Wednesday, May 28, 2025

Meta-analysis using Correlation (r) in Microsoft Excel, and Online calculator (Excel-based)

Meta-analysis using Correlation (r) Usman Zafar Paracha 2.1108 Usman Zafar Paracha Usman Zafar Paracha xᵢ, yᵢ: Individual data points for variables X and Y xᵢ, yᵢ: Individual data points for variables X and Y x̄, ȳ: M... xᵢ, yᵢ: Individual data points for variables X and Yx̄, ȳ: Mean of X and Y respectively Correlation (r) Correlation (r) Correlation (r) r = (Σ(xᵢ - x̄)(yᵢ - ȳ)) / √[Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²] r = (Σ(xᵢ - x̄)(yᵢ - ȳ)) / √[Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²] r = (Σ(xᵢ - x̄)(yᵢ - ȳ)) / √[Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²] formula formula formula Correlation coefficient from each study, ranging from -1 to 1. Correlation coefficient from each study, ranging from -1 to 1. Correlation coefficient from each study, ranging from -1 to 1. illustration Patreon and LinkedIn links LinkedIn Profile /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile /uzparacha /uzparacha /uzparacha Patreon Then then then Usman Zafar Paracha 1 Usman Zafar Paracha Usman Zafar Paracha example example example Usman Zafar Paracha 2 Usman Zafar Paracha Usman Zafar Paracha Suppose we have this data Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.995 Usman Zafar Paracha Usman Zafar Paracha If the correlation (r) is between: If the correlation (r) is between: -1.0 to -0.7, → then it is... If the correlation (r) is between: -1.0 to -0.7, → then it is a Strong Negative Correlation; -0.7 to -0.3, → then it is a Moderate Negative Correlation; -0.3 to 0, → then it is a Weak Negative Correlation;0 to 0.3, → then it is a Weak Positive Correlation;0.3 to 0.7, → then it is a Moderate Positive Correlation; and0.7 to 1.0, → then it is a Strong Positive Correlation Usman Zafar Paracha 4 Usman Zafar Paracha Usman Zafar Paracha r: Correlation coefficient from each study r: Correlation coefficient from each study ln: Natural logari... r: Correlation coefficient from each studyln: Natural logarithm Fisher’s z Fisher’s z Fisher’s z z = 0.5 × ln((1 + r) / (1 - r)) z = 0.5 × ln((1 + r) / (1 - r)) z = 0.5 × ln((1 + r) / (1 - r)) formula.1012 formula formula Fisher’s z transformation of correlation to stabilize variance. Fisher’s z transformation of correlation to stabilize varianc... Fisher’s z transformation of correlation to stabilize variance.Converts r to z to stabilize variance for meta-analysis illustration.1015 Patreon and LinkedIn links.1016 LinkedIn Profile.1017 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1019 /uzparacha /uzparacha /uzparacha Patreon Then.1021 then then Usman Zafar Paracha 1.1022 Usman Zafar Paracha Usman Zafar Paracha example.1023 example example Usman Zafar Paracha 2.1024 Usman Zafar Paracha Usman Zafar Paracha Suppose we have this data.1025 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1026 Usman Zafar Paracha Usman Zafar Paracha These values: These values: - Normalize the distribution of correlation coe... These values:- Normalize the distribution of correlation coefficients (r), especially when conducting meta-analysis or calculating confidence intervals.- Make the sampling distribution of correlations more normally distributed, which improves accuracy in statistical inference.- Enables accurate weighted averages of correlations- Allows calculation of confidence intervals and Z-tests Usman Zafar Paracha 4.1028 Usman Zafar Paracha Usman Zafar Paracha Page break 1 n: Sample size of the study n: Sample size of the study n: Sample size of the study Variance of z Variance of z Variance of z 1 / (n - 3) 1 / (n - 3) 1 / (n - 3) formula.1054 formula formula The variance of Fisher’s z-transformed correlation coefficient... The variance of Fisher’s z-transformed correlation coefficien... The variance of Fisher’s z-transformed correlation coefficient...It is smaller when sample size is larger (more precision), i.e.,The variance of z decreases with larger sample sizes illustration.1057 Patreon and LinkedIn links.1058 LinkedIn Profile.1059 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1061 /uzparacha /uzparacha /uzparacha Patreon Then.1063 then then Usman Zafar Paracha 1.1064 Usman Zafar Paracha Usman Zafar Paracha example.1065 example example Usman Zafar Paracha 2.1066 Usman Zafar Paracha Usman Zafar Paracha Suppose we have this data.1067 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1068 Usman Zafar Paracha Usman Zafar Paracha If Variance ≈ 0.01786 If Variance ≈ 0.01786 ➝ Interpretation: Relatively less preci... If Variance ≈ 0.01786 ➝ Interpretation: Relatively less precise estimate ➝ Likely Reason: Smaller sample sizeIf Variance ≈ 0.00366 ➝ Interpretation: More precise estimate ➝ Likely Reason: Larger sample sizeIf Variance ≈ 0.00209 ➝ Interpretation: Very precise estimate ➝ Likely Reason: Large sample size and high confidence Usman Zafar Paracha 4.1070 Usman Zafar Paracha Usman Zafar Paracha Page break 2 Variance of z: the variance of Fisher’s z-transformed correlation coefficient Variance of z: the variance of Fisher’s z-transformed correla... Variance of z: the variance of Fisher’s z-transformed correlation coefficient Weight Weight Weight 1 / variance of z 1 / variance of z 1 / variance of z formula.1075 formula formula Inverse of variance - Inverse of variance - Reflects importance of each study in po... Inverse of variance -Reflects importance of each study in pooled analysis.A lower variance (i.e., more precise estimate) gives a higher weight.The weight tells how much influence a study has in the final combined estimate. illustration.1078 Patreon and LinkedIn links.1079 LinkedIn Profile.1080 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1082 /uzparacha /uzparacha /uzparacha Patreon Then.1084 then then Usman Zafar Paracha 1.1085 Usman Zafar Paracha Usman Zafar Paracha example.1086 example example Usman Zafar Paracha 2.1087 Usman Zafar Paracha Usman Zafar Paracha Suppose we have this data.1088 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1089 Usman Zafar Paracha Usman Zafar Paracha Study 1: Weight = 56 (less precise) Study 1: Weight = 56 (less precise) Study 2: Weight = 273 (mo... Study 1: Weight = 56 (less precise)Study 2: Weight = 273 (more precise than Study 1)Study 3: Weight = 478 (most precise; contributes the most to pooled estimate)A large, well-conducted study with a small variance will carry more weight and affect the pooled result more than a small study with high uncertainty. Usman Zafar Paracha 4.1091 Usman Zafar Paracha Usman Zafar Paracha Page break 3 Weight: 1 / variance of z (for each study) Weight: 1 / variance of z (for each study) Fisher’s z: Fisher... Weight: 1 / variance of z (for each study)Fisher’s z: Fisher’s z transformation of correlation to stabilize variance (for each study) Weight * Fisher's z Weight * Fisher's z Weight * Fisher's z weight * Fisher’s z weight * Fisher’s z weight * Fisher’s z formula.1096 formula formula Product of weight and Fisher’s z; contributes to the overall weighted average, i.e., Used to compute the weighted average. Product of weight and Fisher’s z; contributes to the overall ... Product of weight and Fisher’s z; contributes to the overall weighted average, i.e., Used to compute the weighted average.It shows the contribution of each individual study to the overall meta-analytic effect size. illustration.1099 Patreon and LinkedIn links.1100 LinkedIn Profile.1101 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1103 /uzparacha /uzparacha /uzparacha Patreon Then.1105 then then example.1107 example example Suppose we have this data.1109 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1110 Usman Zafar Paracha Usman Zafar Paracha Study 1 has a negative Fisher’s z (likely a negative correlation), and contributes -16.11 to the sum. Study 1 has a negative Fisher’s z (likely a negative correlat... Study 1 has a negative Fisher’s z (likely a negative correlation), and contributes -16.11 to the sum.Studies 2 and 3 contribute positively, with larger values suggesting stronger or more heavily weighted correlations. Usman Zafar Paracha 4.1112 Usman Zafar Paracha Usman Zafar Paracha Page break 4 Usman Zafar Paracha 2.1114 Usman Zafar Paracha Usman Zafar Paracha Weight: 1 / variance of z (for each study).1115 Weight: 1 / variance of z (for each study) Fisher’s z: Fisher... Weight: 1 / variance of z (for each study)Fisher’s z: Fisher’s z transformation of correlation to stabilize variance (for each study) Weighted z sum Weighted z sum Weighted z sum Sum of Weight * Fisher's z Sum of Weight * Fisher's z Sum of Weight * Fisher's z formula.1118 formula formula It shows that total impact of each study's Fisher z, considering its weight. It shows that total impact of each study's Fisher z, consider... It shows that total impact of each study's Fisher z, considering its weight. illustration.1121 Patreon and LinkedIn links.1122 LinkedIn Profile.1123 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1125 /uzparacha /uzparacha /uzparacha Patreon Then.1127 then then example.1128 example example Suppose we have this data.1129 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1130 Usman Zafar Paracha Usman Zafar Paracha It reflects the cumulative, weighted effect of all studies in the meta-analysis—before averaging and back-transforming to the correlation scale. It reflects the cumulative, weighted effect of all studies in... It reflects the cumulative, weighted effect of all studies in the meta-analysis—before averaging and back-transforming to the correlation scale. Usman Zafar Paracha 4.1132 Usman Zafar Paracha Usman Zafar Paracha Page break 5 Usman Zafar Paracha 2.1135 Usman Zafar Paracha Usman Zafar Paracha Weight: 1 / variance of z (for each study).1136 Weight: 1 / variance of z (for each study) Weight: 1 / variance of z (for each study) Weight sum Weight sum Weight sum Sum of weights across all studies Sum of weights across all studies Sum of weights across all studies formula.1139 formula formula Total cumulative weight from all studies Total cumulative weight from all studies Total cumulative weight from all studies illustration.1141 Patreon and LinkedIn links.1142 LinkedIn Profile.1143 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1145 /uzparacha /uzparacha /uzparacha Patreon Then.1147 then then example.1148 example example Suppose we have this data.1149 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1150 Usman Zafar Paracha Usman Zafar Paracha 807 represents the total weight; higher weight studies have more influence on the overall result. 807 represents the total weight; higher weight studies have m... 807 represents the total weight; higher weight studies have more influence on the overall result. Usman Zafar Paracha 4.1152 Usman Zafar Paracha Usman Zafar Paracha Page break 6 Weighted z sum: Sum of Weight * Fisher's z Weighted z sum: Sum of Weight * Fisher's z Weight sum: Sum of... Weighted z sum: Sum of Weight * Fisher's zWeight sum: Sum of weights across all studies Weighted Mean of z Weighted Mean of z Weighted Mean of z Weighted z sum / Weight sum Weighted z sum / Weight sum Weighted z sum / Weight sum formula.1158 formula formula It is Pooled Fisher’s z value from all studies It is pooled Fisher’s z value from all studies It is pooled Fisher’s z value from all studies illustration.1160 Patreon and LinkedIn links.1161 LinkedIn Profile.1162 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1164 /uzparacha /uzparacha /uzparacha Patreon Then.1166 then then example.1167 example example Suppose we have this data.1168 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1169 Usman Zafar Paracha Usman Zafar Paracha This shows the pooled Fisher's z value, which is 0.5521. This shows the pooled Fisher's z value, which is 0.5521. This shows the pooled Fisher's z value, which is 0.5521. Usman Zafar Paracha 4.1171 Usman Zafar Paracha Usman Zafar Paracha Page break 7 Excel example data Excel Example Data correlation Excel example data.240 Excel Example Fishers z Excel example data.242 Excel Example variance of z Excel example data.244 Excel example weight Excel example data.246 Excel Example weight and fishers z Excel example data.248 Excel Example weight z sum Excel example data.250 Excel Example weight sum Excel example data.252 Excel Example weighted mean of z e: Euler’s number (~2.71828), base of the natural logarithm e: Euler’s number (~2.71828), base of the natural logarithm z... e: Euler’s number (~2.71828), base of the natural logarithmz: Weighted mean of Fisher’s z Back-transform Fisher’s z to r Back-transform Fisher’s z to r Back-transform Fisher’s z to r r = (e^(2z) - 1)/(e^(2z) + 1) r = (e^(2z) - 1)/(e^(2z) + 1) r = (e^(2z) - 1)/(e^(2z) + 1) formula.1183 formula formula It is pooled correlation estimate It is pooled correlation estimate It is pooled correlation estimate illustration.1185 Patreon and LinkedIn links.1186 LinkedIn Profile.1187 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1189 /uzparacha /uzparacha /uzparacha Patreon Then.1191 then then example.1192 example example Suppose we have this data.1193 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1194 Usman Zafar Paracha Usman Zafar Paracha In this case, the correlation is moderate positive: 0.50. In this case, the correlation is moderate positive: 0.50. In this case, the correlation is moderate positive: 0.50. Usman Zafar Paracha 4.1196 Usman Zafar Paracha Usman Zafar Paracha Page break 8 Excel example data.1198 Excel Example z to r Weight sum: Sum of weights across all studies Weight sum: Sum of weights across all studies Weight sum: Sum of weights across all studies Standard Error of Mean z Standard Error of Mean z Standard Error of Mean z =SQRT(1/Weight sum) =SQRT(1/Weight sum) =SQRT(1/Weight sum) formula.1204 formula formula Standard error of the weighted average z Standard error of the weighted average z Standard error of the weighted average z illustration.1206 Patreon and LinkedIn links.1207 LinkedIn Profile.1208 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1211 /uzparacha /uzparacha /uzparacha Patreon Then.1214 then then example.1215 example example Suppose we have this data.1216 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1217 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 4.1219 Usman Zafar Paracha Usman Zafar Paracha Page break 9 Excel example data.1221 Excel Example SE mean r Standard error is a measure of how much the sample mean z is expected to vary if you repeated the sampling many times. Standard error is a measure of how much the sample mean z is ... Standard error is a measure of how much the sample mean z is expected to vary if you repeated the sampling many times.The "mean z" refers to the average value of some variable z across your sample or study.A standard error of 0.0352 means the estimate of the mean z is quite precise, with an expected variability of about 0.0352 around the true population mean, i.e., If you took many samples and computed the mean z for each, those sample means would typically vary by about 0.0352. Weighted mean of Fisher’s z, Weighted mean of Fisher’s z, Standard error of the weighted m... Weighted mean of Fisher’s z,Standard error of the weighted mean z Z statistic Z statistic Z statistic Weighted mean of Fisher’s z / Standard error of the weighted mean z Weighted mean of Fisher’s z / Standard error of the weighted ... Weighted mean of Fisher’s z / Standard error of the weighted mean z formula.1229 formula formula Test statistic = Mean z / SE; very high, suggesting significance. Test statistic = Mean z / SE; High value, suggest significance. Test statistic = Mean z / SE; High value, suggest significance. illustration.1231 Patreon and LinkedIn links.1232 LinkedIn Profile.1233 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1236 /uzparacha /uzparacha /uzparacha Patreon Then.1239 then then example.1240 example example Suppose we have this data.1241 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1242 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 4.1243 Usman Zafar Paracha Usman Zafar Paracha Page break 10 Excel example data.1245 A Z statistic of 15.6835 is very large in absolute terms. A Z statistic of 15.6835 is very large in absolute terms. Suc... A Z statistic of 15.6835 is very large in absolute terms.Such a high Z value strongly suggests that the result is highly statistically significant — meaning the null hypothesis is almost certainly false under conventional significance levels Excel example Z Weighted mean of Fisher’s z,.1250 Weighted mean of Fisher’s z, Standard error of the weighted m... Weighted mean of Fisher’s z,Standard error of the weighted mean z 95% CI for z (Lower & Upper) 95% CI for z (Lower & Upper) 95% CI for z (Lower & Upper) Lower = (Weighted mean of Fisher’s z) - 1.96 × (Standard error of the weighted mean z) Lower = (Weighted mean of Fisher’s z) - 1.96 × (Standard erro... Lower = (Weighted mean of Fisher’s z) - 1.96 × (Standard error of the weighted mean z)Upper = (Weighted mean of Fisher’s z) + 1.96 × (Standard error of the weighted mean z) formula.1253 formula formula Confidence interval for mean z Confidence interval for mean z Confidence interval for mean z illustration.1255 Patreon and LinkedIn links.1256 LinkedIn Profile.1257 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1260 /uzparacha /uzparacha /uzparacha Patreon Then.1263 then then example.1264 example example Suppose we have this data.1265 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1266 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 4.1267 Usman Zafar Paracha Usman Zafar Paracha Page break 11 Excel example data.1269 We are 95% confident that the true population mean of z lies between 0.4831 and 0.6211. We are 95% confident that the true population mean of z lies ... We are 95% confident that the true population mean of z lies between 0.4831 and 0.6211.95% confidence interval means that if we repeat the sampling process many times, 95% of those confidence intervals would contain the true mean of the variable z. Excel example CI e: Euler’s number (~2.71828), base of the natural logarithm.1274 e: Euler’s number (~2.71828), base of the natural logarithm L... e: Euler’s number (~2.71828), base of the natural logarithmLower: 95% CI for z (Lower)Upper: 95% CI for z (Upper) Back-transform CI to r (Lower & Upper) Back-transform CI to r (Lower & Upper) Back-transform CI to r (Lower & Upper) r_lower = (e^(2 × Lower) - 1) / (e^(2 × Lower) + 1) r_lower = (e^(2 × Lower) - 1) / (e^(2 × Lower) + 1) r_upper =... r_lower = (e^(2 × Lower) - 1) / (e^(2 × Lower) + 1)r_upper = (e^(2 × Upper) - 1) / (e^(2 × Upper) + 1) formula.1277 formula formula Final 95% CI for pooled correlation estimate Final 95% CI for pooled correlation estimate Final 95% CI for pooled correlation estimate illustration.1279 Patreon and LinkedIn links.1280 LinkedIn Profile.1281 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1284 /uzparacha /uzparacha /uzparacha Patreon Then.1287 then then example.1288 example example Suppose we have this data.1289 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1290 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 4.1291 Usman Zafar Paracha Usman Zafar Paracha Page break 12 Excel example data.1293 The CI values of 0.4487 to 0.5519 represent the range of plausible values for the true population correlation after back-transforming from Fisher’s z. The CI values of 0.4487 to 0.5519 represent the range of plau... The CI values of 0.4487 to 0.5519 represent the range of plausible values for the true population correlation after back-transforming from Fisher’s z.It shows that we are 95% confident the true correlation lies between 0.45 and 0.55 Excel Example Back to r Page break 13
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