Chapter 12 ANOVA
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?
1.00
For the independent-measures ANOVA, the F-ratio is
F= MSbetween/MSwithin
An analysis of variance produces SS between = 64 and MS between = 8. In this analysis, how many treatment conditions are being compared?
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.
dfwithin + k =
N
SSbetween can be found by subtraction
SSbetween = SStotal - SSwithin
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?
The F-ratio and the likelihood of rejecting the null hypothesis will increase.
levels of the factor
The individual conditions that make up the factor.
n
The number of scores in each treatment
k
The number of treatment conditions
The assumptions underlying analysis of variance are similar to those for the t test for independent groups:
The observations are independent, and the populations from which the samples are taken are normally distributed and have equal variance. Each of the populations of scores from which the samples are drawn is normally distributed.
G
The sum of all of the scores in the research study
T
The sum of the scores for each treatment condition ΣX
N
The total number of scores in the entire study
What is suggested by a value of 1 for the F-ratio in an ANOVA?
There is no treatment effect and you should fail to reject the null hypothesis.
distribution of F-ratios
all possible F values that can be obtained when the null hypothesis is true.
treatment effect
cause of differences between treatments.
Tukey's HSD test
computation of single value that determines the minimum difference between treatment means necessary for significance.
The MSbetween measures
differences between the treatments by computing the variability of the treatment means or totals. These differences are assumed to be produced by - treatment effects (if they exist). - differences resulting from chance.
total
entire set of scores
k - 1
formula for between-treatments degrees of freedom
When there is no treatment effect (Ho is true), the numerator and the denominator of the F-ratio are
measuring the same variance, and the obtained ratio should be near 1.00.
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).
one; three
eta squared
percentage of variance accounted for by the treatment effect in published reports of ANOVA results
MS =
s squared = SS/df
Each of the two variances in the F-ratio is calculated using the basic formula for sample variance.
s^2= SS/n-1
two-factor design or a factorial design
study that combines two variables
mean square
the average of the squared deviations
In ANOVA, the F test statistic is
the ratio of the between-treatments variance and the within-treatments variance.
In an analysis of variance, the primary effect of large mean differences within each sample is to increase the value for the ______.
variance within treatments
SSBetween
Σ T2/n - G2/N
The effect size for the independent-measures ANOVA is measured by computing eta squared η², the percentage of variance accounted for by the treatment effect.
η² = SSbetween / SSbetween + SSwithin or η² = SSbetween / SStotal
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. SSbetween treatments = 24 and SStotal = 74, find the F-ratio.
3
The test statistic for ANOVA is a ratio of two variances called
F-ratio The variances in the F-ratio are called - mean squares or MS values - Each MS is computed by MS=SS/df
If SSbetween = 125 and SSwithin = 65, what is the effect size, η2, for the corresponding ANOVA?
If SSbetween = 125 and SSwithin = 65 Then SStotal = 125 + 65 = 190. So η2 = SSbetween/SStotal = 125/190 = 66%.
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?
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
F ratio formula
MS between/MS within
As the differences between sample means increase,
MSbetween also increases, and the F-ratio increases.
Increases in sample variability cause
MSwithin to increase and, thereby, decrease the F-ratio
dftotal
N-1
dfwithin
N-k
SStotal
SSbetween + SSwithin
The effect size, or η² (the Greek letter eta squared) and can be found using the formula
SSbetween/SStotal which is the ratio of the variability contributed by the treatment to the total variability
MSbetween
SSbetween/dfbetween
MSwithin
SSwithin/dfwithin
F
The F-ratio
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?
The citation states that dfwithin = 28 and dfbetween = 3. Therefore, the total degrees of freedom is 28 + 3 = 31. The total number of scores must be N = 31 + 1 = 32. Since dfbetween = 3, there must be 4 treatment conditions. If the number of participants in each treatment group is the same, then each group must have 32/4 = 8 participants
The within-treatments variance measures random, unsystematic differences within each of the samples assigned to each of the treatments.
These differences are not due to treatment effects because everyone within each sample received the same treatment; therefore, the differences are sometimes referred to as "error."
For an independent-measures experiment, the F-ratio is 3.1 If dfbetween = 5, and dfwithin = 14 what will the researcher conclude?
Using an F table, the critical value for dfbetween = 5 and dfwithin = 14 is 2.96 for α = .05 and 4.69 for α = .01 Therefore, the null hypothesis will be rejected for .05, but not for .01
n analysis of variance produced an F-ratio with df values 14, 1. If the same data had been evaluated with an independent-measures t test, what would df be?
With df = 14, 1, there were two treatment groups of 8 scores each. The same data in a t test would have df = (8 - 1) + (8 - 1) = 14.
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?
With two therapies and three measurement times, there will be 2 × 3 = 6 different sample means
What will increase the likelihood of rejecting the null hypothesis using ANOVA?
a decrease of SSwithin an increase in the sample sizes
Both the F-ratio and the t statistic compare the _______________________ (numerator) with the _______________________ (the denominator if H₀ is true). If the numerator is sufficiently bigger than the denominator, you conclude that _____________________________________.
actual differences between sample means differences that would be expected if there is no treatment effect there is a significant difference between treatments.
post hoc tests
additional hypothesis test done after an ANOVA to determine whether mean differences are significant.
F = variance between-treatments / variance within-treatments
an increase in the numerator will increase the F -ratio a decrease in the denominator will further increase F. With an increased F-ratio, the likelihood for rejecting the null hypothesis will increase.
When the null hypothesis is true, the F test statistic is
close to 1
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.
ANOVA summary table
diagram showing the source of variability
increasing the size of the variability within treatments (SS) would
increase MSwithin and would reduce the size of the F-ratio.
dfbetween + 1 =
k
dfbetween
k-1
When the null hypothesis is false, the F test statistic is most likely
large
In general, you should reject the null hypothesis for
large values of the F test statistic .
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.
between treatments
term referring to differences from one condition to another.
within treatments
term referring to differences that exist inside the individual conditions.
Which of the following most accurately describes the F-ratio in ANOVA testing?
the ratio of the variance between sample means and the variance expected with no treatment effect.
In ANOVA testing, the F-ratio is the ratio of
the variance between sample means and the variance expected with no treatment effect.
In an analysis of variance, the primary effect of large mean differences from one sample to another is to increase the value for ______.
the variance between treatments.
The null hypothesis states that
there is no treatment effect; in other words, the treatment means do not differ from one another. The term "treatment effect" is used even when there is not actually a treatment.
experimentwise alpha level
total probability of a Type I error accumulated from all individual tests in the experiment.
SSwithin
ΣSS
When the null hypothesis is true for an ANOVA, what is the expected value for the F-ratio?
1.00
ΣT is equal to
G
factor
In analysis of variance, the variable that designates the groups being compared
What is the main advantage of ANOVA testing compared with t testing?
It can be used to compare two or more treatments.
If there is a significant treatment effect, the numerator of the ratio should be
larger than the denominator, and the obtained F value should be much greater than 1.00.
With three or more treatment conditions you need three or more t tests to evaluate all the _________ Each test involves a risk of Type I error. The more tests you do, the more risk there is of a Type I error occurring in any of the tests. The ____________ performs all of the tests simultaneously with a single, fixed ____________.
mean differences ANOVA alpha level
Reducing the size of the mean differences (M) would
reduce the size of MSbetween and would reduce the size of the F-ratio.
The MSwithin measures
variability inside each of the treatment conditions. Because individuals inside a treatment condition are all treated exactly the same, any differences within treatments cannot be caused by treatment effects.
For an analysis of variance, the systematic treatment effects in a study contribute to the _______and appears in the ______ of the F-ratio.
variance between treatments, numerator
The F-ratio is the ratio of
variance between-t / variance within-treatments differences including any treatment effects / differences with no treatment effects
Under what conditions might a post hoc test be performed following ANOVA?
when there are three treatments and the null hypothesis was rejected