1. Welcome to Applied Multiple Regression (Psych 308c)! Spring 2013 MI Professor Dale Berger Dale.Berger@CGU.edu ACB101 Tue 2:00-4:00Dale.Berger@CGU.edu Teaching Associates: 909-621-8084 AlyAlbertina.Lopez@cgu.edu Monday 4:00 -- 5:50 ACB 119Albertina.Lopez@cgu.edu ValValeska.Dubon@cgu.edu Tuesday 4:00 -- 5:50 ACB 208Valeska.Dubon@cgu.edu MaggieMargaret.Burkhart@cgu.edu Wednesday 11:00 -- 12:50 ACB 119Margaret.Burkhart@cgu.edu NicNicolas.Barreto@cgu.eduWednesday 4:00 -- 5:50 ACB 208Nicolas.Barreto@cgu.edu StephenStephen.Weltz@cgu.eduThursday 4:00 -- 5:50 ACB 208Stephen.Weltz@cgu.edu

2. Overview of the course Lecture: conceptual, interpretation, presentation (take notes, rewrite notes, discuss) Homework: conceptual, computer, writing (discuss, critique, check your mastery) Lab/Review sessions: discussion, guidance Sources: Packet, Howell, Sakai, Internet Exams: conceptual (interpretation, explanation)

3. Plan for today Review the packet Brief history of correlation/regression Some vocabulary and notation Univariate and bivariate normal distributions

4. Brief history Helen M. Walker (1891-1983) Studies in the history of statistical method (1929) In the mid-1800s a number of individuals were hovering on the verge of discovering correlation, but the conceptual breakthrough came from Sir Frances Galton (1822-1911). Galton was interested in genetics Cousin of Charles Darwin Collected vast amounts of data Among first to graph data

5. 5 Chart from Sir Francis Galton

6. 6 Galton, F. (1886). Regression Towards Mediocrity in Hereditary Stature, Journal of the Anthropological Institute of Great Britain and Ireland, 15, 246-263. Slope of the line (r) is an index of the strength of the regression Co-relations and their measurement (1888)

7. The Pearsons and Fisher Galton (1822 – 1911) left money to endow a chair in “eugenics” at University College in London – He asked Karl Pearson to hold the first chair in Applied Statistics – Second holder was Sir Ronald Fisher (1890-1962) Department split – Eugenics with Fisher as chair – Applied Statistics with Egon Pearson (1895-1980) as chair Helen Walker (1891-1983) worked with both men – Retired in Claremont, taught statistics in Psychology Department at Claremont Graduate School until Spring 1970 – Four degrees of separation: Galton  Fisher/Pearson  Walker  Berger  YOU

8. 1 2 3 4 5 6 7 76543217654321 Goal: Predict Y from X Y X Example: X = Likert scale of Job Satisfaction; Y = Likert scale of Intention to Stay

9. 1 2 3 4 5 6 7 76543217654321 Model 1: Use Y bar to predict Y i Y X

10. 1 2 3 4 5 6 7 76543217654321 Model 1: Use Y bar to predict Y i Y X The sum of the squared deviations = SS Total = 10. This is the numerator of the variance of Y, an index of the variability of Y to be potentially explained.

11. 1 2 3 4 5 6 7 76543217654321 Model 2: Use regression to predict Y i Y X

12. 1 2 3 4 5 6 7 76543217654321 Model 2: Use regression to predict Y i Y X The sum of the squared errors = SS error = 6.4 ; This model leaves 6.4/10 = 64% of the variance of Y unexplained, so it accounts for 36% of the Y variance.

13. 1 2 3 4 5 6 7 76543217654321 Model 2b: Portion of SS Total explained Y X

14. 1 2 3 4 5 6 7 76543217654321 Model 2b: Portion of SS Total explained Y X The sum of the squared deviations of the regression predictions from the mean = SS reg = 3.6. The proportion of Y variance accounted for by the model is 3.6/10 = 36%.

15. 1 2 3 4 5 6 7 76543217654321 The Regression Model Y X For these data, r =.60, b =.60, and a = 1.60. r 2 =.6 2 =.36 = 36%; Predicted Y = 1.6 +.6 X i

16. 1 2 3 4 5 6 7 76543217654321 The Regression Model Y X Intercept = 1.6 Slope = +.60 1 2.2 4 4.0 7 5.8

17. 17 60 110 65 135 70 160 Example: Use height to predict weight X = Height; Y = Weight -190 + 5x60 = -190 + 300 = 110

18. Height (X) Weight (Y) 160 110 7060 Slope = rise / run = (160-110)/(70-60) = 50 / 10 = 5 Intercept = (predicted Y when X = 0) = -190 rise run

19. 19 General Linear Model (GLM)

20. 20 Applications of Regression Describe relationships r, multiple R 2, models Test hypotheses F test of R and R 2 change, t-test of B Predict Formulas to predict

21. 21 Assumptions depend upon the application Describe a sample: Assume linear relationship R 2 is the proportion of Y variance explained by a linear composite of X variables Test hypotheses: random sampling, normally distributed and independent errors, homogeneity of error variance Predict future: Also need to assume that the system is stable

22. Normal Distribution Z Z

23. Bivariate Normal Distribution

24. Could come from a bivariate normal distribution Bivariate plot (scatterplot)

25. Bivariate data example Data constructed by Anscombe consists of four sets of 11 pairs of X-Y scores. Can these four sets of data be pooled? Bumble applied regression to each set separately and recorded summary statistics. 25 Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17-21.

26. These summary statistics are identical in all four sets Sample Size (N) 11 Mean of X 9.0 Mean of Y 7.5 Correlation 0.816 Linear Equation y′ = 3 +.5x Regression SS 27.50 Residual SS13.75 (df = 9) 26

27. Bumble’s Conclusions: There is a strong linear relationship between X and Y, as is apparent from r =.819, F(1, 9) = 18.00, p=.00217. All four data sets are equivalent and probably were sampled from the same population. What more would you like to know? LOOK at the data!! 27

28. 28 Anscombe’s four data sets

29. 29 Data Set 1

30. 30 Data Set 2

31. 31 Data Set 3

32. 32 Data Set 4

33. 33

34. You can see a lot by just looking ---Yogi Berra 34 If you don’t look, you won’t see it. --- DB

35. Describing distributions What information is needed? – Univariate Shape (Normal?), mean, standard deviation – Bivariate Shape (Bivariate normal? Linear? Correlation?) For X and Y: mean and standard deviation

36. Selected references 36 Anscombe, F. J. (1973). Graphs in statistical analysis, The American Statistician, 27, 17-21. Berger, D. E. et al. (2014). Web Interface for Statistics Education: WISE http://wise.cgu.edu http://wise.cgu.edu Galton, F. (1888). Co-relations and their measurement, chiefly from anthropometric data. Proceedings of the Royal Society, 45, 135-145. Galton, F. (1886). Regression towards mediocrity in hereditary stature. The Journal of the Anthropological Institute of Great Britain and Ireland, 15, 246-263. Walker, H. M. (1929). Studies in the history of statistical method. The Williams and Wilkins Co., Baltimore, 1929