Statistics - Chapter 12 Introduction to Analysis of Variance

formula for within-treatments degrees of freedom

(n-1)=df in each treatment

Three different treatments (A, B, and C) are being studied using ANOVA. The means are computed: MA = 9, MB = 4, MC = 6. If MSwithin = 6.21, n = 7, and q = 3.67, which of the following is true using Tukey's HSD test? I. Treatment A is significantly different from Treatment B. II. Treatment A is significantly different from Treatment C. III. Treatment C is significantly different from Treatment B. a. I only b. II only c. I and II only d. I, II, and III

a. I only

Which of the following most accurately describes the F-ratio in ANOVA testing? a. the ratio of variances b. the ratio of sample mean difference to standard error c. the ratio of sample mean difference to sample variance d. the ratio of variance to sample mean difference

a. the ratio of variances In ANOVA testing, the F-ratio is the ratio of the variance between sample means and the variance expected with no treatment effect. See 12.1: Introduction (An Overview of Analysis of Variance).

post hoc test

additional hypothesis test done after an ANOVA to determine whether mean differences are significant

distribution of F-ratios

all possible F values that can be obtained when the null hypothesis is true

mean square

average of the squared deviations

An analysis of variance produced an F-ratio with df values 1, 14. If the same data had been evaluated with an independent-measures t test, what would df be? a. 2 b. 14 c. 15 d. 16

b. 14

For a posttest following ANOVA, there are four different treatment groups. How many pairwise comparisons must be made to gain a complete understanding of which treatment effects differ significantly from others? a. 4 b. 6 c. 12 d. 24

b. 6

An analysis of variance produces SS between = 64 and MS between = 8. In this analysis, how many treatment conditions are being compared? a. 8 b. 9 c. 16 d. 32

b. 9 Since MS between = SS between/df between, it follows that 8 = 64/df between, so df between = 8. Further, since df between = k - 1, 8 = k - 1, so k = 9. There are 9 treatment conditions being compared. See 12.3: ANOVA Notation and Formulas.

What is the main advantage of ANOVA testing compared with t testing? a. It can be used with populations that have very high variances. b. It can be used to compare two or more treatments. c. It requires a smaller number of subjects. d. There is no advantage. They are simply different tests for different situations.

b. It can be used to compare two or more treatments. The main advantage of ANOVA testing is that it can be used to compare two or more treatments. See 12.1: Introduction (An Overview of Analysis of Variance)

If the variance between treatments increases and the variance within treatments decreases, what will happen to the F-ratios and the likelihood of rejecting the null hypothesis in an ANOVA test? a. The F-ratio will increase, but the likelihood of rejecting the null hypothesis will decrease. b. The F-ratio and the likelihood of rejecting the null hypothesis will increase. c. The F-ratio will decrease, but the likelihood of rejecting the null hypothesis will increase. d. The F-ratio and the likelihood of rejecting the null hypothesis will decrease.

b. The F-ratio and the likelihood of rejecting the null hypothesis will increase.

For an independent-measures experiment, the F-ratio is 3.1. If dfbetween = 5, and dfwithin = 14, what will the researcher conclude? a. The null hypothesis will be rejected with α = .01, but not with α = .05. b. The null hypothesis will be rejected with α = .05, but not with α = .01. c. The null hypothesis will be rejected with both α = .01 and α = .05. d. The null hypothesis will not be rejected with α = .01 or α = .05.

b. The null hypothesis will be rejected with α = .05, but not with α = .01.

A research report concludes that there are significant differences among treatments, with "F(3, 28) = 5.62, p < .01, η2 = 0.28." If the same number of participants was used in all of the treatment conditions, then how many individuals were in each treatment? a. six b. eight c. nine d. cannot determine without additional information

b. eight

A researcher is conducting an ANOVA test to measure the influence of the time of day on reaction time. Participants are given a reaction test at three different periods throughout the day: 7 a.m., noon, and 5 p.m. In this design, there are _______ factor(s) and ______ level(s). a. two; three b. one; three c. two; six d. three; one

b. one; three Correct. In this study, there is only one factor: the time of day. The three times represent the three levels of this factor. See 12.1: Introduction (An Overview of Analysis of Variance).

In an analysis of variance, the primary effect of large mean differences within each sample is to increase the value for the ______. a. variance between treatments b. variance within treatments c. total variance d. Large mean differences will not directly affect any of the three variances.

b. variance within treatments The variance within treatments is a measure of the mean differences within each sample, so the primary effect of large mean differences within each sample will increase the value of the variance within treatments. See 12.2: The Logic of Analysis of Variance.

Under what conditions might a post hoc test be performed following ANOVA? a. when there are two treatments and the null hypothesis was rejected b. when there are three treatments and the null hypothesis was rejected c. when the null hypothesis is not rejected d. all of the above

b. when there are three treatments and the null hypothesis was rejected

An analysis of variance is used to evaluate the mean differences for a research study comparing five treatment conditions with a separate sample of n = 6 in each treatment. If SSbetween treatments = 24 and SStotal = 74, find the F-ratio. a. F = 1 b. F = 2 c. F = 3 d. F = 4

c. F = 3 If SSbetween treatments = 24 and SStotal = 74, then SSwithin treatments = 74 - 24 = 50. In this example, there are 30 scores. With five treatment conditions, dfwithin = N - k = 30 - 5 = 25. dfbetween = k - 1 = 5 - 1 = 4. Therefore, MSbetween = SSbetween/dfbetween = 24/4 = 6, and MSwithin = SSwithin/dfwithin = 50/25 = 2. Finally, F = MSbetween/MSwithin = 6/2 = 3. See 12.3: ANOVA Notation and Formulas.

Which of the following will increase the likelihood of rejecting the null hypothesis using ANOVA? a. a decrease in SSwithin b. an increase in the sample sizes c. both a and b d. none of the above

c. both a and b

treatment effect

cause of differences between treatments

F-ratio

comparison between how much difference exists versus how big the differences are between treatment conditions

pairwise comparison

comparison of individual treatments two at a time

Tukey's HSD test

computation of single value that determines the minimum difference between treatment means necessary for significance

In an ANOVA study on the impact that various forms of cellphone use have on driving speed, a researcher concludes that there are no systematic treatment effects. What was the F-ratio closest to? a. 0 b. ¼ c. ½ d. 1

d. 1 An F-ratio near 1 indicates that the differences between treatments (numerator) are random and unsystematic, just like the differences in the denominator. With an F-ratio near 1, we conclude that there is no evidence to suggest that the treatment has any effect. See 12.2: The Logic of Analysis of Variance.

An analysis of variance is used to evaluate the mean differences for a research study comparing four treatment conditions and seven scores in each sample. How many total degrees of freedom are there? a. 3 b. 7 c. 24 d. 27

d. 27 If there are four treatment conditions with seven scores in each sample, then there are 4 × 7 = 28 scores. The total degrees of freedom is 28 - 1 = 27. See 12.3: ANOVA Notation and Formulas.

If SSbetween = 125 and SSwithin = 65, what is the effect size, η2, for the corresponding ANOVA? a. 34% b. 48% c. 52% d. 66%

d. 66%

An analysis of variance is used to evaluate the mean differences for a research study comparing three treatment conditions and the same number of scores in each sample. If SSbetween treatments = 24 and SSwithin = 72, and F = 4, how many scores are in each sample? a. 30 b. 27 c. 10 d. 9

d. 9 dfbetween = k - 1 = 3 - 1 = 2. Therefore, MSbetween = SSbetween/dfbetween = 24/2 = 12. Since F = 4, it follows that 12/ MSwithin = 4. Therefore, MSwithin = 3. Since MSwithin = SSwithin/dfwithin, it follows that 3 = 72/dfwithin. Therefore, dfwithin = 24. Finally, since dfwithin = N - k, it follows that 24 = N - 3. So N = 27. If all three treatment groups have the same number of participants, there must be 9 participants in each group. See 12.3: ANOVA Notation and Formulas.

ANOVA is to be used in a research study using two therapy groups. For each group, scores will be taken before the therapy, right after the therapy, and one year after the therapy. How many different sample means will there be? a. two b. three c. five d. six

d. six With two therapies and three measurement times, there will be 2 × 3 = 6 different sample means. See 12.1: Introduction (An Overview of Analysis of Variance).

ANOVA summary table

diagram showing the source of variability

total

entire set of scores

analysis of variance (ANOVA)

hypothesis-testing procedure used to evaluate mean differences between two or more treatments or populations

level

individual condition or value that makes up a variable

formula for between-treatments degrees of freedom

k-1

between-treatments variance

measure of how much difference exists between treatment conditions

within-treatments variance

measure of how much difference exists inside each treatment condition

error term

measure of the variance caused by random, unsystematic differences

Scheffé test

method using an F-ratio to evaluate the significance of the difference between two treatment conditions

formula for total degrees of freedom

n-1

eta squared

percentage of variance accounted for by the treatment effect in published reports of ANOVA results

testwise alpha level

risk of a Type I error for an individual hypothesis test

formula for within-treatments sum of squares

ss inside each treatment

formula for between-treatments sum of squares

ss total - ss within

two-factor design or a factorial design

study that combines two variables

single-factor design

study that has only one independent variable

between-treatments

term referring to differences from one condition to another

within-treatments

term referring to differences that exist inside the individual conditions

experimentwise alpha level

total probability of a Type I error accumulated from all individual tests in the experiment

factor

variable that designates the groups being compared

formula for total sum of squares

x2-G2/N