Comparing Several Independent Means

Question 1

What does a one-way Anova tell you?

Answer 1

It compares two or more independent groups. It checks whether there is an actual difference between the groups but it does not tell you which groups differ.

Question 2

What can you do to see which groups differ? (2)

Answer 2

Contrasts tests or posthoc analysis

Question 3

What assumptions does a one way anova have? (4)

Answer 3

1. Continuous variable
2. Random sample
3. Normal distribution
4. Homogeneity of variance (The assumptions that there is equal variance within the groups.)

Question 4

How can the normality assumption and the variance assumption be tested?

Answer 4

The assumption of the normal distribution can be tested using the Shapiro Wilk test and the assumption of homogeneity of variance can be tested using the Levene’s test.

Question 5

What two hypotheses do one way independent ANOVA’s usually have?

Answer 5

h0 = There is no differences between the groups
h𝑎 = There is a difference between the groups

Question 6

What test statistic does the one way independent ANOVA utilise?

Answer 6

F-statistic

Question 7

What ratio and formula does the F-statistic utilise?

Answer 7

It represents a signal to noise ratio of the data and it uses the following formula:
F= MSmodel/MS error

Question 8

What do the degrees of freedom of the F-statistic depend on?

Answer 8

The sample size and the number of groups.

Question 9

What are the three types of variance in a one way independent ANOVA?

Answer 9

Model (between-groups)
Error (within-groups)
Total

Question 10

What is the sum of squares for each of these types of variances?

Answer 10

𝑆𝑆𝑚𝑜𝑑𝑒𝑙 =∑𝑛𝑘(𝑋𝑘 −𝑋)^2
𝑑𝑓𝑚𝑜𝑑𝑒𝑙 =𝑘−1

𝑆𝑆 = ∑𝑠2𝑘(𝑛 −1) 
𝑑𝑓𝑒𝑟𝑟𝑜𝑟 = N-k

𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆𝑆𝑚𝑜𝑑𝑒𝑙 + 𝑆𝑆𝑒𝑟𝑟𝑜𝑟
𝑑𝑓 =𝑁−1 𝑡𝑜𝑡𝑎𝑙

N denotes the sample size, 𝒏𝒌 denotes the sample size per category and k denotes the number of categories.

Question 11

What is the mean square for each of these categories?

Answer 11

𝑀𝑆𝑚𝑜𝑑𝑒𝑙 = 𝑆𝑆𝑚𝑜𝑑𝑒𝑙/ 𝑑𝑓 𝑚𝑜𝑑𝑒𝑙

𝑀𝑆𝑒𝑟𝑟𝑜𝑟 = 𝑆𝑆𝑒𝑟𝑟𝑜𝑟/ 𝑑𝑓 𝑒𝑟𝑟𝑜𝑟

𝑆𝑆𝑡𝑜𝑡𝑎𝑙/ 𝑑𝑓 𝑡𝑜𝑡𝑎𝑙

Question 12

What do the model and error variations represent in terms of explaining the variation?

Answer 12

The model variance represents the variance that can be explained by the experimental manipulation. The error variance represents the variance that cannot be explained by the experimental manipulation.

Question 13

What are contrasts?

Answer 13

Planned comparisons used to investigate a specific hypothesis and compare different parts of data with each other

Question 14

How are contrasts limited?

Answer 14

They can only use one piece of data once

Question 15

Why are contrasts limited in this way?

Answer 15

In order to not inflate the type-I error rate

Question 16

What can solve this problem?

Answer 16

Post-hoc tests are unplanned comparisons that compare all groups with each other and adjusts the alpha level to control for the inflated type-I error rate.

Question 17

What is the difference between the ANOVA and regression between two variables?

Answer 17

There isn’t one

Question 18

What three rules are there to develop a planned contrast?

Answer 18

1. If you have a control group, this is there usually because you want to compare it against any other groups
2. Each contrast must compare only two chunks of variation
3. Once a group has been singled out in a contrast, it can’t be used in another contrast

Question 19

What rules should be followed when applying weight to contrasts?

Answer 19

1. Choose sensible contrasts
2. Groups coded with positive weights will be contrasted against groups coded with negative weights
3. If you add up the weights for a given contrast the result should be 0
4. If a group is not involved in a contrast, automatically assign it a weight of zero, which will eliminate it from the contrast
5. For a given contrast, the weights assigned to the group(s) in one chunk of variation should be equal to the number of groups in the opposite chunk of variation

Question 20

What three criteria decide which post-hoc test is best?

Answer 20

Does the test control the type I error rate?
Does it control the type II error rate?
Is the test robust?

Question 21

When should the REGWQ post-hoc test be used and when shouldn’t it?

Answer 21

It has good power and tight control of the type I error rate so it should be used when you want to test all pairs of means however when group sizes are different this should not be used

Question 22

Comment on the type I error rate and power of the Tukey and Boneferoni test

Answer 22

Control the Type I error rate pretty well but lack statistical power. Boneferoni has more power when number of comparisons are small but Tukey has more power when testing large numbers of means

Question 23

Are post-hoc procedures robust

Answer 23

Small deviances from reality they perform well, perform badly when group sizes are unequal and when population variances are different

Question 24

What tests were designed for when group sizes are different? When should each of these be used?

Answer 24

Hochberg’s GT2 (if group sizes are very different) and Gabriel’s pairwise test (If they are slightly different)

Question 25

When you have equal sample sizes and group variances are similar use ______ or ______

Answer 25

REGWQ; Tukey

Question 26

If you want guaranteed control over type I use _______

Answer 26

Bonferroni

Question 27

If there is any doubt that group variances are equal then use ______

Answer 27

Games-Howell procedure