Analyse de la variance multivariée MANOVA Analyse de la variance multivariée

Quand? Pourquoi? Quand? Dans le cas où plus d’une seule variable dépendante est analysée Pourquoi? l’utilisation de plusieurs tests univariés augmente le . Dans le cas de 10 variables dépendantes le  est approximativement .60! les tests univariés ne tiennent pas compte de la corrélation entre les VD tandis qu’une MANOVA le fait de différences systématiques mais petites peuvent être individuellement non significatives mais une MANOVA fait ressortir l’effet cumulatif Note that MANOVA is more powerful when (a) effect sizes are large and DVs are slightly or even negatively correlated or (b) when effect sizes are small and DVs are highly correlated.

Comment?

Il faut calculer des fonctions discriminantes Variable dépendante Score de la fonction discriminante Accpeu 0.685097 Accint 0.286191 Accatt -0.441880 Accdeg 0.176559 Accirr 0.232894 There will be (whatever is smaller) p - number of DVs or k-1 (levels of the IV) We have here an example for such an discriminant score - the discriminant function combines the IVs so as to maximally discriminate between the groups. The are maximaly K (number of groups) -1 discriminant functions

Il faut calculer des fonctions discriminante There will be (whatever is smaller) p - number of DVs or k-1 (levels of the independent variable) We have here an example for such an discriminant score - the

Quatre manières de poursuivre Phillai-Bartlett trace: La somme des proportions de la variance expliquée par les fonctions discriminantes. Comparable à un ratio de variance expliquée/variance totale. Hotelling’s T2: La somme des eigenvalues pour chaque variate. Wilk’s lambda: Produit de la variance non expliquée pour chaque variate, Roy’s largest Root: La même chose que Hotelling’s T mais pour la 1ere variate seulement

Lequel des indices faut-il prendre? Critères Puissance Robustesse Égalité des tailles d’échantillons Phillai’s Trace Robuste quand les tailles des échantillons sont égales Hotelling’s T2 Le test le plus utilisé quand la variable indépendante a deux niveaux Wilk’s Lambda Le test le plus utilisé quand la VI a plus que deux niveaux Roy’s largest root: Le test le plus puissant mais le moins robuste face aux violations de la normalité For small to moderate sample sizes little difference between measures

Postulats de la MANOVA Indépendance des observations Échantillon aléatoire, mesuré à niveau intervalle Normalité multivariée Homogénéité des matrices de variance - covariance (pour chaque variable il faut que les variances soient homogènes ainsi que les corrélations entre chaque paire de VDs) Levine’s test for homogenity of variances and Box test for homogenity of covariances

Manova avec SPSS

Vérification de l’homogénéité des variances Levine’s test suggests problemes with the homogeneity of variance for fear and irritation, Box’s test suggests problemes With the homogeneity of the covariance matrices. As group size is equal, we will use Philai’s Trace

Moyennes et écart-types

Tests multivariés

Tests univariés We note that the univariate test is significant only for fear. Also if we consider the sums of square and cross-product marices we note that the error cross-products are larger than the model cross-products which can suggest that the relationship between variables is pertinent.

Analyse discriminatoire

Options

Résultats Note that Wilk’s lambda is the same as in the SPSS output (which really has to be the case). The standardized Disciminant function coefficients are like betas. Thus we can see that fear and aggression are the two emotions which are most pertinent to distinguish between the two groups. People who identify with cats are better at detecting fear and people who do not are better at detecting agression. Overall, people who identify with cats are better at detecting all emotions except aggression. The structure matrix tells us something about the importance of the variables for the discrimination between groups. Again fear is very pertinent. The discriminant function coefficients denote the unique (partial) contribution of each variable to the discriminant function(s), while the structure coefficients denote the simple correlations between the variables and the function(s).

Un autre exemple

Manova

Vérification de l’homogénéité des variances

Test multivarié

Tests univariés

Moyennes

Inspection de la matrice des sommes des carrés et des produits croisés The cross-products for the error are higher than those for the model -- suggesting that the relationship between the variables may explain part of the effect

Analyse discriminatoire Only the first function is significant. Recent immigrants are better decoders of happiness but worse at sadness and disgust when compared to French Canadians.The structure matrix suggest that sadness is important for function 1 and disgust for function 2.

Classification

Scatterplot

Analyse Discriminante

Dialectes nonverbaux

Analyse

Résultats

Classification