Multiple Linear Regression for the Supervisor Data

Transcription

1 for the Supervisor Data Rating Complaints Privileges Learn Raises Critical Advance

2 Problem & Data Overview Primary Research Questions: 1. What makes a good or bad supervisor? (Inference) Analysis Setup: 1. What is the response variable Y? Rating 2. What is the predictor (covariate) X? Everything else

3 Rating Complaints Privileges Learn Raises Critical Advance Exploratory Techniques when you have multiple X s Scatterplot Matrix

4 Exploratory Techniques when you have multiple X s Covariance Matrix (usually denoted by ) Rating Complaints Privileges Learn Raises Critical Advance Rating Complaints Privileges Learn Raises Critical Advance

5 Exploratory Techniques when you have multiple X s Correlation Matrix (usually denoted by R) Rating Complaints Privileges Learn Raises Critical Advance Rating Complaints Privileges Learn Raises Critical Advance Example: Correlation between Rating and Raises = 0.59

6 Model Specification Model Setup: y i = i th Supervisor Rating x i1 = i th Supervisor Complaint Score. x i6 = i th Supervisor Advance Score y i = f(x i1,...,x i6 )+ i f(x i1,...,x i6 )=??? Big Question: What is a good mathematical model for the supervisor data?

7 The Multiple Linear Regression Model is written as: y i = x i1 + 2 x i x i6 + i Or, equivalently: i N(0, 2 ) y i N x i1 + 2 x i x i6, 2

8 Note: We are now drawing planes through spheres. y i = x i1 + 2 x i x i6

9 The Multiple Linear Regression Model in Matrix Form: y = = 0 y 1 1 y 2 B A X = y n n 1 C A x 11 x x 16 1 x 21 x x 26 B A 1 x n1 x n2... x n6 = 0 Then the MLR model can be written as: C A y = X + Matrix Multiplication N (0, 2 I) ) y N X, 2 Multivariate I Normal

10 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + Question #1: What is the interpretation of? When all the x s are zero, then y is This is a terrible interpretation. 0 2! px ip + i, i N (0, 0 on average. Side Note: If you center each of your x s, i.e. define x? ip = x ip then 0 is the value of y when all the x s are at their average (more interpretable) and slopes don t change. 2 ) x p

11 The Multiple Linear Regression Model: y i N 0 + px ip, Question #2: What is the interpretation of p? Holding all other x s constant, then as x p goes up by 1 then y goes up by p on average. is the effect of on y. p y i = 0 + x p 2! px ip + i, i N (0, 2 )

12 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + 2! px ip + i, i N (0, 2 ) Question #2: What is the interpretation of? x (x 2 + 1) ( x x 2 ) = 2 p

13 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + 2! px ip + i, i N (0, 2 ) Question #3: What is the interpretation of? Same as before: Is the distance between a dot and the line. Represents unmodelable random error. i

14 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + 2! px ip + i, i N (0, 2 ) Question #4: What is the interpretation of? Same as before: average distance of the dots from the line.

15 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + 2! px ip + i, i N (0, 2 ) Question #5: How do we get estimates ˆ0,..., ˆP? Least squares estimation; or, Maximum likelihood estimation (Stat 340) Bayesian Estimation (Stat 451)

16 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + 2! px ip + i, i N (0, 2 ) Question #5: How do we get estimates ˆ0,..., ˆP? Least squares estimation nx i=1 y i ˆ0 ˆpx ip! 2 = min 0,..., P nx i=1 y i 0 P X px ip! 2

17 Question #5: How do we get estimates ˆ0,..., ˆP? Least squares estimation the individual formulas are ugly (but are quite nice in matrix form) so lets just say ˆ = ˆ0,..., ˆP = coef(lm( )) v u t P n i=1 y i ˆ0 n P 1 P P ˆpx ip 2 We ll just get these from the summary(lm( )) in R.

18 Question #5: How do we get estimates ˆ0,..., ˆP? For the supervisor data: 0 ˆ0 ˆComplaints ˆPrivileges ˆLearn ˆRaises ˆCritical ˆAdvance 1 Interpretation of ˆComplaints : Holding all else constant, if the supervisor s score on the complaints question goes up by 1, the rating goes up by 0.613, on average. 0 = C B C A

19 An Aside on Question #5: Now that we have slope estimates, how can we plot our fitted regression line? Unless P=2, we can t so we use added-variable (partial regression) plots. Added variables plots plot y-residuals vs x-residuals 1. Regress y i on all x ij s except x ip and calculate residuals 2. Regress x ip on all other x ij s and calculate residuals 3. Plot residuals from (1) and (2)

20 Complaints others Rating others Privileges others Rating others Learn others Rating others Raises others Rating others Critical others Rating others Advance others Rating others Added Variable Plots

21 Question #6: Properties of the estimators ˆ0,..., ˆP? 1. The estimators ˆ0,..., ˆP and ˆ are unbiased estimates of 0,..., P and. That is, on average (over lots of repeated sampling), I get the right answer. 2. The accuracy of the estimators ˆ0,..., ˆP and ˆ increase with sample size. So, as sample size goes up the standard errors go down. Note: standard error formulas are ugly so I don t show them to you. Let the computer calculate it for you.

22 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + Question #7: How do we use our model to make predictions? Just insert all x s into the model ŷ = ˆ0 + ˆpx p Note on extrapolation: Predicting outside range of at least 1 covariate. 2! px ip + i, i N (0, 2 )

23 The Multiple Linear Regression Model: y i N 0 + px ip, y i = 0 + 2! px ip + i, i N (0, 2 ) Question #8: What are the assumptions of this model? Linear Independent Normal Equal Variance

24 Question #9: How do we check if the assumptions are valid? Linear Ø Scatterplot Matrix Ø Residuals vs. Fitted Values Ø Residuals vs. X s ˆ i = y i ŷ i = y i ˆ0 Ø Added-variable plots ˆpx p Standardized Residuals Standardized Residuals Standardized Residuals Fitted Values Complaints Learn

25 Question #9: How do we check if the assumptions are valid? Independence Ø You just need to think about this one. In my opinion, there aren t many good graphics to diagnose this. Ø You can use fitted values vs. residual plots to get an idea. Standardized Residuals Fitted Values

26 Question #9: How do we check if the assumptions are valid? Normality Ø Histogram of Residuals Density Standardized Residuals

27 Question #9: How do we check if the assumptions are valid? Equal Variance Ø Residuals vs. Fitted Values Standardized Residuals Fitted Values

28 Question #10: How do we know if our model is any good? Does it fit the data well? Total Sums of Squares (SST) = Sum of Square Errors (SSE) = Sum of Squares from Regression (SSR) = nx (y i ȳ) 2 i=1 nx (y i ŷ i ) 2 i=1 nx (ŷ i ȳ) 2 i=1 R 2 2 [0, 1] = SSR SST =1 SSE SST =0.7326

29 Question #11: How well does my model do at prediction? Cross-Validation Bias = 0.32! I am predicting by about.32 too low. RPMSE = 6.45! My predictions are o by about ±6.45 Coverage = 0.95 Width =

30 End of Supervisor Analysis (see webpage for R and SAS code)