REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

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1 REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I lear regreo, we coder the frequecy dtrbuto of oe varable (Y) at each of everal level of a ecod varable (X). Y kow a the depedet varable. The varable for whch you collect data. X kow a the depedet varable. The varable for the treatmet. Determg the Regreo Equato Oe goal of regreo to draw the bet le through the data pot. The bet le uually obtaed ug mea tead of dvdual obervato. Example Effect of hour of mxg o temperature of wood pulp Hour of mxg (X) Temperature of wood pulp (Y) XY X4 Y39 XY800 X 364 Y,967 6

2 Effect of hour of mxg o temperature of w ood pulp Temperature Hour of m xg The equato for ay traght le ca be wrtte a: Ŷ b + 0 bx where: b o Y tercept, ad b regreo coeffcet lope of the le The lear model ca be wrtte a: where: e redual Y Ŷ Y β + β X + ε Wth the data provded, our frt goal to determe the regreo equato 0 Step. Solve for b (X X)(Y Y) (X X) ( XY) XY SS Cro Product b X SSCP SS ( X) SS X X for the data th example

3 X 4 Y 39 XY,800 X 364 Y,967 ( XY) XY X ( X) (4x39) b 8. The umber calculated for b, the regreo coeffcet, dcate that for each ut creae X (.e., hour of mxg), Y (.e., wood pulp temperature) wll creae 8. ut (.e., degree). The regreo coeffcet ca be a potve or egatve umber. To complete the regreo equato, we eed to calculate b o Y - b X b Therefore, the regreo equato : Ŷ X 0 00 Temperature ( o F) Hour of mxg

4 Aumpto of Regreo. There a lear relatohp betwee X ad Y. The value of X are kow cotat ad preumably are meaure wthout error. 3. For each value of X, Y depedet ad ormally dtrbuted: Y~N(0, σ ). 4. Sum of devato from the regreo le equal zero: ( Y Yˆ ) Sum of quare for error are a mmum. Effect of hour of mxg o temperature of wood pulp 0 00 Temperature Hour of mxg If you quare the devato ad um acro all obervato, you obta the defto formula for the followg um of quare: ( Ŷ Y) ( Y ) Ŷ ( Y Y) Sum Square Due to Regreo Sum Square Due to Devato from Regreo (Redual) Sum Square Total

5 Tetg the hypothe that a lear relatohp betwee X ad Y ext The hypothee to tet that a lear relatohp betwee X ad Y ext are: H o : ß 0 H A : ß 0 Thee hypothee ca be teted ug three dfferet method:. F-tet. t-tet 3. Cofdece terval Method. F-tet The ANOVA to tet H o : $ 0 ca be doe ug the followg ource of varato, degree of freedom, ad um of quare: SOV df Sum of Square Due to regreo ( XY) XY X SSCP ( X) SS X Redual - Determed by ubtracto ( Y) Total - Y SS Y Ug data from the example: X 4 Y 39 XY,800 X 364 Y,967 Step. Calculate Total SS Y ( Y) 39, ,

6 Step. Calculate SS Due to Regreo ( XY) XY X ( X) ( 4x39) , ,59.7 Step 3. Calculate Redual SS SS Devato from Regreo Total SS - SS Due to Regreo Step 4. Complete ANOVA SOV df SS MS F Due to Regreo Due to Reg. MS/Redual MS ** Redual Total The redual mea quare a etmate of σ Y X, read a varace of Y gve X. Th parameter etmate the tattc σ Y X. Step 5. Becaue the F-tet o the Due to Regreo SOV gfcat, we reject H o : ß 0 at the 99% level of cofdece ad ca coclude that there a lear relatohp betwee X ad Y. Coeffcet of Determato - r From the ANOVA table, the coeffcet of varato ca be calculated ug the formula r SS Due to Regreo / SS Total Th value alway wll be potve ad rage from 0 to.0. A r approache.0, the aocato betwee X ad Y mprove. r x 00 the percetage of the varato Y that ca be explaed by havg X the model. For our example: r / We ca coclude that 9.7% (.e x 00) of the varato wood pulp temperature ca be explaed by hour of mxg.

7 Method. t-tet The formula for the t-tet to tet the hypothe H o : ß 0 : b t b where: b the regreo coeffcet, ad b Y X SS X For our example: Step. Calculate Remember that Y X Redual MS [SS Y - (SSCP / SS X)] / (-) b We kow from prevou part of th example: Therefore, SS Y SSCP SS X 70.0 b ( Y X / SS X) SSCP SS Y - SS X - SS X

8 Step. Calculate t tattc b t b Step 3. Look up table t value Table t "/, (-) df t.05/, 4df.776 Step 4. Draw cocluo Sce the table t value (.776) le that the calculated t-value (6.66), we reject H o : ß 0 at the 95% level of cofdece. Thu, we ca coclude that there a lear relatohp betwee hour of mxg ad wood pulp temperature at the 95% level of cofdece. Method 3. Cofdece Iterval The hypothe H o : ß 0 ca be teted ug the cofdece terval: CI b ± t α,( ) df ( b ) For th example: CI b ± t α,( ) df ( b ) 8.± β.476 We reject Ho: ß 0 at the 95% level of cofdece ce the CI doe ot clude 0.

9 SAS Commad opto pageo; data reg; put x y; datale; ; proc reg; model yx/cl clm; ttle 'SAS Output for Lear Regreo Example Cla'; ru;

10 SAS Output for Lear Regreo Example Cla The REG Procedure Model: MODEL Depedet Varable: y Number of Obervato Read 6 Number of Obervato Ued 6 Source DF Aaly of Varace Sum of Square Mea Square F Value Pr > F Model Error Corrected Total Root MSE R- Square Depedet Mea Adj R- Sq Coeff Var Varable DF Parameter Etmate Parameter Etmate Stadard Error t Value Pr > t Itercept x

11 SAS Output for Lear Regreo Example Cla The REG Procedure Model: MODEL Depedet Varable: y Ob Depedet Varable Predcted Value Output Stattc Std Error Mea Predct 95% CL Mea 95% CL Predct Redual Sum of Redual 0 Sum of Squared Redual Predcted Redual SS (PRESS)

Reaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4

Reaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4 CHAPTER Smple Lear Regreo EXAMPLE A expermet volvg fve ubject coducted to determe the relatohp betwee the percetage of a certa drug the bloodtream ad the legth of tme t take the ubject to react to a tmulu.