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Chapter 3: Examining Relationships . 3.1 Scatterplots 3.2 Correlation 3.3 Least-Squares Regression. Relationship Between Fiber Tenacity and Fabric Tenacity. Variable Designations. Which variable is the dependent variable ? Our text uses the term response variable .

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chapter 3 examining relationships
Chapter 3: Examining Relationships

3.1 Scatterplots

3.2 Correlation

3.3 Least-Squares Regression

variable designations
Variable Designations
  • Which variable is the dependent variable?
    • Our text uses the term response variable.
  • Which variable is the independent variable?
    • Explanatory variable
  • Note: Sometimes we do not have a clear explanatory-response variable situation … we may just want to look at the relationship between two variables.
  • Problems 3.1 and 3.4, p. 123
scatterplot 1 relationship between fiber tenacity and fabric tenacity
Scatterplot 1: Relationship Between FiberTenacity and Fabric Tenacity

Note placement of response and explanatory variables. Also note

axes labels and plot title.

problem 3 6 p 125
Problem 3.6, p. 125
  • Type data into your calculator.
  • Examining a scatterplot:
    • Look for the overall pattern and striking deviations from that pattern.
      • Pay particular attention to outliers
    • Look at form, direction, and strength of the relationship.
examining a scatterplot cont
Examining a Scatterplot, cont.
  • Form
    • Does the relationship appear to be linear?
  • Direction
    • Positively or negatively associated?
  • Strength of Relationship
    • How closely do the points follow a clear form?
    • In the next section, we will discuss the correlation coefficient as a numerical measure of strength of relationship.
homework
Homework
  • Reading: pp. 121-135
practice
Practice
  • Problems:
    • 3.11 (p. 129)
    • 3.12 (p. 132)
    • 3.16 (p. 136)
the two plots represent the same data
The two plots represent the same data!
  • Our eye is not good enough in describing strength of relationship.
    • We need a method for quantifying the relationship between two variables.
  • The most common measure of relationship is the Pearson Product Moment correlation coefficient.
    • We generally just say “correlation coefficient.”
correlation coefficient r
Correlation Coefficient, r
  • The correlation, r, is an average of the products of the standardized x-values and the standardized y-values for each pair.
correlation coefficient r1
Correlation Coefficient, r
  • A correlation coefficient measures these characteristics of the linear relationship between two variables, x and y.
    • Direction of the relationship
      • Positive or negative
    • Degree of the relationship: How well do the data fit the linear form being considered?
      • Correlation of (1 or -1) represents a perfect fit.
      • Correlation of (0) indicates no relationship.
interpreting correlation coefficient r
Interpreting Correlation Coefficient, r
  • Correlation Applet: http://www.duxbury.com/authors/mcclellandg/tiein/johnson/correlation.htm
  • Facts about correlation
    • pp.143-144
  • Correlation is not a complete description of two-variable data. We also need to report a complete numerical summary (means and standard deviations, 5-number summary) of both x and y.
outlier or influential point
Outlier, or influential point?
  • Let’s enter the data into our calculators and calculate the correlation coefficient. The data are in the middle two columns of Table 1.10, p. 59.
    • r=?
  • Now, remove the possible influential point. What happens to r?
exercises understanding correlation
Exercises: Understanding Correlation
  • Review “Facts about correlation,” pp. 143-144
  • 3.34, 3.35, and 3.37, p. 149
  • Reading: pp. 149-157
least squares regression
Least Squares Regression
  • Ultimately, we would like to predict elongation by using a more practical measurement, winding tension.
    • A regression line, also called a line of best fit, was found.
  • How was the line of best fit determined?
    • Determine mathematically the distance between the line and each data point for all values of x.
    • The distance between the predicted value and the actual (y) value is called a residual (or error).
least squares regression line of best fit
Least Squares Regression: Line of Best Fit
  • This could be done for each data point. If we square each residual and sum all of the squared residuals, we have:
  • The best-fitting line is the line that has the smallest sum of e2 ... the least squares regression line! That is, the line of best fit occurs when:
least squares regression line
Least-Squares Regression Line
  • With the help of algebra and a little calculus, it can be shown that this occurs when:
exercise 3 12 p 132
Exercise 3.12, p. 132
  • Is there a relationship between lean body mass and resting metabolic rate for females?
    • Quantify this relationship.
  • Find the line of best fit (the least-squares regression, LSR).
  • Use the LSR to predict the resting metabolic rate for a woman with mass of 45 kg and for a woman with mass of 59.5 kg.
interpreting the regression model
Interpreting the Regression Model
  • The slope of the regression line is important for the interpretation of the data:
    • The slope is the rate of change of the response variable with a one unit change in the explanatory variable.
  • The intercept is the value of y-predicted when x=0. It is statistically meaningful only when x can actually take values close to zero.
r 2 coefficient of determination
R2: Coefficient of Determination
  • Proportion of variability in one variable that can be associated with (or predicted by) the variability of the other variable.

1- r2 = 0.28

r = 0.85, r2 = 0.72

residuals
Residuals
  • In regression, we see deviations by looking at the scatter of points about the regression line. The vertical distances from the points to the least-squares regression line are as small as possible, in the sense that they have the smallest possible sum of squares.
  • Because they represent “left-over” variation in the response after fitting the regression line, these distances are called residuals.
examining the residuals
Examining the Residuals
  • The residuals show how far the data fall from our regression line, so examining the residuals helps us to assess how well the line describes the data.
    • Residuals Plot
residuals plot
Residuals Plot
  • Let’s construct a residuals plot, that is, a plot of the explanatory variable vs. the residuals.
    • pp. 174-175
  • The residuals plot helps us to assess the fit of the least squares regression line.
    • We are looking for similar spread about the line y=0 (why?) for all levels of the explanatory variable.
residuals plot interpretation cont
Residuals Plot Interpretation, cont.
  • A curved or other definitive pattern shows an underlying relationship that is not linear.
    • Figure 3.19(b), p. 170
  • Increasing or decreasing spread about the line as x increases indicates that prediction of y will be less accurate for smaller or larger x.
    • Figure 3.19(c), p. 171
  • Look for outliers!
how to create a residuals plot
How to create a residuals plot
  • Create regression model using your calculator.
  • Create a column in your STAT menu for residuals. Remember that a residual is the actual value minus the predicted value:
slide40
HW
  • Read through end of chapter
  • Problems:
    • 3.42 and 3.43 (parts a and b only), p. 165
    • 3.46, p. 173
  • Chapter 3 Test on Monday
regression outliers and influential observations
Regression Outliers and Influential Observations
  • A regression outlier is an observation that lies outside the overall pattern of the other observations.
  • An observation is influential for a statistical calculation if removing it would markedly change the result of the calculation.
    • Points that are outliers in the x direction of a scatterplot are often influential for the least-squares regression line.
      • Sometimes, however, the point is not influential when it falls in line with the remaining data points.
    • Note: An influential point may be an outlier in terms of x, but we label it as “influential” if removing it significantly influences the regression.
practice problems
Practice Problems
  • Problems:
    • 3.56, p. 179
    • 3.74, p. 188
    • 3.76, p. 189
preparing for the test
Preparing for the Test
  • Re-read chapter.
    • Know the terms, big concepts.
  • Chapter Review, pp. 181-182
  • Go back over example and HW problems.
  • Study slides!