Deck 11: Multifactor Analysis of Variance

Question 1

In the three-factor effects model, assume that there are 4 levels for factor A, 2 levels for factor B, 3 levels for factor C, and 4 observations for each combination of levels of the three factors. Then, the three-factor interaction sum of squares (SSABC) has df = __________.

Answer 1

6

Question 2

An experiment in which there are p factors, each at two levels, is referred to as a

Answer 2

factorial experiment.

Question 3

In a three-factor experiment, if the levels of factor A are identified with the rows of a two-way table and the levels of B with the columns of the table, then the defining characteristic of a Latin square design is that every level of factor C appears exactly __________ in each row and exactly __________ in each column.

Answer 3

once, once

Question 4

When several factors are to be studied simultaneously, an experiment in which there is at least one observation for every possible combination of levels is referred to as __________.

Question 5

Consider a experiment with 2 blocks. The price paid for this blocking is that __________ of the factor effects cannot be estimated.

Question 6

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 The parameters are called __________, and is called a __________, whereas are the __________ parameters.

Question 7

The parameters for the fixed effects model with interaction are and Thus the model is The are called the __________ for factor A, whereas the are the __________ for factor B. The are referred to as the __________ parameters.

Question 8

In two-factor ANOVA, when factor A consists of I levels and factor B consists of J levels, there are __________ different combinations (pairs) of levels of the two factors, each called a __________.

Question 9

In the fixed effects model with interaction, assume that there are I levels of factor A, J levels of Factor J, and K observations (replications) for each of the IJ combinations of levels of the two factors. Then SSE (the error sum of squares) has df = __________.

Question 10

In the three-factor fixed effects model, assume that there are 5 levels of factor A, 4 levels of factor B, 3 levels of factor C, and 2 observations for each combination of levels of the three factors. Then the error sum of squares (SSE) has df = __________.

Question 11

A experiment has __________ factors, and each factor has __________ levels.

Question 12

In the fixed effects model with interaction, assume that there are I levels of factor A, J levels of factor B, and K observations (replications) for each of the IJ combinations of levels of the two factors. Then SSAB (the interaction sum of squares) has df =__________.

Question 13

In the three-factor fixed effects model, assume that there are 4 levels for each of the three factors A, B, and C, and 3 observations for each combination of levels of the three factors. Then, the two-factor interaction sum of squares for factors B and C (SSBC) has df = __________.

Question 14

In 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, SSE has __________ degrees of freedom.

Question 15

In the fixed effects model with interaction, assume that there are I levels of factor A, J levels of factor B, and K observations (replications) for each of the IJ combinations of levels of the two factors. Then SST (the total sum of squares) has df = __________.

Question 16

In 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, SST has __________ degrees of freedom.

Question 17

A three-factor experiment, with I levels of factor A, J levels of factor B, and C levels of factor C, in which fewer than IJK observations are made is called an __________.

Question 18

In a two-factor experiment where factor A consists of 5 levels, factor B consists of 4 levels, and there is only one observation on each of the 20 treatments, the critical value for testing the null hypothesis that the different levels of factor B have no effect on true average response at significance level .05 is denoted by __________, and is equal to __________.

Question 19

Assume the existence of I parameters and J parameters such that The model specified by the above equations is called an __________ model because each mean response is the __________ of an effect due to factor A at level and an effect due to factor B at level .

Question 20

In two-factor ANOVA, 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. When additivity does not hold, we say that there is __________ between the different levels of the factors.

Question 21

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.

Question 22

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

Question 23

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.

Question 24

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.

Question 25

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

Question 26

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

Question 27

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

Question 28

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

Question 29

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

Question 30

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.

Question 31

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

Question 32

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.

Question 33

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

Question 34

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

Question 35

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

Question 36

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.

Question 37

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) 11edff9d_a02c_75df_98d9_5f65b54a33be_TB3498_11 $: \beta _ { 1 } = \beta _ { 1 } = \ldots \ldots = \beta _ { 3 } = 0$
C) 11edff9d_a02c_75df_98d9_5f65b54a33be_TB3498_11 $\text { : at least one } \alpha _ { i } \neq 0$
D) $\text { : at least one } \beta _ { \mathrm { j } } \neq 0$

E) None of the above answers are correct.

Question 38

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

Question 39

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

Question 40

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.

Question 41

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends of the type of capping material used or variability in different batches. Each number is a cell total

Question 42

The strength of concrete used in commercial construction tends to vary from one batch to another. Consequently, small test cylinders of concrete sampled from a batch are "cured" for periods up to about 28 days in temperature- and moisture-controlled environments before strength measurements are made. Concrete is then "bought and sold on the basis of strength test cylinders". The accompanying data resulted from an experiment carried out to compare three different curing methods with respect to compressive strength (MPa). Analyze this data.

Question 43

A four-factor ANOVA experiment was carried out to investigate the effects of fabric (A), type of exposure (B), level of exposure (C), and fabric direction (D) on extent of color change in exposed fabric as measured by a spectrocolorimeter. Two observations were made for each of the three fabrics, two types, three levels, and two directions, resulting in MSA = 2207.329, MSB = 47.255, MSC = 491.783, MSD = .044, MSAB = 15.303, MSAC = 275.446, MSAD = .470, MSBC = 2.141, MSBD = .280, MSE = .977, and MST = 93.621 ("Accelerated Weathering of Marine Fabrics," J. Testing and Eval.,. 1992: 139-143). Assuming fixed effects for all factors, carry out an analysis of variance using for all tests and summarize your conclusions.

Question 44

In an experiment to see whether the amount of coverage of light-blue interior paint depends either on the brand of paint or on the brand of roller used, 1 gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of square feet covered).
a. Construct the ANOVA table.
b. State and test hypotheses appropriate for deciding whether paint has any effect on coverage. Use
c. Repeat part (b) for brand of roller.
d. Use Tukey's method to identify significant differences among brands. Is there one brand that seems clearly preferable to the others?

Question 45

Which of the following statements are not true?

A) If the two three-factor interactions BCD and CDE are chosen for confounding, then their generalized interaction is BE.
B) If the two three-factor interactions ABC and CDE are chosen for confounding, then their generalized interaction is ABCDE.
C) When the number p of factors is large, a single replicate of a $2^3$
Experiment can be expensive and time consuming.
D) All of the above statements are true.
E) None of the above statements are true.

Question 46

The accompanying data resulted from a experiment with three replications per combination of treatments designed to study the effects of concentration of detergent (A), concentration of sodium carbonate (B), and concentration of sodium carboxymethyl cellulose (C0 on cleaning ability of a solution in washing tests (a larger number indicates better cleaning ability than a smaller number).
a. After obtaining cell totals
compute estimates of
b. Use the cell totals along with Yate's method to compute the effect contrasts and sums of squares. Then construct an ANOVA table and test all appropriate hypotheses using

Question 47

Which of the following statements are not true?

A) An experiment in which there are p factors, each at two levels, is referred to as a $p ^ { 2 }$
Factorial experiment.
B) A
Factorial experiment provides a simple setting for introducing the important concepts of confounding and fractional replications.
C) A
Experiment, with four factors A, B, C, and D, has 16 different experimental conditions.
D) All of the above statements are true.
E) None of the above statements are true.

Question 48

The current (in ) necessary to produce a certain level of brightness of a television tube was measured for two different types of glass and three different types of phosphor, resulting in the accompanying data: Assuming that both factors are fixed, test at level .01. Then if cannot be rejected, test the two sets of main effect hypotheses.

Question 49

Which of the following statements are true?

A) Blocking is always effective in reducing variation associated with extraneous sources.
B) It is often not possible to carry out all
Experimental conditions of a
Factorial experiment in a homogeneous experimental environment.
C) When the
Experimental conditions are placed in $2 ^r$
Homogeneous blocks (r<p), the price paid for this blocking is that $2 ^ { r } - 1$
Of the factor effects cannot be estimated.
D) All of the above statements are true.
E) None of the above statements are true.

Question 50

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

A) One-factor ANOVA
B) Two-factor ANOVA with interaction
C) Three-factor ANOVA with interactions
D) Latin square design
E) Randomized block design

Question 51

Which of the following statements are true?

A) For experimental situations with more than three factors, there are often no replications, so sums of squares associated with nonconfounded higher-order interactions are usually pooled to obtain an error sum of squares that can be used in the denominators of the various F statistics.
B) One replicate of a $2 ^ { 6 }$
Factorial experiment involves an observation for each of the 64 different experimental conditions.
C) If an experimenter decides to include only
Of the
Possible conditions in the experiment; this is usually called a half-replicate.
D) The first step in selecting half-replicate is to select a defining effect as the nonestimable effect.
E) All of the above statements are true.

Question 52

A data from an experiment to assess the effects of vibration (A), temperature cycling (B), altitude cycling (C), and temperature for altitude cycling and firing (D) on thrust duration are shown below. Use the Yates method to obtain sums of squares and the ANOVA table. Then assume that three- and four-factor interactions are absent, pool the corresponding sums of squares to obtain an estimate of and test all appropriated hypotheses at level .05.

Question 53

The accompanying data table gives observations on total acidity of coal samples of three different types, with determinations made using three different concentrations of ethanolic NaOH.
Type of Coal Additionally,
a. Assuming both effects to be fixed, construct an ANOVA table, test for the presence of
ANOVA table, test for the presence of interaction, and then test for the presence of main effects for each factor (all using level .01).
b. Use Tukey's procedure to identify significant differences among the types of coal.

Question 54

The number of miles useful tread wear (in 1000's) was determined for tires of five different makes of subcompact car (factor A, with I = 5) in combination with each of four different brands of radial tires (factor B, with J = 4), resulting in IJ = 20 observations. The values SSA = 30, SSB = 45, and SSE = 60 were then computed. Assume that an additive model is appropriate.
a. Test
(no differences in true average tire lifetime due to makes of cars) versus
using a level .05 test.
b.
(no differences in "true" average tire lifetime due to brands of tires) versus
using a level .05 test.

Question 55

Because of potential variability in aging due to different castings and segments on the castings, a Latin square design with N = 7 was used to investigate the effect of heat treatment on aging. With A = castings, B = segments, C = heat treatments, summary statistics include and Obtain the ANOVA table and test at level .05 the hypothesis that heat treatment has no effect on aging.

Question 56

A particular county in Indiana employs three assessors who are responsible for determining the value of residential property in the county. To see whether these assessors differ systemically in their assessments, five houses are selected, and each assessor is asked to determine the market value of each house. With factor A denoting assessors (I = 3) and factor B denoting houses (J=5), suppose SSA = 12, SSB = 110, and SSE = 26.
a. Test
states that there are no systemic differences among assessors).
b. Explain why a randomized block experiment with only 5 houses was used rather than a one-way ANOVA experiment involving a total of 15 different houses with each assessor asked to assess 5 different houses (a different group of 5 for each assessor).

Question 57

The output of a continuous extruding machine that coats steel pipe with plastic was studied as a function of the thermostat temperature profile (A, at three levels), type of plastic (B, at three levels), and the speed of the rotating screw that forces the plastic through a tube-forming die (C, at three levels). There were two replications (L = 2) at each combination of levels of the factors, yielding a total of 54 observations on output. The sums of squares were SSA = 14,144.44, SSB = 5511.27, SSC = 244,696.39, SSAB = 1069.62, SSAC = 62.67, SSBC = 331.67, SSE = 3127.50, and SST = 270,024.33.
a. Construct the ANOVA table.
b. Use appropriate F tests to show that none of the F ratios for two- or three-factor interactions is
at level .05.
c. Which main effects appear significant?
d. With
use Tukey's procedure to identify significant
differences among the levels of factor C.

Question 58

The following summary quantities were computed from an experiment involving four levels of nitrogen (A), two times of planting (B), and two levels of potassium (C). Only one observation (N content, in percentage, of corn grain) was made for each of the 16 combinations of levels.
SSA = .22625 SSB = .000025 SSC = .0036 SSAB = .004325
SSAC = .00065 SSBC = .000625 SST = .2384.
a. Construct the ANOVA table.
b. Assume that there are no three-way interaction effects, so that MSABC is a valid estimate of
and test at level .05 for interaction and main effects.
c. The nitrogen averages are
Use Tukey's method to examine differences in percentage N among the nitrogen levels

Question 59

Which of the following statements are not true?

A) Tukey's multiple comparison procedure can be used in two-factor ANOVA but not in three-factor (or more) ANOVA.
B) When several factors are to be studied simultaneously, an experiment in which there is at least one observation for every possible combination of levels is referred to as complete layout.
C) A three-factor experiment, with I levels of factor A, J levels of factor B, and K levels of factor C, in which fewer than IJK observations are made is called an incomplete layout.
D) There are some incomplete layouts in which the pattern of combinations of factors is such that the analysis is straightforward. One such three-factor design is called a Latin square.
E) All of the above statements are true.

Question 60

In an experiment to assess the effects of curing time (factor A) and type of mix (factor B) on the compressive strength of hardened cement cubes, three different curing times were used in combination with four different mixes, with three observations obtained for each of the 12 curing time-mix combinations. The resulting sums of squares were computed to be SSA = 30,763.0, SSB = 34,185.6, SSE = 97,436.8, and SST = 205,966.6.
a. Construct an ANOVA table.
b. Test at level .05 the null hypothesis
(no interaction of factors) against
c. Test at level .05 the null hypothesis
(factor A main effects are absent) against
d. Test
at least one
using a level .05 test.
e. The values of the
Use Tukey's procedure to investigate significant differences among the three curing times.

Question 61

Answer the following questions:
a. In a
experiment, suppose two blocks are to be used, and it is decided to confound the ABCD interaction with the block effect. Which treatments should be carried out in the first block [containing the treatment (1)], and which treatments are allocated to the second block?
b. In an experiment to investigate niacin retention in vegetables as a function of cooking temperature (A), sieve size (B), type of processing (C), and cooking time (D), each factor was held at two levels. Two blocks were used, with the allocation of blocks as given in part (a) to confound only the ABCD interaction with blocks. Use Yate's procedure to obtain the ANOVA table for the accompanying data.
c. Assume that all three-way interaction effects are absent, so that the associated sums of squares
Can be combined to yield an estimate of
and carry out all appropriate test at level .05.

Question 62

In an experiment involving four factors (A,B,C, and D) and four blocks show that at least one main effect or two-factor interaction effect must be confounded with the block effect.