Lq2

• Linear Regression


  A model where a variation of one variable (y) is considered to be a consequence of the other (predictor) variable (x)
• Linear Regression


  • Determines the expected value of a dependent variable (y) for the given values of one or more independent variables (x)
  • Quantifies how mean difference in the outcome (y) changes with a one-unit difference x
• Applicable for Linear Regression


  Relationship between variable X and variable Y can be described by a straight line
• Types of Linear Regression

  • Multiple Linear Regression: effects of two or more independent (X) variables on one dependent (Y) variable are simultaneously considered
  • Simple Linear Regression: involves one independent variable and one dependent variable only
• Simple Linear Regression Equation

  y = mx + b
• Multiple Linear Regression Equation


  y = β0 + β1x1 + ε
• Regression Parameters


  • b (y-intercept): Value of Y when X is 0
  • m (slope): Determines the unit increase in the mean of y, represents increase/decrease in the mean of Y associated with 1 increase/decrease of X
• Coefficient of Determination (r^2)

  Percentage reduction in the variance of variable Y due to variable X
• Assumptions for Linear Regression
• Steps on Simple Linear Regression

  1. Determine whether or not assumptions underlying a linear relationship are met in the data available for analysis

  2. Obtain the equation for the line (y = mx + b) that best fits the sample data

  3. Evaluate the equation to obtain some idea of the strength of the relationship and usefulness of the equation in predicting and estimating

  4. If data appear to conform satisfactorily to the linear model, use equation obtained from the sample data to predict and estimate
• Formulas for Simple Linear Regression
• Formula for Coefficient of Determination (r^2)
• Sample Problem: Relationship between mental ability scores (Y) and % Recommended Dietary Allowance (RDA) for Calorie (X) of Grade 6 pupils


  • Data table with X, Y, x^2, y^2, and xy values
• Steps to solve the sample problem
• Data


  • 1444
  • 1687.2
  • 6
  • 60.7
  • 71
  • 3684.49
  • 5041
  • 4309.7
  • 7
  • 73.1
  • 48
  • 5343.61
  • 2304
  • 3508.8
  • 8
  • 98.8
  • 69
  • 9761.44
  • 4761
  • 6817.2
  • 9
  • 52.6
  • 30
  • 2766.76
  • 900
  • 1578.0
  • 10
  • 85.8
  • 59
  • 7361.64
  • 3481
  • 5062.2
• ∑ x = 737.5
• ∑ y = 502
• ∑ x2 = 57, 250.93
• ∑ y2 = 27, 936
• ∑ XY = 38, 278.9
• ��


  73.75
• ��


  50.2
• Compute the Pearson correlation coefficient to determine the linearity of the relationship of the variables


  1. r = (n(∑ xy) - (∑ x)(∑ y)) / √[(n(∑ x2) - (∑ x)2)(n(∑ y2) - (∑ y)2)]

  2. r = 0.48
• The variables %RDA Calorie Intake and Mental Ability Score has a moderate positive linear relationship. This means that as %RDA Calorie Intake increases, the Mental Ability Score also increases.
• Compute for the slope (m) and y-intercept (b)

  1. m = (n(∑ xy) - (∑ x)(∑ y)) / [(n ∑ x2) - (∑ x)2]

  2. m = 0.47

  3. b = y - mx

  4. b = 15.54
• Equation of the line

  • y = mx + b
  • y = (0.47)x + 15.54
• For every 1% increase in calorie intake, the mental ability score will increase by 0.47 points (based on the slope)
• With an 80% calorie intake


  The expected mental ability score is 53.14 points
• Compute for the coefficient of determination (r2) to determine the percentage reduction in the variance of variable Y due to variable X


  1. r2 = [(n(∑ xy) - (∑ x)(∑ y)) / √[(n(∑ x2) - (∑ x)2)(n(∑ y2) - (∑ y)2)]]2

  2. r2 = 0.2304
• The RDA accounts for 23.04% of the variability in the mental ability score of children