67
4 Lesh et al. 2021 -0.293 33
5 Ravanfar et al. 2022 -0.286 122) Fisher‐Transform and Variances
df$z <- 0.5 * log((1 + df$r) / (1 - df$r))
df$var_z <- 1 / (df$n - 3)
df author r n z var_z
1 Matsuzawa et al. 2008 -0.410 20 -0.43561122 0.05882353
2 Reyes-Madrigal et al. 2019 0.420 10 0.44769202 0.14285714
3 Iwata et al. 2021 -0.080 67 -0.08017133 0.01562500
4 Lesh et al. 2021 -0.293 33 -0.30184486 0.03333333
5 Ravanfar et al. 2022 -0.286 12 -0.29420447 0.111111113) Random‐Effects with DerSimonian–Laird (DL)
library(metafor)
res_dl <- rma(yi = df$z,
vi = df$var_z,
data = df,
method = "DL") # DerSimonian-Laird
summary(res_dl)
Random-Effects Model (k = 5; tau^2 estimator: DL)
logLik deviance AIC BIC AICc
-0.0860 5.5191 4.1721 3.3910 10.1721
tau^2 (estimated amount of total heterogeneity): 0.0124 (SE = 0.0427)
tau (square root of estimated tau^2 value): 0.1113
I^2 (total heterogeneity / total variability): 20.71%
H^2 (total variability / sampling variability): 1.26
Test for Heterogeneity:
Q(df = 4) = 5.0451, p-val = 0.2827
Model Results:
estimate se zval pval ci.lb ci.ub
-0.1750 0.1077 -1.6246 0.1042 -0.3862 0.0361
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- The pooled effect size in z units is around –0.1750 with a p‐value near 0.104.
Back‐transform to r:
pred_dl <- predict(res_dl, transf=transf.ztor)
cat("DerSimonian-Laird (DL) overall correlation:\n")DerSimonian-Laird (DL) overall correlation:print(pred_dl)
pred ci.lb ci.ub pi.lb pi.ub
-0.1733 -0.3681 0.0361 -0.4452 0.1279 4) Random‐Effects with REML (metafor’s Default)
res_reml <- rma(yi = df$z,
vi = df$var_z,
data = df,
method = "REML")
summary(res_reml)
Random-Effects Model (k = 5; tau^2 estimator: REML)
logLik deviance AIC BIC AICc
-0.5464 1.0928 5.0928 3.8654 17.0928
tau^2 (estimated amount of total heterogeneity): 0.0013 (SE = 0.0292)
tau (square root of estimated tau^2 value): 0.0359
I^2 (total heterogeneity / total variability): 2.64%
H^2 (total variability / sampling variability): 1.03
Test for Heterogeneity:
Q(df = 4) = 5.0451, p-val = 0.2827
Model Results:
estimate se zval pval ci.lb ci.ub
-0.1678 0.0911 -1.8419 0.0655 -0.3464 0.0108 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- The overall effect is around –0.1678 with p ≈ 0.066.
Back‐transform to r:
pred_reml <- predict(res_reml, transf=transf.ztor)
cat("REML overall correlation:\n")REML overall correlation:print(pred_reml)
pred ci.lb ci.ub pi.lb pi.ub
-0.1662 -0.3331 0.0108 -0.3449 0.0241 5) Forest Plot (DL Example)
forest(res_dl,
slab = df$author,
transf = transf.ztor,
xlab = "Correlation (r)",
mlab = "Pooled Effect (DL)",
main = "Random-Effects (DerSimonian-Laird)")
Conclusion (Part III)
- Stata’s default (DL) → r ≈ –0.175, p ≈
0.104.
- metafor default (REML) → r ≈ –0.166, p ≈
0.066.
- Both are valid methods for estimating random‐effects variance; the
difference is purely in the \(\tau^2\)
estimator.
- Decide on one and report it clearly (and consistently) in your Methods.
Final Note
Across these three sections:
- We saw how to convert t‐values to partial
correlations (Part I).
- We demonstrated how an overall F test only yields a
multiple R, not a predictor‐specific r (Part
II).
- We highlighted how different random‐effects estimators (DL vs. REML) can produce small differences in pooled effect estimates (Part III).