PSY 429 Experimental Psychology

Class Exercise #7 

Two-Way Independent-Measures Analysis of Variance

Post-Hoc Tests: Hartley's HSD Test

Answers

 

1. The following data are from an experiment examining the extent to which different personality types are affected by distraction on a monotonous task. The number of errors committed by each individual was tabulated. Individuals were selected to represent two different personality types: introverts and extroverts. Half of the individuals in each group were tested on a monotonous task in a relatively quiet, calm room. The individuals in the other half of each group were tested in a noisy room filled with distractions.

a. Identify the independent variables and the levels of each.

1. personality type: introverts and extroverts (2 levels)

2. environment type: quiet and noisy (2 levels)

b. Identify the dependent variable(s).

number of errors made on task

c. State the null hypotheses.

1. The mean score of the introverts is equal to the mean score of the extroverts.

2. The mean score in a quiet environment is equal to the mean score in a noisy environment.

3. No interaction: environmental type does not interact with type of environment.

Another way of stating it is that scores on the task for introverts (or extroverts) do not depend on the type of environment they were tested in.

2. An experiment evaluated computer monitors and whether the size of the monitor or the color of the monitor made a difference in how easy to use it. The study measured the mean ease-of-use ratings given by subjects. The experimenter evaluated three different sizes (9, 12, or 15 inches) and two different colors (amber or green) in a between-subjects design.

a. Identify the independent variables and the levels of each.

1. size of monitor: 9, 12, and 15 inches (3 levels)

2. color of monitor: amber and green (2 levels)

b. Identify the dependent variable(s).

mean satisfaction rating

c. State the null hypotheses.

1. The mean satisfaction scores for the different sizes of monitors are equal. (Another way of stating it would be: Mean satisfaction scores do not differ depending on the size of the monitor.)

2. The mean satisfaction scores for the different colors, amber and green, are equal. (Another way: Participants felt similarly about the monitor regardless of what color it was.)

3. There was no interaction between size and color. (Another way: Satisfaction with a particular size monitor did not depend on what color it was.)

 

Questions 3-10 refer to the following summary table for an experiment.

Source	df	SS	MS	F
A	1	50.00	50.00	10.00
B	2	40.00	20.00	4.00
A X B	2	40.00	20.00	4.00
Error	48	240.00	5.00
Total	53	370.00

3. How many levels of Factor A were there in the experiment?

 2

4. How many levels of Factor B were there in the experiment?

 3

5. What was the total number of subjects in the experiment?

 54

6. How many groups were there in the experiment?

 6

7. Assuming there were equal numbers of subjects in each group, what was the per-group sample size?

 9

8. What is the total amount of between-group variability represented in the table (that is, what is the value of the sum of squares treatment)?

 SS-treatment = 130.00

(SS-treatment consists of SS-A, SS-B, & SS-AxB)

9. State the null and alternative hypotheses for the main effects of Factor A, Factor B, and the interaction.

 Factor A:

1. The mean for group A1 will not differ significantly from the mean of group A2. (null)

2. The mean for group A1 will be significantly different from the mean of group A2. (alternate)

Factor B:

1. The mean for group B1 will not differ significantly from the mean of group B2. (null)

2. The mean for group B1 will be significantly different from the mean of group B2. (alternate)

Interaction:

1. Mean scores for Factor A will not depend on the mean scores for Factor B, thus no significant interaction will be observed. (Another way: The effect of Factor A on the dependent variable will not differ depending on levels of Factor B.) (null)

2. Mean scores for Factor A will depend on the mean scores for Factor B, thus a significant interaction will be observed. (Another way: The effect of Factor A on the dependent variable will differ depending on levels of Factor B.) (alternate)

10. Test the null hypotheses using a level of significance of .05.

 F_A = 10.00

Thus, F(1,48) = 10.00, p < .05. Therefore, there is a significant main effect for Factor A. (The critical F value = 4.00.)

F_B = 4.00

Thus, F(2,48) = 4.00, p < .05. Therefore, there is a significant main effect for Factor B. (The critical F value = 3.15.)

F_AxB = 4.00

Thus, F(2,48) = 4.00, p < .05. Therefore, there is a significant interaction. (The critical F value = 3.15.)

11. Complete the missing entries in the summary table for a 3 x 4 factorial analysis of variance.

12. State the critical values of F that would be used to reject the null hypothesis for the main effect of Factor A, the main effect of Factor B, and the A x B interaction for a two-way independent-measures analysis of variance at an alpha level of .05 under each of the following conditions:

13. Describe when and why post-hoc tests are used.

When: after completing the ANOVA

Why: if you want to obtain information that the ANOVA cannot give you, e.g. Tukey's HSD test is done if you want to find out if there are significant differences between any two means.

14. Explain why you would not need to do post-hoc tests for an experiment with only k = 2 treatment conditions.

 You would not need to do a Tukey HSD test with k = 2 because there are only two groups and the HSD test would not yield new information from the t-test you would normally perform.

15. In an experiment with one independent variable and k = 4 treatment conditons, how many pairwise comparisons would you have to assess using the Tukey HSD test?

 You would perform 6 pairwise comparisons.

M₁ - M₂, M₁ - M₃, M₁ - M₄, M₂ - M₃, M₂ - M₄, M₃ - M₄

16. Conduct Hartley HSD tests on the following data and describe the results using a .05 level of significance.

Mean of group 1 = 3.0; mean of group 2 = 2.0; mean of group 3 = 1.0

 At an alpha level of .05, the critical value of F is 3.88; therefore, we fail to reject the null hypothesis: F(2,12) = 3.75, p > .05.

Even though the F ratio was not significant, there may be two groups which do differ significantly.

HSD = 1.94 using an alpha of .05.

M₁ - M₂= 1.0; Since 1.0 is less than the HSD of 1.94, the HSD test shows that groups 1 and 2 do not significantly differ from each other.

M₁ - M₃ = 2.0; Since 2.0 is greater than the HSD of 1.94, the HSD test shows that groups 1 and 3 significantly differ.

M₂ - M₃= 1.0; Since 1.0 is less than the HSD of 1.94, the HSD test shows that groups 2 and 3 do not significantly differ from each other.

17. An investigator was interested in the effects of gender and academic major on the number of job offers received by college seniors. The data below are from a sample of 20 males and 20 females in which the number of job offers were counted for each participant.

a. Use appropriate tests to evaluate the data.

b. Put the results into an summary table.

c. Describe the results including any descriptive statistics which would help the reader understand the results.

Source	df	SS	MS	F	Significance
Sex	1	2.025	2.025	2.16	p > .05
Major	3	4.475	1.492	1.59	p > .05
Sex x Major	3	20.875	6.958	7.42	p < .001
Error	32	30.00	0.9375
Total	39	57.375

The number of job offers of college graduates as a function of sex (male & female) and undergraduate major (Business, Computer Science, Liberal Arts, & Behavioral Science) was examined in a 2 x 4 independent measures analysis of variance. The sex of college graduates nor their undergraduate major was found to significantly affect the number of job offers, F(1,32) = 2.16, p > .05 and F(3,32) = 1.59, p > .05, respectively. However, a crossover interaction was obtained, F(3,32) = 7.42, p < .001. Job offers for males majoring in Computer Science or Business was significanlty greater than job offers for females in those same majors. On the other hand, the reverse held true for females majoring in the Behavioral Sciences with significantly more job offers for females than for males. Little or no difference in job offers was detected for males and females majoring in the Liberal Arts.

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