Class 28 Notes

Class 28 Get Ready….

Height and Weight • Is CM or Inches the better predictor of KG? – Whichever has the lower standard error • Will also have a variety of better stats

– NOT whichever has the bigger coefficient

• A multiple regression lets you test – H0: all b’s = 0 (nothing in the model matters) – H0: b1=0 given all the other b’s

• When using both CM and INCHES – We reject H0 b1=b2=0 – We fail to reject H0 b1=0 given b2 – We fail to reject H0 b2=0 given b1 • You need either CM or INCHES but not both – Because they are highly correlated

• Regressions ALWAYS go thru the sample averages

Things I expect you will know • How to interpret a regression using p-1 dummy variables – The p possible forecasts will equal the sample average Y for each of the p groups – The intercept is the average of the left-out group – The coefficients are differences in group averages. – The p-value/significance F will match that from ANOVA single factor

Things I expect you will know • How to interpret a residual (error) – It is Y - 𝑌 – It is the distance each Y is from the line. – Positive means above the line. – They measure the difference between actual Y and expected Y (based on the X’s) – The most over-weight girl (for her height) is the girl with the largest positive residual. • Check the box to get residuals.

Things I expect you will know • How to interpret a coefficient in a multiple regression. – It measures the change in expected Y for a unit change in that X keeping all other Xs constant. • If I keep miles and stops constant and change from williams to spencer, expect 0.97 hours less. • If I change from Williams to Spencer, expect 0.33 hours more.

– It is the easy way to answer some questions. • If the previous rating goes from 17.5 to 20, how will the expected ratings change? (by 0.18571 per point)

Things I expect you will know • How to use a regression model to calculate a point forecast. – Plug and chug. • I use SUMPRODUCT • You must know what Xs to plug in. • It is a package deal….you must know and plug in ALL the Xs.

Things I expect you will know • How to use a regression model to calculate a probability. – The question gives you the Y. – You Plug and chug to get the 𝑌. – You calculate t = (Y - 𝑌 )/ standard error – Use t.dist.rt( t , dof) • Dof is n – total number of regression terms.

– Requires the FOUR assumptions.

Things I expect you will know • If the coefficient of X1 changes when X2 is included in the model….. – You know X1 and X2 are correlated. – You can use the two regression results to tell whether X1 and X2 are positively or negatively correlated. • • • • •

Ds was positively correlated with Miles Fact was negatively correlated with Stars Nobel was positively correlated with Yanks Speed was positively correlated with Dcorporate Exam 1 was negatively correlated with Exam 2.

UNDERSTANDING Coefficient Regression Table Constant Fact

13.24615 1.40107

Coefficient Regression Table Constant Fact Stars

Oh…Fact Movies had fewer Stars!

12.568 1.799 1.259

Secret Formula Coefficient Regression Table Constant Fact

13.24615 1.40107

Coefficient

Regress Y on X1

Regression Table Constant Fact Stars

12.568 1.799 1.259

𝑏 − 𝑏1

Regress Y on X1 and X2

𝑏2Movies Oh…Fact had fewer Stars!

Regress Y on X1 and X2

𝑐=

Regress X2 on X1

Secret Formula Coefficient

Coefficient

Regression Table

Regression Table Constant Fact

13.24615 1.40107

Regress Y on X1

Constant Fact Stars

1.40 − 1.80 𝑐= 1.26

𝑐 = −0.32 Regress X2 on X1

12.568 1.799 1.259

Regress Y on X1 and X2 Regress Y on X1 and X2

UNDDERSTANDING Coefficient Regression Table Constant Fact

13.24615 1.40107

Coefficient Regression Table Constant Fact Stars

Oh…Fact Movies had fewer Stars!

12.568 1.799 1.259

UNDERSTANDING Secret Formula Coefficient Regression Table Constant Fact

13.24615 1.40107

Coefficient Regression Table Constant Fact Stars

Fact Movies averaged 0.32 fewer Stars!

12.568 1.799 1.259

Regression is the line through a cloud of points • Scatter-plot the cloud • It is up to YOU to interpret the results. • Don’t assume X causes Y – Y might be causing X – Both might be caused by Z

• Don’t assume better fitting lines are better at forecasting – They usually are not…..too good a fit means too complicated a model…..means poorer performance.

Class 28 Assignment Variable

School

% of Classes Graduation Under 20 Rate Description The name Percentage Percentage of of the of enrollees Classes offered Universit who with