Seven observations on two explanatory variates, X1 and X2, and a response variate Y , are presented in Table 10.6. Table 10.6 Observations of two explanatory variates and a response variate X1 X2 Y 42 7.3 128.0 58 9.2 145.0 93 3.9 101.6 70 4.1 90.9 35 8.4 135.6 61 3.7 85.0 29 8.0 108.7

(a) Fit the model Y = β0 + β1X1 + β2X2 + E to these data, and obtain estimates of β0, β1 and β2.

(b) When this model is fitted to these data, how many dimensions does each of the following have: (i) the data space? (ii) the fixed-effect subspace? (iii) the random-effect subspace?

(c) Obtain the estimated value of Y , and the estimate of the residual effect, for each observation. For Observation 5, the estimated value of Y = 128.17.

(d) What is the contribution to this value of: (i) the constant effect? (ii) the effect of X1? (iii) the effect of X2? It is assumed that E ∼ N(0, σ2 ).

(e) Obtain the maximum likelihood estimate of σ2 and the REML estimate of σ2.

(f) What is the minimum number of observations required to obtain estimates of β0, β1, β2 and σ2? If the number of observations available is one less than this minimum, what estimates can be obtained? What is then the relationship between the estimated and observed values of Y ?