Question 1

In a two-factor experiment where factor A consists of 4 levels, factor B consists of 3 levels, and there is only one observation on each of the 12 treatments, which of the following statements are not true?
A) SST has 12 degrees of freedom
B) SSA has 3 degrees of freedom
C) SSB has 2 degrees of freedom
D) SSE has 6 degrees of freedom
E) None of the above statements are correct.

Accepted Answer

The total degrees of freedom (SST) is calculated as the total number of observations minus one, which is 12 - 1 = 11, not 12.

Question 2

To select a half-replicate of a  factorial experiment  possible treatment conditions), the number of defining effects that must be selected is __________.

Accepted Answer

The answer of To select a half-replicate of a ...

Question 3

The primary interest of designing a randomized block experiment is:
A) to reduce the variation among blocks
B) to increase the between-treatments variation to more easily detect differences among the treatment means
C) to reduce the within-treatments variation to more easily detect differences among the treatment means
D) to increase the total sum of squares
E) All of the above statements are true.

Accepted Answer

The primary interest of designing a randomized block experiment is to reduce the within-treatments variation to more easily detect differences among the treatment means. The use of blocks allows for better control of confounding variables and reduces the impact of variation due to those variables, resulting in a more accurate assessment of treatment effects. Choices A, B, and D are incorrect as they either go against the primary interest of the design or are irrelevant to the goal of detecting treatment differences.

Question 4

Which of the following statements are not true?
A) In a two-factor experiment there are I levels of one factor and J levels of the other factor. When there is more than one observation for at least one (i,j) pair of the IJ combinations of levels of the two factors, a valid estimator of the random error variance $\sigma ^ { 2}$
Cannot be obtained without assuming additivity.
B) In a two-factor experiment, additivity means that the difference in true average responses for any two levels of one of the factors is the same for each level of the other factor.
C) The parameters for the fixed effects model with interaction are $\alpha _ { i } = \mu _ {i } - \mu =$
The effect of factor A at level $i , \beta _ { j } = \mu. j - \mu =$
The effect of factor B at level j, and $\lambda _ { y } = \mu _ { y } - \left( \mu + \alpha _ { i } + \beta _ { j } \right) =$
The interaction of factor A at level i and factor B at level j. Thus, the model is $\mu _ { y } = \mu + \alpha _ { i } + \beta _ { j } + \gamma _ { y }$
This model is additive if and only if $y _ { y } = 0 \text { for all } ( i , j )$
D) All of the above statements are true.
E) None of the above statements are true.

Accepted Answer

The answer of Which of the following statements are not...

Question 5

The following equation SST = SSA + SSB +SSC + SSAB + SSAC + SSBC + SSABC + SSE applies to which ANOVA model?
A) One-factor ANOVA
B) Two-factor ANOVA with interaction
C) Three-factor ANOVA with interactions
D) Randomized block design
E) Latin square design

Accepted Answer

The equation given has terms for all possible main effects and interactions for a three-factor ANOVA.

Question 6

To select a quarter-replicate of a  factorial experiment  possible treatment conditions), the number of defining effects that must be selected is __________.

Accepted Answer

The answer of To select a quarter-replicate of a ...

Question 7

In the three-factor fixed effects model, assume that there are 4 levels of factor A, 2 levels of factor B, 4 levels of factor C, and 3 observations for each combination of levels of the three factors. Then, the number of degrees of freedom for the three-factor interaction sum of squares (SSABC) is
A) 32
B) 10
C) 9
D) 12
E) 13

Accepted Answer

The degrees of freedom for the three-factor interaction (SSABC) is calculated as (a-1)(b-1)(c-1), where a, b, and c are the number of levels for factors A, B, and C, respectively. Thus, (4-1)(2-1)(4-1) = 3*1*3 = 9.

Question 8

The following equation SST = SSA + SSB +SSAB +SSE applies to which ANOVA model?
A) One-factor ANOVA
B) Two-factor ANOVA with interaction
C) Three-factor ANOVA
D) Randomized block design
E) All of the above

Accepted Answer

The presence of the interaction term (SSAB) in the equation indicates that there are two factors being considered in the ANOVA model, and that their effects on the outcome measure are not additive. This corresponds to a two-factor ANOVA with interaction. The other choices (one-factor ANOVA, three-factor ANOVA, randomized block design) do not involve an interaction term in their equations.

Question 9

In a two-factor ANOVA problem, there are 4 levels of factor A, 5 levels of factor B, and 2 observations (replications) for each combination of levels of the two factors. Then, the number of treatments in this experiment is
A) 40
B) 11
C) 10
D) 20
E) 8

Accepted Answer

The number of treatments in a two-factor ANOVA can be calculated by multiplying the number of levels of factor A by the number of levels of factor B (i.e., the number of cells in the design). In this case, there are 4 levels of factor A and 5 levels of factor B, so the number of treatments is 4 x 5 = 20. However, since there are 2 observations for each combination of levels of the two factors, the total number of observations in the experiment is 4 x 5 x 2 = 40.

Question 10

Which of the following statements are not true regarding the model $X _ { y } = \mu + \alpha _ { i} + \beta _ { j} + \varepsilon _ { y }$ where $\sum _ { l = 1 } ^ { \prime } \alpha _ { 1 } = 0 \text { and } \sum _ { j = 1 } ^ { j } \beta _ { j } = 0 ?$
A) $\mu$
Is the true grand mean; that is, the mean response averaged over all levels of both factors A and B.
B) $\alpha _ { i }$
Is the effect of Factor A at level i.
C) $\beta _ { j }$
Is the effect of Factor B at level j.
D) $\varepsilon _ { y }$
Are assumed independent and normally distributed with mean 0 and variance 1.
E) All of the above statements are true.

Accepted Answer

The statement in D is not necessarily true as it assumes a specific distribution and variance for $\varepsilon _ { y }$ without specifying that these conditions must always hold. In general, $\varepsilon _ { y }$ are assumed to be independent and normally distributed with mean 0, but the variance does not have to be 1; it can be any positive value $\sigma^2$.

Question 11

In the fixed effects model with interaction, assume that there are 4 levels of factor A, 3 levels of factor B, and 3 observations for each of the 12 combinations of levels of the two factors. Then, the number of degrees of freedom for the error sum of squares (SSE) is
A) 36
B) 35
C) 24
D) 10
E) 9

Accepted Answer

The total number of observations is 4 x 3 x 3 = 36. Since we have fixed effects for both factors A and B, the degrees of freedom for the model is dfm = (4-1) x (3-1) = 6. The degrees of freedom for the interaction term is dfi = (4-1) x (3-1) = 6 as well. Therefore, the degrees of freedom for error is dfE = (4 x 3 x 3) - dfm - dfi = 24.

Question 12

Which of the following statements are true regarding a two-factor experiment?
A) In some experiments, the levels of either factor may have been chosen from a large population of possible levels, so that the effects contributed by the factor are random rather than fixed.
B) If both factors contribute random effects, the model is referred to as a random effects model.
C) If one factor is fixed, and the other contributes random effects, a mixed effects model results.
D) If both factors are fixed, the model is referred to as a fixed effects model.
E) All of the above statements are true.

Accepted Answer

All the statements accurately describe aspects of two-factor experiments and the classification of models based on the nature of the factors involved: random, fixed, or mixed.

Question 13

Consider a $2^3$ experiment with four blocks. In this case, __________ factor effects are confounded with the blocks.

Accepted Answer

The answer of Consider a $2^3$ experiment with four blocks....

Question 14

In the fixed effects model with interaction, assume that there are 5 levels of factor A, 4 levels of factor B, and 3 observations (replications) for each of the 20 combinations of levels of the two factors. Then the number of degrees of freedom of the interaction sum of squares (SSAB) is
A) 60
B) 20
C) 15
D) 12
E) 59

Accepted Answer

The formula for the degrees of freedom of the interaction sum of squares in a fixed effects model with interaction is dfAB = (a-1)*(b-1)*n, where a is the number of levels of factor A, b is the number of levels of factor B, and n is the number of replications. Substituting the values given in the question, we get dfAB = (5-1)*(4-1)*3 = 12. Therefore, the correct answer is D.

Question 15

In the randomized block design for ANOVA where the single factor of primary interest has I levels, and b blocks are created to control for extraneous variability in experimental units or subjects, the number of degrees of freedom for SSE (error sum of squares) is given by
A) Ib-1
B) (I-1) (b-1)
C) I-1
D) b-1
E) I+b-1

Accepted Answer

In the randomized block design, the degrees of freedom for SSE is equal to the total number of experimental units or subjects minus the total number of treatments minus the total number of blocks, which is (I*b) - I - b + 1 = (I-1) (b-1).

Question 16

Which of the following statements are not true?
A) The model specified by $X _ { y } = \alpha _ { ? } + \beta _ { j } + \varepsilon _ { y  } \text { and } \mu _ { y } = \alpha _ { ?} + \beta _ { j } ( i = 1 , \ldots \ldots , I \text { and } j = 1 , \ldots \ldots , J )$
Is called an additive model.
B) The model $X _ { y} = \mu + \alpha _ { i } + \beta _ { j } + \varepsilon _ { \mathrm { ij } } \text { where } \sum _ { i = 1 } ^ { \mathrm { I } } \alpha _ {i } = 0 , \sum _ { j=1} ^ { j } \beta _ { j } = 0 \text {, and the } \varepsilon _ { y } { } ^ { \prime } s$
Are assumed independent, normally distributed with mean 0 and common variance $\sigma ^ { 2 }$
Is an additive model in which the parameters are uniquely determined.
C) In two-way ANOVA, when the model is additive, additivity means that the difference in mean responses for two levels of one of the factors is the same for all levels of the other factor.
D) All of the above statements are true.
E) None of the above statements are true.

Accepted Answer

The statements provided describe characteristics of additive models and conditions related to two-way ANOVA correctly, indicating that they are true.

Question 17

A two-factor experiment where factor A consists of I levels, factor B consists of J levels, and there is only one observation on each of the IJ treatments, can be represented by the model $X _ { 4 } = \mu + \alpha _ { i } + \beta _ { j } + \varepsilon _ { y } \text { where } \sum _ { i = 1 } ^ { j} \alpha _ { i } = 0 \text { and } \sum _ { j = 1 } ^ { j } \beta _ { j } = 0$ . Which of the following is the correct form in testing the null hypothesis that the different levels of factor A have no effect on true average response?
A)  : $\alpha _ { 1 } = \alpha _ { 2 } = \ldots \ldots = \alpha _ { j } = 0$
B) $ : \beta _ { 1 } = \beta _ { 1 } = \ldots \ldots = \beta _ { 3 } = 0$
C) $ \text { : at least one } \alpha _ { i } \neq 0$
D) $ \text { : at least one } \beta _ { \mathrm { j } } \neq 0$

E) None of the above answers are correct.

Accepted Answer

The answer of A two-factor experiment where factor A consists...

Question 18

In the three-factor fixed effects model, assume that there are 3 levels for each of the three factors A, B, and C, and 2 observations for each combination of levels of the three factors. Then the number of degrees of freedom for the error sum of squares (SSE) is
A) 54
B) 27
C) 11
D) 18
E) 16

Accepted Answer

The answer of In the three-factor fixed effects model, assume...

Question 19

In the fixed effects model with interaction, assume that there are 3 levels of factor A, 2 levels of factor B, and 3 observations for each of the six combinations of levels of the two factors. Then the critical value for testing the null hypothesis of no interaction between the levels of the two factors at the .05 significance level is
A) 3.49
B) 3.89
C) 3.00
D) 3.55
E) 3.11

Accepted Answer

The critical value for testing the null hypothesis of no interaction between the levels of the two factors in a fixed effects model can be found using an F-distribution table or calculator, given the degrees of freedom for the interaction and the significance level. Here, the degrees of freedom for the interaction term are calculated as (levels of A - 1) * (levels of B - 1) = (3-1) * (2-1) = 2. With 3 observations for each combination, the total number of observations is 3 * 6 = 18, leading to degrees of freedom for error as total observations minus the number of groups, which is 18 - 6 = 12. Looking up the critical value for F with (2, 12) degrees of freedom at the .05 significance level typically yields a value around 3.89.

Question 20

The three-factor fixed effects model, with the same number of observations for each combination of levels I, J, and K of the three factors A, B, and C, respectively, is represented by  . Which of the following statements are true?
A) The restrictions necessary to obtain uniquely defined parameters are that the sum over any subscript of any parameter on the right-hand side of the above equation equals 0.
B) The parameters $\alpha _ { i } ^ { \prime } s , \beta _j ' s , \text { and } \delta _ { k} ^ { \prime } s$
Are called the main effects of the factors A, B, and C, respectively.
C) The parameters 
Are called two-factor interactions.
D) The parameter 
Is called a three-factor interaction.
E) All of the above statements are true.

Accepted Answer

The answer of The three-factor fixed effects model, with the...

Quiz 11: Multifactor Analysis of Variance

In a two-factor experiment where factor A consists of 4 levels, factor B consists of 3 levels, and there is only one observation on each of the 12 treatments, which of the following statements are not true?

To select a half-replicate of a factorial experiment possible treatment conditions), the number of defining effects that must be selected is __________.

The primary interest of designing a randomized block experiment is:

Which of the following statements are not true?

The following equation SST = SSA + SSB +SSC + SSAB + SSAC + SSBC + SSABC + SSE applies to which ANOVA model?

To select a quarter-replicate of a factorial experiment possible treatment conditions), the number of defining effects that must be selected is __________.

In the three-factor fixed effects model, assume that there are 4 levels of factor A, 2 levels of factor B, 4 levels of factor C, and 3 observations for each combination of levels of the three factors. Then, the number of degrees of freedom for the three-factor interaction sum of squares (SSABC) is

The following equation SST = SSA + SSB +SSAB +SSE applies to which ANOVA model?

In a two-factor ANOVA problem, there are 4 levels of factor A, 5 levels of factor B, and 2 observations (replications) for each combination of levels of the two factors. Then, the number of treatments in this experiment is

In the fixed effects model with interaction, assume that there are 4 levels of factor A, 3 levels of factor B, and 3 observations for each of the 12 combinations of levels of the two factors. Then, the number of degrees of freedom for the error sum of squares (SSE) is

Which of the following statements are true regarding a two-factor experiment?

Consider a 232^323 experiment with four blocks. In this case, __________ factor effects are confounded with the blocks.

In the fixed effects model with interaction, assume that there are 5 levels of factor A, 4 levels of factor B, and 3 observations (replications) for each of the 20 combinations of levels of the two factors. Then the number of degrees of freedom of the interaction sum of squares (SSAB) is

In the randomized block design for ANOVA where the single factor of primary interest has I levels, and b blocks are created to control for extraneous variability in experimental units or subjects, the number of degrees of freedom for SSE (error sum of squares) is given by

Which of the following statements are not true?

In the three-factor fixed effects model, assume that there are 3 levels for each of the three factors A, B, and C, and 2 observations for each combination of levels of the three factors. Then the number of degrees of freedom for the error sum of squares (SSE) is

The three-factor fixed effects model, with the same number of observations for each combination of levels I, J, and K of the three factors A, B, and C, respectively, is represented by . Which of the following statements are true?

Consider a $2^3$ experiment with four blocks. In this case, __________ factor effects are confounded with the blocks.