Orthogonal polynomial contrasts. Recall the Facebook

Chapter , Problem 12.50

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Orthogonal polynomial contrasts. Recall the Facebook friends study. In Example 12.16 (page 664) we used a contrast to assess the linear trend between the social attractiveness score and number of Facebook friends. With orthogonal polynomial contrasts, we can assess the contributions of different polynomial trends to the overall pattern. Given the 5 equally spaced levels of the factor in this problem, we can investigate up to a quartic (x4 ) trend. The derivation of the coefficients is beyond the scope of this book, so we will just investigate the trends here. The coefficients for the linear, quadratic, and cubic trends follow: FRIENDS Trend a1 a2 a3 a4 a5 Linear 22 21 0 12 Quadratic 2 21 22 21 2 Cubic 21 2 0 22 1 (a) Plot the ai versus i for the linear trend. Describe the pattern. Suppose that all the mi were constant. What would the value of c equal? (b) Plot the ai versus i for the quadratic trend. Describe the pattern. Suppose that all the mi were constant. What would the value of c equal? Suppose that mi 5i (that is, a linear trend). What would the value of c equal? (c) Construct the sample contrasts for the quadratic and cubic trends using the Facebook data. (d) Test the hypotheses that there is a quadratic trend and that there is a cubic trend. Combine these results with the earlier linear trend results. What do you conclude?