2.4.3 Le coefficient de corrélation multiple (ou coefficient de
corrélation de 1.0 entre les deux variables. Que ceci vous suggère-t-il lorsque vous etudiez un jeu de données et êtes à la recherche de corrélations fortes?
Introduction à la régression multiple
La quantité. R est encore appelée coefficient de corrélation multiple entre Y et les variables explicatives c'est le coefficient de corrélation usuel entre
Cours 12 : Corrélation et régression
Test sur le coefficient de corrélation de Pearson. Corrélation multiple. ... Pouvoir tester si une corrélation est significativement différente de.
Le rapport de corrélation multiple et ses applications
In this research the multiple corrélation ratio which extends to k variables. (k > 2) the Pearson' s corrélation ratio
Le rapport de corrélation multiple et ses applications
In this research the multiple corrélation ratio which extends to k variables. (k > 2) the Pearson' s corrélation ratio
Régression multiple : principes et exemples dapplication
La première repose sur la connaissance des coefficients de corrélation linéaire simple de toutes les paires de variables entre elles de la moyenne arithmétique
Régressions et corrélations multiples en hydrologie
The automatic computers make it possible now to use on a larger scale in hydrology the multiple regressions and correlations. The main outlines of their theory
Corrélation simple et corrélation multiple
corrélation multiple symbolisée par R… … pour mesurer la relation entre trois variables continues ou plus (c'est-à-dire entre une variable.
Sample size planning for multiple correlation: reply to Shieh (2013)
In most multiple regression analyses a point estimate of the squared multiple correlation is reported and is often given primary.
Multiple Correlation Coefficient
The multiple correlation coefficient generalizes the standard coef- ficient of correlation. It is used in multiple regression analysis to.
[PDF] 243 Le coefficient de corrélation multiple (ou coefficient de
Dans le calcul de corrélations simples tous les facteurs sont confondus Très souvent on est intéressé à éliminer l'effet (linéaire) d'une ou de plusieurs
[PDF] Cours 12 : Corrélation et régression
Un coefficient de corrélation multiple s'interprète de la même façon qu'un r régulier dans le cas d'un problème à deux variables De plus il est aussi possible
[PDF] Le rapport de corrélation multiple et ses applications - Numdam
Dans cet article on introduit le rapport de corrélation multiple qui généralise à k (k > 2) caractères le rapport de corrélation de Pearson Particulièrement on
[PDF] Régressions et corrélations multiples en hydrologie - Horizon IRD
Les régressions et corrélations multiples sont d'un grand intérêt en hydrologie pour étudier les relations entre les variables hydrologiques
[PDF] Introduction à la régression multiple
La quantité R est encore appelée coefficient de corrélation multiple entre Y et les variables explicatives c'est le coefficient de corrélation usuel entre
[PDF] Régression linéaire multiple ou modèle gaussien
La quantité R est appelée coefficient de corrélation multiple entre Y et les variables explicatives c'est le coefficient de corrélation usuel entre y et sa
[PDF] Régression multiple : principes et exemples dapplication
La première repose sur la connaissance des coefficients de corrélation linéaire simple de toutes les paires de variables entre elles de la moyenne arithmétique
[PDF] Régression multiple - Free
Régression multiple - corrélation multiple et partielle 1 Daniel Borcard 2001-2006 Legendre et Legendre fournie en pdf sur la page web du cours
[PDF] Analyse de corrélation - Gilles HUNAULT
Ce support décrit les méthodes statistiques destinées à quantifier et tester la liaison entre 2 variables quantitatives : on parle d'analyse de corrélation dans
[PDF] Résumé du Cours de Mod`eles de Régression - UniNE
10 jan 2011 · Le coefficient de corrélation est la covariance divisée par les deux écart-types appelée le coefficient de corrélation multiple
Comment calculer le coefficient de corrélation multiple ?
Le coefficient de corrélation multiple correspond au coefficient de corrélation entre les valeurs réelles de la variable aléatoire dépendante et les valeurs estimées par l'équation de régression. En résumé, le coefficient de corrélation multiple R est le cosinus de l'angle ? fait par y et y^.Quand utiliser la régression linéaire multiple ?
L'analyse par régression linéaire multiple est une des solutions qui existe pour observer les liens entre une variable quantitative dépendante et n variables quantitatives indépendantes.Quels sont les différents types de corrélation ?
De façon générale, on va parler de corrélation linéaire ou non-linéaire. Pour une corrélation linéaire, on va y rattacher le concept de droite de régression. Du côté du sens, on définit une corrélation positive lorsque les deux ensembles varient dans le même sens.- Équation de régression multiple
Le nombre de variables indépendantes peut croître jusqu'à n et la constante b avec chaque variable indique sa valeur numérique. Le but de la constante a est de désigner la valeur de la variable dépendante dans le cas où toutes les valeurs de la variable indépendante tournent à zéro.
Multiple Correlation Coefficient
Hervé Abdi1
1 Overview
Themultiplecorrelation coefficient generalizes the standard coef- ficient of correlation. It is used in multiple regression analysis to assess the quality of the prediction of the dependent variable. It correspondsto the squared correlationbetween the predicted and the actual values of the dependent variable. It can also be inter- preted as the proportion of the variance of the dependent variable explained by the independent variables. When the independent of the squared coefficients of correlation between each indepen- dent variable and the dependent variable. This relation does not hold when the independent variables are not orthogonal. Thesig- nificance of a multiple coefficient of correlation can be assessed withanFratio. Themagnitudeofthemultiplecoefficientofcorre- lation tends to overestimate the magnitude of the population cor- relation,butitispossibletocorrectforthisoverestimation. Strictly speaking we should refer to this coefficient as thesquaredmulti- ple correlation coefficient, but current usage seems to ignore the1In: Neil Salkind (Ed.) (2007).Encyclopedia of Measurement and Statistics.
Thousand Oaks (CA): Sage.
Address correspondence to: Hervé Abdi
Program in Cognition and Neurosciences, MS: Gr.4.1,The University of Texas at Dallas,
Richardson, TX 75083-0688, USA
E-mail:herve@utdallas.edu http://www.utd.edu/≂herve 1Hervé Abdi: Multiple Correlation Coefficient
adjective squared," probably because mostly its squared value is considered.2 Multiple Regression framework
In linear multiple regression analysis, the goal is to predict, know- ing the measurements collected onNsubjects, a dependent vari- ableYfrom a set ofJindependent variables denoted {X1,...,Xj,...,XJ} . (1) for the independent variables (this matrix is called augmented be- vector of observations for the dependent variable. These two ma- trices have the following structure. 1... y n... y (2)The predicted values of the dependent variable
?Yare collected in a vector denoted ?yand are obtained as: y=Xbwithb=? X TX? -1XTy. (3)The regression sum of squares is obtained as
SS regression=bTXTy-1N(1
Ty)2(4)
(with1Tbeing a row vector of 1"s conformable withy).
The total sum of squares is obtained as
SS total=yTy-1N(1
Ty)2. (5)
2Hervé Abdi: Multiple Correlation Coefficient
The residual (or error) sum of squares is obtained as SS error=yTy-bTXTy. (6)
The quality of the prediction is evaluated by computing the multiple coefficient of correlation denotedR2Y.1,...,J. This coeffi-
cient is equal to the squared coefficient of correlation between the dependent variable (Y) and the predicted dependent variable (?Y). relation is to divide the regression sum of squares by the total sum of squares. This shows thatR2Y.1,...,Jcan also be interpreted as the
proportion of variance of the dependent variable explainedby the independent variables. With this interpretation, the multiple co- efficient of correlation is computed as R 2Y.1,...,J=SSregression
SSregression+SSerror=SSregressionSStotal. (7)
2.1 Significance test
In order to assess the significance of a givenR2
Y.1,...,J, we can com-
pute anFratio as F=R2Y.1,...,J
1-R2Y.1,...,J×N-J-1J. (8)
Under the usual assumptions of normality of the error and of in- dependence of the error and the scores, thisFratio is distributed under the null hypothesis as a Fisher distribution withν1=Jand2=N-J-1 degrees of freedom.
2.2 Estimating the population correlation:
shrunken and adjustedR Just like its bivariate counterpartr, the multiple coefficient of cor- relation is adescriptivestatistic which always overestimates the population correlation. This problem is similar to the problem of the estimation of the variance of a population from a sample. 3Hervé Abdi: Multiple Correlation Coefficient
Table 1:A set of data. The dependent variableYis to be predicted from two orthogonal predictorsX1andX2(data from Abdiet al.,2002). These data are the results of an hypothetical experiment on
retroactive interference and learning.Yis the number of sentences remembered from a set of sentences learned,X1is the number of learning trials, andX2is the number of interpolated lists learned.Number of
interpolated lists (T)Number of
learning trials (X)2 4 8235 21 639 31 8
440 34 1852 42 26
861 58 4673 66 52
In order to obtain a better estimate of the population, the value R 2 Y.1,...,Jneeds to be corrected. The corrected value ofR2Y.1,...,Jgoes
under different names:corrected R,shrunken R, oradjusted R(th- ere are some subtle differences between these different appella- tions, but we will ignore them here) and we denote it by ?R2Y.1,...,J.
There are several correction formulas available, the one most of- ten used estimates the value of the population correlation as R2Y.1,...,J=1-??1-R2
Y.1,...,J??N-1
N-J-1??
. (9) 4Hervé Abdi: Multiple Correlation Coefficient
3 Example 1:
Multiple correlation coefficient
with orthogonal predictors When the independent variables are pairwise orthogonal, the im- portance of each of them in the regression is assessed by comput- ing the squared coefficient of correlation between each of the in- dependentvariablesandthedependentvariable. Thesumofthese squared coefficients of correlation is equal to the multiplecoeffi- cient of correlation. We illustrate this case with the data from Ta- ble1. Inthisexample, thedependentvariable(Y)isthenumberor sentences recalled by participants who learned a list of unrelated sentences. The first independent variable or first predictor,X1is the number of trials used to learn the list. It takes the values 2,4, and 8. It is expected that recall will increase as a function of the
numberoftrials. Thesecondindependentvariable,X2isthenum- ber of additional interpolated lists that the participantsare asked to learned. It takes the values 2, 4, and 8. As a consequence of retroactive inhibition, it is expected that recall will decrease as a function of the number of interpolated lists learned.Using Equation 3, we found that
?Ycan be obtained fromX1 andX2as ?Y=30+6×X1-4×X2. (10) Using these data and Equations 4 and 5, we find that SS regression=5824,SStotal=6214, andSSerror=390 . (11) This gives the following value for the multiple coefficient of corre- lation: R 2Y.1,...,J=SSregression
SStotal=58246214=.9372 . (12)
In order to decide if this value ofR2
Y.1,...,Jis large enough to be con-
sidered significant, we compute anFratio equal to F=R2Y.1,...,J
1-R2 Y.1,...,J×N-J-1J=.93721-.9372×152=111.93 . (13) 5Hervé Abdi: Multiple Correlation Coefficient
Such a value ofFis significant at all the usual alpha levels, and therefore we can reject the null hypothesis. BecauseX1andX2are orthogonal to each other (i.e.,their cor- to the sum of the squared coefficients of correlation betweenthe independent variables and the dependent variable: R 2Y.1,...,J=.9372=r2
Y,1+r2
Y,2=.6488+.2884 . (14)
A better estimate of the population value of the multiple coef- ficient of correlation can obtained as R2Y.1,...,J=1-??1-R2
Y.1,...,J??N-1
N-J-1??
=1-(1-.9372)1715=.9289 . (15)4 Example 2:
Multiple correlation coefficient
with non-orthogonal predictors When the independent variables are correlated, the multiple coef- ficient of correlation is not equal to the sum of the squared corre- lation coefficients between the dependent variable and the inde- pendent variables. In fact, such a strategy wouldoverestimatethe contribution of each variable because the variance that they share would be counted several times. For example, consider the data given in Table 2 where the de- X1andX2. The prediction of the dependent variable (using Equa-
tion 3) is found to be equal toY=1.67+X1+9.50X2; (16)
this gives a multiple coefficient of correlation ofR2Y.1,...,J=.9866.
1.X2= .7500, betweenX1andYis equal torY.1=.8028, and betweenX2 andYis equal torY.2=.9890. It can easily be checked that the 6Hervé Abdi: Multiple Correlation Coefficient
Table 2:A set of data. The dependent variableYis to be predicted from two correlated (i.e.,non-orthogonal) predictors:X1andX2 (data from Abdiet al.,2002).Yis the number of digits a child can remember for a short time (the "memory span"),X1is the age of the child, andX2is the speech rate of the child (how many words the child can pronounce in a given time). Six children were tested.Y(Memory span)142330503967
X1(age)44771010
X2(Speech rate)122436
multiple coefficient of correlation is not equal to the sum ofthe squared coefficients of correlation between the independent vari- ables and the dependent variables: R 2Y.1,...,J=.9866?=r2
Y.1+r2
Y.2=.665+.9780=1.6225 . (17)
Using the data from Table 2 along with Equations 4 and 5, we find that SS regression=1822.00,SStotal=1846.83, andSSerror=24.83 . (18) This gives the following value for the multiple coefficient of corre- lation: R 2Y.1,...,J=SSregression
SStotal=1822.001846.83=.9866 . (19)
In order to decide if this value ofR2
Y.1,...,Jis large enough to be con-
sidered significant, we compute anFratio equal to F=R2Y.1,...,J
1-R2 Y.1,...,J×N-J-1J=.98661-.9866×32=110.50 . (20) Such a value ofFis significant at all the usual alpha levels, and therefore we can reject the null hypothesis. A better estimate of the population value of the multiple coef- ficient of correlation can obtained as R2Y.1,...,J=1-??1-R2
Y.1,...,J??N-1
N-J-1??
=1-(1-.9866)52=.9776 . (21) 7Hervé Abdi: Multiple Correlation Coefficient
References
[1] Abdi, H., Dowling, W.J., Valentin, D., Edelman, B., & Posa- mentier M. (2002).Experimental Design and research meth- ods. Unpublished manuscript. Richardson: The University ofTexas at Dallas, Program in Cognition.
[2] Cohen, J., & Cohen, P. (1983).Applied multiple regres- sion/correlation analysis for the behavioral sciences(2nd edi- tion). Hillsdale (NJ): Erlbaum. [3] Darlington,R.B.(1990).Regressionandlinearmodels.NewYork:McGraw-Hill.
[4] Pedhazur, E.J. (1997).Multiple regression in behavioral re- search. (3rd edition) New York: Holt, Rinehart and Winston, Inc. 8quotesdbs_dbs35.pdfusesText_40[PDF] exercice fonction cout de production
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