Sample questions

1.

A researcher is investigating the effects of two factors, X₁ and X₂, each at 2 levels, on a response variable Y. A balanced two-factor factorial design is used with 1 replicate. The researcher will fit the linear model

$$Y = \beta_{0} + \beta_{1}X_{1} + \beta_{2} X_{2} + \varepsilon,$$

with the usual assumptions about the error term. The two values of X₁ and X₂ are 0 and 1.

Display of Design Used to Fit Two Variable Model

				X1
X2			0		1
	0		$\bullet$		$\bullet$
	1		$\bullet$		$\bullet$

(a)

Write out the X matrix for the model

$${\underline{Y}} = {\underline{X\beta}} + {\underline{\varepsilon}}.$$

(b)

If the error variance $\sigma^{2} = 1.0$, find the variance for each of the estimates of $\beta_{0}$, $\beta_{1}$, and $\beta_{2}$, and show that the covariance of the estimates of $\beta_{1}$ and $\beta_{2}$ {Cov($\beta_{1}$, $\beta_{2}$)} is equal to zero.

(c)

What are the practical implications of this covariance being equal to zero? Would this tend to happen in an observational study?

2.

You have a consulting client who is conducting a two factor experiment. Factor A is at 2 levels and Factor B is at 2 levels. For some reason, uneven replication was used. The data appear below. This table indicates that 2 observations were taken at the (1,1) (A,B) combination and that one observation was taken at the (2,1) (A,B) combination.

				Factor A
			1		2
Factor B
	1		12, 10		14
	2		14		16, 18

(a)

Give a numerical value that represents the effect of Factor A. That is, what is the effect on the mean of changing Factor A from level 1 to level 2?

(b)

Give a numerical value that represents the effect of Factor B. That is, what is the effect on the mean of changing Factor B from level 1 to level 2?

(c)

Do you think the estimates given in parts (a) and (b) are stochastically independent? If not, why not? If you were to compute sum of squares for Factor A and sum of squares for Factor B, would they be independent?

(d)

The client wants you to fit a two factor ANOVA model containing A and B main effects and the AB interaction term, in addition to the usual error term. The model is symbolically represented as $Y = A + B + AB + \varepsilon$. You are to use regression models (as opposed to Anova models) to test the hypothesis of no factor A main effect for this model. Write out the model(s) you need to do this, defining any necessary independent variables. For each model you want to fit, write the Y vector, the Beta vector, and the X matrix that would be used to fit the model. You do not need to do any computations. Do include the degrees of freedom involved in the test.

3.

You are fitting a multiple regression model, but there is some autocorrelation in the residuals, and possible heteroskedasticity. What effect does this have on the parameter estimates regarding bias and variance?

4.

In linear models, sums of squares can be written as quadratic forms, such as SS₁ = $\bf y^{\prime}Ay$ and SS₂ = $\bf y^{\prime}By$, where y denotes a random vector with covariance matrix $\Sigma$. What conditions must hold for SS₁ and SS₂ to be independent?

5.

Consider a simple one-way ANOVA with t treatments and a total of n observations. Total variation is decomposed into variation between treatments, SST, and error variation, SSE, with SSTOTAL = SST + SSE.

(a)

The degrees of freedom for treatments = and the degrees of freedom for error = .

(b)

Where do degrees of freedom ``come from"? Explain briefly.

6.

The multivariate normal distribution model is the foundation of many multivariate techniques, and also of regression analysis. Consider the bivariate case with two variables, X and Y, with means $\mu_{X}$ and $\mu_{Y}$, variances $\sigma^{2}_{X}$ and $\sigma^{2}_{Y}$, and correlation coefficient $\rho$. What is the conditional mean of Y given X in terms of these parameters? The conditional variance of Y given X?

7.

A farmer raises pigs, and wants to fatten them up as much as possible. The farmer wants to compare weight gains of adult pigs using two diets, A and B. Several pigs are selected at random to use diet A, and several use diet B. The pigs are weighed at the end of weeks 1, 2, and 3. The data are displayed are the weight of each pig at different points in time.

Diet A pigs

Weight in pounds at the end of each week.

	Week 1	Week 2	Week 3
Pig #1	455	460	465
Pig #2	387	390	395
...

goes on for more pigs

Diet B pigs

Weight in pounds at the end of each week.

	Week 1	Week 2	Week 3
Pig #1	415	417	414
Pig #2	501	503	499
...

goes on for more pigs

Let ${\bf {\mu_{A}}}$ denote the vector of mean weights at weeks 1, 2, and 3 for pigs on diet A: ${\bf {\underline{\mu}}^{\prime}_{A} = [\mu_{1A}, \mu_{2A}, \mu_{3A}]}$. The vector ${\bf {\underline{\mu}}_{B}}$ is similarly defined. State the hypothesis you would test to see if the changes in weight are equal for the two diets.

8.

The model typically fit to one-way Anova type of data is $Y_{ij} = \mu + \tau_{i} + \varepsilon_{ij}$, where $\mu$ denotes an overall mean, $\tau_{i}$ denotes the effect of the i^th treatment, and $\varepsilon_{ij}$ denotes the random error term, $i=1,2 \ldots t$. The $\varepsilon_{ij}$ are assumed to be iid N(0, o²).

(a)

When the model parameters $\mu$ and $\tau_{i}$ are estimated via least squares, what types of restrictions are typically placed on the $\tau_{i}$? How is a generalized inverse used in parameter estimation for the one-way ANOVA model? Set up the matrix approach to parameter estimation for the case when t=3 and there are 2 observations per treatment. Write out the X-matrix and a generalized inverse for it.

(b)

If you were analyzing data with a one-way ANOVA model, what role could a non-central Chi-squared distribution play?

(c)

What is meant by a ``random effects model"? What model parameter descriptions need to be changed?

9.

Consider the simple linear regression model $Y = \beta_{0} + \beta_{1}X_{1} + \varepsilon$, where the $\varepsilon$ are iid $N(0,\sigma^{2})$.

(a)

Assuming that this is the correct model for these data, prove that the ordinary least squares estimator of the slope $\beta_{1}$ is unbiased.

(b)

Suppose that you wanted to force the intercept to be zero, thus throwing $\beta_{0}$ out of the model. Would the least squares estimate of $\beta_{1}$ change?

(c)

What does the Gauss-Markov Theorem state with regards to the least squares estimates of $\beta_{0}$ and $\beta_{1}$?

10.

Data on heights and weights of 111 girls were obtained and a simple linear regression model of weight on height was fit by least squares. A portion of the resulting ANOVA table is displayed below.

Source	DF	SS	MS	F-value
Regression		21506
Error
Total		38121

Thus you are given only SSR and SSDTOT.

(a)

Fill in the other entries in the ANOVA table (DF, SS, MS, F).

(b)

Consider the model

$$Y_{i} = \beta_{0} + \beta_{1}X_{i} + \varepsilon_{i}$$

where the $\varepsilon_{i}$ are iid $N(0,\sigma^{2})$, Y = weight and X = height. Test $H_{0}: \beta_{1} = 0$ vs. $H_{1}: \beta_{1} \neq 0$. Use a 0.01 level of significance.

(c)

If the estimate of $\beta_{1}$ is b₁ = 4.16, find SE(b₁).

(d)

In addition to height and weight, the age of each of the 111 girls was recorded. The variable AGE is added to the model, and the model can be stated as WEIGHT = $\beta_{0} + \beta_{1}$ HEIGHT $+ \beta_{2}$ AGE. The residual sum of squares from this two-variable model is 15689. Conduct a partial F-test of $H_{0}: \beta_{2} = 0$ vs. $H_{1}: \beta_{2} \neq 0$ at a 0.01 level of significance.

11.

Given 3 factors A, B, and C, construct a 2^3-1 design with highest possible resolution.

(a)

Specify the 4 runs using +/- notation.

(b)

Give a defining relation.

(c)

What is the design resolution?

(d)

Identify all aliases.

12.

An experimenter constructed a 2^5-2 fractional factorial design. The design generators are D = AB and E = BC. After analyzing data from an experiment conducted with this design, he found that factor A was important and decided to perform a second 2^5-2exactly the same as the first, except that the signs are changed in the ``A" column of the design matrix. That is, a ``+" was changed to a ``-" and vice-versa.

(a)

How many runs does the first design contain?

(b)

Give a set of generators for the second design.

(c)

What is the resolution of the second design?

(d)

What is the defining relation of the combined design? (The combined design is the first and second design taken collectively as a single design.)

(e)

What is the resolution of the combined design?

13.

A pharmaceutical company claims that Compound E improves the vision of nearsighted people. It is desired to test this claim in a clinical trial on n = 2k patients. For each patient the vision in each eye will be measured before each treatment and again after 10 weeks of treatment. The response variable, Y, is a quantitative scale variable.

The statistical objective is to estimate the difference in mean levels of vision improvement between the Compound E treatment and the placebo.

One possible design is to randomly select and treat k patients (n eyes) with ointment containing Compound E and the other k patients with a placebo ointment. An eye is selected at random for each patient. A two sample t-test can be done to test for a difference in mean levels, or a confidence interval constructed to estimate the mean difference.

An alternative design would be to have each patient use both Compound E and the placebo. For each patient, a coin flip dictates which eye would receive which treatment. A paired difference t-test could be used to test for a difference or to construct a confidence interval for the mean difference.

Define appropriate notation and find the variance of the effect estimate under each design. Under what conditions is the paired design better than the two independent samples design?

14.

Increased arterial blood pressure in the lungs frequently leads to the development of heart failure in patients with chronic obstructive pulmonary disease (COPD). A cardiologist collected data on 19 mild to moderate COPD patients. The data include the invasive measure of systolic pulmonary arterial pressure (Y) and three potential non-invasive predictor variables. Two were obtained by using radionuclide imaging: the emptying rate of blood into the pumping chamber of the heart (X₁), the ejection rate of blood pumped out out of the heart into the lungs (X₂), and a measure of blood gas (X₃).

The next 2 pages contain output from a regression package (omitted).

For the model $Y = \beta_{0} + \beta_{1}X_{1} + \beta_{2}X_{2} + \beta_{3}X_{3}$:

(a)

Test the hypothesis $H_{0}: \beta_{1} = \beta_{3} = 0$ vs. H₁: not both of $\beta_{1}$, $\beta_{3} = 0$. Use a 0.05 level of significance.

(b)

Test the hypothesis $H_{0}: \beta_{2} = -0.5$ vs. $H_{1}: \beta_{2} \neq -0.5$ at a 0.05 level of significance.

(c)

Calculate r²_Y2 and $r^{2}_{Y2 \bullet 13}$.

(d)

Identify the best two-variable model and calculate its R².

(e)

Two patients have (X₁,X₂,X₃) readings of (33,25,41) and (33,26,41), respectively. Estimate their expected difference in Y using a $95\%$ confidence interval.

15.

Continuing from problem 14: In further analysis of these same data, a backwards elimination procedure is done. All first and second order terms for all 3 predictors were considered. The model selected by the search procedure included X₁, X₂, and their cross product X₁X₂. Output from the fitting of this model is displayed on the next page (omitted).

(a)

Are there any indications that serious multicollinearity problems exist? What criteria do you use?

(b)

Use studentized deleted residuals and identify any outlying Yobservations. Use the Bonferroni outlier test procedure with $\alpha = 0.1$.

(c)

Identify any outlying X observations. What criteria are you using?

(d)

Are there any cases that appear to be overly-influential on the parameter estimates? State the criteria you use.