### 6.8.3. Summary of Effect Sizes

The summary effect size type is selected from the next dialogue. The program will estimate a fixed effect model and a random effects model for the mean effect size using a weighted average. By default, for each individual study the inverse of its variance is used as the weight for both models. It is also possible to select the Mantel-Haenszel weights for 2 x 2 tables.

**Run analysis: **

**0: Standard:**
Individual effect sizes and their standard errors are displayed with both fixed
and random effects models.

**1: One study
removed**** with fixed effect:** For each study, that study is removed and
the fixed effect summary for the rest of the studies is displayed in its place.

**2: One study
removed with random effects**

**3: Cumulative**** with fixed effect:** For each study, the fixed effect summary for all
studies up to and including that study is displayed in its place. If the
studies are already sorted in chronological order, then the cumulative analysis
provides the summary effects of all studies known to a particular study on its
publication date.

**4: Cumulative with
random effects**

**LB/UB Symmetry: **When both lower and upper bounds are
given for a confidence interval, they must be consistent with each other.
UNISTAT will calculate the central tendency using both LB and UB. If the two
values are within 10% of each other they will be deemed consistent and the
results will be displayed. Otherwise, the current study will be reported as
having invalid data. You can adjust the consistency level by entering a
different multiplier.

**Imputed r:** Calculation of effect size for paired
samples requires knowing the correlation between them. However, this may not be
readily available in most studies. In such cases, the program assumes a
correlation of 0.5 by default. This value can be changed by the user. See
section 6.8.2.4. Paired Samples
for details.

**Use t-test for correlation:** If the t-statistic was
originally reported for a study based on the correlation coefficient, then
enter a non-zero value for this box. See section 6.8.2.4. Paired Samples
for details.

**Standard Error for Hedges’ g****:** You can choose
one of two alternative methods for calculating the standard error of Hedges’ g.
See section 6.8.2.3. Unpaired Samples for details.

**Display relative or absolute weights:** In Results and Forest Plot output, the weights can be displayed as relative
weights in percentage terms or as absolute weights as used in calculations.

**Ratio or Log(Ratio):** This option will be available
when one of odds, Peto odds, risk, rate and hazard ratio is selected as the summary effect size.

**Weights: Inverse Variance or Mantel-Haenszel:** The
inverse variance method is the default for all types of effect size.
Mantel-Haenszel weights can be calculated for the fixed effect model when one
of odds or risk ratios or risk difference is selected as the summary effect
size.

**Heterogeneity with inverse variance weights:** When
Mantel-Haenszel weights are selected, the heterogeneity statistics are
calculated with Mantel-Haenszel weights by default (see 6.8.4.2. Tests). You can,
however, override this and force using inverse variance weights.

#### 6.8.3.1. Fixed Effect Model

Inverse variance method

The weighted mean of individual effect sizes M is calculated as:

where the weight Wi is the reciprocal of the effect size variance Vi for a study:

and the variance of the mean effect size M is:

The Z-statistic is:

The relative weights are:

and the standardised residuals:

Mantel-Haenszel method for 2 x 2 tables

**Odds
ratio****:**

The summary odds ratio calculated as:

where the Mantel-Haenszel weight W_{MHi} is:

and the variance Ln(OR_{MH}) is:

where:

**Risk
ratio****:**

The Mantel-Haenszel weight W_{MHi} is calculated as:

and the variance Ln(RR_{MH}) is:

where:

**Risk
difference****:**

The Mantel-Haenszel weight W_{MHi} is:

and the variance Ln(RD_{MH}) is:

where:

#### 6.8.3.2. Random Effects Model

Define the between-studies variance as introduced by DerSimonian R, Laird N. (1986):

where df = n – 1, Cochran’s Q is:

and:

Next, determine the total variance for each study as the sum
of within-study variance V_{i} and the between-studies variance of τ^{2}:

The mean effect size, its variance, relative weights and standardised residuals are then computed as in the inverse variance method.