Ignoring compounding measurement error leads to overconfidence in reliability

---

This vignette was motivated by the students projects I mentored this past year (2025-2026). Many students seek to measure new things, explicitly or implicitly, in their research projects. However, measuring new things is effortful and challenging to do well. When the effects found in studies with new or adapted measures are small and uncertain, there is a temptation to draw conclusions around the constructs being measured (e.g., “perhaps there is no relationship between x and y in the world”), but this conclusion is neither defensible statistically, nor psychometrically without a through understanding of the instruments employed. This vignette is intended to highly why understanding the measurement properties of measures is so important, and just how much sampling error we can see in small studies when error comes from multiple sources.

Part 1. Compounding sources of measurement error

Every measurement mixes a true score (what we want to know) with error (random or systematic noise that distorts measurement). Classical Test Theory writes this as observed = true + error. But “error” is rarely one thing and often comes from different sources:

• Inter-rater: different raters score the same performance differently.
• Intra-rater: the same rater scores the same performance inconsistently.
• Test-retest: the same person scores differently on a different day, even when the target of measurement should be stable.
• Alternate-forms: two supposedly equivalent versions of a test return different scores.

Reliability characterizes this measurement error: how much of what we observe is real, and how much is noise?

\[\rho = \dfrac{\text{Var(True)}}{\text{Var(True)} + \text{Var(Error)}}\]

Most instruments report reliability as separate correlations, each against established benchmarks¹

• Less than 0.50: Poor reliability
• 0.50 to 0.75: Moderate reliability
• 0.75 to 0.90: Good reliability
• Greater than 0.90: Excellent reliability

If each of these reliability coefficients separately achieves a sufficient value (e.g., all are >=0.75 or have “good” reliability), then the conclusion is often that the test is reliable. However, measurement error compounds across the different sources of error, such that the overall reliability of the instrument falls well below any one reliability estimate. The extent to which it compounds depends on the degree to which the sources of error overlap, but because error sources are characterized separately, this is rarely known.

If we start from the assumption that the four sources are completely independent, we can visualizes how error compounds. Scores below use the T-score scale (mean 50, SD 10), the same scale used by many standardized instruments. A perfectly reliable test would produce scores with exactly that spread. However, each source of measurement error affects the sample drawn, and returns a more realistic distribution given the population and measurement instrument.²

The dashed grey curve is the true score distribution: the real variability across people. Each Redraw simulates a fresh sample of N respondents and plots their observed scores as a solid curve. The four sources here are treated as fully independent, the most conservative case. True overall reliability is usually at least this good, and often better, since real sources of error typically share some of the same underlying cause. Unless a dedicated study tells you otherwise, though, assuming independence is the safer default.

---

Part 2: When Both Variables Are Unreliable

Part 1 looked at one unreliable variable. But most applied research asks about bivariate reliationships. “Does physical function correlation with quality of life?” or “Do people with higher self-efficacy report greater well-being?”. However, if both X and Y are measured with error, the correlation you’d observe is not only more uncertain - but it is likely to be substantially smaller than the true correlation. This idea, that poor reliability attenuates correlations has been well established for well over a century,³ but (at least in my experience), does not get anywhere near the attention it deserves in today’s statistics education or use.

\[\rho_{\text{observed}} = \rho_{\text{true}} \times \sqrt{\rho_{xx} \times \rho_{yy}}\]

Below, X and Y each both have sources of test-retest and inter-rater error. Set a true correlation below, choose both dimensions of reliability for X and Y, and examine how far the observed correlation falls from the true relationship.⁴

The dashed grey line is the true correlation: what you’d see if X and Y were measured perfectly. Its slope matches your true-correlation slider exactly, since both axes use the same T-score scale no matter how much error inflates either variable’s spread. Each Redraw plots a fresh sample and a line through it with slope set to the observed correlation r. As reliability drops, the solid line has a tendency to be flatter than the true correlation. The shaded band is the 95% CI on r, shown as the range of slopes consistent with that draw.

The attenuation factor √(relₓ·relᵧ) is the fraction of the true correlation you should expect to recover, on average, given the current reliability settings. The same independence caveat from Part 1 applies here: X’s and Y’s sources are treated as fully independent, so the attenuation shown is a conservative estimate. Real attenuation is often somewhat milder, for the same reasons.

Takeaway

• Meeting individual reliability benchmarks is not the same as having a reliable instrument. If four sources each achieve “good” reliability (≥0.75), the overall instrument may still fall well below that threshold once error compounds.

• The independence assumption used here is conservative and real error sources likely share some underlying cause, so true overall reliability is usually somewhat better than these simulations suggest. But without a dedicated joint study, we can’t know by how much.

• Poor reliability doesn’t only increase uncertainty, it also results in bias. In this case, it pulls correlations systematically toward zero. A weak observed correlation between two constructs can reflect poor measurement rather than a weak relationship.

• The cost of unreliability compounds when studying relationships. A study can have adequate power on paper and still be underpowered in practice if the reliability of both variables was not factored in when planning the study.

Footnotes

1. These benchmarks are not intended as blanket thresholds across all instruments, but are often used and reported as such. Koo, T. K., & Li, M. Y. (2016). A guideline of selecting and reporting intraclass correlation coefficients for reliability research. Journal of chiropractic medicine, 15(2), 155-163.

2. Each simulated draw generates a sample of \(N\) people. For each person, a true score \(T_i \sim N(0,1)\) is drawn, then four independent error terms are added — one for each reliability source — where each error term is \(e \sim N(0, \sigma^2)\) with \(\sigma = \sqrt{1/\rho - 1}\), derived by inverting the single-source reliability identity \(\rho = 1/(1 + \sigma^2)\). The observed score is \(X_i = T_i + e_{\text{rater}} + e_{\text{intra}} + e_{\text{occasion}} + {\text{forms}}\), and overall reliability across the four independent sources is \(\rho_{\text{overall}} = 1/(1 + \sum \sigma^2_k)\). Scores are displayed on the T-score scale (\(\mu = 50\), SD \(= 10\)).

3. Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72–101. https://doi.org/10.2307/1412159

4. Each simulated draw generates a sample of \(N\) bivariate observations. True scores are drawn from a bivariate normal distribution: \(T_x \sim N(0,1)\) and \(T_y = \rho_{\text{true}} \cdot T_x + \sqrt{1 - \rho_{\text{true}}^2} \cdot Z\) where \(Z \sim N(0,1)\), which preserves the specified true correlation. Each variable then accumulates independent error from two sources — test-retest and inter-rater — using the same \(\sigma = \sqrt{1/\rho - 1}\) conversion as Part 1, giving \(X_i = T_{x_i} + e_{\text{tr},x} + e_{\text{ir},x}\) and \(Y_i = T_{y_i} + e_{\text{tr},y} + e_{\text{ir},y}\). The observed correlation \(r\) is computed from the \(N\) \((X, Y)\) pairs, and the fit line drawn through the sample has slope fixed to \(r\) rather than the OLS slope — keeping it directly comparable to the true-correlation reference line regardless of asymmetries in \(X\) and \(Y\) reliability.