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Chapter 3: Examining Relationships

3.1 Scatterplots 3.2 Correlation 3.3 Least-Squares

Regression

y = 3.9951x + 4.5711R2 = 0.9454

181920212223242526

3.5 4.0 4.5 5.0

Fiber Tenacity, g/den

Fa

bri

c T

ena

city

, lb

/oz/

yd^

2

2

Relationship Between Fiber Tenacity and Fabric Tenacity

Fiber Tenacity,g/den

Fabric Tenacity,lb/oz/yd2

3.6 19.0

3.9 20.5

4.1 20.8

4.3 21.0

4.8 23.0

5.0 24.9

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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

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Scatterplot 1: Relationship Between Fiber Tenacity and Fabric Tenacity

181920212223242526

3.5 4.0 4.5 5.0

Fiber Tenacity, g/den

Fabr

ic T

enac

ity, l

b/oz

/yd^

2

Note placement of response and explanatory variables. Also note axes labels and plot title.

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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.

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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.

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Scatterplot for 3.6

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Problem 3.9, p. 129

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Tips for Drawing Scatterplots

•  p. 128

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0

10

20

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60 70 80 90 100 110

Year (67=year 1967)

Inco

me

(Tho

usan

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Black Hispanic White Asian

Adding a Categorical Variable to a Scatterplot

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Homework

•  Reading: pp. 121-135

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Practice

•  Problems: –  3.11 (p. 129) –  3.12 (p. 132) –  3.16 (p. 136)

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Figure 3.6, p. 136

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Which shows the strongest

relationship?

800

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30 40 50 60

200

600

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2200

0 20 40 60 80 100 120

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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.”

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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.

⎟⎟⎠

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⎞⎜⎜⎝

⎛ −−

= ∑= y

in

i x

i

syy

sxx

nr

111

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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.

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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.

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Exercise 3.25, p. 146

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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?

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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

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Relationship Between Winding Tension and Yarn Elongation

y = -0.0759x + 9.4455R2 = 0.732

6.06.57.07.58.08.59.0

10 15 20 25 30 35Winding Tension, g

Elongation%

24 (e)error yyresidual^

i =−=

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).

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∑∑=

−=n

1i

2^

i2 )y(ye

•  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:

minimum )y(yen

1i

2^

i2 =−=∑∑

=

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:

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A Residual (Figure 3.11, p. 151)

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bxa +=^y

Least-Squares Regression Line

•  With the help of algebra and a little calculus, it can be shown that this occurs when:

x

y

ssrb =

xbya −=

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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.

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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.

30 r = 0.85, r2 = 0.72

1- r2 = 0.28

R2: Coefficient of Determination

•  Proportion of variability in one variable that can be associated with (or predicted by) the variability of the other variable.

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Exercise 3.45, p. 166

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Exercise 3.45, p. 166

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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.

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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

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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.

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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!

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Figures 3.19 (a-c), pp. 170-171

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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"

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Residuals Plot for 3.45

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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

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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.

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Practice Problems

•  Problems: –  3.56, p. 179 –  3.74, p. 188 –  3.76, p. 189

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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!