Chapter 9: Statistical Inference and the Relationship between Two Variables

Transcription

1 Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter, you wll Be able to obtan a smple lnear regresson model and use t to make predctons. Be able to calculate the coeffcent of determnaton and to nterpret tests of regresson coeffcents. Be able to calculate correlatons among varables. Understand how regresson and correlaton dffer and when the use of each s approprate. STA 231: Bostatstcs 1

2 Introducton Regresson and Correlaton are all statstcal technques that use the dea that one varable say, may be related to one or more varables through an equaton. Here we consder the relatonshp of two varables only n a lnear form, whch s called lnear regresson and lnear correlaton; or smple regresson and correlaton. The relatonshps between more than two varables, called multple regresson. Regresson analyss s helpful n assessng specfc forms of the relatonshp between two quanttatve varables, and the ultmate objectve when ths method of analyss s employed to predct or estmate the value of one varable correspondng gven value of another varable. Correlaton s a statstcal technque concerned wth measurng the strength of the lnear relatonshp between two varables. STA 231: Bostatstcs 2

3 Scatter Dagram Rectangular coordnate. Two quanttatve varables. One varable s called ndependent (X) and the second s called dependent (Y). Ponts are not joned. No frequency table. Example: The followng table represents the weght and blood pressure of ten patents. Weght (kg) SBP (mmhg) STA 231: Bostatstcs 3

4 Scatter Plots: are a graph showng you the relatonshp between two factors or varables. It can show you f one varable effects another. Ths can be a very effectve tool to fnd out f you change one thng n a process wll t affect another. To see f there s a cause and effect relatonshp between two factors or varables. Whch varable to put on the horzontal axs (x) and whch to put on the vertcal axs (y). Ths s your choce, but many put the potental cause on the horzontal axs (x) and the effect on the vertcal (y) axs. The pattern of data s ndcatve of the type of relatonshp between your two varables: Postve relatonshp Negatve relatonshp No relatonshp STA 231: Bostatstcs 4

5 Type of Relatonshp between Varables STA 231: Bostatstcs 5

6 STA 231: Bostatstcs 6

7 Examples STA 231: Bostatstcs 7

8 Smple Correlaton Coeffcent (r) It s also called Pearson's correlaton or product moment correlaton coeffcent. It measures the nature and strength between two varables of the quanttatve type. The Values of r ranges between +1 and -1 perfect postve correlaton r = +1 perfect negatve correlaton r = -1 no lnear relatonshp r = 0 The sgn of r denotes the nature of assocaton. whle the value of r denotes the strength of assocaton. If the sgn s +ve ths means the relaton s drect (an ncrease n one varable s assocated wth an ncrease n the other varable and a decrease n one varable s assocated wth a decrease n the other varable). Whle f the sgn s -ve ths means an nverse or ndrect relatonshp (whch means an ncrease n one varable s assocated wth a decrease n the other). STA 231: Bostatstcs 8

9 Pearson s coeffcent of correlaton r = +1 r = -1 r = 0 r = 0.6 STA 231: Bostatstcs 9

10 The Values of r ranges between -1 and +1 Indrect (Negatve Correlaton) Drect (Postve Correlaton) Strong Moderate Week Week Moderate Strong - sgn means an nverse or ndrect relatonshp. + sgn means the relaton s drect relatonshp. If r = Zero ths means no assocaton or correlaton between the two varables. If 0 < r < 0.25, weak correlaton. If 0.25 r < 0.75, moderate correlaton. If 0.75 r < 1, strong correlaton. If r = ± l, perfect correlaton. STA 231: Bostatstcs 10

11 Computaton of the Pearson s correlaton coeffcent (r) n S XX (X 1 n 2 2 (Y Y) Y ( SYY Y) /n S XY n 1 n (XX)(YY) n n X) 2 X2 ( X ) 2/n 1 1 n 1 n XY ( n 1 X)( n 1 Y)/n r r n 1 (X X)(Y Y) n n X) 2 (Y Y) 2 (X 1 1 S S XY XX n n n nxy ( X)( Y ) n n n n n X2 ( X ) 2 n Y2( Y ) S YY STA 231: Bostatstcs 11

12 Example 1: A sample of 6 chldren was selected, data about ther age n years and weght n klograms was recorded as shown n the followng table. It s requred to fnd the correlaton between age and weght. Seral No Age (years) Weght (Kg) These two varables are of the quanttatve type, one varable (Age) s called the ndependent and denoted as (X) varable and the other (weght) s called the dependent and denoted as (Y) varable to fnd the relaton between age and weght compute the smple correlaton coeffcent usng the followng formula. STA 231: Bostatstcs 12

13 r n n n nxy ( X)( Y ) n n n n n X2 ( X ) 2 n Y2( Y ) No. Age (years) (x) Weght (Kg) (y) X 2 Y 2 xy Total X = 41 Y= 66 x 2 = 291 y 2 = 742 xy = 461 r ( 6).( 461) ( 41).( 66) 2 ( 6).( 291) ( 41). ( 6).( 742) ( 66) ( 65).( 96) 2. Drect strong correlaton between age and weght STA 231: Bostatstcs 13

14 EXAMPLE 2 : Relatonshp between Anxety and Test Scores Anxety (X) Test score (Y) X 2 Y 2 XY X = 32 Y = 32 X 2 = 230 Y 2 = 204 XY=129 (6)(129) (32)(32) 6(230)32 6(204) (356)(200) r Indrect strong correlaton between anxety and test scores

15 Exercse STA 231: Bostatstcs 15

16 Regresson Analyss Regresson: technque concerned wth predctng some varables by knowng others The process of predctng varable Y usng varable X. Uses a varable (X) to predct some outcome varable (Y). Tells you how values n Y change as a functon of changes n values of X. Correlaton and Regresson Correlaton descrbes the strength of a lnear relatonshp between two varables Lnear means straght lne Regresson tells us how to draw the straght lne descrbed by the correlaton STA 231: Bostatstcs 16

17 Smple Lnear Regresson Smple regresson uses the relatonshp between the two varables to obtan nformaton about one varable by knowng the values of the other. The equaton showng ths type of relatonshp s called smple lnear regresson equaton. The best-ft lne for a certan set of data, the regresson lne makes the sum of the squares of the resduals smaller than for any other lne. y = β 0 + β 1 x +ϵ ˆ ˆ ˆ Y β0 β1 x Resdual e Y Yˆ STA 231: Bostatstcs 17

18 Smple Lnear Regresson: Suppose that we are nterested n a varable Y, but we want to know about ts relatonshp to another varable X or we want to use X to predct (or estmate) the value of Y that mght be obtaned wthout actually measurng t, provded the relatonshp between the two can be expressed by a lne. X s usually called the ndependent varable and Y s called the dependent varable. We assume that the values of varable X are ether fxed or random. By fxed, we mean that the values are chosen by researcher--- ether an expermental unt (patent) s gven ths value of X (such as the dosage of drug or a unt (patent) s chosen whch s known to have ths value of X. By random, we mean that unts (patents) are chosen at random from all the possble unts,, and both varables X and Y are measured. We also assume that for each value of x of X, there s a whole range or populaton of possble Y values and that the mean of the Y populaton at X = x, denoted by µ y/x, s a lnear functon of x. That s, µ y/x = β 0 +β 1 x. β 0 and β 1 are called the populaton regresson coeffcents (parameters). STA 231: Bostatstcs 18

19 Estmaton of the regresson coeffcents β 0 and β 1 We select a sample of n observatons (x, y ) from the populaton, By usng the least squares method (a procedure that mnmzes the vertcal devatons of plotted ponts surroundng a straght lne) we are able to construct a best fttng straght lne to the scatter dagram ponts and then formulate a regresson equaton n the form of: Yˆ βˆ βˆ x 0 1 Where, and βˆ1 n n n n XY ( X)( Y n n n X ( X ) ) βˆ Yβˆ X 0 1 S S XY XX STA 231: Bostatstcs 19

20 Regresson equaton descrbes the regresson lne mathematcally Intercept Slope STA 231: Bostatstcs 20

21 Example 1 A sample of 6 persons was selected, the values of ther age ( x varable) and ther weght s demonstrated n the followng table. Fnd the regresson equaton related weght to age and what s the predcted weght when age s 8.5 years. No. Age x Weght y xy x 2 y Total

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23 x , y ˆ β 1 n n n n XY ( X)( Y) ( 461) ( 41)( 66) n 2 n 2 6( 291) 41 n X 1 ( X ) 1 βˆ 0 βˆ X11 ( )( 683. ) 469. Y 1 Y ˆ x ŷ (8.5) * Kg ŷ (7.5) * Kg

24 Example 2 The followng are the age (n years) and systolc blood pressure of 20 apparently healthy adults. Age (x) B.P (y) Age (x) B.P (y) Fnd the regresson equaton. 2. What s the predcted blood pressure for a man agng 25 years?

25 Calculatons x=852, y=2630 x 2 =41678, xy= ˆ β 1 n n n n XY ( X)( Y) ( ) ( 852)( 2630) n 2 n ( ) 852 n X 1 ( X ) 1 βˆ 0 βˆ X131. 5( )(. ) Y 1 Y ˆ x For age 25: B.P = * 25= = mmhg STA 231: Bostatstcs 25

26 EXAMPLE 3 nvestgators at a sports health centre are nterested n the relatonshp between oxygen consumpton and exercse tme n athletes recoverng from njury. Approprate mechancs for exercsng and measurng oxygen consumpton are set up, and the results are presented below: X Exercse Tme (mn) Y Oxygen Consumpton STA 231: Bostatstcs 26

27 STA 231: Bostatstcs 27

28 Calculatons x=27.5, y=8630 x 2 =96.25, xy=25750 ˆ β 1 n n n n XY ( X)( Y) ( ) ( 275. )( 8630) n 2 n (. ) 275. n X 1 ( X ) 1 βˆ 0 βˆ X863 ( )( 275. ) 594 Y 1 Y ˆ x For Exercse tme 2.8 mn: Oxygen Consumpton = * 2.8= 868 STA 231: Bostatstcs 28