Homework 3

# Homework 3 - HOMEWORK III JAN DE LEEUW 1 P ROBLEM Consider...

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HOMEWORK III JAN DE LEEUW 1. P ROBLEM Consider a table with n rows and m columns and an unequal number of observations in each cell. We can index the observations as x i jk with i = 1 , ··· , n and j = 1 , ··· , m and k = 1 , ··· ,‘ i j . An example of such a table will look like the following -1.3690510 0.8803977 -0.3459382 -0.1190956 0.4348185 0.2042370 0.4662728 1.1817889 -0.3297959 -0.6008543 -0.6795710 -0.9254206 This gets coded as the data structure 1 1 -1.3690510 1 1 0.8803977 1 1 -0.3459382 1 2 -0.1190956 1 2 0.4348185 2 1 0.2042370 2 2 0.4662728 2 2 1.1817889 2 3 -0.3297959 2 3 -0.6008543 2 3 -0.6795710 2 3 -0.9254206 The first two columns only indicate which cell in the design the observation comes from. They are not for computing, only for indexing. Date : November 10, 2010 — 21h 8min — Typeset in T IMES R OMAN . 1

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2 JAN DE LEEUW We want to fit a main effect approximation of the form x i jk μ + α i + β j using least squares. Thus we want to minimize the loss function σ ( μ , α , β ) = n i = 1 m j = 1 i j k = 1 ( x i jk - μ - α i - β j ) 2 . In the balanced case (in which i j = for all i , j ) the solution is ˆ μ = x ••• , ˆ α i = x i •• - x ••• , ˆ β j = x j - x ••• , where bullets denote averages. Thus x
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