Chapter 3: Examining Relationships 3.1 Scatterplots 3.2 Correlation 3.3 Least-Squares Regression Fabric Tenacity, lb/oz/yd^2 26 25 24 23 22 21 20 19 18 y = 3.9951x + 4.5711 R 2 = 0.9454 3.5 4.0 4.5 5.0 Fiber Tenacity, g/den 1
Relationship Between Fiber Tenacity and Fabric Tenacity Fiber Tenacity, g/den Fabric Tenacity, lb/oz/yd 2 3.6 19.0 3.9 20.5 4.1 20.8 4.3 21.0 4.8 23.0 5.0 24.9 2
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 explanatoryresponse variable situation we may just want to look at the relationship between two variables. Problems 3.1 and 3.4, p. 123 3
Scatterplot 1: Relationship Between Fiber Tenacity and Fabric Tenacity Fabric Tenacity, lb/oz/yd^2 26 25 24 23 22 21 20 19 18 3.5 4.0 4.5 5.0 Fiber Tenacity, g/den Note placement of response and explanatory variables. Also note axes labels and plot title. 4
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. 5
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. 6
Scatterplot for 3.6 7
Problem 3.9, p. 129 8
Tips for Drawing Scatterplots p. 128 9
Adding a Categorical Variable to a Scatterplot Income (Thousands of Year 2000 Dollars) 60 50 40 30 20 10 0 60 70 80 90 100 110 Year (67=year 1967) Black Hispanic White Asian 10
Homework Reading: pp. 121-135 11
Practice Problems: 3.11 (p. 129) 3.12 (p. 132) 3.16 (p. 136) 12
Figure 3.6, p. 136 13
1600 1500 1400 1300 1200 1100 1000 Which shows the strongest relationship? 900 800 30 40 50 60 2200 1800 1400 1000 600 200 0 20 40 60 80 100 120 14
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. 15
16 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. = = y i n i x i s y y s x x n r 1 1 1
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. 17
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 twovariable data. We also need to report a complete numerical summary (means and standard deviations, 5-number summary) of both x and y. 18
Exercise 3.25, p. 146 19
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? 20
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Exercises: Understanding Correlation Review Facts about correlation, pp. 143-144 3.34, 3.35, and 3.37, p. 149 Reading: pp. 149-157 22
Relationship Between Winding Tension and Yarn Elongation Elongation% 9.0 8.5 8.0 7.5 7.0 6.5 y = -0.0759x + 9.4455 R 2 = 0.732 6.0 10 15 20 25 30 35 Winding Tension, g 23
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). residual = ^ yi y = error (e) 24
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: e 2 = n i= 1 (y i The best-fitting line is the line that has the smallest sum of e 2... the least squares regression line! That is, the line of best fit occurs when: ^ y) 2 e 2 n = (y i= 1 i ^ y) 2 = minimum 25
A Residual (Figure 3.11, p. 151) 26
Least-Squares Regression Line With the help of algebra and a little calculus, it can be shown that this occurs when: b = r s s y x a = y bx ^ y = a + bx 27
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. 28
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. 29
R 2 : Coefficient of Determination Proportion of variability in one variable that can be associated with (or predicted by) the variability of the other variable. 1- r 2 = 0.28 r = 0.85, r 2 = 0.72 30
Exercise 3.45, p. 166 31
Exercise 3.45, p. 166 32
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. 33
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 34
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. 35
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! 36
Figures 3.19 (a-c), pp. 170-171 37
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: residual = y! y " 38
Residuals Plot for 3.45 39
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 40
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. 41
Practice Problems Problems: 3.56, p. 179 3.74, p. 188 3.76, p. 189 42
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! 43