The fitted relation between relative compaction and collapse time is expressed as

where a is the intercept and b is the slope of the RC-log₁₀T relation. The slope represents the rate of change of RC with respect to logarithmic collapse time.

The slope may be obtained from any two points on the fitted line:

An engineering transition interval between two collapse times T₁ and T₂ is therefore:

For the practical SCT anchors T = 15 min and T = 60 min:

Thus, ΔRC (15 → 60), as defined in Equation (4), is a projection of the fitted slope over a specified interval rather than an independent descriptor. Logarithms are expressed in base 10 throughout. The choice of logarithmic base affects only the numerical scaling of the fitted slope and does not influence interpretation.

The fitted relation provides three complementary descriptors:

RC@15: indicator of a behavioural threshold.b: sensitivity of RC to logarithmic time.ΔRC (15 → 60): projected transition interval.

Robustness is evaluated using leave-one-out regression. Slopes b_−i are computed after removing each observation in turn.

The index I_b defined in Equation (6) quantifies overall sensitivity to individual observations, while Δb_i (Equation (7)) identifies specific influential points:

The index I_b quantifies overall sensitivity of the fitted slope to individual observations, while Δb_i identifies specific influential points. Low values of I_b indicate a stable fitted relationship, whereas higher values indicate increasing dependence on individual observations. In the present examples, values of I_b on the order of 0.2 distinguish relatively stable from point-sensitive datasets and are therefore proposed as an indicative threshold for interpretation. I_b = 0.2 corresponds approximately to a situation where the total leave-one-out slope spread approaches 20% of the fitted slope magnitude. Further validation across broader datasets is required before general application.

Beyond this level, changes in the interpreted compaction window become sufficiently large that engineering interpretation may begin to depend noticeably on individual observations rather than the overall material response. The threshold should therefore be viewed as practical engineering guidance rather than a strict statistical limit. For elevated point sensitivity, the practitioner should examine the individual influence values Δb_i to identify which observations control the fitted slope. If a single point strongly steepens the regression, it may be appropriate to assess the alternative, flatter relation obtained when that point is removed and to consider whether this represents a more conservative interpretation for engineering purposes.

The slope b should be determined from the fitted regression rather than from individual point pairs. This approach uses the full dataset, reduces the effect of local scatter and provides a consistent basis for comparison between materials. It also allows interpretation even when T = 60 min is not strongly constrained by the achieved data.

Relative compaction RC depends on the chosen compaction reference (for example Modified Proctor or Standard Proctor). Changing reference shifts absolute RC values, and therefore RC@15, but does not affect b or ΔRC (15 → 60), which depend only on differences along the fitted relation. This distinction is important when comparing datasets obtained using different compaction standards.

Material A represents a high-correlation case (R² ≈ 0.95) with slope b ≈ 7.97 and a comparatively wide compaction window ΔRC (15 → 60) ≈ 4.7%. This indicates relatively low sensitivity to densification across the tested range, and a stable interpretation of collapse behaviour with respect to compaction. Leave-one-out analysis yields a low influence index (I_b ≈ 0.11), with the largest absolute point influence of about 5.8%, confirming that the fitted slope is highly stable with respect to removal of individual observations. The variation in slope is limited, indicating that the fitted relation is representative of the dataset rather than controlled by specific points.

Material D represents a contrasting case with lower correlation (R² ≈ 0.61), slope b ≈ 2.76, and a narrower compaction window ΔRC (15 → 60) ≈ 1.7%. This indicates higher sensitivity to densification. Leave-one-out analysis yields I_b ≈ 0.20, indicating elevated point sensitivity. Removal of individual observations produces noticeable changes in slope, with the largest absolute point influence of about 11.7%, reflecting the greater dependence of the fitted relation on individual observations, and implying that the inferred transition between collapsible and crack-holding behaviour may shift depending on the dataset.

This effect is illustrated in Figure 2, where removal of the most influential observation (Point 5) results in a reduction of the fitted slope from b ≈ 2.76 to b ≈ 2.44. The corresponding flattening of the fitted relationship leads to a reduced compaction window ΔRC (15 → 60), demonstrating how interpretation of collapse behaviour may shift depending on individual observations. This highlights the practical importance of identifying influential data points when interpreting SCT results.

The comparison shows that regression quality and robustness are related but not identical. The present examples show that moderate R² does not necessarily invalidate a fitted trend, while robustness must be evaluated independently, as high correlation alone does not guarantee low point sensitivity. Material A combines high correlation with low sensitivity, whereas Material D exhibits both weaker correlation and greater dependence on individual observations. This highlights the need to evaluate robustness explicitly rather than relying solely on regression metrics.

The quantitative comparison of slope, compaction window and influence index is provided in Table 1, while detailed variation in leave-one-out slopes is summarised in Table 2.

Table 2. Summary of leave-one-out slope variation for representative datasets.

The interpretation of fitted relationships should be supported by both graphical representation (Figures 1-3) and quantitative measures (Tables 1-3). Figure 1 defines the compaction window concept, Figure 2 presents fitted behaviour, and Figure 3 quantifies robustness, while the tables provide corresponding numerical measures.

Table 3. Most influential observations for Materials A and D based on leave-one-out analysis.

From an engineering perspective, the fitted relationship describes how achieved compaction governs collapse behaviour and therefore crack-stopping ability. The slope b (Equation (2)) defines sensitivity, while the compaction window ΔRC (15 → 60) (Equation (4)) expresses the rate of transition between practical collapse-time anchors. A narrow window implies that small variations in achieved density may shift a material from collapsible to crack-holding behaviour, whereas a broader window indicates a more gradual transition. The reliability of this interpretation therefore depends directly on the stability of the fitted slope.

Established statistical influence measures, such as Cook’s distance and DFFITS, as well as robust regression techniques, provide formal tools for identifying influential observations in fitted relationships. However, these approaches are not always directly interpretable in terms of engineering quantities. The present framework complements such methods by expressing sensitivity directly in terms of slope variation and the resulting change in interpreted behavioural thresholds, providing a transparent and practically applicable measure of robustness. From a statistical perspective, fitted relationships derived from limited datasets may appear stable while being influenced by individual observations. This sensitivity is quantified using leave-one-out analysis, where the robustness index I_b measures overall slope stability and Δb_i identifies specific influential points. It identifies vulnerable data points, quantifies how ΔRC (15 → 60) may shift due to those points, and supports conservative interpretation through alternative, flatter fits when sensitivity is elevated. In the case studies, Material A shows low Δb_i and stable ΔRC, indicating a reliable fit, whereas Material D exhibits influential points that materially alter the slope, requiring targeted scrutiny and cautious interpretation. The practical impact of point sensitivity is illustrated in Figure 2, where removal of a single influential observation produces a measurable change in slope and corresponding compaction window, demonstrating how statistical sensitivity translates directly into engineering interpretation. Influential observations should not be interpreted solely as statistical anomalies. In geotechnical testing, such points may reflect genuine transitions in material behaviour, for example associated with changes in moisture conditions or structural collapse mechanisms. The present framework therefore identifies sensitivity in the fitted relationship but does not prescribe removal of observations; rather, it highlights cases where interpretation should be supported by experimental judgement.

This distinction is critical when fitted relationships are used to define behavioural thresholds. High correlation does not guarantee robustness, and moderate correlation does not invalidate a trend. For point-sensitive datasets, interpretation should therefore consider plausible alternative fits derived from removal of influential observations.

For datasets exhibiting both moderate regression quality and elevated point sensitivity, interpretation should be undertaken conservatively. A practical approach is to consider the most negative Δb_i, corresponding to a flatter fitted relation, as a lower-bound estimate of behaviour. Reductions in fitted slope on the order of 10% - 15% may produce comparable reductions in the derived compaction window ΔRC (15 → 60).

In practical terms, this may correspond to density differences on the order of tens of kg/m³, which are within the range of normal construction variability. Consequently, materials exhibiting narrow compaction windows and elevated sensitivity may transition from collapsible to crack-holding behaviour under routine variations in achieved density. In such cases, apparent collapsibility may be overestimated if robustness is not explicitly considered.

The combined use of graphical representation, fitted relationships, and robustness metrics provides a consistent framework for interpretation. Engineering behaviour (ΔRC), statistical stability (I_b and Δb_i), and practical consequence (density variation) are directly linked, allowing informed and transparent assessment of collapse behaviour from limited datasets.

The proposed framework is intentionally simple. Its novelty lies not in the statistical method itself, but in its direct application to engineering interpretation. By linking slope sensitivity to a physically meaningful descriptor (ΔRC) and to practical decision-making, the method provides a transparent tool for assessing confidence in fitted relationships derived from limited datasets.

Proposedprocedure:

1) Evaluaterobustness

Fit the RC-log(T) relationship and compute I_b and Δb_i using leave-one-out analysis. Classify the dataset as robust (low I_b) or point-sensitive (elevated I_b). In practical implementation, regression fitting in common software environments may use either base-10 or natural logarithms. The choice of logarithmic base does not affect interpretation, provided consistency is maintained. Where natural logarithms are used, the fitted slope should be converted to the equivalent base-10 slope to ensure consistency with the definition of ΔRC (15 → 60). This is achieved by dividing the slope obtained from ln(T) by ln(10), i.e.b = b_ln/ln(10), where b_ln is the slope obtained from the RC-ln(T) relation. This conversion ensures that derived quantities remain directly comparable and physically interpretable within the SCT framework.

2) Identifyandverifyinfluentialobservations

For point-sensitive datasets, examine Δb_i to identify controlling observations. Check data quality, test conditions, and representativeness. Where warranted, repeat laboratory tests or review field data to confirm or refute the influence of these points.

3) Adoptadefensibleinterpretation

If influential points cannot be conclusively resolved, adopt a conservative interpretation by considering alternative fits (e.g. a flatter slope corresponding to the most negative Δb_i). Translate the resulting change in slope into ΔRC (15 → 60) using Equation (4) and reassess the implied transition between collapsible and crack-holding behaviour.

4) Updateengineeringdecisions

Where reduced ΔRC indicates a narrow compaction window, adjust specifications and procedures accordingly (e.g. tighter compaction control, revised acceptance criteria, or additional verification testing) to account for construction variability and reduce the risk of unintended transition to crack-holding behaviour.

A similar procedure may be applied in other geotechnical contexts where fitted relationships are used to define project-specific criteria, allowing robustness assessment to be linked directly to governing parameters and acceptance thresholds relevant to the problem at hand.