Assignment I Questions - Applied Regression Analysis | STAT 333, Assignments of Statistics

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Statistics 333 Assignment 1 Due Sept. 19, 2003

1. For the simple linear regression model Y

i

= β 0

• β 1

X

i

• ε i

, i = 1,... , n , consider the least

squares fits

ˆ

Y

i

= b 0

• b 1

X

i

and residuals given by e

i

= Y

i

−

ˆ

Y

i

= Y

i

− ( b 0

• b 1

X

i

), i = 1,... , n.

Since b

0

= Y − b 1

X , it is easy to see (1/ n )

Σ

n

i = 1

ˆ

Y

i

= Y and hence also that

Σ

n

i = 1

( Y

i

−

ˆ

Y

i

) = 0.

Similarly, also show that the residuals e

i

satisfy the additional ‘orthogonality’ constraint

Σ

n

i = 1

X

i

( Y

i

−

ˆ

Y

i

) = 0.

2. For the simple linear regression model in Exercise 1, under the standard model assump-

tions for the random errors ε

i

, show that the least squares estimator b

1

= S

xy

/ S

xx

and

Y = (1/ n )

Σ

n

i = 1

Y

i

have zero covariance, i.e., Cov( b

1

, Y ) = 0 , where S

xy

=

Σ

n

i = 1

( X

i

− X ) Y

i

.

3. For the simple linear regression model, the estimator of the error variance σ

2

= Var( ε i

) is

given by

S

2

=

n − 2

n

i = 1

Σ

( Y

i

−

ˆ

Y

i

)

2

≡

n − 2

SSE.

Show that this estimator S

2

is unbiased for σ

2

, i.e., prove that E ( S

2

) = σ

2

.

Note: To prove this result, use the following approach. First, verify the identity

Y

i

− ( β 0

• β 1

X

i

) = [ Y

i

− ( b 0

• b 1

X

i

) ] + [ Y − ( β

0

• β 1

X ) ] + [ ( b

1

− β 1

) ( X

i

− X ) ].

Then verify that the sum of squares of elements on the left-hand side,

Σ

n

i = 1

[ Y

i

− ( β 0

• β 1

X

i

) ]

2

, is equal to the sum of the 3 sums of squares of individual elements

on the right-hand side, due to ‘orthogonality’ in cross-terms (Exer. 1). Hence, verify that

Σ

n

i = 1

[ Y

i

− ( β 0

• β 1

X

i

) ]

2

= SSE + n [ Y − ( β

0

• β 1

X ) ]

2

+ S

xx

( b 1

− β 1

)

2

.

Finally, equate the expected values of both sides of the sums of squares relation, to get

n σ

2

= E [ SSE ] + n E [ Y − ( β

0

• β 1

X ) ]

2

+ S

xx

E [ ( b

1

− β 1

)

2

] , ‘evaluate’ the other expected

value terms, and solve for E [ SSE ]. Also use definitions and known results for Var( b

1

)

etc., for instance, E {[ Y

i

− ( β 0

• β 1

X

i

) ]

2

} = Var( Y

i

) ≡ σ

2

by definition of variance.

4. Consider the zero or no intercept model given by Y

i

= β 1

X

i

• ε i

, i = 1,... , n , with the errors

ε i

being independent, normal r.v.’s with mean 0 and variance σ

2

.

i) Derive the least squares estimator b

1

of β

1

, and also derive the variance of this estimator.

ii) For an arbitrary fixed value X

0

of X , establish that a 100(1 − α )% confidence interval for

the mean response value E ( Y | X

0

) = β 1

X

0

is given by

b 1

X

0

± t

( α /2)

n − 1

S

√

X

2

0

/

Σ

n

i = 1

X

2

i

,

where S

2

=

Σ

n

i = 1

( Y

i

− b 1

X

i

)

2

/( n − 1) provides an unbiased estimator of σ

2

with n − 1 df.

Note: You can conclude that ( b

1

− β 1

)/ se( b

1

) has the t

n − 1

distribution.

5. An experiment was conducted to study the mass of a tracer material exchanged between

the main flow of an open channel and the "dead zone" caused by a sudden open channel

expansion. Researchers need this information to improve the water quality modeling capa-

bility of a river. It is important to determine the exchange constant K for varying flow

conditions. The value of K describes the exchange process when a dead zone appears. In

a study, values of the Froude Numbers ( N

F

) were used to predict K. Numbers are func-

tions of upstream channel velocity and water depth. The data collected were as follows,

with the negative sign of the K values indicating "flushing", the direction of mass transfer

out of the dead zone:

Obs N

F

K

1 0.012500 -0.

2 0.023750 -0.

3 0.025625 -0.

4 0.030000 -0.

5 0.033125 -0.

6 0.038125 -0.

7 0.038125 -0.

8 0.038125 -0.

9 0.041250 -0.

10 0.043125 -0.

11 0.045000 -0.

12 0.046875 -0.

13 0.047500 -0.

14 0.050000 -0.

15 0.051250 -0.

16 0.056250 -0.

17 0.062500 -0.

18 0.068750 -0.

19 0.077500 -0.

20 0.044000 -0.

These data are stored in the file: /u/r/e/reinsel/stat333/exchange.dat ; also on webpage.

Perform and show calculations for (ii)-(v) below by ‘direct calculations’ on a calculator or

computer; then use regression in Minitab or other software to confirm calculations.

i) Construct a scatter plot of K versus N

F

, and provide some relevant comments.

ii) Use the least squares method to fit the model K

i

= β 0

• β 1

( N

F

)

i

• ε i

.

iii) Compute S

2

, R

2

, and obtain the basic analysis of variance (ANOVA) table. Provide

some brief interpretation for these results, e.g., in terms of the amount of variation of the K

values explained by the fitted regression (i.e., the variable N

F

).

iv) Obtain the standard errors for the least squares estimates b

0

and b

1

, and under the usual

normal theory model assumptions, give a 95% confidence interval for β

1

.

v) Determine the explicit form for the 95% confidence interval of a mean response

E ( K | N

F

) = β 0

• β 1

( N

F

) for any fixed value N

F

, and evaluate for three selected and rea-

sonable values of N

F

.

vi) Obtain the residuals e

i

= K

i

−

ˆ

K

i

, and plot these versus the fitted values

ˆ

K

i

as well as

against the predictor variable ( N

F

)

i

. Does this plot tend to confirm the basic assumptions

of the linear regression model, or does it suggest violation of any assumption? Explain.