Ch. 15

Question 1

What is ANOVA?

Answer 1

Analysis of Variance, comparing multiple variances of multiple groups simultaneously

Question 2

What is the null hypothesis of ANOVA?

Answer 2

Population means are the same for all treatments

Question 3

What is the alternate hypothesis of ANOVA?

Answer 3

At least one mean is different from the others

Question 4

What are “mean squares”?

Answer 4

Means squares are a special name for the measures of variation used in ANOVA

Question 5

What is the group mean square?

Answer 5

It is a value proportional to the observed amount of variation among the group sample means.

From pg. 466, “The group mean square represents variation among the sampled individuals belonging to different groups. It will on average be similar to the erro mean square if the population means are equal.”

Question 6

What is the error mean square?

Answer 6

Estimates the variance among subjects taht belong to the same group

Question 7

What is the F ratio?

Answer 7

The Ratio of MS_groups to MS_error

Question 8

What should F equal if the null hypothesis is true?

Answer 8

1 (no difference in variations)

Question 9

What should F be if the null hypothesis is false?

Answer 9

A value greater than 1 (beause the real differences among group means will enlarge MS_group)

Question 10

Degrees of freedom for groups?

Answer 10

df = k - 1, where k is the number of groups

Question 11

Degrees of freedom for error sum?

Answer 11

df_error = sum of (n-1) = N - k

where N is total number of data points in all groups combined, and k is the number of groups

(i.e. N - k is equal to the sum of the df of each individual group)

Question 12

What is the test statistic for ANOVA?

Answer 12

F-ratio

Question 13

Degrees of freedom for F-ratio?

Answer 13

There are 2, the numerator’s df (MS_group df which is k-1) and the denominator’s df (MS_error df which is N-k)

thus want F_{(k-1), (N-k), 0.05}

Note that numerator (k-1) always goes first

Question 14

What is R²?

Answer 14

It measure the fraction of the variation in Y that is explained by group differences.

Question 15

How to understand R²?

Answer 15

The greater the value of R², the larger the effect of the variation in groups (variation caused by explanatory variable) on the total variable.

Can consider result as a percentage effect of treatment variation compared to total variation (pg. 469)

Question 16

ANOVA w/ two groups vs. t-sample t-test

Answer 16

Two sample t-test and ANOVA give identical resutls. However, t-test can work with hypothesised differences between means, unlike ANOVA testing u₁=u₂ (ex u₁ - u₂ = 10). Also, Wlech’s t-test allows for

Question 17

Assumptions of ANOVA?

Answer 17

Random Samples

Variable is normaly distributed in each population (robust, esp. w/ large sample means

Variance is the same in all populations (same 10-fold boundary as t-test)

Question 18

Kruskal-Wallis test

• What is it?
• When to use it?
• Power?

Answer 18

Non-parametric test similar to ANOVA; based of Mann-Whitney U-test and has the same limitations/assumptions.

Similar power to ANOVA when samples are large, but little power when small. Use ANOVA if able

Question 19

Probability of a Type 1 error in N tests? (equation)

Answer 19

1 - (1 - a)^N

Question 20

Bonferroni correction?

Answer 20

Basically tells you to use a small alpha value

a’ = a/(number of tests)

Question 21

Planned comparion?

Answer 21

A comparison between means planned during the design of the study, identified before the data are examined (since ANOVA doesn’t tell you what means are different, just that one of the means are different, at least)

Gives more power than just a two-sample t-test

Question 22

Unplanned comparison

Answer 22

One of multiple comparisons carried out to help determine where the differences in the means lie

Question 23

Tukey-Kramer test

What is it and when is it done?

Answer 23

Done after finding variation among groups in single-facter ANOVA

Compares all group means w/ all other group means

Question 24

Steps of Tukey-Kramer?

Answer 24

1. Order group means in ascending magnitude
2. create null hypotheses
3. Perform test

Question 25

Probability of making a Type-1 error in Tukey-Kramer?

Answer 25

The error throughout the course of testing all pairs of means is no greater than significance level alpha (that you set)

This is because a compensation for data dregging was built in

Question 26

Why not do t-test w/ Bonferroni correction instead of a Tukey-Kramer?

Answer 26

Tukey-Kramer has more power as it uses more info from the data, so it is better than t-tests.

Also has a built in adjustment for the number of tests.

Question 27

Assumptions of Tukey-Kramer?

Answer 27

Same as ANOVA, Random sample, all normal distriubtions, and equal variances.

However, not as robust as ANOVA since only 1 pair tested at a time.

P-value of Tukey-Kramer is exact when the design is balanced (n1=n2=n3), but conservative if not balanced (i.e actual probability of Type 1 error smaller than stated crit. value)

Question 28

Fixed vs Random effects

Answer 28

Fixed effects have treatments chosen by the experimenter, while random effects have a random sample of treatments from all possible treatments

Single-factor ANOVA is not affected by fixed or random effects

Question 29

Advantage of random-effects ANOVA?

Answer 29

Results are generalizable, unlike fixed-effects that can only talk about the groups included in the study .