ASTM E3076-18e1

Standard Practice for Determination of the Slope in the Linear Region of a Test Record

SIGNIFICANCE AND USE
5.1 It is often necessary to determine the slope of a linear region within a test record, and for standardization purposes, it is desirable to have a method for determining the slope that is not subjective. There are numerous ASTM standard test methods where the test procedure or analysis requires that slope be determined, but the procedures for doing so are not well defined. Ideally, if multiple laboratories analyze the same data for determination of slope, they should produce the same result. The objective of this standard practice is to eliminate the linear-fit as a source of variability in test results.
SCOPE
1.1 This practice presents an automated, objective linear-fitting method for determining the slope of the linear portion of a test record. The method assumes that there is a linear region early in the test record where the value of the y variable increases roughly in proportion to the x variable and the slope of the record decreases after the linear region. The practice determines the best linear fit to the data based on the least normalized residual and provides metrics for evaluating the quality of the test record and the quality of the resultant fit.  
1.2 Data quality metrics are applied that evaluate the level of noise and the digital resolution of the data to determine if the test record is adequate for a linear regression analysis. Fit quality metrics use analysis of residuals in the vicinity of the fit range to determine if the test record is adequately linear and the fit range is sufficiently large.  
1.3 For test records that meet the data and fit quality metrics, the practice determines a repeatable slope without the need for operator input that is independent of operator judgment. For test records that fail one or more of the quality metrics, it is recommended that the analyst evaluate the fit to determine if it is acceptable.  
1.4 This practice represents a general purpose approach that is applicable for any test standard or method in which a linear fit is desired. It is intended that this practice can be called upon by standard test methods when slope must be determined.  
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.6 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 14-Nov-2018
• ICS: 03.120.30 - Application of statistical methods
• ICS: 17.020 - Metrology and measurement in general
• Technical Committee: E08 - Fatigue and Fracture
• Drafting Committee: E08.03 - Advanced Apparatus and Techniques

Relations

• Refers: ASTM E1942-98(2018)e1 - Standard Guide for Evaluating Data Acquisition Systems Used in Cyclic Fatigue and Fracture Mechanics Testing
• Effective Date: 01-Jun-2018
• Refers: ASTM E2443-05(2018)e1 - Standard Guide for Verifying Computer-Generated Test Results Through The Use Of Standard Data Sets
• Effective Date: 01-Jun-2018
• Refers: ASTM E1942-98(2010)e1 - Standard Guide for Evaluating Data Acquisition Systems Used in Cyclic Fatigue and Fracture Mechanics Testing
• Effective Date: 01-Nov-2010
• Refers: ASTM E2443-05(2010) - Standard Guide for Verifying Computer-Generated Test Results Through The Use Of Standard Data Sets
• Effective Date: 01-Nov-2010
• Refers: ASTM E1942-98(2010) - Standard Guide for Evaluating Data Acquisition Systems Used in Cyclic Fatigue and Fracture Mechanics Testing
• Effective Date: 01-Nov-2010
• Refers: ASTM E2443-05(2010)e1 - Standard Guide for Verifying Computer-Generated Test Results Through The Use Of Standard Data Sets
• Effective Date: 01-Nov-2010
• Refers: ASTM E2443-05 - Standard Guide for Verifying Computer-Generated Test Results Through The Use Of Standard Data Sets
• Effective Date: 15-Jun-2005
• Refers: ASTM E1942-98(2004) - Standard Guide for Evaluating Data Acquisition Systems Used in Cyclic Fatigue and Fracture Mechanics Testing
• Effective Date: 01-May-2004
• Refers: ASTM E1942-98e1 - Standard Guide for Evaluating Data Acquisition Systems Used in Cyclic Fatigue and Fracture Mechanics Testing
• Effective Date: 29-Dec-1999

Overview

ASTM E3076-18e1: Standard Practice for Determination of the Slope in the Linear Region of a Test Record provides a rigorous, automated, and objective methodology for identifying and quantifying the slope within the linear region of a test record. The standard aims to reduce subjectivity and laboratory-to-laboratory variability in slope determination for a broad array of test methods, supporting greater consistency in data analysis and reporting. Developed by ASTM, this standard is widely applicable across disciplines where characterization of linear behavior in empirical test data is crucial.

Key applications include testing scenarios involving cyclic fatigue, fracture mechanics, or material deformation, where an accurate, repeatable determination of the linear slope is vital to assessing material properties and ensuring reliable quality control. ASTM E3076-18e1 sets out guidelines for digital data quality assessment and fit validation, fostering robust, operator-independent results.

Key Topics

• Automated Linear Fitting: Utilizes an objective, algorithmic approach for identifying the optimal linear region in test data, minimizing the potential for user-induced variability.
• Normalization and Residual Analysis: Test records are normalized to facilitate quality metrics that are independent of units or scale. Residual analysis is employed to assess and validate linearity.
• Data Quality Metrics: Measurements of noise and digital resolution are applied to ensure the dataset is suitable for linear regression, reducing susceptibility to poor data quality.
• Fit Quality Metrics: Automated checks for curvature and evaluation of the fit range size in the linear region confirm the reliability of the regression.
• Reporting Requirements: Detailed reporting is mandated, including specifics on lab identification, data metrics, fit metrics, final slope, and intercept-enabling reproducibility and traceability.

Applications

ASTM E3076-18e1 is a general-purpose practice applicable to any testing standard or method that necessitates a consistent method for slope determination in the linear region of a test record. Typical use cases include:

• Materials Testing: Tensile, compressive, or fatigue tests where initial linear response is analyzed to determine mechanical properties such as modulus or stiffness.
• Quality Control: Automated slope determination allows for rigorous, repeatable assessment of production samples, ensuring compliance with specifications.
• Comparative Studies: Standardized, operator-independent results facilitate reliable comparison of data across labs, studies, or production batches.
• Fracture Mechanics: Used in conjunction with standards like ASTM E1942 and E2443 to enhance the reliability of slope-dependent evaluations in cyclic fatigue and similar domains.

The practice is particularly valuable for laboratories and testing environments seeking to minimize subjective analyst input and enhance data integrity across repeated or multi-site studies.

Related Standards

Several ASTM standards and documents are referenced within and complement ASTM E3076-18e1:

• ASTM E1942 – Guide for Evaluating Data Acquisition Systems Used in Cyclic Fatigue and Fracture Mechanics Testing
• ASTM E2443 – Guide for Verifying Computer-Generated Test Results Through The Use Of Standard Data Sets
• ASTM E3076-DS1(2018) – Benchmark Data Set to Evaluate Computer Implementation of the Algorithm in ASTM E3076

These related documents support best practices in data acquisition, verification, and computational analysis-key components in ensuring the validity and reproducibility of linear slope determination.

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ASTM E3076-18e1 enhances confidence in test record analysis by providing a standardized, automated approach for determining the slope in the linear region, streamlining data processing, and promoting global best practices in laboratory testing. For organizations seeking rigorous standardization in slope analysis, adherence to this practice will drive consistency, reliability, and quality in test outcomes.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
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Designation: E3076 − 18
Standard Practice for
Determination of the Slope in the Linear Region of a Test
Record
This standard is issued under the fixed designation E3076; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Footnote 3 was editorially corrected in May 2023.
1. Scope Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical
1.1 This practice presents an automated, objective linear-
Barriers to Trade (TBT) Committee.
fitting method for determining the slope of the linear portion of
a test record. The method assumes that there is a linear region
2. Referenced Documents
early in the test record where the value of the y variable
2.1 ASTM Standards:
increases roughly in proportion to the x variable and the slope
E1942 Guide for Evaluating Data Acquisition Systems Used
of the record decreases after the linear region. The practice
in Cyclic Fatigue and Fracture Mechanics Testing
determines the best linear fit to the data based on the least
E2443 Guide for Verifying Computer-Generated Test Re-
normalized residual and provides metrics for evaluating the
sults Through The Use Of Standard Data Sets
quality of the test record and the quality of the resultant fit.
2.2 ASTM Data Set:
1.2 Data quality metrics are applied that evaluate the level
E3076-DS1(2018) File 01 Benchmark Data Set to Evaluate
of noise and the digital resolution of the data to determine if the
Computer Implementation of the Algorithm in Standard
test record is adequate for a linear regression analysis. Fit
E3076.
quality metrics use analysis of residuals in the vicinity of the fit
3. Terminology
range to determine if the test record is adequately linear and the
fit range is sufficiently large.
3.1 Definitions:
3.1.1 digital resolution, n—the precision of stored informa-
1.3 For test records that meet the data and fit quality metrics,
the practice determines a repeatable slope without the need for tion resulting from a discrete digital representation of analog
operator input that is independent of operator judgment. For data.
test records that fail one or more of the quality metrics, it is
3.1.2 linear region, n—a region of the test record where the
recommended that the analyst evaluate the fit to determine if it
underlying physics indicate that the dependent variable would
is acceptable.
increase in proportion to the independent variable if there were
no noise in the test record.
1.4 This practice represents a general purpose approach that
is applicable for any test standard or method in which a linear
3.1.3 normalized, n—data with global minima of 0 and
fit is desired. It is intended that this practice can be called upon
global maxima of 1 in x and y.
by standard test methods when slope must be determined.
3.1.4 residual, n—the difference between the linear fit and
1.5 This standard does not purport to address all of the
the original test record y-values at a given x-value.
safety concerns, if any, associated with its use. It is the
3.1.5 test record, n—the basic raw data from a data set
responsibility of the user of this standard to establish appro-
priate safety, health, and environmental practices and deter- 4. Summary of Practice
mine the applicability of regulatory limitations prior to use.
4.1 This practice is intended for the analysis of test records
1.6 This international standard was developed in accor-
where the y variable increases roughly in proportion to the x
dance with internationally recognized principles on standard-
variable over a region early in the test record.
ization established in the Decision on Principles for the
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
This practice is under the jurisdiction of ASTM Committee E08 on Fatigue and contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
Fracture and is the direct responsibility of Subcommittee E08.03 on Advanced Standards volume information, refer to the standard’s Document Summary page on
Apparatus and Techniques. the ASTM website.
Current edition approved Nov. 15, 2018. Published April 2019. DOI: 10.1520/ This data set is available for download from ASTM at
E3076–18E01 https://www.astm.org/get-involved/technical-committees/adhoc-e08.html
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
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E3076 − 18
4.2 The linear-fitting algorithm utilizes analysis of residuals is desirable to have a method for determining the slope that is
coupled with computational effort to numerically determine the not subjective. There are numerous ASTM standard test
most linear region in a given test record. The analysis of methods where the test procedure or analysis requires that
residuals methodology is more robust than conventional least- slope be determined, but the procedures for doing so are not
squares fitting techniques based on a correlation coefficient well defined. Ideally, if multiple laboratories analyze the same
because the latter is not useful when the underlying physics data for determination of slope, they should produce the same
indicate that the behavior is linear. result. The objective of this standard practice is to eliminate the
linear-fit as a source of variability in test results.
4.3 The test record is offset and normalized so that quality
metrics can be applied that are independent of scale and
6. Procedure
engineering units.
6.1 The procedural description of this standard practice is
4.4 Metrics on noise level and digital resolution in the
assisted through demonstration of an example linear-fit made
normalized data set are applied to evaluate the quality of the
to a tensile test record. The test record is entered into arrays for
test record.
force and extension where the dependent variable (y) is force
4.5 The test record is evaluated to determine the most linear
and the independent variable (x) is extension, and the indices
region using a tangency point approach to truncate the test
start from 1.
record. This approach assumes that there is a linear region early
6.1.1 This test record is designated as File 01 and is
in the test record and that the slope decreases after the linear
available from ASTM International. The test record is plotted
region.
in Fig. 1. See Guide E2443.
4.6 Analysis of residuals is used to evaluate the quality of
6.2 Limit the range of the test record that will be searched
the resulting fit.
for the most linear region by determining if the test record has
4.7 The linear-fitting method presented in this practice is
a local maximum or a knee above which the slope shows a
objective and fully automated for the case where the test record
significant decrease. This behavior is common in many types
passes all quality metrics. If a test record fails one or more
of materials testing. A tangent construction is used to find the
quality metrics, the analyst should examine the test record and
local maximum or knee of the record and establish the upper
the fit to determine whether it is acceptable based on the
end of the data used in the search. The process is illustrated in
application or calling standard.
Fig. 2 and described in the following sub-sections.
6.2.1 Shift the test record such that the starting x and y
5. Significance and Use
values are zeroed by subtracting the x-value for the first point
5.1 It is often necessary to determine the slope of a linear
(xshift) from the x-values for all points in the test record and
region within a test record, and for standardization purposes, it
subtracting the y-value for the first point (yshift) from the
y-values for all points in the test record. Record the xshift and
yshift values so that the final intercept and fit range from linear
S. M. Graham and M. A. Adler, “Determining the Slope and Quality of Fit for
regression can be corrected. For this example, xshift =
the Linear Part of a Test Record,” J Testing and Evaluation, v 39, n 2, pp. 260–268,
5.792E-5 and yshift = 0.1914.
March 2011.
FIG. 1 Plot of Original Test Record File 01
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E3076 − 18
FIG. 2 Tangent Construction Used to Truncate Data Set
6.2.2 Find the point in the shifted test record where the example, the tangency point has an index of 220 with coordi-
y-value just exceeds 5 % of the maximum y-value and desig- nates (0.06989, 20.60).
nate the x and y values (x , y ). For this example, x = 0.01016
1 1 1
6.3 Truncate all data beyond the tangent point, and normal-
and y = 3.012.
ize all y-values by dividing by the tangency-point y-value, and
6.2.3 Create an offset point with the same x-value and a
normalize all x-values by dividing by the tangency point
y-value equal to y plus 15 % of the maximum y-value, (xoffset,
x-value. The resultant data is now normalized and has a range
yoffset) = (x , y +0.15 * max(y)). For this example, yoffset =
1 1
from 0 to 1 in both x and y, as shown in Fig. 3. This facilitates
11.98.
the development of fit criteria that are independent of the data
6.2.4 Starting with the first point where the y-value is
maxima and the units of measure.
greater than yoffset, find the point in the test record where the
line from that point to the offset point has the largest slope and
6.4 First Data Quality Metric—Check for excessive noise in
record this as the tangent point (xtangent, ytangent). For this the data, which is the first metric on data quality. The allowable
FIG. 3 Shifted, Truncated and Normalized Test Record
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E3076 − 18
noise is 0.005, which translates to 0.5 % of the normalized 6.5.8 In this example, for the x-variable z = 2, relative
tangency point x and y values. (See Guide E1942.) x-resolution is 0.67, percentage of data in that bin is 15.6 % and
6.4.1 Determine the number of data points in the normalized the x zeroth bin is 6.4 %. For the y-variable z = 1, relative
test record, N. y-resolution is 0.33, percentage of data in that bin is 4.6 % and
6.4.2 Calculate ∆x 5x 2x for i = 1 to N – 1. the y zeroth bin is 90.8 %. Both of these digital resolutions are
i i11 i
sufficient for analysis of residuals since they meet the criteria
¯
6.4.3 Calculate the mean of thex values, ∆x. Then
specified in section 6.5.6.
calculate the difference betweenx and the meanx for each
6.5.9 The allowable resolution of 3δ is based on good
¯
point, ∆xr 5∆x 2∆x for i = 1 to N – 1. The normalized noise is
i i
experimental practice. It corresponds to a data-acquisition
¯
the standard deviation of the ∆xr values. Relative noise is the
system with a 12-bit digitizer and a full-scale value of the
standard deviation divided by 0.005 such that a value greater
transducer that is three times the value at the tangent point.
than one indicates excessive noise.
This is reasonable for 12-bit systems and not particularly
6.4.4 Repeat 6.4.1 to 6.4.3 for the y-values. stringent for more modern 16-bit and higher systems. A low
6.4.5 The results for the example test record shown in the
data acquisition rate can appear as low resolution, so if the test
table below indicate that the noise levels in x and y are record fails the digital resolution metric the data acquisition
acceptable. rate should also be examined.
6.5.10 Note that if the digital resolution is not sufficient,
x y
analysis of residuals will not work well. When the digital
Standard Deviation
(Normalized Noise) 0.001626 0.0007219
resolution is too low, successive points in the data have the
Allowable Noise 0.005 0.005
same x or y value followed by a relatively large jump
Relative Noise 0.325 0.144
proportional to the system resolution. This results in ∆∆x or
6.5 Second Data Quality Metric—Check the digital
∆∆y equal to zero, and therefore an increase in the percentage
resolution, which is the second metric on the quality of the
in the zeroth bin. This will present itself as diagonal lines on a
data, as follows:
residuals plot. Therefore, a high percentage of data in the
6.5.1 The optimum digital resolution, δ, for the normalized
zeroth bin can be an indicator of either high resolution or low
data set is δ5 . resolution. The differentiating factor is the bin number greater
than 0 with the largest percentage. This is why the
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