Deck 14: One-Factor Fixed-Effects Ancova With Single Covariate

Question 1

A researcher wanted to examine whether people remembered information better with auditory cues or with visual cues. Ten participants were randomly assigned to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from a screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words remembered.
-For which of the following situations is it appropriate to use ANCOVA?

A) In each group, half of the participants had a time limit to write the words down. Whether a time limit was imposed was also included in the model.
B) The participants' socioeconomic status (low, middle, or high) was expected to have an effect on their performance and was also included in the model.
C) Participants took a vocabulary test before the experiment and the scores of the test were included in the model.
D) Participants took a vocabulary test after the experiment and the scores of the test were included in the model.

Answer 1

C

Question 2

A researcher wanted to examine whether people remembered information better with auditory cues or with visual cues. Ten participants were randomly assigned to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from a screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words remembered.
-Suppose the researcher used IQ as the covariate. He then found that the mean IQ is slightly different across two groups, and people with higher IQ tended to remember more words than people with lower IQ. It seems likely that

A) the assumption of independence was violated.
B) the assumption of homogeneity of variance was violated.
C) the assumption of independence of covariate and treatment was violated.
D) the assumption of homogeneity of slopes was violated.
E) there is no indication of assumption violation.

Answer 2

E

Question 3

A researcher wanted to examine whether people remembered information better with auditory cues or with visual cues. Ten participants were randomly assigned to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from a screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words remembered.
-Suppose out of budget concerns, the researcher included only five participants in each group, tested each participant twice, and used each time as separate observations. It seems likely that

A) the assumption of independence was violated.
B) the assumption of homogeneity of variance was violated.
C) the assumption of independence of covariate and treatment was violated.
D) the assumption of homogeneity of slopes was violated.
E) there is no indication of assumption violation.

Answer 3

A

Question 4

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.

-If ANCOVA is used, which group will have the highest adjusted mean height?

A) Group 1 ( )
B) Group 2 ( )
C) Group 3 ( )
D) All groups would have the same adjusted mean.
E) Any of the above situations is possible (not enough information provided).

Question 5

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.

-If there is a substantial, positive correlation between the thickness of soil (X) and the height of daylilies (Y), which group will have the highest adjusted mean height?

A) Group 1 ( )
B) Group 2 ( )
C) Group 3 ( )
D) All groups would have the same adjusted mean.
E) Any of the above situation is possible.

Question 6

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.

-If there is no correlation between X and Y, the MS_with for ANCOVA as compared to that for ANOVA will be

A) less
B) the same
C) greater
D) unpredictably different

Question 7

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.

-If there is a substantial negative correlation between X and Y, the error variation for ANCOVA as compared to that for ANOVA will be

A) less
B) the same
C) greater
D) unpredictably different

Question 8

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.

-If the correlation between X and Y is 0.5 for soil type 1, 0.1 for soil type 2, and -0.5 for soil type 3, it seems likely that

A) the assumption of normality is violated.
B) the assumption of homogeneity of variance is violated.
C) the assumption of linearity is violated.
D) the assumption of homogeneity of slopes is violated.
E) there is no indication of assumption violation.

Question 9

Matt has generated an ANCOVA. In testing the assumptions, he reviews the scatterplot of the covariate (X) and the dependent variable (Y) for each group and overall. For which assumption(s) is Matt likely reviewing evidence?

A) Homogeneity of regression slopes.
B) Homogeneity of variance.
C) Linearity.
D) All of the above.

Question 10

In ANCOVA, suppose Y is the dependent variable, X is the covariate, and the factor has two levels. Also all assumptions of ANCOVA are met. Which of the following situations is the most desirable?

A) r_XY = 0.5; =10, =12, =11.
B) r_XY = 0.1; =10, =12, =11.
C) r_XY = -0.1; =10, =20, =15.
D) r_XY = -0.5; =10, =20, =15.

Question 11

In an one-factor ANCOVA, suppose the factor has two levels (groups). Scatterplots of the dependent variable (Y) and the covariate (X) are generated (where group 1 is indicated by "•" and group 2 by "o"). Which of the following graphs shows the most desirable situation?
a. b. c. d.

Question 12

Which of the following conditions about the dependent variable (Y) and the covariate (X) is required for ANCOVA?

A) Both X and Y are measured on interval or ratio scale.
B) Both X and Y are measured in the same unit.
C) Both X and Y are independent of the independent variable.
D) Both X and Y are measured without error.

Question 13

An experiment was conducted to compare four different weight loss programs. Twenty subjects participated in each program, and their weights were recorded before and after the program. If both the pretest weight and age are used as covariates, what are the degrees of freedom for the error term?

A) 74
B) 75
C) 76
D) 79

Question 14

A study was conducted to compare five new textbooks. Thirty teachers were asked to rate the textbooks they used in the previous semester, and then they were randomly assigned to one of five new textbook groups (n = 6). At the end of the semester, teachers' ratings of the new textbooks (Y) were collected. What may be an appropriate covariate for this study?

A) Students' achievement at the end of the semester.
B) Students' rating of the new textbook at the end of the semester.
C) Teachers' rating of the old textbook.
D) Teachers' ethnicity.

Question 15

In the study described in Question 14, suppose an appropriate covariate has been chosen. What are the degrees of freedom for the error term?

A) 4
B) 24
C) 25
D) 29

Question 16

If the dependent variable, math score (Y), is about the same for persons with higher scores in writing (X) and for those with lower scores in writing, one would expect that

A) adding the covariate will not substantially reduce error variation.
B) the adjusted mean for each group will be the same.
C) ANCOVA will be more powerful than ANOVA.
D) the covariate and factor will be independent of each other.

Question 17

In which of the following situations will the adjusted means always equal the unadjusted means in ANCOVA?

A) All groups have the same unadjusted mean.
B) All groups have the same mean on the covariate.
C) The dependent variable and covariate are highly correlated.
D) Regression of the dependent variable on the covariate with b_w = 0.6.

Question 18

In reviewing the assumptions, Susan runs an ANOVA using the covariate X as the dependent variable and the factor A as the independent variable. For this test, which of the following situations is the most desirable?

A) p = 0.01
B) p = 0.05
C) p = 0.1
D) p = 0.5

Question 19

Now using Y as the dependent variable, Susan runs another ANOVA to evaluate if there is any interaction between the covariate X and the factor ِ. For , which of the following situations is the most desirable?

A) p = 0.01
B) p = 0.05
C) p = 0.1
D) p = 0.5

Question 20

In which of the following situations is the assumption of linearity violated?

A) Trend analysis shows that there exists a significant quadratic trend.
B) There is no association between the independent variable and the covariate.
C) The distribution of the covariate is seriously skewed in all groups.
D) The scatterplot of the dependent variable and the covariate is U shaped for all groups.

Question 21

If both ANOVA and ANCOVA are conducted on the same data set, which part of the summary table will always have the same value?

A) SS_betw
B) SS_with
C) SS_total
D) MS_betw
E) MS_with

Question 22

Using the same data set, which part of the ANCOVA summary table will always have a smaller value as compared to that for ANOVA?

A) df_betw
B) MS_betw
C) df_with
D) MS_with
E) df_total

Question 23

In which of the following situations is ANOVA more powerful than ANCOVA?

A) The adjusted means are the same as unadjusted means.
B) The unadjusted means all have the same value.
C) The correlation between the covariate and the dependent variable is strong.
D) The correlation between the covariate and the dependent variable is weak.

Question 24

In a randomized experiment (true experiment), if the covariate contains considerable measurement error, the use of ANCOVA would likely result in

A) too much adjustment of means.
B) less of the covariate effect being removed from the dependent variable.
C) an F test that is too liberal.
D) biased estimates of the treatment effects.

Question 25

If the regression slopes of the dependent variable (Y) on the covariate (X) are substantially different across the groups, one would expect that

A) the adjusted means will be biased.
B) Y is independent of X.
C) the factor is not independent of X.
D) there is a modest effect with equal n's in quasi-experiment.

Question 26

A researcher wanted to examine if soil type had any effects on the heights of daylilies (Y). The thickness of the soil (X) in the pot was used as the covariate X. Given the data that follow, where there are three types of soil (n = 10 in each group), (a) calculate the adjusted mean values assuming that b_w = 0.5, and (b) determine what effects the adjustment had on the posttest results.

Question 27

A market researcher wanted to know whether different package designs for the same yogurt product would affect consumers' purchase intention. There were four different versions of package designs. Each package was viewed by eight participants, who then rated their likelihood to purchase the product. A one-factor ANCOVA was used to analyze the data where the covariate was the participants' liking of the yogurt product in general. Complete the following ANCOVA summary table ( $\alpha$ = .05):
$\begin{array}{ccccccc}\text { Source } & S S & d f & M S & F & \text { Critical Value } & \text { Decision } \\\hline \text { Between adjusted } & 10.5 & - & - & - & - & - \\\text { Within adjusted } & - & - & - & & \\\text { Covariate } & -& - & 5.6 & - & - &-\\\text { Total } & 35&- & & & \\\hline\end{array}$

Question 28

Mike wanted to examine whether people remembered information better with auditory cues or with visual cues. He recruited 14 participants and randomly assigned seven participants to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from the screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words correctly remembered (Y), and the covariate was the participants' score on a memory test (X) administered before the experiment. Using the data below, conduct an ANOVA on Y, an ANCOVA on Y using X as a covariate, and compare the results ( $\alpha$ = .05). Determine the unadjusted and adjusted means.
$\begin{array}{l}\text { Auditory cue (headphone) }\\\begin{array}{|c|c|}\hline X \quad\quad\quad\quad& Y \quad\quad\quad\quad\\\hline 80 & 10 \\\hline 75 & 9 \\\hline 95 & 16 \\\hline 60 & 4 \\\hline 70 & 11 \\\hline 65 & 8 \\\hline 90 & 15 \\\hline\end{array}\end{array}$$\begin{array}{l}\text { Visual cue (screen) }\\\begin{array}{|c|c|}\hline X\quad\quad\quad\quad & Y\quad\quad\quad\quad \\\hline 75 & 13 \\\hline 90 & 20 \\\hline 65 & 8 \\\hline 70 & 9 \\\hline 80 & 15 \\\hline 85 & 13 \\\hline 75 & 12 \\\hline\end{array}\end{array}$

Question 29

Barbara wants to know whether students will learn most effectively with soft music as background sound, as opposed to loud music or no music at all. In the following table are three independent random samples (different background sounds) of paired values on the covariate (X; pretest score) and the dependent variable (Y; posttest score). Conduct an ANOVA on Y, an ANCOVA on Y using X as a covariate, and compare the results ( $\alpha$ = .05). Determine the unadjusted and adjusted means.
$\begin{array}{c}{\text { Soft music }} \quad \quad {\text { Loud music }} \quad \quad {\text { No music }} \\\begin{array}{|c|c|c|c|c|c|}\hline X \quad \quad& Y \quad \quad& X \quad \quad& Y \quad \quad& X \quad \quad& Y \quad \quad\\\hline 75 & 89 & 50 & 57 & 64 & 72 \\\hline 42 & 49 & 43 & 52 & 70 & 79 \\\hline 34 & 52 & 56 & 62 & 32 & 48 \\\hline 61 & 70 & 48 & 65 & 65 & 87 \\\hline 45 & 66 & 59 & 64 & 67 & 80 \\\hline 70 & 83 & 27 & 35 & 42 & 53 \\\hline 75 & 90 & 65 & 71 & 71 & 85 \\\hline 58 & 73 & 54 & 68 & 74 & 82 \\\hline\end{array}\end{array}$

Question 30

Dr. Green conducted an ANCOVA to determine whether hearing-impaired children who were taught sign language early on would develop better language skills compared to those who did not learn sign language. The data set contains two independent random samples (children who learned sign language and those who did not) of paired values on the covariate (X; child's hearing) and the dependent variable (Y; language skills measured when the child was three years old). Dr. Green also examined the data to see if the assumptions of ANCOVA were met. The following table and figures are the selected output from Dr. Green's analysis ( $\alpha$ = .05):
$\text { Tests of Between-Subjects Efects }$
$\text { Dependent Variable: Language }$
$\begin{array}{cccccc}\hline \text { Source } & \text { Type III Sum of Squares } & \text { df } & \text { Mean Square } & \text { F } & \text { Sig. } \\\hline \text { Sign } & 22681.088 & 1 & 22681.088 & 5.963 & .020 \\\text { Hearing } & 3084.607 & 1 & 3084.607 & .811 & .374 \\\text { Sign * Hearing } & 23130.912 & 1 & 23130.912 & 6.082 & .019 \\\text { Error } & 136920.190 & 36 & 3803.339 & & \\\text { Corrected Total } & 162488.375 & 39 & & & \\\hline\end{array}$ (a) What assumption is being evaluated here?
(b) Was this assumption satisfied? If not, what effect might it have on the results of ANCOVA?