Sunday, June 8, 2025

Meta-analysis using Odds Ratio (OR) in Microsoft Excel, and Online calculator (Excel-based)

Meta-analysis using Odds Ratio Usman Zafar Paracha 2.1108 Usman Zafar Paracha Usman Zafar Paracha 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 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 OR = 1: No association between exposure and outcome; OR < 1: Lower odds; and OR > 1: Higher odds. If OR = 1: No association between exposure and outcome; OR < ... If OR = 1: No association between exposure and outcome; OR < 1: Lower odds; and OR > 1: Higher odds.In the first two studies, the odds are lower, and in the second two studies odds are higher. SE shows Standard Error SE shows Standard Error a shows exposure and outcome, c shows... SE shows Standard Errora shows exposure and outcome,c shows non-exposure and outcome,b shows exposure and no outcome,d shows no exposure and no outcome Variance Variance Variance SE = (1/a) + (1/b) + (1/c) + (1/d) SE = (1/a) + (1/b) + (1/c) + (1/d) Variance = SE² SE = (1/a) + (1/b) + (1/c) + (1/d)Variance = SE² formula.1012 formula formula Variance measures the spread or uncertainty in the estimated effect size. A smaller variance means more precision, and a larger variance means less precision. Variance measures the spread or uncertainty in the estimated ... Variance measures the spread or uncertainty in the estimated effect size. A smaller variance means more precision, and a larger variance means less precision. 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 Suppose we have this data.1025 Suppose we have this data Suppose we have this data Smaller variance means more precise (less uncertain / less variable) effect size estimate. The meta-analysis "trusts" or "relies on" more precise studies more heavily. The studies with smaller variances will carry more weight in the combined estimate of Smaller variance means more precise (less uncertain / less va... Smaller variance means more precise (less uncertain / less variable) effect size estimate. The meta-analysis "trusts" or "relies on" more precise studies more heavily. The studies with smaller variances will carry more weight in the combined estimate of effect size.Larger variance means less precise (more uncertain / more variable) effect size estimate. The studies with larger variances will carry less weight. Usman Zafar Paracha 4.1028 Usman Zafar Paracha Usman Zafar Paracha Page Break 2 w_i is the weight assigned to the ith study, w_i is the weight assigned to the ith study, Var is the varia... w_i is the weight assigned to the ith study,Var is the variance of the effect size of the study Weight for Each Study Weight for Each Study Weight for Each Study w_i = 1 / Var w_i = 1 / Var w_i = 1 / Var formula.1054 formula formula It shows weight for each study. It shows weight for each study. In this case, inverse-varianc... It shows weight for each study.In this case, inverse-variance weighting method has been used as it gives more weight to studies with smaller variance (i.e., more precise estimates). 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 Higher weight (e.g., 49.78) means the study’s estimate has a smaller variance (more precise), so it will have a greater impact on the combined or pooled effect size. This study (Study 4)is the most influential study and contributes approximately more tha Higher weight (e.g., 49.78) means the study’s estimate has a ... Higher weight (e.g., 49.78) means the study’s estimate has a smaller variance (more precise), so it will have a greater impact on the combined or pooled effect size. This study (Study 4)is the most influential study and contributes approximately more than four times than that of Study 1 (11.61).Lower weight (e.g., 11.61) means the study’s estimate is less precise (larger variance), so it has less influence on the pooled estimate. Usman Zafar Paracha 4.1070 Usman Zafar Paracha Usman Zafar Paracha Page Break 3 ln(OR)_i: natural log of odds ratio in study I ln(OR)_i: natural log of odds ratio in study I w_i: the weigh... ln(OR)_i: natural log of odds ratio in study Iw_i: the weight of the i-th study (usually the inverse of the variance). Weighted Effect Weighted Effect Weighted Effect w_i * ln(OR)_i w_i * ln(OR)_i w_i * ln(OR)_i formula.1075 formula formula It refers to the contribution of each study’s effect size to the pooled (overall) effect, adjusted by its weight. It refers to the contribution of each study’s effect size to ... It refers to the contribution of each study’s effect size to the pooled (overall) effect, adjusted by its weight. 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 Negative outcomes favor control (e.g., worse outcome), and Positive outcomes favor treatment (e.g., better outcome). Negative outcomes favor control (e.g., worse outcome), and Po... Negative outcomes favor control (e.g., worse outcome), and Positive outcomes favor treatment (e.g., better outcome).Larger absolute values show greater influence on the overall meta-analytic effect size. Usman Zafar Paracha 4.1091 Usman Zafar Paracha Usman Zafar Paracha Page Break 4 ln(OR)_i: natural log of odds ratio in study i.1093 ln(OR)_i: natural log of odds ratio in study i w_i: the weigh... ln(OR)_i: natural log of odds ratio in study iw_i: the weight of the i-th study (usually the inverse of the variance). Pooled Effect Size (Fixed Effects Model) Pooled Effect Size (Fixed Effects Model) Pooled Effect Size (Fixed Effects Model) Pooled ln(OR)= ∑ (w_i * ln(OR)_i) / ∑ w_i Pooled ln(OR)= ∑ (w_i * ln(OR)_i) / ∑ w_i Pooled ln(OR)= ∑ (w_i * ln(OR)_i) / ∑ w_i formula.1096 formula formula The Pooled Effect Size (Fixed Effects Model) is a way to combine results from multiple studies (typically in a meta-analysis) into a single summary effect size, assuming that all studies share a common true effect size. That is, the variation between stu The Pooled Effect Size (Fixed Effects Model) is a way to comb... The Pooled Effect Size (Fixed Effects Model) is a way to combine results from multiple studies (typically in a meta-analysis) into a single summary effect size, assuming that all studies share a common true effect size. That is, the variation between studies is assumed to be due only to sampling error (not due to true differences in effect sizes across studies). 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 Log-transformed average of effect sizes from all studies Log-transformed average of effect sizes from all studies Log-transformed average of effect sizes from all studies Usman Zafar Paracha 4.1112 Usman Zafar Paracha Usman Zafar Paracha Page Break 5 Usman Zafar Paracha 2.1114 Usman Zafar Paracha Usman Zafar Paracha w_i = the weight of the i-th study (usually the inverse of the variance). w_i = the weight of the i-th study (usually the inverse of th... w_i = the weight of the i-th study (usually the inverse of the variance).∑ w_i = sum of all study weights Variance of Pooled Effect Size Variance of Pooled Effect Size Variance of Pooled Effect Size Var = 1 / ∑ w_i Var = 1 / ∑ w_i Var = 1 / ∑ w_i formula.1118 formula formula The variance of pooled effect size shows how uncertain or “spread out” that estimate is. The variance of pooled effect size shows how uncertain or “sp... The variance of pooled effect size shows how uncertain or “spread out” that estimate is.This shows the certainty of average effect after combining multiple studies. 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 Usman Zafar Paracha 2.1130 Usman Zafar Paracha Usman Zafar Paracha A smaller variance indicates that the pooled effect size estimate is relatively precise — there is less variability or uncertainty around the average effect. Conversely, a larger variance means more uncertainty. A smaller variance indicates that the pooled effect size esti... A smaller variance indicates that the pooled effect size estimate is relatively precise — there is less variability or uncertainty around the average effect. Conversely, a larger variance means more uncertainty.A low variance suggests confidence in the pooled effect size, meaning the studies generally agree on the size of the effect. Usman Zafar Paracha 4.1132 Usman Zafar Paracha Usman Zafar Paracha Page Break 6 Usman Zafar Paracha 2.1135 Usman Zafar Paracha Usman Zafar Paracha Mean OR is the overall mean effect size across all studies included in the meta-analysis - The Pooled Effect Size (Fixed Effects Model) Mean OR is the overall mean effect size across all studies in... Mean OR is the overall mean effect size across all studies included in the meta-analysis - The Pooled Effect Size (Fixed Effects Model)SE is the standard error of Pooled Effect Size Z Z Z Z = Mean OR / SE Z = Mean OR / SE Z = Mean OR / SE formula.1139 formula formula It is used to test the significance of the overall effect size estimate. It is used to test the significance of the overall effect siz... It is used to test the significance of the overall effect size estimate. 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 A large absolute value of Z indicates that the mean effect size is significantly different from zero. A large absolute value of Z indicates that the mean effect si... A large absolute value of Z indicates that the mean effect size is significantly different from zero.You can compare this Z-value against a standard normal distribution to get a p-value. Usman Zafar Paracha 4.1152 Usman Zafar Paracha Usman Zafar Paracha Page Break 7 Pooled Effect Size: The pooled effect size estimate (for example, Mean OR). Pooled Effect Size: The pooled effect size estimate (for exam... Pooled Effect Size: The pooled effect size estimate (for example, Mean OR).SE: The standard error of the pooled effect size estimate. This measures how much variability there is in the pooled estimate.Z: The critical value from the standard normal distribution for the desired confidence level (e.g., 1.96 for 95% confidence). Confidence Interval (CI) for a Pooled Effect Size Confidence Interval (CI) for a Pooled Effect Size Confidence Interval (CI) for a Pooled Effect Size CI = Pooled Effect Size ± Z × SE of Pooled Effect Size CI = Pooled Effect Size ± Z × SE of Pooled Effect Size CI = Pooled Effect Size ± Z × SE of Pooled Effect Size formula.1158 formula formula It is a way to express the uncertainty around the estimated overall effect size (often denoted as d_pooled) that comes from combining multiple studies (like in meta-analysis). It is a way to express the uncertainty around the estimated o... It is a way to express the uncertainty around the estimated overall effect size that comes from combining multiple studies (like in meta-analysis). 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 means you are 95% confident that the true effect size lies between -0.02913 and 0.34711. This means you are 95% confident that the true effect size li... This means you are 95% confident that the true effect size lies between -0.02913 and 0.34711.The 95% CI includes zero (i.e., the lower bound is negative and the upper bound is positive), it indicates that:The effect is not statistically significant at the 0.05 level.In other words, we cannot confidently say that there is a real effect or association — it could be zero (no effect). Usman Zafar Paracha 4.1171 Usman Zafar Paracha Usman Zafar Paracha Page Break 8 a shows exposure and outcome, a shows exposure and outcome, c shows non-exposure and outcom... a shows exposure and outcome,c shows non-exposure and outcome,b shows exposure and no outcome,d shows no exposure and no outcome Odds Ratio Odds Ratio Odds Ratio =(a/c)/(b/d) =(a/c)/(b/d) =(a/c)/(b/d) syntax syntax syntax The Odds Ratio is a measure of association between an exposure/treatment and an outcome. It tells us how much more likely (or less likely) the outcome is to occur in the exposed/treated group compared to the non-exposed/untreated group. The Odds Ratio is a measure of association between an exposur... The Odds Ratio is a measure of association between an exposure/treatment and an outcome. It tells us how much more likely (or less likely) the outcome is to occur in the exposed/treated group compared to the non-exposed/untreated group. illustration Usman Zafar Paracha 1.1180 Usman Zafar Paracha Usman Zafar Paracha Excel example data Excel example data OR Page Break 1 Patreon and LinkedIn links.1186 LinkedIn Profile.1187 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1189 /uzparacha /uzparacha /uzparacha Patreon Then.1191 then then Usman Zafar Paracha 1.1192 Usman Zafar Paracha Usman Zafar Paracha example.1193 example example Suppose we have this data.1194 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1195 Usman Zafar Paracha Usman Zafar Paracha For interpretation, values are converted back to OR using the syntax =EXP(OR). So, the findings can be interpreted to be like that above. For interpretation, values are converted back to OR using the... For interpretation, values are converted back to OR using the syntax =EXP(OR). So, the findings can be interpreted to be like that above. Ln(OR) shows Log natural of Odds Ratio Ln(OR) shows Log natural of Odds Ratio Ln(OR) shows Log natural of Odds Ratio Natural log transform OR Natural log transform OR Natural log transform OR =ln(OR) =ln(OR) =ln(OR) syntax.1200 syntax syntax This is ln(OR), the natural logarithm of the Odds Ratio from each individual study. Calculated as LN(OR). It's used because ln(OR) is normally distributed and suitable for statistical operations like averaging. This is ln(OR), the natural logarithm of the Odds Ratio from ... This is ln(OR), the natural logarithm of the Odds Ratio from each individual study. Calculated as LN(OR). It's used because ln(OR) is normally distributed and suitable for statistical operations like averaging. illustration.1202 Usman Zafar Paracha 1.1203 Usman Zafar Paracha Usman Zafar Paracha Excel example data for ln(OR) Excel example outcome for ln(OR) Excel example data for variance Excel example outcome for variance Excel example data for weight Excel example outcome for weight Excel example data for weighted effect Excel example outcome for weighted effect Excel example data for mean OR Excel example outcome for mean OR Suppose we have this data.1205 Suppose we have this data Suppose we have this data Excel example data for variance of pooled effect size Excel example outcome for variance of pooled effect size Excel example data for Z Excel example outcome for Z Excel example data for CI Excel example outcome for CI EXP: Exponential function(to convert from log scale) EXP: Exponential function(to convert from log scale) EXP: Exponential function(to convert from log scale) Mean OR (not log transformed) Mean OR (not log transformed) Mean OR (not log transformed) EXP(Mean OR) EXP(Mean OR) EXP(Mean OR) formula.1212 formula formula It shows combined odds ratio across studies It shows combined odds ratio across studies It shows combined odds ratio across studies illustration.1214 Patreon and LinkedIn links.1215 LinkedIn Profile.1216 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1219 /uzparacha /uzparacha /uzparacha Patreon Then.1222 then then example.1223 example example Suppose we have this data.1224 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1225 Usman Zafar Paracha Usman Zafar Paracha OR > 1 suggests a positive association between treatment and outcome. OR > 1 suggests a positive association between treatment and ... OR > 1 suggests a positive association between treatment and outcome.If OR = 1, no difference; OR < 1 suggests a negative association. Usman Zafar Paracha 4.1227 Usman Zafar Paracha Usman Zafar Paracha Page Break 9 Excel example data for Mean OR (not log transformed) Excel example outcome for Mean OR (not log transformed) EXP: Exponential function(to convert from log scale).1232 EXP: Exponential function(to convert from log scale) EXP: Exponential function(to convert from log scale) 95% Confidence Interval – Mean OR (not log transformed) 95% Confidence Interval – Mean OR (not log transformed) 95% Confidence Interval – Mean OR (not log transformed) EXP(Lower 95% CI) EXP(Lower 95% CI) EXP(Lower 95% CI) EXP(Lower 95% CI)EXP(Lower 95% CI) formula.1235 formula formula It shows lower and upper bound of confidence interval It shows lower and upper bound of confidence interval It shows lower and upper bound of confidence interval illustration.1237 Patreon and LinkedIn links.1238 LinkedIn Profile.1239 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1242 /uzparacha /uzparacha /uzparacha Patreon Then.1245 then then example.1246 example example Suppose we have this data.1247 Suppose we have this data Suppose we have this data Usman Zafar Paracha 2.1248 Usman Zafar Paracha Usman Zafar Paracha The pooled odds ratio is 1.17 (95% CI: 0.97 to 1.42). The pooled odds ratio is 1.17 (95% CI: 0.97 to 1.42). This su... The pooled odds ratio is 1.17 (95% CI: 0.97 to 1.42).This suggests a potential treatment effect, but not statistically significant at the 0.05 level since the CI includes 1. Usman Zafar Paracha 4.1250 Usman Zafar Paracha Usman Zafar Paracha Page Break 10 Excel example data for CI - Mean OR (not log transformed) Excel example outcome for CI - Mean OR (not log transformed)
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