Term 2 Lecture 4

Question 1

ANOVA is one of the most widely used statistical techniques for the analysis of experiments.

Using ANOVA, we can test hypotheses about
the means of _______

Answer 1

three or more groups.

Question 2

Could we do a series of t-tests instead

Answer 2

this would result in three pairwise comparisons. However, with large numbers of groups, the number of comparisons becomes very large

As well as being laborious, this would increase the risk of committing a Type 1 error: the greater the number of t-tests we conduct, the greater is the risk of obtaining at
least one significant result even if the null hypothesis is
true for all pairwise comparisons (i.e. even if there are no differences between groups)

Question 3

What does ANOVA offer?

what does a null hypothesis look like for ANOVA?

Answer 3

offers a single test of the null hypothesis that
all group means are equal (in the population).

In mathematical notation, we write the null hypothesis as follows:

H0: μ1 = μ2 = μ3,where μ1, μ2, and μ3 are the (population) means of the three
groups.

In the example of anorexia treatments, this null
hypothesis would imply that neither of the two treatments have an effect in a population of anorexic girls.

Question 4

What does an alternative H for ANOVA look like?

(i, j)

i = 1, 2, 3 denotes the experimental group (CBT, FT, or
Control)
j = 1, 2, 3, …, ni denotes individuals within each group

Answer 4

The alternative hypothesis, H1, is that at least one group mean is different from one or more of the others. In mathematical notation, we might write this as:
H1: μi ≠ μj for at least one pair (i, j).

Question 5

What is variance?

Answer 5

a numerator, which is the sum of squared deviations
from a mean, or sum of squares (SS);
• a denominator, which is equal to the degrees of
freedom of the SS.

Question 6

What is ANOVA?

Answer 6

𝑆𝑆𝑇𝑜𝑡𝑎𝑙 = 𝑆𝑆𝐵𝑒𝑡𝑤𝑒𝑒𝑛 + 𝑆𝑆𝑊𝑖𝑡ℎ𝑖𝑛
This equation expresses that we have two sources of variation in
the data:
1. the differences between groups: SSBetween
(also called SSTreat [Treatment])
2. the variation of scores within each group: SSWithin
(also called SSRes [Residual])

Question 7

What is the test statistic for ANOVA?

Answer 7

The test statistic for ANOVA is called F, which is the ratio of SSBetween to SSWithin, taking account of their associated degrees of freedom.

𝐹 = 𝑆𝑆𝐵𝑒𝑡𝑤𝑒𝑒𝑛 Τ𝑑𝑓1
      𝑆𝑆𝑊𝑖𝑡ℎ𝑖𝑛 Τ𝑑𝑓2
Or, specifying the degrees of freedom:
𝐹 = 𝑆𝑆𝐵𝑒𝑡𝑤𝑒𝑒𝑛 Τ(𝑘−1)
       𝑆𝑆𝑊𝑖𝑡ℎ𝑖𝑛 Τ(𝑁−𝑘)

where k is the number of groups, and N is the total number of cases
(participants).

Question 8

If the null hypothesis is correct, we would expect F to be

evidence against the null hypothesis.F would be

Answer 8

small

large

Question 9

How do you report ANOVA?

Answer 9

F2,69 = 5.422, p = 0.006 < α, therefore we reject H0

. We conclude that the group means are not all the same.

Question 10

What are effect sizes in ANOVA? (IMPORTANT LEARN FORMULA)

PES

Answer 10

Partial eta squared

There are different measures of effect sizes for ANOVA. A simple one is η2, which measures the proportion of the Total Variance that is
‘explained’ by the statistical model.
!!!!! 𝜂2 =𝑆𝑆𝐵𝑒𝑡𝑤𝑒𝑒𝑛/𝑆𝑆𝑇𝑜𝑡𝑎𝑙 =

This means that 13.6% of the overall variation in weight change among all the girls is due to the girls having received different treatments. The remainder of the variation is due to other factors (e.g. individual
differences between the girls, and/or random variation)

Question 11

What are the assumptions of ANOVA (3)

Answer 11

Independence
– We assume that the observations are independent of one another (if dealing with experiments, participants should be randomly assigned to experimental conditions)

Normal distribution
— We assume that the sampling distributions of all group means are normal

Homogeneity of variance (homoscedasticity)
—-We assume that all group (population) variances are equal. That is, for three groups, the assumption is: σ1 = σ2 = σ3

Question 12

what test is used to assess homogeneity of variance?

Answer 12

Levene’s Test of Equality of Variances for the

independent-samples t-test, we can test homogeneity of variances for ANOVA.

Question 13

when is homogeneity not assummed?

Answer 13

If the Levene Statistic is significant,

then we must conclude that the three group variances are not all the same, and that the assumption of homogeneity of variance is violated.

Question 14

what happens of hov is violated?

2 alternatives

Answer 14

If we cannot assume homogeneity of variances, we may use one of two alternatives to the ordinary F-test: “Welch’s F” or “BrownForsythe F”

These tests are “robust”, that is, less sensitive to violations of the “homogeneity of variances” assumption than the ordinary F-statistic.

They should not be used when the homogeneity assumption holds, because they tend to be less powerful than the ordinary F-statistic.