You can use this matrix to specify other models including ones without a constant term. Definition It is useful for investigating whether one or more observations are outlying with regard to their X values, and therefore might be excessively influencing the regression results.. It is also simply known as a projection matrix. The diagonals of the hat matrix indicate the amount of leverage (influence) that observations have in a least squares regression. 1 Hat Matrix 1.1 From Observed to Fitted Values The OLS estimator was found to be given by the (p 1) vector, b= (XT X) 1XT y: The predicted values ybcan then be written as, by= X b= X(XT X) 1XT y =: Hy; where H := X(XT X) 1XT is an n nmatrix, which \puts the hat on y" and is therefore referred to as the hat matrix. 2 Moderate leverage if h ii2[0:2;0:5) and high leverage if h ii2[0:5;1]. Alternatively, model can be a matrix of model terms accepted by the x2fx function. The hat matrix provides a measure of leverage. Last week, in our STT5100 (applied linear models) class, I’ve introduce the hat matrix, and the notion of leverage. Leverage V Residuals matrix hat X X X X H 1 \u02c6 \u02c6 1 j n jiji Yh Y HYY n i. In [23]: Hat Matrix and Leverage Hat Matrix Purpose. Use hatvalues(fit).The rule of thumb is to examine any observations 2-3 times greater than the average hat value. The hat matrix diagonal is a standardized measure of the distance of ith an observation from the centre (or centroid) of the x space. 3 Draw a histogram, stem-and-leaf, or other plot of h ii. (d) Explain the concept of leverage, both in intuitive terms and in terms of the hat matrix. If m=1, a vector of diagonal entries of the ‘hat’ matrix. ... hat: a vector containing the diagonal of the 'hat' matrix. A leverage value greater than 2m / n is considered high and should be examined. It is useful for investigating whether one or more observations are outlying with regard to their X values, and therefore might be excessively influencing the regression results.. Hat Matrix and Regression Diagnostics See: “The Hat Matrix in Regression and ANOVA,” Hoaglin and Welsch (1978) Examples of things to note: • Point 4 is a bivariate outlier - and H4,4 is largest, just exceeds 2p/n=6/10. Now let’s look at the leverage’s to identify observations that will have potential great influence on regression coefficient estimates. It describes the influence each response value has on each fitted value. H Hat values, H I. Definition Pages 16. • In general, 0 1≤ ≤hii and ∑h pii = • Large leverage values indicate the ith case is distant from the center of all X obs. Second one is the lowess regression line for that. A projection matrix known as the hat matrix contains this information and, together with the Studentized residuals, provides a means of identifying exceptional data points. I don't know of a specific function or package off the top of my head that provides this info in a nice data … Hat Matrix and Leverage Hat Matrix Purpose. Leverage v residuals matrix hat x x x x h 1 ˆ ˆ 1 j. 6 When n is large, Hat matrix is a huge (n * n). The hat matrix is used to project onto the subspace spanned by the columns of \(X\). In statistics, the projection matrix (), sometimes also called the influence matrix or hat matrix (), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). • Points 1 and 3 have relatively high leverage - extremes in the scatter of points. hat: a vector containing the diagonal of the ‘hat’ matrix. Along the way I show you that positive semi definite matrix has non negative diagonal elements. Points that have high leverage and large residuals are particularly influential. Role of Hat matrix in diagnostics of Regression Analysis involves the observation in the predictor variable X as H=X(XX)-1 X', Helps in identifying outlier. In a classical regression model, (in a matrix form), the ordinary least square estimator of parameter is The prediction can then be writtenwhere is called the hat matrix. The leverage is just hii from the hat matrix. Rousseeuw and Zomeren 22 (p 635) note that ‘leverage’ is the name of the effect, and that the diagonal elements of the hat matrix ( h ii ,), as well as the Mahalanobis distance (see later) or similar robust measures are diagnostics that try to quantify this effect. Notation. Hat Matrix Diagonal (Leverage) The diagonal elements of the hat matrix are useful in detecting extreme points in the design space where they tend to have larger values. necessary and suﬃcient conditions on the space of design matrix, under which the corresponding Hat matrix elements get desired extreme values. 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