Chapter 12 Correlation
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1 Chapter Correlatio Correlatio is very similar to regressio with oe very importat differece. Regressio is used to explore the relatioship betwee a idepedet variable ad a depedet variable, whereas correlatio is used to cosider a relatioship betwee two depedet variables. To illustrate this differece, let us cosider the relatioships betwee stature, femur legths, ad humerus legths o te me from the Uiversity of New Mexico documeted skeletal collectio. These data are preseted i Table.. Table.. Stature ad log boe legths o 0 males from the UNM documeted skeletal collectio. Stature Femur legth Humerus Legth We kow that stature is, at least i part, a fuctio of the legth of the femur. We ca therefore explore the relatioship betwee the idepedet X variable, femur legth, ad the depedet variable Y, stature. Figure. presets this relatioship graphically, as well as the regressio equatio ad r.
2 Figure.. Stature ad log boe legths of te males y 3.08x R Stature Femur Legth Here we see that there is ideed a liear relatioship betwee femur legth ad stature, ad that we could predict a ew Y for ay give X. What about the relatioship betwee stature ad humerus legth? Is stature a fuctio of humerus legth? No, the legth of oe s upper arm cotributes i o way to overall stature. These two variables, however, may be correlated, both beig a fuctio of oe or more additioal variables. Stature ad femur legths may both be the products of factors such as geetics ad utritio for example, as well as a wide array of other evirometal factors. We therefore might expect a coectio betwee stature ad humerous legth such that idividuals with large statures would also ted to have large humerus legths. That is, the two variables might be correlated with each other because they both are the
3 products of uderlyig shared variable(s). Correlatio is the appropriate statistical procedure with which to explore this possibility. Figure.. presets the relatioship visually. Figure.. Relatioship betwee humerus legth ad stature Humerus legth Stature We see that there is a geeral positive liear relatioship, but how do we estimate the stregth of it? This is determied by computig r, the Pearso s product momet correlatio coefficiet. To begi calculatio, ote that we o loger have a idepedet variable X ad a depedet variable Y, but rather two depedet variables Y ad Y. Therefore, it does t matter which variable is depicted o the horizotal or vertical axes. To calculate correlatio, we solve for the followig quatities, where Y stature ad Y humerus legth. Quatity : Y 709.8
4 Quatity : Y 973. Quatity 3: Y 33 Quatity 4: Y Quatity 5: Y Y Quatity 6, the Sum of Squares of Y : ( Y ) y Y (Quatity ) Quatity - (709.8) Quatity 7, the Sum of Squares of Y : ( Y ) y Y (Quatity 3) Quatity 4 - (3) Quatity 8, the Sum of Products: y y Y Y ( Y )( Y ) (Quatity )(Quatity 3) Quatity (709.8)(33) Quatity 9, the Pearso s Product Momet Correlatio Coefficiet: r yy y y Quatity , (Quatity 6)(Quatity 7) ( )(7.38).6698 Pearso s r ca rage from, a perfect egative correlatio i which oe variable decreases as the other icreases, to, a perfect positive correlatio i which both variables icrease. Values close to zero idicate that there is o correlatio betwee the two variables of iterest. Yet, how close to zero is close eough to idicate o relatioship?
5 A sigificace test tells us the aswer. I correlatio, we are actually testig the ull hypothesis: H o :ρ0, where ρ (vocalized as rho or roe) is the populatio parameter of the correlatio coefficiet, r. We use a t-test to determie if our r value is sigificatly differet from 0 i stadard deviatio uits to determie whether or ot there is a sigificat correlatio. We set the level of rejectio (alpha) at.05. The stadard error of the correlatio coefficiet is: S r r The t test: r 0 t r r r As t.05[8].306, we reject H 0. The correlatio is sigificat. Spearma s Rak Order Correlatio Coefficiet May times we wish to examie relatioships betwee variables that are ot ratio or iterval scales of measuremet. Istead, we may have ordial level data that allows us to be certai oly of the order amog variates, ot distace. I these situatios we ca use Spearma s Rak Order Correlatio Coefficiet to explore the relatioship betwee ordial variables. Cosider the followig data preseted i Table.. These are rak order fish boe abudaces from two sites i the lower Illiois Valley preseted by Boie Styles (995).
6 Taxo Newbridge Newbridge Carli % Carli rak (R -R ) (R -R ) % rak (R ) (R ) Bullhead Bowfi Buffalo River Catfish Bass Sufish Pike Redhorse Freshwater drum Crappie While fish boe couts are cotiuous, differet sizes of the fish ad differeces i boe preservatio, amog other factors, require us to be suspicious of the actual couts. We really are t sure if the actual couts are truly represetative of the populatio of iterest. However, we might be iterested i kowig whether there is a relatioship, either positive or egative, betwee the frequecies of various fish species such that sites with a the remais of oe type of fish also are likely (or ulikely) to have the remais of aother type of fish. Further, we might be comfortable with the relative frequecies of the fish boes as a reflectio of their use at the sites, eve if we do t thik these frequecies accurately reflect the total umber of fish used at the site. As a result, we ca accurately create a ordial measure of the relative percetages of the fish i each site. To determie if there is a correlatio betwee the relative frequecies of fish at the Newbridge ad Carli sites, we ca use the Spearma s rak order correlatio coefficiet rather tha Pearso s. It is calculated as follows:
7 Spearma s r is calculated below: 6 (R ( R ) ) rs 6(35.5) 3 0(00 ) Values of Spearma s r are iterpreted i the same maer as Pearso s. Values of idicate a perfect egative correlatio, values of idicate a perfect positive correlatio. Values close to zero idicate o relatioship. The formula t r 0 for the t-test r itroduced above is used to evaluate the sigificace of Spearma s r whe >0. Critical values for 0 are preseted i Table.3. Table.3. Critical values for Spearma s r whe 0. N Sigificace level (oetailed test)
8 The ull hypothesis is H 0 : r 0 ad the level of rejectio (alpha) is set at.05. Comparig our Spearma s r to the critical value listed i Table.3, we fid that we must reject the ull hypothesis ad ca coclude that there is a positive correlatio betwee the fish assemblages from the Newbridge ad Carli sites. Ethobotaical aalysis also leds itself to Spearma s r. Styles (985) presets data o botaical samples from the Newbridge ad Carli sites. These data are preseted i table.4. We ca evaluate the ull hypothesis H 0 : r 0 to determie if there is a correlatio i the relative frequecies of the various botaical remais betwee the sites. The level of rejectio (alpha) is set at.05. Table.4. Ethobotaical data for the Newbridge ad Carli sites. Seed Group Newbridge Newbridge Carli % Carli rak (R -R ) (R -R ) % rak (R ) (R ) Starchy cultivated (?) Misc Oily cultivated Starchy ocultivated Sumac Fleshy fruits Weed seeds N of seeds 5,009,868 We calculate Spearma s r as follows: 6 (R ( R ) ) rs 6() 7 7(49 )
9 We see that there is a positive relatioship, ad that the structure of the botaical assemblages of the sites of Newbridge ad Carli are similar. Comparig to the critical value of.74 listed i Table.3 for N8 idicates that this correlatio is sigificat. r s Now let us compare Newbridge ad aother site, Weitzer. These date are preseted i Table.5. Agai we testig the hypothesis H 0 : r 0 ad the level of rejectio (alpha) is set at.05. Table.5. Paleobotaical iformatio from the Newbridge ad Weitzer sites. Seed Group Newbridge Newbridge Weitzer Weitzer (R -R ) (R -R ) % rak (R ) % rak (R ) Starchy cultivated (?) Misc Oily cultivated Starchy ocultivated Sumac Fleshy fruits Weed seeds N of seeds 5,009,868 Spearma s r is calculated below: 6 (R ( R ) ) rs 6() 6 7(49 ) We see that this relatioship is much weaker that that betwee Newbridge ad Carli. Comparig r s to the critical value of.74 listed i Table.3 for N8 idicates that this correlatio is ot sigificat, ad that we caot reject the ull hypothesis.
10 Pearso s Product Momet Correlatio Coefficiet ad Spearma s Rak Order Correlatio Coefficiet provide excellet tools for examiig relatioships betwee variables that are ot liked i ay causal ways. Similarly, we ow proceed with a examiatio of associatio betwee two or more categorical variables. That is the subject of Chapter 3.
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