Side notes fitting the full model this way gives the

Info icon This preview shows pages 38–41. Sign up to view the full content.

View Full Document Right Arrow Icon
side notes : Fitting the full model this way gives the same slopes and intercepts for the four regions as if you had fit four separate regression lines. The MSE from the full model is the PAGE 38
Image of page 38

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2.3 Qualitative Independent Variables c circlecopyrt HYON-JUNG KIM, 2017 pooled variance estimate from the individual regressions: MSE= SSE 1 + SSE 2 + SSE 3 + SSE 4 n 1 + n 2 + n 3 + n 4 8 If the variances are different for the different groups, then it is best to do separate regressions so that you get separate variance estimates. Sometimes pointing out that the variances are different is just as important as determining whether the slopes are different. If the variances are similar for the different groups, or if a transformation will make the variance fairly homogeneous, then it is more efficient to fit the single regression model than to fit four separate regressions. There are other ways of dealing with the non-full rank design matrix. In particular, if we code the data with 1,0, or -1 for each region as follows, Z 1 = 1 if region 1 0 if region 2 or 3 1 region 4 Z 2 = 1 if region 2 0 if region 1 or 3 1 region 4 Z 3 = 1 if region 3 0 if region 1 or 2 1 region 4, the resulting design matrix for this model is X = 1 1 0 0 X 15 1 1 0 0 X 25 1 0 1 0 X 35 1 0 1 0 X 45 1 0 0 1 X 55 1 0 0 1 X 65 1 0 0 1 X 75 1 1 1 1 X 85 1 1 1 1 X 95 1 1 1 1 X 10 5 and β = β 0 β 1 β 2 β 3 β 4 β 5 The equations for each region are region1: Y i = β 0 + β 1 + β 4 (Pop) i + ǫ i PAGE 39
Image of page 39
2.3 Qualitative Independent Variables c circlecopyrt HYON-JUNG KIM, 2017 region2: Y i = β 0 + β 2 + β 4 (Pop) i + ǫ i region3: Y i = β 0 + β 3 + β 4 (Pop) i + ǫ i region4: Y i = β 0 ( β 1 + β 2 + β 3 ) + β 4 (Pop) i + ǫ i Interaction Effects A regression model contains interaction effects if the response function is not additive and cannot be separated into distinct functions of each of the individual predictors. Two predic- tors interact if the effect on the response variable of one predictor depends on the value of the other. Example. Some researchers were interested in comparing the effectiveness of three treatments (A,B,C) for severe depression and collected a random sample of n = 36 severely depressed individuals. Y i = measure of the effectiveness of the treatment for individual i X i 1 = age (in years) of individual i X i 2 = 1 if individual i received treatment A and 0, if not. X i 3 = 1 if individual i received treatment B and 0, if not.
Image of page 40

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 41
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern