Linear Regression Analysis

Authors: Mary Ann Fiene, MLS(ASCP) and Alan K. Reichert, PhD
Reviewer: Joshua J. Cannon, MS, MLS(ASCP)CMSHCM
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Continuing Education Credits

P.A.C.E.® Contact Hours: 1 hour(s)

Florida Board of Clinical Laboratory Science CE Credit Hours

Florida Board of Clinical Laboratory Science CE - Supervision/Administration, Quality Control/Quality Assurance, and Safety: 1 hour(s)

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The purpose of this course is to demonstrate how to use linear regression to predict the value of one variable, given the value of the other variable and the experimental data concerning the relationship between the variables.

Objectives

  • Define linear regression and explain how it is used.
  • Given data points that fall on a straight line, find the equation for the line.
  • Use the regression equation to predict the value of a dependent variable give the value of the independent variable.
  • Explain what is meant by the phrase "line of best fit."
  • Given a set of data, determine the best fit using the least squares method.
  • Define and calculate standard error of estimate.
  • Explain the difference between a, alpha, b, and beta, as applied to regression analysis, and describe why confidence intervals are calculated for the slope and y-intercept.

Course Outline

  • Introduction to Regression Analysis
    • Predicting a Value
    • A Regression Analysis Example
    • A Regression Analysis Example, continued
    • Calculating the Y-Intercept
    • Prediction Using the Resulting Equation
    • Given the following creatinine standards:mg/dLAbsorbance30.1460.2690.38What is the correct form of the regression line?
    • Given the data and linear regression line you calculated on the previous question, what is the expected absorbance of a 10 mg/dL sample?
    • True or False: You should make a scatterplot of your data before you calculate the regression line.
    • A scatterplot of the data below is shown on the right, confirming a linear relationship. Given the following data, calculate the regression line.xy2 9.24 8.46 7.68 6.810 6.0
  • Introduction to Least Squares Method of Best Fit
    • Introduction to Least Squares Method
    • Introduction to Least Squares Method, continued
    • The Least Squares Line
    • Standard Error of Estimate
    • Calculate the sum of squares for line B. To do this, you must calculate the difference y- , and the squared difference (y-)2 for each point, and then sum the squared differences. You may find it useful to make a chart similar to this one. Some of the data has been filled in for you: The equation for line B is = x. Pointxyy-(y-)21 10 5.0 10 -5.0 252 18 24.0 183 38 27.5 384 50 60.0 505 63 50.0 63 W
    • Using the sum of squares from the previous question (440.25), calculate the Standard Error of Estimate for line B to the nearest thousandth using the formula on the right.Pointxyy-(y-)21 10 5.0 10 -5.0 25.02 18 24.0 18 6.0 36.03 38 27.5 38 -10.5 110.254 50 60.0 50 10.0 100.05 63 50.0 63 -13.0 169.0
  • Least Squares Calculation
    • Determining the Least Squares Line
    • Formulae for Determining the Slope and Intercept
    • Calculating the Standard Error of Estimate
    • Correlation Coefficient
    • Example Regression Line Calculation
    • Using the Least Squares Formulae
    • Determining Se and r2
    • Data for Questions
    • Using the data, calculate the total of the (x-)(y-) values. What is the total (rounded to the nearest whole number)?PointRef. Method (x)Test Method (y)x-y-(x-)(y-)(x-)21 3 6 2 14 18 3 31 34 4 5 6 5 24 29 6 16 21 7 32 39 8 10 11 9 40 42 10 6 5 11 18 21 12 29 34 13 10 16 14 43 48 15 19 27 Total
    • Using the same data, calculate the total of the (x-)2 values. What is the total?The average of x is 20, and the average of y is 23.8.PointRef. Method (x)Test Method (y)x-y-(x-)(y-)(x-)21 3 6 -17-17.8 302.6 2 14 18 -6 -5.8 34.83 31 34 11 10.2112.2 4 5 6 -15 -17.8267 5 24 29 4 5.220.8 6 16 21 -4 -2.811.2 7 32 39 12 15.2182.4 8 10 11 -10-12.8 1289 40 42 20 18.2364 10 6 5 -14 -18.8263.2 11
    • Using the formulas below and the information from the previous questions (shown again in the table below), what are the slope and y-intercept of the least squares regression line for this data?PointRef. Method (x)Test Method (y)x-y-(x-)(y-)(x-)21 3 6 -17-17.8 302.6 2892 14 18 -6 -5.8 34.8 363 31 34 11 10.2112.2 121 4 5 6 -15 -17.8267 255 5 24 29 4 5.220.8 166 16 21 -4 -2.811.2 167 32 39 1
    • What is the Standard Error of Estimate for this regression line, using the shortcut form of the equation shown below:a = 2.4b = 1.070= 23.8PointRef. Method (x)Test Method (y)x-y-(x-)(y-)(x-)21 3 6 -17-17.8 302.6 2892 14 18 -6 -5.8 34.8 363 31 34 11 10.2112.2 121 4 5 6 -15 -17.8267 255 5 24 29 4 5.220.8 166 16 21 -4 -2.811.2 167 32 39 12 15.2182.4 1448 10 11 -10-12.8 128 1009 40 42 20
  • Calculation of Confidence Intervals for Least Squares
    • Confidence Intervals for Slope and Intercept Parameters
    • Calculating Confidence Intervals
    • Formulae for Confidence Intervals
  • References
    • References

Additional Information

Level of instruction: Intermediate
Intended audience: This course is appropriate for laboratory professionals and for students in clinical laboratory science programs who want a review of the statistics that are analyzed for assessment of quality control.
Author Information:  Mary Ann Fiene, MLS(ASCP), has authored several articles on the subjects of curriculum development, competency evaluation, and job restructuring. Her articles have appeared in the Journal of Allied Health, American Journal of Medical Technology (now published as Clinical Laboratory Science), and Medical Laboratory Observer. Ms. Fiene was affiliated as an educator with the Kettering Medical Center School of Medical Technology.
Alan Reichert, PhD, is a professor of finance at Cleveland State University in Ohio.
Reviewer Information: Joshua J. Cannon, MS, MLS(ASCP)CMSHCM received his Bachelor of Science and Master of Science in Medical Laboratory Science from Thomas Jefferson University in Philadelphia, PA. He holds Medical Laboratory Scientist and Specialist in Hematology certifications through the ASCP Board of Certification. He was a professor at Thomas Jefferson University for seven years before transitioning into his current role as Education Developer at MediaLab. His areas of expertise and professional passions include clinical hematology and interprofessional education.
About the Course: This course is part of a series of courses adapted for the web by MediaLab Inc. under license from Educational Materials for Health Professionals Inc. Dayton OH, 45420. Copyright EMHP. The course was reviewed and revised in 2024.