Psychology dissertation E-value analysis quantifies how strong unmeasured confounding would need to be to explain away an observed association. It does not prove causality or repair a biased design. Used carefully, it turns the vague phrase “residual confounding may remain” into a transparent sensitivity question.
This guide explains the E-value on the risk-ratio scale, shows how to calculate and interpret it, covers confidence-limit E-values and benchmarking, and identifies situations where another sensitivity analysis is more informative. Examples are framed for psychology dissertations using observational data.
Table of Contents
What psychology dissertation E-value analysis asks
An E-value is the minimum strength of association that an unmeasured confounder would need to have with both the exposure and the outcome, on the risk-ratio scale and conditional on measured covariates, to move an observed association to a specified target. The usual target is the null.
VanderWeele and Ding introduced the measure in their 2017 E-value paper. They recommended reporting an E-value for the adjusted point estimate and for the confidence-interval limit closest to the null. The first addresses the magnitude needed to explain away the estimate; the second addresses the magnitude needed to make the interval include the null.
A large E-value means stronger joint confounding associations would be required. A small E-value means comparatively weak unmeasured confounding could move the result to the target. Neither label is meaningful without comparing the E-value with plausible confounders in the relevant setting.
| Quantity | Question answered | What it does not establish |
|---|---|---|
| Point-estimate E-value | How strong must confounding be to move the estimate to the target? | That the target is impossible or unlikely |
| Confidence-limit E-value | How strong must confounding be to move the interval to include the target? | That sampling error and bias are interchangeable |
| Benchmark comparison | Are the required associations larger than measured covariate associations? | That measured variables bound all unmeasured variables |
| Bias contour | Which combinations of exposure-confounder and confounder-outcome associations suffice? | The actual values of those unknown associations |
Decide whether the E-value matches the research question
The E-value is designed for effect estimates from observational studies that may be distorted by unmeasured confounding. It is most natural when a dissertation asks a causal question, has already adjusted for a defensible measured confounder set, and reports an association that can be expressed on or translated to the risk-ratio scale.
It is not needed to interpret a properly randomised treatment assignment under ideal implementation, although non-adherence, loss to follow-up and other biases may still need assessment. It is also not a general robustness score for prediction models, factor analyses, qualitative studies or purely descriptive prevalence estimates.
Start with the estimand and causal structure. Define the exposure, outcome, target population, time zero, follow-up and effect measure. Use a directed acyclic graph and the confounding variables guide to justify measured adjustment. Sensitivity analysis comes after credible design and analysis, not instead of them.
Understand the two confounding associations
The E-value summarises two relationships. First, the unmeasured confounder must differ across exposure groups. Second, the confounder must predict the outcome after accounting for exposure and measured covariates. Both are expressed as risk-ratio associations under the bounding framework.
The reported E-value is the minimum value required for both relationships when they are taken to be equally strong. If one relationship is weaker than the E-value, the other must be stronger. A contour plot can show asymmetric combinations and is often more informative than one threshold.
“An unmeasured confounder would need a risk ratio of 3.4” is incomplete. State that it would need associations of at least 3.4 with both exposure and outcome, above and beyond measured covariates, to explain away the estimate, if the two associations were equal. Avoid implying that the confounder itself has an observed risk ratio.
Calculate an E-value for a risk ratio
For an observed risk ratio above one, the point-estimate E-value is calculated as:
E-value = RR + square root of [RR × (RR − 1)].
Suppose an adjusted risk ratio is 2.00. The E-value is 2 + √(2 × 1), which is approximately 3.41. Under the method’s assumptions, an unmeasured confounder would need risk-ratio associations of at least 3.41 with both exposure and outcome, conditional on measured covariates, to reduce the observed risk ratio to one when those associations are equally strong.
For a protective association below one, take the reciprocal before applying the formula. An adjusted risk ratio of 0.50 therefore has the same null E-value as a risk ratio of 2.00. Always preserve the original direction when explaining the substantive result.
The official EValue vignette demonstrates calculations for harmful and protective associations and shows how to target a non-null value. The software should support transparent computation, but the dissertation still needs to justify the input scale and interpretation.
Calculate the confidence-limit E-value
Select the adjusted confidence-interval limit closest to the null. For a risk ratio above one, this is normally the lower limit. For a protective risk ratio below one, it is normally the upper limit. Apply the same formula after taking the reciprocal where necessary.
Imagine an adjusted risk ratio of 2.00 with a 95% confidence interval from 1.20 to 3.33. The point E-value is about 3.41. The E-value for the lower confidence limit of 1.20 is about 1.69. Thus substantially weaker unmeasured confounding could move the interval to include the null than would be required to move the point estimate itself.
If the confidence interval already includes the null, its confidence-limit E-value is one. Report that directly. Do not hide it by calculating only the point-estimate E-value. The confidence-limit value is often the more cautious result because it combines the observed association’s uncertainty with the sensitivity threshold.
| Input | Illustrative value | Correct interpretation |
|---|---|---|
| Adjusted risk ratio | 2.00 | Observed association after measured adjustment |
| 95% confidence interval | 1.20 to 3.33 | Sampling-uncertainty interval under the fitted analysis |
| Point E-value | 3.41 | Equal confounding associations needed to move the estimate to one |
| Limit E-value | 1.69 | Equal confounding associations needed to move the interval to include one |
These values are mathematical illustrations, not results from a real psychology study.
Handle odds ratios, hazard ratios and continuous outcomes carefully
Odds ratios
An odds ratio can approximate a risk ratio when the outcome is rare under the relevant design and follow-up. When the outcome is common, entering the odds ratio as though it were a risk ratio can exaggerate the E-value. Use an appropriate conversion based on the study design and outcome prevalence, and report the conversion.
Case-control sampling requires particular care. The interpretation of an odds ratio depends on how controls were sampled. The methodological note on E-values and incidence-density sampling explains why standard risk-ratio calculations cannot be applied mechanically to every odds ratio or hazard ratio.
Hazard ratios
A hazard ratio is not a risk ratio. A direct approximation may be reasonable only under conditions such as a relatively rare outcome and a defensible proportional-hazards interpretation. When events are common or hazards vary materially, use the conversion supported by the EValue software or choose a sensitivity method designed for the estimand.
Mean differences and standardised effects
Psychology frequently reports mean differences, standardised mean differences and regression coefficients. E-value methods use approximate transformations for some continuous-outcome measures. Those transformations add assumptions and should not be presented as exact. State the original scale, the conversion method and why it is reasonable.
Where possible, estimate an interpretable probability or risk contrast directly rather than converting a standardised effect solely to obtain an E-value. A method should clarify the scientific question, not force every outcome into one metric.
Use software reproducibly
The current CRAN EValue package provides functions for unmeasured confounding, selection bias, measurement error, meta-analysis and combined biases. Choose the function that matches the effect measure, specify whether an outcome is rare only when that claim is defensible, and save the code and package version.
For a risk ratio, the workflow records the estimate, lower and upper confidence limits and the target value. For an odds ratio or hazard ratio, it records the relevant prevalence or rare-outcome assumption. Reproduce the output in a clean session and check it against a hand calculation for a simple risk-ratio example.
Do not paste software output without explanation. Report the inputs, transformation, point E-value, confidence-limit E-value and interpretation. If the package returns a warning, resolve the underlying scale or direction issue rather than suppressing it.
Benchmark the E-value against plausible confounders
An E-value has little context on its own. Compare it with associations observed for well-measured covariates that are substantively related to the exposure and outcome. For example, a study of social-media exposure and depressive symptoms might benchmark prior depression, family adversity, sleep difficulty and offline social support.
Use adjusted associations on comparable scales and time points. A baseline confounder’s association with exposure should be assessed separately from its association with outcome. Do not compare an E-value with a correlation coefficient, odds ratio or standardised mean difference without a valid translation.
Measured covariates are not guaranteed to be the strongest possible unmeasured confounders. Benchmarking therefore informs plausibility rather than proving robustness. Explain whether an omitted construct could plausibly combine several risks, be measured more strongly than observed proxies or act differently across subgroups.
A psychology dissertation example
Suppose a longitudinal observational study examines whether sustained exposure to online harassment predicts clinically elevated distress six months later. The analysis adjusts for baseline distress, age, gender, prior victimisation, offline support and platform use. It estimates an adjusted risk ratio of 1.80 with a 95% confidence interval from 1.15 to 2.82.
The point E-value is approximately 3.00. The lower-limit E-value is approximately 1.57. The point estimate would therefore require stronger equal associations with exposure and outcome to be fully explained away than the confidence interval would require to include the null.
The researcher benchmarks these thresholds against baseline distress, the strongest measured confounder. If baseline distress has adjusted risk-ratio associations near 2.2 with future exposure and 2.5 with the outcome, an omitted factor of comparable strength is not obviously implausible. The conclusion should remain cautious despite the point E-value.
The dissertation reports both E-values, the formulas or software, measured benchmarks and limitations. It does not claim the association is causal. The example is hypothetical and uses rounded figures only to demonstrate interpretation.
Distinguish the E-value from other sensitivity analyses
The E-value answers a threshold question: how strong would unmeasured confounding need to be? It does not specify a particular omitted variable, estimate a corrected effect under chosen bias parameters or provide the probability that the effect is causal.
Negative controls can reveal certain bias structures when a credible control is available. Quantitative bias analysis specifies plausible confounder prevalence and outcome associations to produce corrected estimates over scenarios. Rosenbaum bounds assess hidden bias in matched observational studies. Tipping-point analyses are useful for missing outcomes and departures from missing-data assumptions.
Use the method that matches the threat. A dissertation using propensity score matching may combine balance diagnostics with Rosenbaum-style sensitivity analysis. A g-computation analysis may use an E-value for its final relative estimate while separately testing model specification and intervention support.
| Method | Main question | Best use |
|---|---|---|
| E-value | What minimum confounding strength reaches a target? | Concise sensitivity threshold for relative effects |
| Bias-factor analysis | What corrected effects follow specified bias parameters? | Scenario-based quantitative adjustment |
| Negative control | Is a bias mechanism detectable through a control relation? | Design-based evidence about selected biases |
| Rosenbaum bounds | How sensitive is matched inference to hidden treatment odds? | Matched observational studies |
| Missing-data tipping point | Which missing outcomes would change the conclusion? | Attrition or item nonresponse |
Know what an E-value cannot do
An E-value does not address selection bias, measurement error, model misspecification, interference, reverse causation or outcome reporting choices unless an extended method explicitly models those biases. The CRAN package’s multiple-bias guidance treats confounding, selection and differential misclassification as distinct parameters.
The E-value also does not account for whether a proposed unmeasured confounder is temporally plausible. A variable measured after exposure may be a mediator rather than a confounder. Nor does the measure tell researchers which confounders to collect in future studies.
Large E-values can coexist with poor design. Selective samples, unreliable measures, inappropriate adjustment, weak temporal ordering or post hoc analyses can invalidate causal claims independently of unmeasured confounding. Present the E-value within a broader risk-of-bias assessment.
Report E-values without overstating robustness
State the adjusted effect estimate and interval first. Identify the scale, target and direction. Then report both E-values with a complete definition. Include any conversion, rare-outcome assumption or non-null target. Benchmark against named measured covariates where possible.
A proportionate statement is: “The adjusted risk ratio was 1.80. The E-value was 3.00 for the point estimate and 1.57 for the confidence limit closest to the null. An unmeasured confounder associated with both exposure and outcome by risk ratios of 1.57 each could move the interval to include the null, conditional on measured covariates.”
Avoid phrases such as “the result passed the E-value test,” “confounding was ruled out” or “the E-value confirms causality.” Reporting recommendations emphasise context, measured benchmarks and clear distinction between the point and confidence-limit values.
Quality checklist
- Define a causal estimand and justify why unmeasured confounding is the target bias.
- Report the adjusted effect and confidence interval on the original scale.
- Use the correct E-value function or documented conversion.
- Calculate values for both the point estimate and confidence limit closest to the null.
- State whether the outcome is rare and justify that assumption if used.
- Benchmark both confounding associations against relevant measured covariates.
- Show asymmetric combinations with a contour plot when helpful.
- Assess selection, measurement, missingness and modelling biases separately.
- Share code, package version, inputs and any rounding decisions.
- Use cautious language that does not equate sensitivity with proof.
Frequently asked questions
What is an E-value in simple terms?
It is a threshold describing how strongly an unmeasured confounder would need to relate to both exposure and outcome, beyond measured covariates, to move an observed association to a chosen target.
Is a higher E-value always better?
A higher value indicates greater resistance to unmeasured confounding under the method’s framework, but “large” depends on the topic, benchmarks, effect scale and other biases. It is not a universal quality score.
Why report two E-values?
The point-estimate E-value concerns explaining away the estimate. The confidence-limit E-value concerns moving the interval to include the null and is usually smaller. Both answer useful but different questions.
Can I calculate an E-value for an odds ratio?
Yes, using a method that accounts for whether the outcome is rare and for the study design. Do not automatically treat a common-outcome odds ratio as a risk ratio.
Can an E-value prove causation?
No. It quantifies sensitivity to one bias mechanism. Causal interpretation still depends on design, measurement, temporal ordering, modelling, selection and other assumptions.
What if the confidence interval includes the null?
The confidence-limit E-value is one. Report it transparently alongside the point-estimate E-value and avoid presenting the result as statistically incompatible with the null.
Should every psychology dissertation report an E-value?
No. Use it when an observational causal estimate and unmeasured confounding are central. Other questions and bias mechanisms require different sensitivity analyses.
Conclusion
Psychology dissertation E-value analysis makes one aspect of residual-confounding sensitivity explicit. A strong dissertation uses the correct effect scale, reports point and confidence-limit values, benchmarks them against plausible covariates and keeps the result within a comprehensive bias assessment.
If you need methodological support, choose ethical guidance that helps you verify the effect measure, assumptions, calculations and reporting. Consultation should complement supervision, protect confidential data and strengthen your independent reasoning rather than produce hidden analysis.
