eval 6970: meta-analysis effect sizes and precision: part ii dr. chris l. s. coryn spring 2011
TRANSCRIPT
EVAL 6970: Meta-AnalysisEffect Sizes and Precision:Part II
Dr. Chris L. S. CorynSpring 2011
Agenda
β’ Effect sizes based on binary dataβ’ Effect sizes based on correlationsβ’ Converting among effect sizesβ’ Precisionβ’ Review questionsβ’ In-class activity
2 2 Tables for Binary Data
β’ For risk ratios, odds ratios, and risk differences data are typically represented in 2 2 tables with cells A, B, C, and D
Events Non-Events N
Treated A B n1
Control C D n1
Risk Ratios
β’ The risk ratio is the ratio of two risks where
Risk Ratio=π΄ /π1πΆ /π2
Risk Ratios
β’ For the purpose of meta-analysis, computations are conducted using a log scale where
log Risk Ratio=ln(Risk Ratio)
Risk Ratios
β’ With variance
β’ And standard error
π log Risk Ratio=1π΄β1π1
+1πΆβ1π2
ππΈ log Risk Ratio=βπ log Risk Ratio
Odds Ratios
β’ The odds ratio is the ratio of two odds where
Odds Ratio=π΄π·π΅πΆ
Odds Ratios
β’ For the purpose of meta-analysis, computations are conducted using a log scale where
logOdds Ratio=ln(OddsRatio)
Odds Ratios
β’ With variance
β’ And standard error
π logOdds Ratio=1π΄
+1π΅
+1πΆ
+1π·
ππΈ logOdds Ratio=βπ logOdds Ratio
Risk Difference
β’ The risk difference is the difference between two risks where
β’ For risk differences all computations are performed on the raw units
Risk Difference=( π΄π1 )β( πΆπ2 )
Risk Difference
β’ With variance
β’ And standard error
π π ππ π π·πππππππππ=AB
π13 +
CD
π23
ππΈRisk Difference=βπ Risk Difference
Correlation Coefficient r
β’ The estimate of the correlation population parameter is r where
β’ With variance
π=βπ=1
π
( ππβπ ) (π πβπ )
ββπ=1
π
(π πβπ )2ββπ=1
π
(π πβπ )2
π π=(1βπ2 )2
πβ1
Correlation Coefficient r
β’ For meta-analyses, r is converted to Fisherβs z
β’ The transformation of r to z is
π§=0.5Γ ln( 1+π1βπ )
Correlation Coefficient r
β’ With variance
β’ And standard error
π π§=1
πβ3
ππΈπ§=βπ π§
Converting Among Effect Sizes
β’ Often, different studies report different effect sizes (if at all) and for a meta-analysis all effect sizes need to be converted to a common index
β’ Meta-Analysis 2.0 will automate this process and many effect sizes calculators are also useful
Converting from Odds Ratio to d
β’ To convert from the log odds ratio to d
β’ With variance of
π=logOdds RatioΓ β3π
π π=π logOdds RatioΓ3
π 2
Converting from d to Odds Ratio
β’ To convert from d to the log odds ratio
β’ With variance of
logOdds Ratio=πΓπβ3
π logOdds Ratio=π πΓπ 2
3
Converting from r to d
β’ To convert from r to d
β’ With variance of
π=2π
β1βπ 2
π π=4π π
(1βπ2 )2
Converting from d to r
β’ To convert from d to r
β’ Where a is a correction factor when
π=π
βπ=π
π=(π1+π2 )2
π1π2
Converting from d to r
β’ With variance
π π=π2π π
(π2+π )3
Precision
β’ Provides the context for computing standard errors
β’ Precision includes variance, standard error, and confidence intervals
β’ With variance the standard error can be computedβ For different effect size metrics, the
computation of differs
Confidence Intervals
β’ Assuming that an effect size is normally distributed
β’ And
β’ 1.96 is the Z-value corresponding to confidence limits of 95% (with error of 2.5% at either end of the distribution)
πΏπΏπ=π β1.96ΓππΈπ
ππΏπ=π+1.96ΓππΈπ
Review Questions
1. When is it appropriate to use the risk ratio?
2. When is it appropriate to use the odds ratio?
3. When is it appropriate to use the risk difference?
4. When is it appropriate to use r?5. What factors affect precision and
how?
Todayβs In-Class Activity
β’ Individually, or in your working groups, download βData Sets 1-6 XLSXβ from the course Websiteβ Calculate the appropriate effects sizes,
standard deviations, variances, and standard errors for Data Sets 5 and 6
β Calculate the 95% confidence intervals (i.e., LL and UL) for Data Sets 1, 2, 3, 4, 5, and 6
β Be certain to save your work as we will use these data again