Chapter 1

Question 0

What is a functional relation?

Answer 0

X-Independent variable and Y-Dependent variable then the functional relation is Y=F(X)

Question 1

What is Regression Analysis?

Answer 1

Stat methodology that utilizes the relation between two or more quantitative variables so that a response/outcome variable can be predicted from the others.

Question 2

What is a Statistical Relation?

Answer 2

Not exact like a functional relation. In example, book used scatter plot to show a linear trend…there is variation in the points on the graph. Relation could be curvilinear (not linear).

Question 3

What is a regression model?

Answer 3

formal means of expressing the two essential ingredients of a statistical relation:

1. A tendency of the response variable Y to vary with the predictor variable X in a systematic fashion
2. A scattering of points around the curve of statistical relationship

Question 4

2 characteristics embodied in a regression model by postulating that:

Answer 4

1. There is a probability distribution of Y for each level of X
2. The means of these probability distributions vary in some systematic fashion with X.

Question 5

What is a major consideration in the selection of Predictor Variables during the construction of regression models?

Answer 5

The extent to which a chosen variables contributes to reducing the remaining variation in Y after allowance is made for the contributions of other predictor variables that have tentatively been included in the regression model.

Question 6

List some other considerations in selecting Predictor Variables.

Answer 6

1. The importance of the variable as a casual agent in the process under analysis
2. The degree to which observations on the variable can be attained more accurately, or quickly, or economically than on competing variables
3. The degree to which the variable can be controlled

Question 7

How is the scope of a model determined?

Answer 7

Either by the design of the investigation or by the range of data at hand.

Question 8

What are the 3 major purposes of Regression Analysis?

Answer 8

1. Description
2. Control
3. Prediction

Question 9

T/F: No matter how strong the statistical relation between X and Y, there is a cause-effect pattern implied by the regression model.

Answer 9

F: No matter how strong the statistical relation between X and Y, no cause-and effect pattern is necessarily implied by the regression model

Question 10

Define the basic regression model:

Answer 10

Y_i = B_0 + B_1X_i + e_1
note: Y_i is the value of the response variable in the i-th trial, B_0 and B_1 are parameters (Beta), X_i is a known constant, namely, the value of the predictor variable in the i-th trial, e_i is a random error term with mean E[e_i]=0 and variance V[e_i]=(simga)^2

Question 11

Why is this model: Y_i=B_0+B_1X_i+e_i called a first order model?

Answer 11

The regression model is said to be simple, linear in the parameters, and linear in the predictor variable. Simple because there is only one predictor variable, linear because no parameter is squared, multiplied or divided by another parameter.

Question 12

E[Y_i] where Y_i = simple regression model

Answer 12

E[Y_i] = E[B_0+B_1X_i+e_i] = B_0 + B_1X_i + E[e_i] = B_0 + B_1X_i
because E[e_i]=0

Question 13

V[Y_i] =

Answer 13

(sigma)^2

because the error term e_i has constant variance, (sigma)^2

Question 14

T/F: Since the error terms, say e_i and e_j, in a regression model are assumed to be uncorrelated then Y_i and Y_j (any two responses) are also uncorrelated.

Answer 14

T

Question 15

What are the regression coefficients in a simple regression model?

Answer 15

B_0, B_1 are the parameters

Question 16

What is B_1 in the simple regression model?

Answer 16

Slope of the regression line. It is the change in the mean of the probability distribution of Y per unit increase in X.

Question 17

What is the parameter B_0 in the simple regression model?

Answer 17

Y intercept of the regression line.

Question 18

What happens when the scope of the simple regression model includes X = 0?

Answer 18

B_0 gives the mean of the probability distribution of Y at X = 0.

Question 19

What does B_0 mean when the scope of the model does not cover X = 0?

Answer 19

B_0 does not have any particular meaning as a separate term in the regression model.

Question 20

How is observational data obtain?

Answer 20

Observational data is obtained from non-experimental studies. Theses studies do not control the explanatory or the predictor variable(s) of interest.

Question 21

What is one major limitation of observational data?

Answer 21

They often do not provide adequate information about cause and effect relationships.

Question 22

When control over the explanatory variable(s) is exercised through random assignments the resulting experimental data provide much stronger information about cause and effect relationships than observational data. Why?

Answer 22

The reason is that randomization tends to balance out of any other variables that might effect the response variable.

Question 23

T/F: Control over the explanatory variable(s) consists of assigning a treatment to each of the experimental units by means of randomization.

Answer 23

T

Question 24

Describe Completely Randomized Design.

Answer 24

Makes randomized assignments of treatments to experimental units (or vice versa). Useful when experimental units are homogeneous. The design is flexible because it accommodates any number of treatments and permits different sample sizes for different treatments.

Question 25

What is the Method of Least Squares?

Answer 25

A way to find “good” estimators of the regression parameters B_0 and B_1.

Question 26

How do you use the Method of Least Squares?

Answer 26

For the observations (X_i,Y_i) for each case, the MLS considers the deviation of Y_i from its expected value: Y_i - (B_o +B_1X_i).
The MLS requires that we consider the sum of the n squared deviations: Q = (sum from i=1 to n) (Y_i - B_0 - B_1X_i)^2

Question 27

What is another way to describe the Method of Least Squares?

Answer 27

The estimators of B_0 and B_1 are those b_0 and b_1 that minimize the criterion Q for the given sample observations (X_1,Y_1), …,()X_n,Y_n)

Question 28

What are the point estimators of B_0 and B_1?

Answer 28

b_0 and b_1 (page 17 of applied linear models)

Question 29

What is the Gauss-Markov Theorem?

Answer 29

Under the conditions of the simple regression model, the least squares estimators b_0 and b_1 are unbiased and have a minimum variance among all unbiased linear estimators (b_0 and b_1 sampling distributions are less variable than any other estimators).
E[b_0]=B_0 and E[b_1]=B_1

Question 30

What is the estimate of the regression function?

Answer 30

Y(hat) = b_0 + b_1X

Y hat is the value of the estimated regression function at the level X of the predictor variable.

Question 31

What is the difference between the response and the mean response?

Answer 31

We call a value of the response variable a response and E[Y] the mean response. Mean response is the mean of the probability distribution of Y corresponding to the level X of the predictor variable. Y hat is a point estimator of the mean response when the level of the predictor variable is X. Y hat is an unbiased estimator of E[Y], with minimum variance in the class of unbiased estimators (by Markov Theorem).

Question 32

Define residual.

Answer 32

The i-th residual is the difference between the observed value Y_i and the corresponding fitted value Y_i hat. e_i = Y_i - Y_i hat
e_i = Y_i - (b_0 +b_1X_i)
The magnitude of a residual is represented by the vertical deviation of the Y_i observation from the corresponding point on the estimated regression function.

Question 33

Distinguish between the model error term value epsilon = Y_i - E[Y_i] and the residual e_i = Y_i - Y_i hat.

Answer 33

The model error term is the vertical deviation from the unknown true regression line and hence is unknown. The residual is the vertical deviation of Y_i from the fitted value Y_i hat on the estimated line and is known. Residuals are highly useful for studying whether a given regression model is appropriate for the data at hand.

Question 34

What is the sum of the residuals?

Answer 34

0

Question 35

What is the sum of the squared residuals?

Answer 35

a minimum

Question 36

What does the sum of the observed values Y_i equal?

Answer 36

The sum of the fitted values, Y_i hat.

Question 37

What is the sum of the weighted residuals when the residual in the i-th trial is weighted by the level of the predictor variable in the i-th trial.

Answer 37

Sum from i=1 to n of X_ie_i = 0

Question 38

What point does the regression line always go through?

Answer 38

(X(bar), Y(bar))

Question 39

T/F: The sum of the weighted residuals is 1 when the residual in the i-th trial is weighted by the fitted value of the response variable for the ith trial.

Answer 39

F: Sum from i=1 to n of Yhat_i e_i =0

Question 40

Why does the variance, (sigma)^2, of the error terms in a simple regression model need to be estimated?

Answer 40

To obtain an indication of the variability of the probability distributions of Y.

Question 41

What is the variance (sigma)^2 of a single population estimated by?

Answer 41

s^2, the sample variance. The sample variance is also known as the mean square.

Question 42

T/F: s^2, the sample variance, is an unbiased estimator of the the variance (sigma)^2 of an infinite population.

Answer 42

True

Question 43

What is E[MSE], where MSE is the error mean square/residual mean square

Answer 43

(sigma)^2 because MSE is an unbiased estimator of (sigma)^2 for the regression model.

Question 44

Describe the Method of Maximum likelihood.

Answer 44

This method chooses as estimates those values of the parameters that are most consistent with the sample data. Way to obtain the estimators of the parameters B_0, B_1, and (sigma)^2. The method of maximum likelihood chooses the maximum likelihood estimate that value of mu for which the likelihood value is largest.