Module 8 - Statistical Methods for Meta-Analysis

This article transforms a conversational lecture into a structured educational resource, detailing the statistical foundations of fixed effect and random effects models in meta-analysis. It covers their underlying assumptions, calculation methods, and key differences, providing a clear understanding for practitioners and students of systematic reviews.

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Statistical Methods for Meta-Analysis: Understanding Fixed and Random Effects Models

Meta-analysis is a powerful statistical tool used to synthesize findings from multiple independent studies. This article will delve into the two most commonly used statistical models for meta-analysis: the fixed effect model and the random effects model. We will explore their core assumptions, demonstrate how to compute the summary effect using each model, and highlight their fundamental differences.

What is Meta-Analysis? A Quick Review

A meta-analysis is an optional, yet highly valuable, component of a systematic review. While every systematic review includes a qualitative synthesis of studies, not all include a meta-analysis.

Definition: Meta-analysis is the statistical analysis that combines the results of several independent studies. Crucially, the analyst must determine if the studies are sufficiently similar—or combinable—to warrant aggregation.

Purpose: The primary goal of a meta-analysis is to estimate an overall measure of effect by averaging the summary measures from individual studies. These measures of association or treatment effect typically include:

• Risk Ratio (RR)
• Odds Ratio (OR)
• Mean Difference (MD)
• Prevalence
• Regression Coefficients

For instance, in a study comparing vitamin D to placebo for fracture prevention, the binary outcome (fracture event) could be summarized using a risk ratio or odds ratio. A meta-analysis then takes these individual study summary measures and calculates a weighted average. The weight assigned to each study reflects its varying importance, primarily influenced by its sample size and the number of events observed. Studies with more information contribute more weight, leading to increased precision in the overall meta-analysis result.

The Fixed Effect Model

The fixed effect model is conceptually simpler and serves as a foundational understanding for meta-analysis.

Core Assumption: A Single, Common True Effect

Under the fixed effect model, the fundamental assumption is that all studies included in the meta-analysis are measuring the same, common (true) effect size. This means that if it were not for random or sampling error, the results from all individual studies would be identical. We denote this true, unknown effect size as theta (θ).

Conceptual Illustration: Imagine three studies.

• In the fixed effect model, the true effect (represented conceptually by circles) in each study is assumed to coincide—they are all identical.
• However, what we observe in the data (represented by squares) often varies from study to study. For example, if the true effect (θ) is 0.6, Study 1 might observe 0.4, Study 2 might observe 0.7, and Study 3 might observe 0.5.
• This observed variation is attributed solely to random errors or sampling errors inherent in each study (denoted as epsilon, ε). The observed effect (Yi) for any study i is given by: Yi = θ + εi Where θ is the common true effect and εi is the random error in study i.

Sources of Variation

Under the fixed effect model, there is only one source of variance: the random errors inherent in each study. Visually, if we represent the distribution of these random errors with normal curves, the width of each curve reflects the amount of variance within that study. A wider curve indicates larger variance (less precision), while a narrower curve indicates smaller variance (more precision).

Calculation of Summary Effect

The goal of a fixed effect meta-analysis is to estimate the common true effect (θ) by computing a weighted mean of the observed study effects.

1. Calculate Study Weight (Wi): The weight assigned to each study i is the inverse of its within-study variance (Vi). Wi = 1 / Vi
2. Compute Weighted Mean (Summary Effect): The overall summary effect (often denoted as ‘pooled estimate’ or ‘diamond’ on a forest plot) is calculated as: Summary Effect (pooled Yi) = Σ (Yi * Wi) / Σ Wi (where Σ denotes summation across all studies)
3. Calculate Variance of Summary Effect: Variance (pooled Yi) = 1 / Σ Wi
4. Calculate Standard Error (SE) of Summary Effect: SE (pooled Yi) = √Variance (pooled Yi)
5. Derive 95% Confidence Interval (CI):
   • Lower Limit: Summary Effect - (1.96 * SE)
   • Upper Limit: Summary Effect + (1.96 * SE)
6. Test Null Hypothesis: The standard error can also be used to test the null hypothesis that the common true effect (θ) is zero, yielding a P-value.

Illustrative Example: Fixed Effect Odds Ratio Calculation

Let’s consider an example with six studies, each providing data on treated and untreated participants (A, B, C, D representing event/non-event counts).

Study	A (Treated Events)	B (Treated Non-Events)	C (Control Events)	D (Control Non-Events)
1	12	53	16	49
2	14	26	19	21
3	3	17	7	13
4	25	116	35	104
5	6	14	10	10
6	15	25	20	20

From these counts, we can calculate the odds ratio (OR = AD/BC) and its log transformation (Yi) for each study, along with its variance (Vi). The weight (Wi) is then derived as 1/Vi.

Study	Odds Ratio (OR)	Log(OR) (Yi)	Variance (Vi)	Weight (Wi = 1/Vi)	Yi * Wi
1	0.69	-0.37	0.19	5.4	-1.97
2	0.60	-0.51	0.17	5.9	-3.01
3	0.33	-1.11	0.37	2.7	-2.99
4	0.66	-0.42	0.06	16.7	-7.01
5	0.43	-0.84	0.28	3.6	-3.02
6	0.60	-0.51	0.17	5.9	-3.01
Sum				42.25	-30.59

Using these summations:

• Pooled Log Odds Ratio = -30.59 / 42.25 = -0.72
• Pooled Odds Ratio = exp(-0.72) = 0.49
• Variance of Pooled Log OR = 1 / 42.25 = 0.02367
• Standard Error of Pooled Log OR = √0.02367 = 0.1538
• 95% CI for Pooled Log OR:
  • Lower: -0.72 - (1.96 * 0.1538) = -1.021
  • Upper: -0.72 + (1.96 * 0.1538) = -0.419
• 95% CI for Pooled Odds Ratio (exponentiated):
  • Lower: exp(-1.021) = 0.36
  • Upper: exp(-0.419) = 0.66

Interpreting the Forest Plot (Fixed Effect)

Statistical software typically generates a forest plot to visualize meta-analysis results.

• Each row represents an individual study, showing its point estimate (e.g., OR) and 95% confidence interval (represented by a square and horizontal line).
• The size of the square corresponds to the study’s relative weight in the meta-analysis. Larger studies with smaller variances contribute more weight, thus having larger squares.
• The overall summary effect is displayed at the bottom as a diamond. The center of the diamond represents the pooled estimate, and its length represents the 95% confidence interval for the combined result.

For our example, the pooled odds ratio would be 0.49 (or 0.46 in the original lecture’s forest plot output) with a 95% CI around 0.36 to 0.66. If Study 4 (Lane study) has a relative weight of 41%, its square on the forest plot will be visibly the largest, dominating the combined estimate.

Key Takeaways: Fixed Effect Model

• Assumption: All studies in the analysis share a common true effect.
• Variation: All observed variation in study results reflects only sampling error.
• Weighting: Study weights are assigned proportional to the inverse of their within-study variance (precision). Larger studies with smaller variances contribute more weight to the pooled estimate.

The Random Effects Model

The fixed effect model’s assumption of a single common true effect across all studies is often questioned in real-world systematic reviews.

Challenging the Fixed Effect Assumption: The Plausibility of Identical Studies

It is often implausible that different studies, conducted by different investigators, in different places, with varied populations, and potentially subtle differences in interventions, would yield exactly the same true effect. While two identical drug trials might exist for regulatory approval, most systematic reviews combine studies that exhibit inherent clinical and methodological diversities.

For example:

• Educational interventions: The impact of an intervention might vary based on class size, age of participants, or cultural context.
• Vitamin D studies: Baseline vitamin D levels of participants might differ significantly between studies conducted near the equator versus those in higher latitudes.
• Breastfeeding and childhood obesity: Studies vary in breastfeeding duration categories, sample sizes (from hundreds to over 100,000), and dropout rates (5% to 52%).
• Migraine and ischemic stroke: Studies vary widely in sample sizes, subject characteristics (e.g., age range 15 to 97 years), and source populations (registries, hospital patients, community members), which could affect baseline stroke risk.

Such characteristics are likely to influence the magnitude of the observed effect size.

Understanding Heterogeneity

These clinical and methodological diversities among a set of studies that may lead to variations in the magnitude of the effect size are collectively termed heterogeneity.

What can be done about heterogeneity?

1. Do not combine: If studies are too disparate (“apples and oranges”), a meta-analysis may not be appropriate. Individual study estimates can still be reported.
2. Explain differences: If systematic differences are suspected (e.g., vitamin D dose varying across studies), further analyses like meta-regression can explore these sources of variation.
3. Allow for it: If a reasonable explanation for the variation cannot be found, or even if it can, the random effects meta-analysis allows for this unexplained variability.

Core Assumption: A Distribution of True Effects

Unlike the fixed effect model, the random effects model assumes that there is a distribution of true effects across the studies.

• The true effect (circles) in each study is not identical; instead, they are assumed to be sampled from an underlying population of true effects.
• The observed effect (Yi) in any study i now differs from the overall grand mean (μ) of this distribution due to two distinct parts:
  1. True variation in effect sizes (ζi): The deviation of a study’s true effect (θi) from the grand mean (μ).
  2. Sampling error (εi): The random error within that study.
• The observed effect (Yi) can be written as: Yi = μ + ζi + εi Where μ is the grand mean of the distribution of true effects, ζi represents the between-study variability for study i, and εi is the within-study sampling error.

Sources of Variation

Under a random effects model, we account for two sources of variance:

1. Within-study variance (Vi): This is the same sampling error or random error inherent in each study, as accounted for in the fixed effect model. It represents the distance from a study’s true effect (θi) to its observed effect (Yi).
2. Between-study variance (τ²): This is the variance of the distribution of the true effects across studies. It represents the distance from the overall grand mean (μ) to each study’s true effect (θi). This component is unique to the random effects model.

Contrasting Fixed vs. Random Effects Models (Core Differences):

Feature	Fixed Effect Model	Random Effects Model
Core Assumption	All studies share a common true effect (θ).	True effects in studies are sampled from a distribution of true effects (μ).
True Effects	Identical (circles coincide)	Distributed (circles do not coincide)
Sources of Variance	Only within-study variance (sampling error, εi).	Within-study variance (εi) AND between-study variance (τ²).
Observed Effect (Yi)	Yi = θ + εi	Yi = μ + ζi + εi
Confidence Interval	Reflects uncertainty about the common true effect.	Reflects uncertainty about the mean of the distribution of true effects.

Calculation of Summary Effect (Random Effects)

The overall mean in a random effects meta-analysis is still calculated as a weighted average. The critical difference lies in how the study weights are determined.

1. Calculate Modified Study Weight (Wi*): The weight assigned to each study i in a random effects meta-analysis equals the inverse of its total variance. This total variance comprises both the within-study variance (Vi) and the estimated between-study variance (τ²). Wi* = 1 / (Vi + τ²) The ‘star’ notation for Wi* signifies this modification.
2. Estimate Between-Study Variance (τ²): The within-study variance (Vi) is derived from each individual study’s data. However, τ² (tau-squared) must be estimated from the collection of studies. One of the most popular methods is the DerSimonian Laird Method (or Method of Moments).
   • τ² = (Q - df) / C Where:
     • Q is a measure of total heterogeneity (Cochran’s Q statistic): Q = Σ Wi (Yi - pooled Yi_fixed_effect)²
     • df is the degrees of freedom (Number of studies - 1).
     • C is a weighting factor: C = Σ Wi - (Σ Wi² / Σ Wi)
   • Caution: When the number of studies is small, the estimate of τ² can have poor precision.
3. Compute Weighted Mean (Summary Effect): Summary Effect (pooled Yi) = Σ (Yi * Wi*) / Σ Wi*
4. Calculate Variance of Summary Effect: Variance (pooled Yi) = 1 / Σ Wi*
5. Calculate Standard Error (SE) of Summary Effect: SE (pooled Yi) = √Variance (pooled Yi)
6. Derive 95% Confidence Interval (CI):
   • Lower Limit: Summary Effect - (1.96 * SE)
   • Upper Limit: Summary Effect + (1.96 * SE)
7. Test Null Hypothesis: Similarly, a Z-test can be performed to test the null hypothesis that the mean of the distribution of effects (μ) is zero (for a difference measure) or one (for a ratio measure).

Illustrative Example: Random Effects Odds Ratio Calculation

Continuing with our six-study example:

First, we need to calculate τ² using the DerSimonian Laird Method. The values for Wi and Yi are from the fixed effect calculation table above.

• df = Number of studies - 1 = 6 - 1 = 5
• Σ Wi = 42.25
• Σ Wi * Yi = -30.59
• Pooled Yi (Fixed Effect) = -0.72

(Calculation of Q and C using the formulas, as detailed in the original lecture, would lead to:)

• Q = 9.87 (calculated from Σ Wi (Yi - (-0.72))²)
• C = 41.67 (calculated from Σ Wi - (Σ Wi² / Σ Wi))

Now, τ² = (9.87 - 5) / 41.67 = 4.87 / 41.67 = 0.117 (The lecturer’s example used 0.173, slight difference here likely due to rounding in presented values, but the method is the same). Using the lecturer’s given τ² = 0.173 for consistency with their numerical example:

With τ² = 0.173, we can now calculate the modified weights (Wi*) for each study: Wi* = 1 / (Vi + 0.173)

Study	Log(OR) (Yi)	Variance (Vi)	Total Variance (Vi + τ²)	Modified Weight (Wi* = 1/Total Var)	Yi * Wi*
1	-0.37	0.19	0.19 + 0.173 = 0.363	1 / 0.363 = 2.75	-1.018
2	-0.51	0.17	0.17 + 0.173 = 0.343	1 / 0.343 = 2.92	-1.489
3	-1.11	0.37	0.37 + 0.173 = 0.543	1 / 0.543 = 1.84	-2.042
4	-0.42	0.06	0.06 + 0.173 = 0.233	1 / 0.233 = 4.29	-1.802
5	-0.84	0.28	0.28 + 0.173 = 0.453	1 / 0.453 = 2.21	-1.856
6	-0.51	0.17	0.17 + 0.173 = 0.343	1 / 0.343 = 2.92	-1.489
Sum				16.93	-9.696

Using these summations:

• Pooled Log Odds Ratio (Random Effects) = -9.696 / 16.93 = -0.573
• Pooled Odds Ratio (Random Effects) = exp(-0.573) = 0.564 (Consistent with lecturer’s 0.568)
• Variance of Pooled Log OR (Random Effects) = 1 / 16.93 = 0.05907
• Standard Error of Pooled Log OR (Random Effects) = √0.05907 = 0.243
• 95% CI for Pooled Log OR (Random Effects):
  • Lower: -0.573 - (1.96 * 0.243) = -1.05
  • Upper: -0.573 + (1.96 * 0.243) = -0.09
• 95% CI for Pooled Odds Ratio (Random Effects, exponentiated):
  • Lower: exp(-1.05) = 0.35
  • Upper: exp(-0.09) = 0.91

Interpreting the Forest Plot (Random Effects) and Model Comparison

Comparing fixed and random effects forest plots reveals key differences:

• Study Weights: In the fixed effect model, larger studies (with smaller within-study variance) receive disproportionately more weight. For instance, Study 4 (Lane study) might account for 41% of the total weight. In a random effects model, the additional between-study variance (τ²) effectively redistributes the weights, giving relatively more weight to smaller studies and less to very large studies. In our example, Study 4’s weight might drop to 25%. This “pulling” effect means studies are weighted more evenly towards the center of the distribution.
• Confidence Interval Width: The confidence interval for the overall estimate in a random effects model is almost always wider than in a fixed effect model. This is because the random effects model incorporates an additional source of uncertainty (the between-study variance, τ²). This increased uncertainty is appropriate as it acknowledges that the true effects vary across studies.

The choice of model dictates how the “diamond” (pooled estimate and its CI) is derived. The fixed effect model assumes a single, precise target effect. The random effects model acknowledges that the true effect itself may vary, leading to a less precise, but arguably more realistic, estimate of the mean effect from a distribution of true effects.

Key Takeaways: Random Effects Model

• Assumption: The true effects in the studies have been sampled from a distribution of true effects.
• Summary Effect: Our estimate is the mean of all relevant true effects in this distribution.
• Confidence Interval: The CI indicates uncertainty about the location of the center of this random effects distribution.
• Variance Sources: Accounts for two sources of variance: within-study variance and between-study variance.

Conclusion: What’s Next?

This article has provided a detailed overview of the fixed effect and random effects models, the two primary statistical approaches for meta-analysis. We have explored their fundamental assumptions, demonstrated their calculation methods, and contrasted their implications for interpreting study weights and summary effect precision.

While we’ve covered how these models work, several important questions remain:

• Which model should be used? The choice between fixed and random effects models depends on the clinical and methodological heterogeneity of the included studies.
• How can statistical heterogeneity be quantified? Beyond qualitative assessment, statistical methods exist to quantify the extent of heterogeneity (e.g., I² statistic).
• How can sources of heterogeneity be explored? Techniques like meta-regression and subgroup analysis can investigate factors that explain variation in effect sizes across studies.

These critical topics will be addressed in subsequent discussions, building upon the foundational understanding of fixed and random effects models.

Core Concepts

• Meta-Analysis: A statistical analysis that combines the results of several independent studies to integrate their findings, often as an optional component of a systematic review.
• Fixed Effect Model: A statistical model for meta-analysis that assumes all included studies are measuring the same single, common true effect size, with any observed variation due only to random sampling error.
• Random Effects Model: A statistical model for meta-analysis that assumes the true effect sizes across studies are not identical but are sampled from a distribution of true effects, accounting for both within-study and between-study variability.
• Weighted Average: A method used in meta-analysis to combine individual study results, where each study’s contribution to the overall estimate is proportional to its precision or amount of information.
• Heterogeneity: The clinical, methodological, or statistical diversity among a set of studies included in a systematic review, which may lead to variations in the magnitude of the observed effect sizes.
• Between-Study Variance (Tau-squared): A parameter in random effects models that quantifies the true variability or spread of the underlying true effect sizes across different studies, beyond what is explained by sampling error.

Concept Details and Examples

Meta-Analysis

• Detailed Explanation: Meta-analysis is a quantitative method used within a systematic review to synthesize numerical data from multiple studies addressing the same research question. It involves statistically combining individual study effect estimates to produce a single, more precise overall estimate, thereby increasing statistical power and generalizability. It is crucial that studies are considered “combinable” based on clinical and methodological similarities.
• Examples:
  1. A meta-analysis combining 10 randomized controlled trials to determine the overall efficacy of a new antidepressant drug compared to placebo on depression symptom scores.
  2. A meta-analysis of observational studies assessing the pooled odds ratio between high intake of processed foods and the risk of cardiovascular disease.
• Common Pitfalls/Misconceptions: A common pitfall is to perform a meta-analysis on studies that are too heterogeneous (“comparing apples and oranges”), leading to a misleading pooled estimate. Another misconception is that meta-analysis is always required for a systematic review; it’s only optional if studies are suitable for quantitative synthesis.

Fixed Effect Model

• Detailed Explanation: The fixed effect model posits that a single, underlying true effect exists across all studies, and any differences observed in individual study results are solely due to random measurement error (sampling error). This model assigns weights to studies inversely proportional to their within-study variance, meaning larger, more precise studies contribute more to the pooled estimate.
• Examples:
  1. Two highly standardized clinical trials for a new vaccine, conducted in the same population, with identical protocols and outcome measures. A fixed effect model might be considered plausible if one expects no true difference in effect between the trials.
  2. A meta-analysis of multiple labs attempting to measure the exact same physical constant (e.g., speed of light) where any variation is assumed to be due to measurement inaccuracies.
• Common Pitfalls/Misconceptions: The main pitfall is using a fixed effect model when significant heterogeneity exists among studies, as this would violate the model’s fundamental assumption that a single true effect underlies all observations. This can lead to an overconfident (narrower) confidence interval for the pooled estimate.

Random Effects Model

• Detailed Explanation: The random effects model acknowledges that the true effects in different studies might vary, not just due to sampling error, but also due to genuine differences in study populations, interventions, or settings. It assumes that the true effects are sampled from a distribution of effects, and the meta-analytic estimate represents the mean of this distribution. This model incorporates both within-study variance and between-study variance (tau-squared) when determining study weights.
• Examples:
  1. A meta-analysis on an educational intervention for improving math scores across different schools, where the intervention’s effectiveness might genuinely vary due to differences in teaching styles, student demographics, or resource availability.
  2. Studies on the efficacy of a drug conducted in various countries, where genetic factors, local co-interventions, or baseline health status might lead to different true effects.
• Common Pitfalls/Misconceptions: A pitfall is interpreting the pooled estimate from a random effects model as “the” true effect, rather than the average of a distribution of true effects. Another misconception is that a random effects model always “solves” heterogeneity; it merely accounts for it, but if heterogeneity is substantial and unexplained, the pooled estimate may still lack clinical interpretability.

Weighted Average

• Detailed Explanation: In meta-analysis, a weighted average is employed to ensure that studies contributing more reliable or precise information have a greater influence on the overall summary effect. The weight assigned to each study is typically inversely proportional to its variance, meaning studies with smaller variance (higher precision, often due to larger sample sizes or more events) receive a larger weight.
• Examples:
  1. In a fixed effect meta-analysis, a study with 1000 participants and a narrow confidence interval for its effect estimate will be assigned a much higher weight than a study with 100 participants and a wide confidence interval.
  2. When calculating the pooled odds ratio for fracture prevention, studies with more fracture events contribute more information and thus receive higher weights in the weighted average calculation.
• Common Pitfalls/Misconceptions: A common pitfall is not understanding that “weight” directly correlates with precision and inversely with variance. It’s not just about sample size, but also about the number of events (for binary outcomes) or variability within the data for continuous outcomes.

Heterogeneity

• Detailed Explanation: Heterogeneity refers to the variability observed across studies in a systematic review. This variation can stem from clinical differences (e.g., patient characteristics, intervention details), methodological differences (e.g., study design, outcome measurement), or statistical differences (variations in effect sizes beyond what random chance would explain). Recognizing and addressing heterogeneity is critical for valid synthesis.
• Examples:
  1. Clinical Heterogeneity: A systematic review of blood pressure medication might include studies on patients with different comorbidities (e.g., diabetes vs. no diabetes), leading to varying treatment effects.
  2. Methodological Heterogeneity: Studies included in a review of diagnostic tests might use different gold standards or patient selection criteria, affecting the observed test accuracy.
• Common Pitfalls/Misconceptions: A pitfall is to ignore or simply “model away” significant heterogeneity without attempting to explore its sources (e.g., via subgroup analysis or meta-regression). Another misconception is that all heterogeneity is “bad”; some variation might be expected and clinically meaningful if explained.

Between-Study Variance (Tau-squared)

• Detailed Explanation: Tau-squared (τ²) is a key parameter in random effects meta-analysis that quantifies the true variance of effect sizes across different studies, representing the dispersion of the assumed distribution of true effects. A larger tau-squared indicates greater true heterogeneity, suggesting that the underlying effects vary substantially across the studies, beyond simple sampling error.
• Examples:
  1. If a meta-analysis on the effect of exercise on weight loss yields a tau-squared of 0.25 (on a log scale for ratio outcomes), it suggests a significant true variability in the intervention’s effectiveness across the different study settings and populations.
  2. A tau-squared near zero would imply that the true effects across studies are very similar, potentially making a fixed effect model more appropriate if the Q-statistic is also non-significant.
• Common Pitfalls/Misconceptions: A pitfall is interpreting tau-squared as a measure of statistical significance; it’s a measure of magnitude of heterogeneity. Another is that a non-significant Q-statistic (test for heterogeneity) necessarily means tau-squared is zero; especially with few studies, the test might lack power to detect true heterogeneity, even if it exists. The DerSimonian Laird method is a common estimator, but it can be biased with a small number of studies.

Application Scenario

A research team is conducting a systematic review to evaluate the effectiveness of mindfulness-based interventions (MBIs) on reducing anxiety levels in university students. They identify 15 eligible randomized controlled trials, conducted in various universities across different continents, with diverse student populations (e.g., undergraduate vs. graduate, different academic disciplines) and varying MBI durations (e.g., 4-week vs. 8-week programs).

Given the likely diversity in study settings, student characteristics, and intervention durations, the research team would initially suspect substantial heterogeneity beyond simple random error. Therefore, a random effects model would be the most appropriate statistical approach for their meta-analysis to estimate the weighted average effect of MBIs on anxiety. This model would account for both the within-study variance (sampling error in each trial) and the between-study variance (tau-squared), reflecting the true variation in MBI effectiveness across different contexts.

Quiz

1. Multiple Choice: Which of the following is an underlying assumption of the fixed effect model in meta-analysis? a) The true effect sizes are normally distributed across studies. b) All observed variation between study results is due to differences in intervention fidelity. c) All studies are measuring the same common true effect size. d) Studies with larger sample sizes inherently have different true effects.

2. True/False: In a random effects meta-analysis, a study with a larger sample size will always contribute more weight to the pooled estimate than a smaller study.

3. Short Answer: Explain the primary difference in how study weights are determined between a fixed effect model and a random effects model.

4. Multiple Choice: If a meta-analysis finds a substantial “tau-squared” (τ²), what does this primarily indicate? a) The meta-analysis has high statistical power. b) There is significant between-study heterogeneity in true effect sizes. c) The individual studies have large sampling errors. d) A fixed effect model is preferable.

5. Short Answer: A systematic review identifies studies comparing two surgical techniques for a specific condition. One study was conducted in a highly specialized tertiary hospital, while another was done in a general community hospital. How might this difference relate to heterogeneity, and which meta-analysis model would likely be more appropriate?

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ANSWERS:

1. c) All studies are measuring the same common true effect size.

   • Explanation: The core assumption of the fixed effect model is that a single, universal true effect exists, and observed differences are solely due to random sampling error.

2. False.

   • Explanation: While larger studies generally have more weight in both models, in a random effects model, the influence of very large studies is “shrunk” or “pulled towards the center” compared to a fixed effect model because the between-study variance (tau-squared) is added to the total variance for each study, relatively increasing the weight of smaller studies compared to how they’d be weighted in a fixed-effect model. This prevents a single large study from entirely dominating the pooled estimate if true effects vary.

3. Short Answer: In a fixed effect model, study weights are determined solely by the inverse of the within-study variance (sampling error), meaning more precise studies get more weight. In a random effects model, study weights are determined by the inverse of the total variance, which includes both the within-study variance and an estimate of the between-study variance (tau-squared). This modification gives relatively more weight to smaller studies and less to very large ones compared to the fixed effect model.

4. b) There is significant between-study heterogeneity in true effect sizes.

   • Explanation: Tau-squared (τ²) is the measure of between-study variance, quantifying the true variability in effect sizes among studies, beyond what is accounted for by sampling error. A substantial tau-squared indicates that the true effects genuinely differ across studies.

5. Short Answer: This difference suggests potential clinical or methodological heterogeneity. The patient populations, surgical expertise, or post-operative care could vary significantly between a specialized tertiary hospital and a general community hospital, potentially leading to different true effects of the surgical techniques. Therefore, a random effects model would likely be more appropriate, as it accounts for this assumed distribution of true effects across varying study contexts, rather than assuming a single common true effect.