# Solve one-way ANOVA problem using excel with a real example

Part I Jackson, even-numbered Chapter Exercises, pp. 335-337.

2. How many independent variables are in a 4 x 6 factorial design? How many conditions (cells) are in this design?

There are two independent variables in a 4 x 6 design. One variable has four levels, and the other has six levels. There would be 24 conditions in a 4 x 6 design.

4. What is the difference between a cell (condition) mean and the means used to interpret a main effect?

A cell or condition mean represents the performance in one single condition of an experiment. The row and column mean (used to interpret main effects) represent the performance collapsed across all levels of each independent variable. Thus, there would be two or more cell means contributing to each row or column mean.

6. What is the difference between a complete factorial design and an incomplete factorial design?

In a full factorial design, all levels of each independent variable are paired with all levels of every other independent variable. In an incomplete factorial design, there is more than one independent variable; however, there is not a perfect pairing of all levels of each independent variable with all levels of every other independent variable.

8. Explain the difference between a two- way ANOVA and a three- way ANOVA.

A "two-way" ANOVA is used to analyze data from experiments in which there are two independent variables, whereas as "three-way" ANOVA is used to analyze data from experiments in which there are three independent variables (McDonald, 2014).

10. Complete each of the following ANOVA summary tables. Also, answer the following questions for each of the ANOVA summary tables:

(1)Source df SS MS F

A 1 60

B 2 40

AxB 2 90

Error 30

Total 35 390

(2)Source df SS MS F

A 2 40

B 3 60

AxB 6 150

Error 72

Total 8 400

(3)Source df SS MS F

A 1 10

B 1 60

AxB 1 20

Error 36

Total 39 150

(1)Source df Ss MS F

A 1 60 608.996

B 2 40 202.999

AxB 2 90 456.75

Error 30 200 6.67

Total 35 390

a. What is the factorial notation? 2 x 3

b. How many conditions were in the study? 6

c. How many subjects were in the study? 36

d. Identify significant main effects and interaction effects.

A: Yes, p < .01

B: No

A X B: Yes, p < .05

(2)Source df SS MS F

A 2 40 209.62

B 3 60 209.62

AxB 6 150 2512.02

Error 72 150 2.08

Total 83 400

a. What is the factorial notation? 3 x 4

b. How many conditions were in the study? 12

c. How many subjects were in the study? 84

d. Identify significant main effects and interaction effects.

A: Yes, p < .01

B: Yes, p < .01

A X B: Yes, p < .01

(3)Source df SS MS F

A 1 10 105.99

B 1 60 6035.93

AxB 1 20 2011.98

Error 36 60 1.67

Total 39 150

a. 2 x 2

b. 4

c. 40

d. A - Yes, p < .05

B - Yes, p < .01

AxB - Yes, p < .01

12. A researcher is attempting to determine the effects of practice and gender on a timed task. Participants in an experiment are given a computerized search task. They search a computer screen of various characters and attempt to find a particular character on each trial. When they find the designated character, they press a button to stop a timer. Their reaction time ( in seconds) on each trial is recorded. Subjects practice for 2, 4, or 6 hours and are either female or male. The reaction time data for the 30 subjects appear here.

Women Men

2 Hours 12 11

13 12

12 13

11 12

11 11

4 Hours 10 8

10 8

10 10

8 10

7 9

6 Hours 7 5

5 6

7 8

6 6

7 8

Source df SS MS F

Gender 0.027

Practice 140.60

Gender Practice 0.073

Error 28.00

Total 168.70

a. Complete the ANOVA summary table. ( If your instructor wants you to calculate the sums of squares, use the preceding data to do so.)

Source df SS MS F

Gender 1 0.027 0.027 0.016

Practice 2 140.60 70.30 42.10

Gender x Practice 2 0.073 0.036 0.022

Error 24 28

Total 29 168.70

b. Are the values of Fobt significant At α = .05? At α = .01?

The only significant F-ratio is that Practice, F (2, 24) = 42.10, p < .01.

c. What conclusions can be drawn from the F- ratios?

There is no significant effect of gender on reaction time. There is a significant effect of practice on reaction time such that as practice increased, reaction time decreased. There is no significant interaction effect.

d. What is the effect size, and what does this mean?

The effect size (2) is .01% for Gender (gender accounts for less than .01 % of the variability in reaction time scores), 83% for Practice (amount of time spent practicing accounts for 83% of the variability in reaction time scores), and .04% for the interaction (the interaction of gender and practice accounts for .04% of the variability in reaction time scores).

e. Graph the means.

Part II

1. Explain the difference between multiple independent variables and multiple levels of independent variables. Which is better?

In the design of the experiment, each researcher must decide the dependent variable and independent variable. Thre is usually only one dependent variable. However, for the independent variable, we can have many. Each independent variable can be treated as a parameter. Multiple independent variables mean multiple parameters. For example, if the weight of the person is the dependent variable, then the age, sex and feet size can be independent variables. In this case, we have three (3) independent variables. However, we can assign multiple levels for one independent variable. For the same example, sex can be assigned to male and female and age can be allocated as junior and senior. In most cases, multiple levels of independent variables are better. There are many reasons to claim this: 1. Usually the more there are independent variables, the more money we spend. 2. Usually the more there are independent variables, the less reliability the model is because we need to test each independent variable first instead of testing level for one independent variable.

2. What is blocking and how does it reduce "noise"? What is a disadvantage of blocking?

In the design of the experiment, we sometimes divide the population into several groups in which all units are homogeneous in each cluster. Such group is called the blocking. A requirement for blocking is that each unit must be randomly allocated to each cluster. In this manner,it make the parameters more reliable and reduce the noise error caused by between groups. Although the units are randomly assigned to each cluster, it is still hard to ensure that there are homogenous units in each blocking is the main advantage of blocking.

3. What is a factor? How can the use of factors benefit a design?

A factor is an Independent variable studied. In the design of the experiment, each researcher must decide the dependent variable and independent variable. In other words, the independent variable is controlled by the research. Such independent variable is called the factor. The levels of the factor are also decided by the research. The purpose of a design is to build a model with one dependent variable and relevant factors and decide their relationships. Hence, the use of factors will benefit a design if best factors (including their levels) are included in the model.

4. Explain main effects and interaction effects.

In the design of the experiment, when there are more than two factors in the design, we must check both main effects and interaction effects. The main effect refers to the impact of a single factor. In other words, it is the individual contribution to the model from a single factor. However, the interaction effect refers to the effect of two factors. When interaction effect exists, the contribution of factors to the model will significantly reduce.

5. How does a covariate reduce noise?

A covariate is a pre-measure of the test in the design of the experiment. It can be a pre-test although it does not have to be but pre-test should be the best choice. We can also have multiple covariates in one design. First, we use to plot the scores to show the relationship between pre-test and post-test and fit the trend line for both tests. Then, we want to reduce the noise by adjusting the post-test scores for pre-test variability it is a job called analysis of covariance

6. Describe and explain three trade-offs present in experiments.

The three trade-offs present in experiments : 1. Bias: When we randomly select samples from a population, the bias will exist especially for large samples. As we know, to make the design valid, we try to make the sample size as large as possible so that the sample can be representative of the population. However, when the sample size turns larger, the randomness will be reduced, and bias will increase. Hence, we may pay attention to the procedure of randomness. 2. Interaction effort: As discussed, interaction effect refers to the impact between any two factors. If we remove any one factor from the model, we may lose the key factor. However, if we leave the key element in the model, there are interaction effects. Usually, we still keep all factors in the model but make some transformations to the relevant factors. 3. Outliers: Sometimes when we build the initial model, we find that some points are far from the trend line. These points are called outliers. If we just delete the outliers, the resulting model may be not accurate (Jackson, 2012).

References

Jackson, S. L. (2012). Research methods and statistics: A critical thinking approach (4th ed.). Belmont, CA: Wadsworth Cengage Learning.

McDonald, J.H. 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. This web page contains the content of pages 173-179. http://www.biostathandbook.com/twowayanova.html

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#### Solution Preview

Part I Jackson, even-numbered Chapter Exercises, pp. 335-337.

2. How many independent variables are in a 4 x 6 factorial design? How many conditions (cells) are in this design?

There are two independent variables in a 4 x 6 design. One variable has four levels, and the other has six levels. There would be 24 conditions in a 4 x 6 design.

4. What is the difference between a cell (condition) mean and the means used to interpret a main effect?

A cell or condition mean represents the performance in one single condition of an experiment. The row and column mean (used to interpret main effects) represent the performance collapsed across all levels of each independent variable. Thus, there would be two or more cell means contributing to each row or column mean.

6. What is the difference between a complete factorial design and an incomplete factorial design?

In a full factorial design, all levels of each independent variable are paired with all levels of every other independent variable. In an incomplete factorial design, there is more than one independent variable; however, there is not a perfect pairing of all levels of each independent variable with all levels of every other independent variable.

8. Explain the difference between a two- way ANOVA and a three- way ANOVA.

A "two-way" ANOVA is used to analyze data from experiments in which there are two independent variables, whereas as "three-way" ANOVA is used to analyze data from experiments in which there are three independent variables (McDonald, 2014).

10. Complete each of the following ANOVA summary tables. Also, answer the following questions for each of the ANOVA summary tables:

(1)Source df SS MS F

A 1 60

B 2 40

AxB 2 90

Error 30

Total 35 390

(2)Source df SS MS F

A 2 40

B 3 60

AxB 6 150

Error 72

Total 8 400

(3)Source df SS MS F

A 1 10

B 1 60

AxB 1 20

Error 36

Total 39 150

(1)Source df Ss MS F

A 1 60 608.996

B 2 40 202.999

AxB 2 90 456.75

Error 30 200 6.67

Total 35 390

a. What is the factorial notation? 2 x 3

b. How many conditions were in the study? 6

c. How many subjects were in the study? 36

d. Identify significant main effects and interaction effects.

A: Yes, p < .01

B: No

A X B: Yes, p < .05

(2)Source df SS MS F

A 2 40 209.62

B 3 60 209.62

AxB 6 150 2512.02

Error 72 150 2.08

Total 83 400

a. What is the factorial notation? 3 x 4

b. How many conditions were in the study? 12

c. How many subjects were in the study? 84

d. Identify significant main effects and interaction effects.

A: Yes, p < .01

B: Yes, p < .01

A X B: Yes, p < ...

#### Solution Summary

The solution gives detailed steps on solving one-way ANOVA problem using excel with a real example. Detailed analysis is followed afterwards.