12.68 Orthogonal polynomial contrasts. Recall the Facebook friends study (page 648). Previous research has shown that the bigger one’s social network, the higher one’s social attractiveness. In fact, the relationship between the number of friends and social attractiveness is approximately linear. A reasonable question to ask is whether this is same sort of pattern exists within an online social network. With orthogonal polynomial contrasts, we can assess the contributions of different polynomial trends to the overall pattern. Given the five equally spaced levels of the factor in this study, 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:
| Trend | a1 | a2 | a3 | a4 | a5 |
|---|---|---|---|---|---|
| Linear | −2 | −1 | 0 | 1 | 2 |
| Quadratic | 2 | −1 | −2 | −1 | 2 |
| Cubic | −1 | 2 | 0 | −2 | 1 |
(a) Plot the versus i for the linear trend. Describe the pattern. Suppose that all the μi were constant. What would the value of ψ equal?
(b) Plot the versus i for the quadratic trend. Describe the pattern. Suppose that all the μi were constant. What would the value of ψ equal? Suppose that μi = 5i (that is, a linear trend). What would the value of ψ equal?
(c) Test the hypotheses that there is a linear, quadratic, and cubic trend. What do you conclude?