Partial F Test

This content discusses the Partial F Test conducted by Nina Gunnes on May 20, 2021, during the Spring 2021 lecture. The lecture likely covers statistical analysis, hypothesis testing, and relevant methodologies to understand the data presented. It sheds light on the significance of the test results and their implications in the given context.

• Statistical Analysis
• Hypothesis Testing
• Data Interpretation
• Lecture Series
• Spring 2021

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1. Partial F Test Nina Gunnes May 20, 2021 05/20/2021 Spring 2021 - Lecture 9 1

2. Partial F test Comparing two linear regression models Reduced model: ? = ?0+ ?1?1+ + ????+ ? Full model: ? = ?0+ ?1?1+ + ????+ ??+1??+1+ + ??+???+?+ ? Testing if ? extra regression coefficients in full model are equal to 0 Null hypothesis: ?0:??+1= ??+2= = ??+?= 0 Defining the residual sum of squares for each regression model ??? = ?=1 ?? ?? Amount of variance not explained by the regression model 2 ? 05/20/2021 Spring 2021 - Lecture 9 2

3. Partial F test, cont. Defining the test statistic ?????????? ??????? ????????? ?????? ?????????? ??????? ??????? ? ? = = ????????????? ? ? ? 1 Test statistic following a Fisher distribution under ?0: ?~??1,?2 Degrees of freedom for numerator: ?1= ????????? ??????= ? Degrees of freedom for denominator: ?2= ??????= ? ? + ? + 1 Null hypothesis rejected if P value is below significance level 05/20/2021 Spring 2021 - Lecture 9 3

4. Example 3.11 in Laake et al. (2007) Data on systolic blood pressure 32 subjects A continuous outcome variable Systolic blood pressure (in mmHg): ??? Three exposure variables Age (in years): ????? Body mass index: ??? Smoking (0 non-smoker, 1 smoker): ? ????? 05/20/2021 Spring 2021 - Lecture 9 4

5. Example 3.11 in Laake et al. (2007), cont. Described in Chapter 3 in the book by Laake et al. (2007) Data and variable description available for download https://www.med.uio.no/imb/personer/vit/veierod/boker/epidemiologiske- og-kliniske-forskningsmetoder/datafiler/ 05/20/2021 Spring 2021 - Lecture 9 5

6. How to do it in Stata 05/20/2021 Spring 2021 - Lecture 9 6

7. How to do it in Stata, cont. Not necessarily a linear effect of body mass index Categorizing body mass index as by its quartiles Category 1: first quartile Category 2: second quartile Category 3: third quartile Category 4: fourth quartile Defining dummy variables for each category First category as the reference 05/20/2021 Spring 2021 - Lecture 9 7

8. How to do it in Stata, cont. 05/20/2021 Spring 2021 - Lecture 9 8

9. How to do it in Stata, cont. Fitting the reduced linear regression model ? = ?0+ ?1 ????? + ?2 ? ????? Fitting the full linear regression model ? = ?0+ ?1 ????? + ?2 ? ????? + ?3 ???_??2 + ?4 ???_??3 + ?5 ???_??4 Effect of (categorized) body mass index on systolic blood pressure? Testing the constraint that three regression coefficients are all zero ?0:?3= ?4= ?5= 0 05/20/2021 Spring 2021 - Lecture 9 9

10. How to do it in Stata, cont. Reduced model Full model 05/20/2021 Spring 2021 - Lecture 9 10

11. How to do it in Stata, cont. Obtaining relevant information from the Stata output Residual sum of squares and degrees of freedom Defining quantities for the reduced model ??????????= 2461.954 ?????????= 29 Defining quantities for the full model ???????= 1778.775 ??????= 26 05/20/2021 Spring 2021 - Lecture 9 11

12. How to do it in Stata, cont. Calculating the test statistic ? = 3.33 Finding the P value ? ?3,26> 3.33 = 0.035 Effect of body mass index Choosing the full model 2461.954 1778.775 1778.775 26 29 26 = 05/20/2021 Spring 2021 - Lecture 9 12

13. References Goldstein H. Introduction to F-testing in linear regression models. Lecture notes. November 2015. Accessed May 19, 2021. https://www.uio.no/studier/emner/sv/oekonomi/ECON4130/h15/lecture- note-on-f-test-2015.pdf. Laake P, Hjart ker A, Thelle DS, Veier d MB. Epidemiologiske og kliniske forskningsmetoder. Oslo: Gyldendal akademisk; 2007. https://www.med.uio.no/imb/forskning/publikasjoner/boker/2007/epide miolgiske-kliniske-forskningsmetoder.html. StataCorp. 2017. Stata 15 Base Reference Manual. College Station, TX: Stata Press. 05/20/2021 Spring 2021 - Lecture 9 13