Measuring the Strength of Association

Transcription

1 Stat 3000 Statstcs for Scentsts and Engneers Measurng the Strength of Assocaton Note that the slope s one measure of the lnear assocaton between two contnuous varables t tells ou how much the average of the outcome varable changes wth respect to a one-unt ncrease n the eplanator varable. However, the estmated slope tells ou nothng about the varablt of the ponts about the lne. Correlaton s a measure of the strength of assocaton between two varables that reflects the degree of varablt around the ftted lne. It s another popular summar statstc for llustratng the degree to whch varables are lnearl assocated.

2 Stat 3000 Statstcs for Scentsts and Engneers The slope tself does not alwas reflect the strength of assocaton For eample, note that n the two plots below we observe two data sets wth appromatel the same estmated slope. However, the assocaton n the frst case looks much stronger, as the cloud of ponts more tghtl clusters about the regresson lne.

3 Stat 3000 Statstcs for Scentsts and Engneers The Correlaton Coeffcent The correlaton ρ s another populaton parameter that we can estmate from the data. We tpcall use r to denote our estmate of ρ. The so-called correlaton coeffcent r has several mportant features: r has a range of 1 to 1. It s an nde, and has no unts. The closer r s to 1, the stronger the postve lnear assocaton (r 1 ndcates perfect postve correlaton). The closer r s to 1, the stronger the negatve lnear assocaton (r 1 ndcates perfect negatve correlaton). An r close to zero ndcates weak lnear assocaton. If r 0, ths means no lnear assocaton. r measures lnear assocaton onl. Two varables can be hghl correlated n a nonlnear wa, nevertheless eldng r close to 0.

4 Stat 3000 Statstcs for Scentsts and Engneers Computng r Our estmated correlaton coeffcent s gven b, ) ( ) ( ) )( ( YY XX XY n n n S S S r Where S XY, S XX, and S YY respectvel represent the sums of squares for X and Y, for X, and for Y.

5 Stat 3000 Statstcs for Scentsts and Engneers Eample X.G Plots llustratng varous values of r:

6 Stat 3000 Statstcs for Scentsts and Engneers Eample X.A Engneers were nterested n the effects of salt dstrbuton on the roadwas wth salt concentraton n adjacent waterwas. The gathered data at 0 locatons, measurng the roadwa area at each ste along wth the salt concentraton n the nearb rver. The data are shown on the followng slde. We would lke to know whether or not greater roadwa area s assocated wth hgher average salt concentraton.

7 Stat 3000 Statstcs for Scentsts and Engneers Eample X.A (cont d) Salt Roadwa Salt Roadwa Obs Concentraton Area Obs Concentraton Area

8 Stat 3000 Statstcs for Scentsts and Engneers Eample X.A (cont d) SALT CONCENTRATION ROADWAY AREA

9 Stat 3000 Statstcs for Scentsts and Engneers Eample X.A (cont d) Gven the summar statstcs below, and that what s the correlaton coeffcent between salt concentraton and roadwa area? n 0 n

10 Stat 3000 Statstcs for Scentsts and Engneers It turns out that the statstc Inference for r t r n 1 r appromatel follows a t n- dstrbuton. To carr out a test of H 0 : ρ 0 versus the alternatve hpothess H A : ρ 0, we smpl compare the value of t to quantles of the t n- dstrbuton that s, the p-value for ths test s gven b P(t n- t ).

11 Stat 3000 Statstcs for Scentsts and Engneers Eample X.B Carr out a test of the null hpothess that the salt concentraton and road area are not correlated, versus the alternatve hpothess that the are correlated. What s the p-value of ths test? Interpret ths result n words.