Simple Linear Regression Analysis - Revision history

Lisa Hacker at 18:47, 15 September 2023

2023-09-15T18:47:30Z

← Older revision		Revision as of 18:47, 15 September 2023
Line 368:		Line 368:
	==Confidence Intervals in Simple Linear Regression==		==Confidence Intervals in Simple Linear Regression==

	A confidence interval represents a closed interval where a certain percentage of the population is likely to lie. For example, a 90% confidence interval with a lower limit of <math>A\,\!</math> and an upper limit of <math>B\,\!</math> implies that 90% of the population lies between the values of <math>A\,\!</math> and <math>B\,\!</math>. Out of the remaining 10% of the population, 5% is less than <math>A\,\!</math> and 5% is greater than <math>B\,\!</math>. (For details refer to the ~~[[Life_Data_Analysis_Reference_Book\|~~ Life ~~Data Analysis Reference Book]]~~.) This section discusses confidence intervals used in simple linear regression analysis.		A confidence interval represents a closed interval where a certain percentage of the population is likely to lie. For example, a 90% confidence interval with a lower limit of <math>A\,\!</math> and an upper limit of <math>B\,\!</math> implies that 90% of the population lies between the values of <math>A\,\!</math> and <math>B\,\!</math>. Out of the remaining 10% of the population, 5% is less than <math>A\,\!</math> and 5% is greater than <math>B\,\!</math>. (For details refer to the Life data analysis reference.) This section discusses confidence intervals used in simple linear regression analysis.

	===Confidence Interval on Regression Coefficients===		===Confidence Interval on Regression Coefficients===

Chuck Smith at 22:51, 9 August 2018

2018-08-09T22:51:30Z

← Older revision		Revision as of 22:51, 9 August 2018
Line 1:		Line 1:
	{{Template:Doebook\|3}}		{{Template:Doebook\|3}}
	Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.		Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.

	This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.		This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.

Chuck Smith at 22:18, 9 August 2018

2018-08-09T22:18:53Z

← Older revision		Revision as of 22:18, 9 August 2018
Line 1:		Line 1:
	{{Template:Doebook\|3}}		{{Template:Doebook\|3}}
	Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all Weibull++ DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.		Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.

	This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.		This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.

Richard House: /* Coefficient of Determination (R^2 ) */

2017-08-10T15:58:05Z

Coefficient of Determination (R^2 )

← Older revision		Revision as of 15:58, 10 August 2017
Line 485:		Line 485:


	Therefore, 98% of the variability in the yield data is explained by the regression model, indicating a very good fit of the model. It may appear that larger values of <math>{{R}^{2}}\,\!</math> indicate a better fitting regression model. However, <math>{{R}^{2}}\,\!</math> should be used cautiously as this is not always the case. The value of <math>{{R}^{2}}\,\!</math> increases as more terms are added to the model, even if the new term does not contribute significantly to the model. Therefore, an increase in the value of <math>{{R}^{2}}\,\!</math> cannot be taken as a sign to conclude that the new model is superior to the older model. Adding a new term may make the regression model worse if the error mean square, <math>M{{S}_{E}}\,\!</math>, for the new model is larger than the <math>M{{S}_{E}}\,\!</math> of the older model, even though the new model will show an increased value of <math>{{R}^{2}}\,\!</math>. In the results obtained from DOE++, <math>{{R}^{2}}\,\!</math> is displayed as R-sq under the ANOVA table (as shown in the figure below), which displays the complete analysis sheet for the data in the preceding [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| table]].		Therefore, 98% of the variability in the yield data is explained by the regression model, indicating a very good fit of the model. It may appear that larger values of <math>{{R}^{2}}\,\!</math> indicate a better fitting regression model. However, <math>{{R}^{2}}\,\!</math> should be used cautiously as this is not always the case. The value of <math>{{R}^{2}}\,\!</math> increases as more terms are added to the model, even if the new term does not contribute significantly to the model. Therefore, an increase in the value of <math>{{R}^{2}}\,\!</math> cannot be taken as a sign to conclude that the new model is superior to the older model. Adding a new term may make the regression model worse if the error mean square, <math>M{{S}_{E}}\,\!</math>, for the new model is larger than the <math>M{{S}_{E}}\,\!</math> of the older model, even though the new model will show an increased value of <math>{{R}^{2}}\,\!</math>. In the results obtained from the DOE folio, <math>{{R}^{2}}\,\!</math> is displayed as R-sq under the ANOVA table (as shown in the figure below), which displays the complete analysis sheet for the data in the preceding [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| table]].

	The other values displayed with are S, R-sq(adj), PRESS and R-sq(pred). These values measure different aspects of the adequacy of the regression model. For example, the value of S is the square root of the error mean square, <math>MS_E\,\!</math>, and represents the "standard error of the model." A lower value of S indicates a better fitting model. The values of S, R-sq and R-sq(adj) indicate how well the model fits the observed data. The values of PRESS and R-sq(pred) are indicators of how well the regression model predicts new observations. R-sq(adj), PRESS and R-sq(pred) are explained in [[Multiple Linear Regression Analysis]].		The other values displayed with are S, R-sq(adj), PRESS and R-sq(pred). These values measure different aspects of the adequacy of the regression model. For example, the value of S is the square root of the error mean square, <math>MS_E\,\!</math>, and represents the "standard error of the model." A lower value of S indicates a better fitting model. The values of S, R-sq and R-sq(adj) indicate how well the model fits the observed data. The values of PRESS and R-sq(pred) are indicators of how well the regression model predicts new observations. R-sq(adj), PRESS and R-sq(pred) are explained in [[Multiple Linear Regression Analysis]].

Richard House: /* Confidence Interval on New Observations */

2017-08-10T15:57:35Z

Confidence Interval on New Observations

← Older revision		Revision as of 15:57, 10 August 2017
Line 458:		Line 458:


	The 95% limits on <math>{{\hat{y}}_{p}}\,\!</math> are 189.9 and 207.2, respectively. In ~~DOE~~++, confidence and prediction intervals can be calculated from the control panel. The prediction interval values calculated in this example are shown in the figure below as Low Prediction Interval and High Prediction Interval, respectively. The columns labeled Mean Predicted and Standard Error represent the values of <math>{{\hat{y}}_{p}}\,\!</math> and the standard error used in the calculations.		The 95% limits on <math>{{\hat{y}}_{p}}\,\!</math> are 189.9 and 207.2, respectively. In Weibull++ DOE folios, confidence and prediction intervals can be calculated from the control panel. The prediction interval values calculated in this example are shown in the figure below as Low Prediction Interval and High Prediction Interval, respectively. The columns labeled Mean Predicted and Standard Error represent the values of <math>{{\hat{y}}_{p}}\,\!</math> and the standard error used in the calculations.


	[[Image:doe4_11.png\|center\|786px\|Calculation of prediction intervals in ~~DOE~~++.\|link=]]		[[Image:doe4_11.png\|center\|786px\|Calculation of prediction intervals in Weibull++.\|link=]]

	==Measures of Model Adequacy==		==Measures of Model Adequacy==

Richard House: /* F Test */

2017-08-10T15:56:59Z

F Test

← Older revision		Revision as of 15:56, 10 August 2017
Line 361:		Line 361:


	Assuming that the desired significance is 0.1, since the <math>p\,\!</math> value < 0.1, then <math>{{H}_{0}}:{{\beta }_{1}}=0\,\!</math> is rejected, implying that a relation does exist between temperature and yield for the data in the preceding [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| table]]. Using this result along with the scatter plot of the above [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| figure]], it can be concluded that the relationship that exists between temperature and yield is linear. This result is displayed in the ANOVA table as shown in the following figure. Note that this is the same result that was obtained from the <math>t\,\!</math> test in the section [[Simple_Linear_Regression_Analysis#Tests\|t Tests]]. The ANOVA and Regression Information tables in ~~DOE~~++ represent two different ways to test for the significance of the regression model. In the case of multiple linear regression models these tables are expanded to allow tests on individual variables used in the model. This is done using extra sum of squares. Multiple linear regression models and the application of extra sum of squares in the analysis of these models are discussed in [[Multiple_Linear_Regression_Analysis\| Multiple Linear Regression Analysis]].		Assuming that the desired significance is 0.1, since the <math>p\,\!</math> value < 0.1, then <math>{{H}_{0}}:{{\beta }_{1}}=0\,\!</math> is rejected, implying that a relation does exist between temperature and yield for the data in the preceding [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| table]]. Using this result along with the scatter plot of the above [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| figure]], it can be concluded that the relationship that exists between temperature and yield is linear. This result is displayed in the ANOVA table as shown in the following figure. Note that this is the same result that was obtained from the <math>t\,\!</math> test in the section [[Simple_Linear_Regression_Analysis#Tests\|t Tests]]. The ANOVA and Regression Information tables in Weibull++ DOE folios represent two different ways to test for the significance of the regression model. In the case of multiple linear regression models these tables are expanded to allow tests on individual variables used in the model. This is done using extra sum of squares. Multiple linear regression models and the application of extra sum of squares in the analysis of these models are discussed in [[Multiple_Linear_Regression_Analysis\| Multiple Linear Regression Analysis]].

Richard House: /* t Tests */

2017-08-10T15:55:53Z

t Tests

← Older revision		Revision as of 15:55, 10 August 2017
Line 199:		Line 199:
	Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value < 0.1, <math>H_0 : \beta_1=0\,\!</math> is rejected indicating that a relation exists between temperature and yield for the data in the preceding [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| table]]. Using this result along with the scatter plot, it can be concluded that the relationship between temperature and yield is linear.		Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value < 0.1, <math>H_0 : \beta_1=0\,\!</math> is rejected indicating that a relation exists between temperature and yield for the data in the preceding [[Simple_Linear_Regression_Analysis#Simple_Linear_Regression_Analysis\| table]]. Using this result along with the scatter plot, it can be concluded that the relationship between temperature and yield is linear.

	In ~~DOE~~++, information related to the <math>t\,\!</math> test is displayed in the Regression Information table as shown in the following figure. In this table the <math>t\,\!</math> test for <math>\beta_1\,\!</math> is displayed in the row for the term Temperature because <math>\beta_1\,\!</math> is the coefficient that represents the variable temperature in the regression model. The columns labeled Standard Error, T Value and P Value represent the standard error, the test statistic for the test and the <math>p\,\!</math> value for the <math>t\,\!</math> test, respectively. These values have been calculated for <math>\beta_1\,\!</math> in this example. The Coefficient column represents the estimate of regression coefficients. The Effect column represents values obtained by multiplying the coefficients by a factor of 2. This value is useful in the case of two factor experiments and is explained in [[Two_Level_Factorial_Experiments\| Two Level Factorial Experiments]]. Columns Low Confidence and High Confidence represent the limits of the confidence intervals for the regression coefficients and are explained in [[Simple_Linear_Regression_Analysis#Confidence_Interval_on_Regression_Coefficients\|Confidence Interval on Regression Coefficients]].		In Weibull++ DOE folios, information related to the <math>t\,\!</math> test is displayed in the Regression Information table as shown in the following figure. In this table the <math>t\,\!</math> test for <math>\beta_1\,\!</math> is displayed in the row for the term Temperature because <math>\beta_1\,\!</math> is the coefficient that represents the variable temperature in the regression model. The columns labeled Standard Error, T Value and P Value represent the standard error, the test statistic for the test and the <math>p\,\!</math> value for the <math>t\,\!</math> test, respectively. These values have been calculated for <math>\beta_1\,\!</math> in this example. The Coefficient column represents the estimate of regression coefficients. The Effect column represents values obtained by multiplying the coefficients by a factor of 2. This value is useful in the case of two factor experiments and is explained in [[Two_Level_Factorial_Experiments\| Two Level Factorial Experiments]]. Columns Low Confidence and High Confidence represent the limits of the confidence intervals for the regression coefficients and are explained in [[Simple_Linear_Regression_Analysis#Confidence_Interval_on_Regression_Coefficients\|Confidence Interval on Regression Coefficients]].

Richard House: /* Calculation of the Fitted Line Using Least Square Estimates */

2017-08-10T15:55:21Z

Calculation of the Fitted Line Using Least Square Estimates

← Older revision		Revision as of 15:55, 10 August 2017
Line 115:		Line 115:


	In DOE++, fitted values and residuals can be calculated. The values are shown in the figure below.		In DOE folios, fitted values and residuals can be calculated. The values are shown in the figure below.

Richard House: /* Simple Linear Regression Analysis */

2017-08-10T15:54:45Z

Simple Linear Regression Analysis

← Older revision		Revision as of 15:54, 10 August 2017
Line 11:		Line 11:


	This data can be entered in DOE++ as shown in the following figure:		This data can be entered in the DOE folio as shown in the following figure:


	[[Image:doe4_1.png\|center\|530px\|Data entry in DOE++ for the observations.\|link=]]		[[Image:doe4_1.png\|center\|530px\|Data entry in the DOE folio for the observations.\|link=]]

Richard House at 15:53, 10 August 2017

2017-08-10T15:53:57Z

← Older revision		Revision as of 15:53, 10 August 2017
Line 1:		Line 1:
	{{Template:Doebook\|3}}		{{Template:Doebook\|3}}
	Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in ~~DOE~~++ includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all ~~DOE~~++ calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE++ also ~~includes~~ a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.		Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all Weibull++ DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.

	This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.		This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.

← Older revision		Revision as of 22:51, 9 August 2018
Line 1:		Line 1:
	{{Template:Doebook\|3}}		{{Template:Doebook\|3}}
	Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.		Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.

	This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.		This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.

← Older revision		Revision as of 22:18, 9 August 2018
Line 1:		Line 1:
	{{Template:Doebook\|3}}		{{Template:Doebook\|3}}
	Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all Weibull++ DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.		Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all [https://koi-3QN72QORVC.marketingautomation.services/net/m?md=Rw01CJDOxn%2FabhkPlZsy6DwBQ%2BaCXsGR Weibull++] DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.

	This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.		This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.

← Older revision		Revision as of 15:53, 10 August 2017
Line 1:		Line 1:
	{{Template:Doebook\|3}}		{{Template:Doebook\|3}}
	Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in ~~DOE~~++ includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all ~~DOE~~++ calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE++ also ~~includes~~ a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.		Regression analysis is a statistical technique that attempts to explore and model the relationship between two or more variables. For example, an analyst may want to know if there is a relationship between road accidents and the age of the driver. Regression analysis forms an important part of the statistical analysis of the data obtained from designed experiments and is discussed briefly in this chapter. Every experiment analyzed in a Weibull++ DOE foilo includes regression results for each of the responses. These results, along with the results from the analysis of variance (explained in the [[One Factor Designs]] and [[General Full Factorial Designs]] chapters), provide information that is useful to identify significant factors in an experiment and explore the nature of the relationship between these factors and the response. Regression analysis forms the basis for all Weibull++ DOE folio calculations related to the sum of squares used in the analysis of variance. The reason for this is explained in [[Use_of_Regression_to_Calculate_Sum_of_Squares\|Appendix B]]. Additionally, DOE folios also include a regression tool to see if two or more variables are related, and to explore the nature of the relationship between them.

	This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.		This chapter discusses simple linear regression analysis while a [[Multiple_Linear_Regression_Analysis\|subsequent chapter]] focuses on multiple linear regression analysis.