Set L and Set M
Increasing the size of the sample variances:
Increases in sample variability cause MSwithin to increase and, thereby, decrease the F-ratio.
Eta squared (n^2)
indicates the proportion of variance accounted for by the differences between treatments in the ANOVA.
levels of the factor
individual conditions that make up the factor (dependent)
SSbetween
SSbetween = = Σ(T^2/n)−G2/N
F
The F-ratio
If the SS for treatment I were changed to SS = 1400(before ss=1005), what would happen to the size of the F-ratio (increase or decrease)? Explain your answer.
The F-ratio would decrease because increasing the variability within treatments would increase MSwithin.
If the mean for treatment III were changed to M = 25 (before M=35), what would happen to the size of the F-ratio (increase or decrease)? Explain your answer.
The F-ratio would decrease because reducing the size of the mean differences would reduce MSbetween
What is the principal reason why you should use ANOVA instead of several t tests to evaluate mean differences when an experiment consists of three or more treatment conditions?
With three or more treatment conditions, you need three or more t tests to evaluate all the mean differences. Each test involves a risk of a Type I error, equal to the alpha (α) specified for that test. The more tests you do, the more risk there is of a Type I error. The ANOVA performs all of the tests simultaneously with a single, fixed alpha level. **Multiple t tests accumulate the risk of a Type I error.
ANOVA
is always a one-tailed test.
The F-ratio (F test statistic)
is calculated by dividing the between-treatments variance by the within-treatment variance.
Within-treatments variance
is caused by random unsystematic variance caused by individual differences or sampling error. provides a measure of how big the differences are when H₀ is true.
A large F-ratio
is evidence for the existence of systematic treatment effects; that is, there are significant differences between treatments (allowing you to reject the null hypothesis). Null hypothesis is false
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."
k
The number of treatment conditions
factor.
the variable that designates the groups being compared (indep)
ANOVA avoids...
An ANOVA avoids the problem of an inflated experimentwise alpha level.
F ratio near 1.00
An F-ratio near 1.00 indicates that the differences between treatments (numerator) are random and unsystematic, just like the differences in the denominator. With an F-ratio near 1.00, you conclude that there is no evidence to suggest that the treatment has any effect. Null hypothesis is true
Increasing the differences between the sample means:
As the differences between sample means increase, MSbetween also increases, and the F-ratio increases.
Describe the similarities between an F-ratio and a t statistic.
Both the F-ratio and the t statistic compare the actual differences between sample means (numerator) with the differences that would be expected if there is no treatment effect (the denominator if H₀ is true). If the numerator is sufficiently bigger than the denominator, you conclude that there is a significant difference between treatments.
hypothesis for an ANOVA
Null: is always that populations defined by the different factor levels all have equal means. Alternative: not all of the population means are equal.
Which of the following are assumptions underlying independent-measures analysis of variance?
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. ANOVA is robust; that is, it is acceptable to use an ANOVA if there are small deviations from these assumptions, provided the samples are of equal size.
the effect of Eta squared (n^2)
The effect size, or η², can be found using the formula SSbetween/SStotal, which is the ratio of the variability contributed by the treatment to the total variability.
n
The number of scores in each treatment
What value is expected for the F-ratio, on average, if the null hypothesis is true in an ANOVA?
The numerator of the F-ratio measures all differences between samples , and the denominator measures only random differences . If there is no treatment effect, differences between samples are due to only random differences , so the numerator and denominator measurethe same sources of variability and should be about equal and have a ratio close to 1
The testwise alpha level
The risk of a Type I Error, or alpha level, for an individual hypothesis test.
G
The sum of all of the scores in the research study
T
The sum of the scores for each treatment condition
N
The total number of scores in the entire study
alternative hypothesis
states that at least one of the treatment means is different. In this case, at least one of the age groups has an effect on the amount of alcohol consumed.
null hypothesis
states that there is no treatment effect; in other words, the treatment means do not differ from one another. In this case, the age groups have no effect on the amount of alcohol consumed.
When you evaluated the mean difference from the independent-measures study comparing only two samples, the ANOVA and the t test resulted in...
the same conclusion
The experimentwise alpha level
the total probability of a Type I error that is accumulated from all of the individual tests in the experiment
Between-treatments variance
you are measuring differences that could be caused by a systematic treatment effect or could simply be random and unsystematic mean differences caused by sampling error.