ASTM E3080-19
(Practice)Standard Practice for Regression Analysis with a Single Predictor Variable
Standard Practice for Regression Analysis with a Single Predictor Variable
ABSTRACT
This practice covers regression analysis of a set of data to define the statistical relationship between two numerical variables for use in predicting one variable from the other. This practice is restricted in scope to consider only a single numerical response variable and a single numerical predictor variable. The objective is to obtain a regression model for use in predicting the value of the response variable Y for given values of the predictor variable X.
SIGNIFICANCE AND USE
4.1 Regression analysis is a procedure that uses data to study the statistical relationships between two or more variables (1, 2).3 This practice is restricted in scope to consider only a single numerical response variable and a single numerical predictor variable. The objective is to obtain a regression model for use in predicting the value of the response variable Y for given values of the predictor variable X.
4.2 A regression model consists of: (1) a regression function that relates the mean values of the response variable distribution to fixed values of the predictor variable, and (2) a statistical distribution that describes the variability in the response variable values at a fixed value of the predictor variable.
4.2.1 The regression analysis utilizes either experimental or observational data to estimate the parameters defining a regression model and their precision. Diagnostic procedures are utilized to assess the resulting model fit and can suggest other models for improved prediction performance.
4.3 The information in this practice is arranged as follows.
4.3.1 Section 5 gives a general outline of the steps in the regression analysis procedure. The subsequent sections cover procedures for estimation of specific regression models.
4.3.2 Section 6 assumes a straight line relationship between the two variables. This is also known as the simple linear regression model or a first order model. This model should be used as a starting point for understanding the XY relationship and ultimately defining the best fitting model to the data.
4.3.3 Section 7 considers a proportional relationship between the variables, where the ratio of one variable to the other is constant. The intercept is constrained to be zero. This model is useful for single point calibration, where a reference material is run periodically as a standard during routine testing to correct for drift in instrument performance over a given range of test results.
4.3.4 Section 8 di...
SCOPE
1.1 This practice covers regression analysis of a set of data to define the statistical relationship between two numerical variables for use in predicting one variable from the other.
1.2 The regression analysis provides graphical and calculational procedures for selecting the best statistical model that describes the relationship and for evaluation of the fit of the data to the selected model.
1.3 The resulting regression model can be useful for developing process knowledge through description of the variable relationship, in making predictions of future values, in relating the precision of a test method to the value of the characteristic being measured, and in developing control methods for the process generating values of the variables.
1.4 The system of units for this practice is not specified. Dimensional quantities in the practice are presented only as illustrations of calculation methods. The examples are not binding on products or test methods treated.
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 ...
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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.
Designation: E3080 − 19 An American National Standard
Standard Practice for
1
Regression Analysis with a Single Predictor Variable
This standard is issued under the fixed designation E3080; 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.
1. Scope E456 Terminology Relating to Quality and Statistics
E2586 Practice for Calculating and Using Basic Statistics
1.1 This practice covers regression analysis of a set of data
to define the statistical relationship between two numerical
3. Terminology
variables for use in predicting one variable from the other.
3.1 Definitions—Unless otherwise noted, terms relating to
1.2 The regression analysis provides graphical and calcula-
quality and statistics are as defined in Terminology E456.
tional procedures for selecting the best statistical model that
3.1.1 degrees of freedom, n—the number of independent
describes the relationship and for evaluation of the fit of the
data points minus the number of parameters that have to be
data to the selected model.
estimated before calculating the variance. E2586
1.3 The resulting regression model can be useful for devel-
3.1.2 predictor variable, X, n—a variable used to predict a
oping process knowledge through description of the variable
response variable using a regression model.
relationship, in making predictions of future values, in relating
the precision of a test method to the value of the characteristic
3.1.2.1 Discussion—Also called an independent or explana-
being measured, and in developing control methods for the
tory variable.
process generating values of the variables.
3.1.3 regression analysis, n—a statistical procedure used to
1.4 The system of units for this practice is not specified. characterize the association between two or more numerical
Dimensional quantities in the practice are presented only as variables for prediction of the response variable from the
predictor variable.
illustrations of calculation methods. The examples are not
binding on products or test methods treated.
3.1.3.1 Discussion—In this practice, only a single predictor
1.5 This standard does not purport to address all of the
variable is considered.
safety concerns, if any, associated with its use. It is the
3.1.4 residual, n—the observed value minus fitted value,
responsibility of the user of this standard to establish appro-
when a regression model is used.
priate safety, health, and environmental practices and deter-
3.1.5 response variable, Y, n—a variable predicted from a
mine the applicability of regulatory limitations prior to use.
regression model.
1.6 This international standard was developed in accor-
3.1.5.1 Discussion—Also called a dependent variable.
dance with internationally recognized principles on standard-
2
ization established in the Decision on Principles for the
3.1.6 sample coeffıcient of determination, r,n—square of
Development of International Standards, Guides and Recom-
the sample correlation coefficient.
mendations issued by the World Trade Organization Technical
3.1.7 sample correlation coeffıcient, r, n—a dimensionless
Barriers to Trade (TBT) Committee.
measure of association between two variables estimated from
the data.
2. Referenced Documents
3.1.8 sample covariance, s ,n—an estimate of the associa-
2
xy
2.1 ASTM Standards:
tion of the response variable and predictor variable calculated
E178 Practice for Dealing With Outlying Observations
from the data.
3.2 Definitions of Terms Specific to This Standard:
1
This practice is under the jurisdiction ofASTM Committee E11 on Quality and
3.2.1 intercept, β , n—of a regression model, the value of
0
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
the response variable when the value of the predictor variable
Statistics.
Current edition approved Sept. 1, 2019. Published January 2020. Originally
is equal to zero.
approved in 2016. Last previous edition approved in 2017 as E3080 – 17. DOI:
3.2.2 regression model parameter, n—a descriptive constant
10.1520/E3080-19.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
defining a regression model that is to be estimated.
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
3.2.3 residual standard deviation, σ, n—of a regression
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. model, the square root of the residual variance.
Copyright © ASTM Internationa
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E3080 − 17 E3080 − 19 An American National Standard
Standard Practice for
1
Regression Analysis with a Single Predictor Variable
This standard is issued under the fixed designation E3080; 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.
1. Scope
1.1 This practice covers regression analysis methodology for estimating, evaluating, and using the simple linear regression
model of a set of data to define the statistical relationship between two numerical variables.variables for use in predicting one
variable from the other.
1.2 The regression analysis provides graphical and calculational procedures for selecting the best statistical model that describes
the relationship and for evaluation of the fit of the data to the selected model.
1.3 The resulting regression model can be useful for developing process knowledge through description of the variable
relationship, in making predictions of future values, in relating the precision of a test method to the value of the characteristic being
measured, and in developing control methods for the process generating values of the variables.
1.4 The system of units for this practice is not specified. Dimensional quantities in the practice are presented only as illustrations
of calculation methods. The examples are not binding on products or test methods treated.
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.
2. Referenced Documents
2
2.1 ASTM Standards:
E178 Practice for Dealing With Outlying Observations
E456 Terminology Relating to Quality and Statistics
E2586 Practice for Calculating and Using Basic Statistics
3. Terminology
3.1 Definitions—Unless otherwise noted, terms relating to quality and statistics are as defined in Terminology E456.
2
3.1.1 coeffıcient of determination, r , n—square of the correlation coefficient.
3.1.1 degrees of freedom, n—the number of independent data points minus the number of parameters that have to be estimated
before calculating the variance. E2586
3.1.3 residual, n—observed value minus fitted value, when a model is used.
3.1.2 predictor variable, X, n—a variable used to predict a response variable using a regression model.
3.1.2.1 Discussion—
Also called an independent or explanatory variable.
3.1.3 regression analysis, n—a statistical procedure used to characterize the association between two or more numerical
variables for prediction of the response variable from the predictor variable.
1
This practice is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling / Statistics.
Current edition approved Nov. 1, 2017Sept. 1, 2019. Published January 2018January 2020. Originally approved in 2019.2016. Last previous edition approved in 20162017
as E3080 – 16.17. DOI: 10.1520/E3080-17.10.1520/E3080-19.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’sstandard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 ----------------------
E3080 − 19
3.1.3.1 Discussion—
In this practice, only a single predictor variable is considered.
3.1.4 residual, n—the observed value minus fitted value, when a regression model is used.
3.1.5 response variable, Y, n—a variable predicted from a regression model.
3.1.5.1 Discussion—
Also called a dependent variable.
2
3.1.6 sample coeffıcient of determination, r , n—square of the sample correlation coefficient.
3.1.7 sample correlation coeffıcient, r, n—a dimensionless measure of association
...
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E3080 − 19 An American National Standard
Standard Practice for
1
Regression Analysis with a Single Predictor Variable
This standard is issued under the fixed designation E3080; 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.
1. Scope E456 Terminology Relating to Quality and Statistics
E2586 Practice for Calculating and Using Basic Statistics
1.1 This practice covers regression analysis of a set of data
to define the statistical relationship between two numerical
3. Terminology
variables for use in predicting one variable from the other.
3.1 Definitions—Unless otherwise noted, terms relating to
1.2 The regression analysis provides graphical and calcula-
quality and statistics are as defined in Terminology E456.
tional procedures for selecting the best statistical model that
3.1.1 degrees of freedom, n—the number of independent
describes the relationship and for evaluation of the fit of the
data points minus the number of parameters that have to be
data to the selected model.
estimated before calculating the variance. E2586
1.3 The resulting regression model can be useful for devel-
3.1.2 predictor variable, X, n—a variable used to predict a
oping process knowledge through description of the variable
response variable using a regression model.
relationship, in making predictions of future values, in relating
the precision of a test method to the value of the characteristic
3.1.2.1 Discussion—Also called an independent or explana-
being measured, and in developing control methods for the
tory variable.
process generating values of the variables.
3.1.3 regression analysis, n—a statistical procedure used to
1.4 The system of units for this practice is not specified. characterize the association between two or more numerical
variables for prediction of the response variable from the
Dimensional quantities in the practice are presented only as
illustrations of calculation methods. The examples are not predictor variable.
binding on products or test methods treated.
3.1.3.1 Discussion—In this practice, only a single predictor
1.5 This standard does not purport to address all of the
variable is considered.
safety concerns, if any, associated with its use. It is the
3.1.4 residual, n—the observed value minus fitted value,
responsibility of the user of this standard to establish appro-
when a regression model is used.
priate safety, health, and environmental practices and deter-
3.1.5 response variable, Y, n—a variable predicted from a
mine the applicability of regulatory limitations prior to use.
regression model.
1.6 This international standard was developed in accor-
3.1.5.1 Discussion—Also called a dependent variable.
dance with internationally recognized principles on standard-
2
ization established in the Decision on Principles for the
3.1.6 sample coeffıcient of determination, r , n—square of
Development of International Standards, Guides and Recom-
the sample correlation coefficient.
mendations issued by the World Trade Organization Technical
3.1.7 sample correlation coeffıcient, r, n—a dimensionless
Barriers to Trade (TBT) Committee.
measure of association between two variables estimated from
the data.
2. Referenced Documents
2 3.1.8 sample covariance, s , n—an estimate of the associa-
xy
2.1 ASTM Standards:
tion of the response variable and predictor variable calculated
E178 Practice for Dealing With Outlying Observations
from the data.
3.2 Definitions of Terms Specific to This Standard:
1
This practice is under the jurisdiction of ASTM Committee E11 on Quality and
3.2.1 intercept, β , n—of a regression model, the value of
0
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
the response variable when the value of the predictor variable
Statistics.
Current edition approved Sept. 1, 2019. Published January 2020. Originally is equal to zero.
approved in 2016. Last previous edition approved in 2017 as E3080 – 17. DOI:
3.2.2 regression model parameter, n—a descriptive constant
10.1520/E3080-19.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or defining a regression model that is to be estimated.
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
3.2.3 residual standard deviation, σ, n—of a regression
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. model, the square root of the residual variance.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 --------------------
...
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