1. Stat 512 – Lecture 16 Two Quantitative Variables (Ch. 9)

2. Last Time With one quantitative response and one qualitative explanatory, can use one-way ANOVA to compare the population/true treatment means This procedure is easily extendable to any number of qualitative explanatory variables  EV 1: Adoptive SES H 0 :  Adopt High =  Adopt Low  EV 2: Biological SES H 0 :  Bio High =  Bio Low Two-way ANOVA, General Linear Model

3. Two-way ANOVA The “effect” of the SES of the adoptive parents is statistically significant (p-value =.010) even after adjusting for the stronger “effect” of the SES of the biological parents (p-value =.001)

4. Last Time When have multiple explanatory variable (“factors”), can also consider the interaction between these variables

5. Last Time When have “paired” or “dependent” samples, the blocking variable can be incorporated into the model as well Example: Chip melting times  H 0 :  B  MC  SS H a : at least one  differs  Compare n 1 =11 n 2 =11 n 3 =17 “completely randomized design” 37 subjects butterscotch milk chocolate semi-sweet Compare melting times random “randomized block design” 13 subjects b-mc-ss b-ss-mc mc-b-ss Compare melting times random ss-b-mc ss-mc-b mc-ss-b

6. Last Time “Repeated measures” analyses are just like taking the differences first in “paired samples.” If you want to compare results within blocks or within subjects (instead of across and ignoring the pairing), include that variable in the ANOVA

7. Practice Problem With random assignment to distinct groups (milked by machine or human), will consider independent With any correspondence, relationship between units, will consider dependent  Litter mates  “Split plots”  Both calculators on sample problem

8. Example 12.6: Positive and Negative Influences on Children (p. 463) “Children are exposed to many influences in their daily lives. What kind of influence does each of the following have on children? 1. Movies, 2. Programs on network television, 3. Rock music”  -2=very negative, -1=negative, 0=neutral, 1=positive, 2= very positive Research question: Are the population mean responses identical for the three influences?  H 0 :    TV  R H a : at least one  differs

9. Example 12.6: Positive and Negative Influences on Children Influence SubjectMoviesTVRock 101 2100 301-2 4201 50 6-2 7 0 801 9 10101 1111 12 -2

10. Example 12.6: Positive and Negative Influences on Children While different people do seem to tend to give significantly different ratings (p-value =.003), once we adjust for that, we do not have super convincing evidence of an “influence effect” (p-value =.101).

11. Example 1: Airline Costs Best prediction of cost?  Sample mean, 295.2 Another explanatory variable  Distance

12. Describing the association between two quantitative variables Moderate, positive, linear relationship

13. Describing the association between two quantitative variables Which is stronger? r =.444 r = -.265

14. Describing the association between two quantitative variables Moderate, positive, linear relationship r =.439

15. Modeling the relationship How decide on the best line Residual = observed - predicted

16. Example 2: Height vs. Foot Length Least Squares Regression applet The “least squares line” finds the equation for the line that minimizes the sum of the squared residuals  Trying to minimize “prediction errors”  Using squared residuals means there will be a unique equation that does this

17. Example 2: Height vs. Foot Length Interpretation of slope: For each additional cm in height, we predict an additional 1.033 inches taller Interpretation of intercept: If someone has 0cm foot, predict 38.302 inches tall!  Not always meaningful in every context!

18. “Resistance” Remove or change point and see if line changes dramatically The least squares line is not resistant to extreme observations  Especially those that are extreme in the explanatory variable (often a stronger determinant than the size of the residual)

19. R2R2 If predict everyone to have the same height, lots of “unexplained” variation (SSE = 475.75) If take explanatory variable into account, much less “unexplained” variation (SSE = 235)

20. Example 1: Airline costs Each flight has a ‘set up’ cost of $151 and each additional mile of travels is associated with an predicted increase in cost of about 7 cents. 19.3% of the variability in airfare is explained by this regression on distance (still lots of unexplained variability) Might investigate further while the cost for ACK was so much higher than expected

21. For Thursday PP 14 in Blackboard by 3 pm Finishing up HW 7 Continue reading in Ch. 9 By next Tuesday – another project report  Narrowed in on 2 “research questions” and which statistical methods you think will answer them…