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Johnson BT, Huedo-Medina TB. Meta-Analytic Statistical Inferences for Continuous Measure Outcomes as a Function of Effect Size Metric and Other Assumptions [Internet]. Rockville (MD): Agency for Healthcare Research and Quality (US); 2013 Apr.

## Meta-Analytic Statistical Inferences for Continuous Measure Outcomes as a Function of Effect Size Metric and Other Assumptions [Internet].

Show details- Bias
The extent to which the observed UMD or SMD differs from the parametric value. Positive values of bias imply over-estimations of the parametric effect size and negative values imply under-estimations.

- Change-score metric
The difference between two repeated measures compared with the variability of change scores.

- Coverage
The proportion of replications for which the 95% confidence interval for each index did not include the null value, ES = SMD = UMD = 0.

- Effect size (ES)
The magnitude or degree of the association between two variables. In the current investigation, comparisons between groups, across time, or both, are used (e.g., standardized mean difference; unstandardized mean difference).

- Efficiency
A measure of the optimality of an estimator that reaches the closest value to the parameter with the minimum variance. To the extent that efficiency is positive, statistical power is maximized to detect the parametric value.

- Mean square error (MSE)
A measure of the average of the square of the errors that evaluates the quality of an estimator in terms of its variation and unbiasedness. Bias and efficiency are, in effect, components of the MSE.

- One-group repeated-measures design
A study methodology in which a single sample is observed at two or more time points (e.g., before and after a treatment).

- Pooled standard deviation
The sample-size weighted mean standard deviations of two or more groups.

- Raw-score metric
A metric that compares a mean or mean difference between conditions or times with the variability of scores within each condition.

- Sampling variance of effect size for one-group repeated-measures design
The variance of the sampling distribution, which is the distribution of values that result from repeated random samples of the same size using a repeated-measures design (see Table 3 for extant estimations of this statistic).

- Sampling variance of effect size for two independent groups
The same as the preceding one but for studies following a two-groups design, so using two independent samples (see Table 4 for extant estimations of this statistic).

- Standardized mean difference (SMD) effect size
The difference of two means divided by the pooled standard deviation.

- Standardized mean difference (SMD) for one-group repeated-measures design
The effect size comparing two means at different times for the same group relative to the standard deviation (see Table 1 for extant estimations of this statistic).

- Standardized mean difference (SMD) for two independent groups
The effect size reflecting the change in means for two independent groups (repeated-measures, between-groups version) or the comparison of the means at post-test for two independent groups (between-groups version). (See Table 2 for extant estimations of this statistic.)

- Two-groups repeated measures design
A study methodology in which two groups (or arms; e.g., treatment and control) are observed at two or more times.

- Unstandardized mean difference (UMD) effect size
The difference of the two means in their original metric or scale.

## Following are the Greek terms that appear in this report

Term | Definition |
---|---|

δ | The parametric difference between two groups |

$\widehat{\delta}$_{j} | The sample estimate of population parameter δ for the j_{th} replication |

${\mu}_{post}^{E}$ | Parametric mean at post-test for the experimental group |

${\mu}_{Pre}^{E}$ | Parametric mean at pretest for the experimental group |

${\mu}_{Post}^{C}$ | Parametric mean at post-test for the control group |

${\mu}_{Pre}^{C}$ | Parametric mean at pretest for the control group |

${\sigma}_{Pre}^{2}$ | Parametric variance at pretest |

${\sigma}_{Post}^{2}$ | Parametric variance at post-test |

σ_{Pre,Post} | Covariance between pre- and post-test |

ρ | Parametric correlation |

ρ_{pre-post} | Parametric correlation between pre-and post-test |

σ_{C} | Parametric standard deviation of the control group |

σ_{E} | Parametric standard deviation of the experimental group |

τ^{2} | Between-study variance |

## Following are the Latin abbreviations used in this report

Term | Definition |
---|---|

d_{b} | Standardized mean difference proposed by Becker (See Table 1, No. 3 for details and elements of the equation) |

d_{b_nonr} | Becker’s standardized mean difference, excluding the correlation factor 2(1 – r) in its variance estimation |

df | Degrees of freedom |

d_{g} | Standardized mean difference proposed by Gibbons (See Table 1, No. 4 for details and elements of the equation) |

d_{hb} | Standardized mean difference proposed by Hedges (See Table 2, No. 6 for details and elements of the equation) |

d_{hw} | Standardized mean difference proposed by Huedo-Medina & Johnson (See Table 1, No. 5 for details and elements of the equation) |

d_{hw_nonr} | Hedges’ standardized mean difference using the within-study degrees of freedom and excluding the correlation factor 2(1 – r) in its variance estimation. |

d_{s1} | Standardized mean difference proposed by Shadish (See Table 2, No. 10 for details and elements of the equation) |

d_{s2} | Standardized mean difference proposed by Shadish using the standard deviation from ANOVA results (See Table 2, No. 11 for details and elements of the equation) |

d_{s3} | Standardized mean difference proposed by Shadish using the standard deviation from ANCOVA results (See Table 2, No. 12 for details and elements of the equation) |

d_{tch} | Standardized mean difference based on t-test for change-score metric (See Table 1, No. 2 for details and elements of the equation) |

d_{tra} | Standardized mean difference based on t-test for raw-score metric (See Table 1, No. 1 for details and elements of the equation) |

ES | Effect size |

HAM-D | Hamilton rating scale of depression |

k | Number of studies |

M HAM-D | Mean score on the HAM-D |

M_{C} | Mean for control group |

M_{E} | Mean for experimental group |

mmHg | Millimeters of mercury (used in measures of blood pressure) |

MSE | Mean square error |

N | Total sample size |

n | Group sample size |

OR | Odds ratio |

r | Estimated correlation |

Rns | The number of replications |

SD | Standard Deviation |

SE_{UMD} | The standard error for the UMD |

SE_{SMD} | The standard error for the SMD |

SMD | Standardized mean difference (d) |

UMD | Unstandardized mean difference (Equation 19, Table 5) |

var (_{one-g}d)_{b} | Variance estimate for one group design with repeated measures of the standardized mean difference proposed by Becker (See Table 3, No. 13 for details and elements of the equation) |

var (_{one-g}d)_{g} | Variance estimate for one group design with repeated measures of the standardized mean difference proposed by Gibbons (See Table 3, No. 14 for details and elements of the equation) |

var (_{two-g}d)_{b} | Variance estimate for two group design with repeated measures of the standardized mean difference as a function of two effect sizes proposed by Becker (See Table 4, No. 16 for details and elements of the equation) |

var (_{two-g}d)_{b_t} | Variance estimate for two group design with repeated measures of the standardized mean difference proposed by Hedges (See Table 4, No. 15 for details and elements of the equation) |

var (_{two-g}d)_{g} | Variance estimate for two group design with repeated measures of the standardized mean difference as a function of two effect sizes proposed by Gibbons (See Table 4, No. 18 for details and elements of the equation) |

var (_{two-g}d)_{g_t} | Variance estimate for two group design with repeated measures of the standardized mean difference proposed by Gibbons (See Table 4, No. 17 for details and elements of the equation) |

Var_{UMD} | The variance estimate for the UMD |

Var_{SMD} | The variance estimate for the SMD |

Y^{C} | Control outcome |

Y^{E} | Experimental outcome |

- Glossary of Terms - Meta-Analytic Statistical Inferences for Continuous Measure ...Glossary of Terms - Meta-Analytic Statistical Inferences for Continuous Measure Outcomes as a Function of Effect Size Metric and Other Assumptions

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