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. $\begin{array}{cccc} \hline \text { soil type } & n & \text { heights of daylilies (Y) } & \text { thickness of soil }(X) \\ \hline 1 & 10 & \bar{Y}_{1}=20 & \bar{X}_{1}=21 \\ 2 & 10 & \bar{Y}_{2}=25 & \bar{X}_{2}=18 \\ 3 & 10 & \bar{Y}_{3}=30 & \bar{X}_{2}=15 \\ \hline \end{array}$ -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 ( $\bar{Y}_{1}^{\prime}$ ) B) Group 2 ( $\bar{Y}_{2}^{\prime}$ ) C) Group 3 ( $\bar{Y}_{3}^{\prime}$ ) D) All groups would have the same adjusted mean. E) Any of the above situation is possible. A) Group 1 ( $\bar{Y}_{1}^{\prime}$ ) B) Group 2 ( $\bar{Y}_{2}^{\prime}$ ) C) Group 3 ( $\bar{Y}_{3}^{\prime}$ ) D) All groups would have the same adjusted mean. E) Any of the above situation is possible.

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

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

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Question 1

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(a) The assumption of homogeneous slopes is being evaluated.
(b) The interaction between signing (factor) and hearing (covariate) is significant (F = 6.082, p = .019). Moreover, the scatterplots of Y vs. X suggest that the regression slopes differ between two groups (i.e., children who learned sign language and those who did not). Therefore, the assumption of homogeneous slopes seems to be violated.
With heterogeneous slopes, the size of the group differences in Y (i.e., children's language skills) will depend on the value of X (i.e., children's hearing). For example, learning to sign may be helpful for children with worse hearing, but not for children with better hearing. In this case a straightforward interpretation of the ANCOVA would be misleading. The use of $\beta$ _w can yield biased adjusted means and can affect the F test. Considering the study is probably not a randomized experiment, the violation of this assumption may have a modest effect on the results of F test.

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 $F_{A \times X}$ , which of the following situations is the most desirable?

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

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Question 3

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

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Question 4

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

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Question 5

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?

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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. soil type n heights of daylilies (Y) thickness of soil (X) 1 10 =20 =21 2 10 =25 =18 3 10 =30 =15 -If there is no correlation between X and Y, the MS_with for ANCOVA as compared to that for ANOVA will be

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Question 7

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

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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. soil type n heights of daylilies (Y) thickness of soil (X) 1 10 =20 =21 2 10 =25 =18 3 10 =30 =15 -If there is a substantial negative correlation between X and Y, the error variation for ANCOVA as compared to that for ANOVA will be

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Question 9

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?

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Question 10

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

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Question 11

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

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Question 12

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

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Question 13

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

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Question 14

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

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Question 15

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.

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Question 16

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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?

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Question 17

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. soil type n heights of daylilies (Y) thickness of soil (X) 1 10 =20 =21 2 10 =25 =18 3 10 =30 =15 -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?

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Question 18

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. Soft music Loud music No music X Y X Y X Y 75 89 50 57 64 72 42 49 43 52 70 79 34 52 56 62 32 48 61 70 48 65 65 87 45 66 59 64 67 80 70 83 27 35 42 53 75 90 65 71 71 85 58 73 54 68 74 82

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Question 19

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?

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Question 20

Showing 1 - 20 of 30

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 $F_{A \times X}$ , which of the following situations is the most desirable?

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

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

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?

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

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

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

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

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

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

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.

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?

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?

Introduction

Data Representation

Univariate Population Parameters and Sample Statistics

Normal Distribution and Standard Scores

Introduction to Probability and Sample Statistics

Inferences About a Single Mean

Inferences About the Difference Between Two Means

Inferences About Proportions

Inferences About Variances

Bivariate Measures of Association

One-Factor Anova: Fixed-Effects Model

Multiple Comparison Procedures

Factorial Anova: Fixed-Effects Model

Random- and Mixed-Effects Analysis of Variance Models

Hierarchical and Randomized Block Analysis of Variance Models

Simple Linear Regression

Multiple Regression

Logistic Regression

Filters

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

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 FA×XF_{A \times X}FA×X​ , which of the following situations is the most desirable?

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

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

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?

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

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

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

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

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

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

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.

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?

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?

Introduction

Data Representation

Univariate Population Parameters and Sample Statistics

Normal Distribution and Standard Scores

Introduction to Probability and Sample Statistics

Inferences About a Single Mean

Inferences About the Difference Between Two Means

Inferences About Proportions

Inferences About Variances

Bivariate Measures of Association

One-Factor Anova: Fixed-Effects Model

Multiple Comparison Procedures

Factorial Anova: Fixed-Effects Model

Random- and Mixed-Effects Analysis of Variance Models

Hierarchical and Randomized Block Analysis of Variance Models

Simple Linear Regression

Multiple Regression

Logistic Regression

Filters

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 $F_{A \times X}$ , which of the following situations is the most desirable?