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Unformatted text preview: Chapter 5 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 43 Chapter 3: Diagnostics Section 3.1: Outlying xvalues can be found via boxplot (or a scatterplot!) Useful for assessing extrapolation. More advanced method in Sec. 10.3 for multiple predictors. Section 3.2: Recall r i = Y i ˆ Y i is i th residual. The (externally) studentized residual t i (defined on p. 396) has a t n 3 distribution. Section 3.3: Plots to consider 1 Plot of r i vs. ˆ Y i or r i vs. x i : nonlinearity means line β + β 1 x i inappropriate. Nonconstant variance means var( i ) = σ 2 not appropriate. Outlying observations (very large or small residuals) can indicate several potential problems (later). 2 Histogram or boxplot of r i , normal probability plot of r i to check normality. Expect one outlier out of 150 observations for truly normal data in boxplot. There are also formal tests for normality (later). 2 / 43 5.1 Matrices A matrix is a rectangular array of numbers. Here’s an example: A = 2 . 3 1 . 4 17 22 . 5 √ 2 . This matrix has dimensions 2 × 3. The number of rows is first, then the number of columns. We can write the n × p matrix X abstractly as X = x 11 x 12 x 13 ··· x 1 p x 21 x 22 x 23 ··· x 2 p x 31 x 32 x 33 ··· x 3 p . . . . . . . . . . . . . . . x n 1 x n 2 x n 3 ··· x np . 3 / 43 Other notation Another notation that is common is A = [ a ij ] n × m for an n × m matrix A with element a ij in the i th row and j th column. The matrix X on the previous page would then be written X = [ x ij ] n × p . 4 / 43 5.1 (cont’d) Transpose The transpose of a matrix A takes the matrix A and makes the rows the columns and the columns the rows. Precisely, if A = [ a ij ] n × m then A is the m × n matrix with elements a ij = a ji . For example: If A = 1 2 3 4 5 6 , then A = 1 4 2 5 3 6 . Question: what is ( A ) ? 5 / 43 5.2 Matrix addition If two matrices A = [ a ij ] n × m and B = [ b ij ] n × m have the same dimensions, you can add them together, element by element, to get a new matrix C = [ c ij ] n × m . That is, C = A + B is the matrix with elements c ij = a ij + b ij . For example,  1 2 5 7 10 20 + 1 2 3 4 1 2 =  1 + 1 2 + 2 5 + 3 7 + 4 10 + 1 20 + 2 = 8 11 9 22 . 6 / 43 5.3 Multiplying a matrix times a number Multiplying a matrix A = [ a ij ] by a number b yields the matrix C = A b with elements c ij = a ij b . For example, ( 2)  1 2 5 7 10 20 =  1( 2) 2( 2) 5( 2) 7( 2) 10( 2) 20( 2) = 2 4 10 14 20 40 ....
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 Fall '11
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 Linear Algebra, Statistics, Regression Analysis, Yi

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