Lecture 41- ANOVA

Question 1

When is an ANOVA used?

Answer 1

Comparing means on continuous responses between multiple groups (more than 3)

Question 2

What is ANOVA doing in terms of the signal to noise ratio?

Answer 2

Signal= difference between treatment means (each group)
Noise= The difference within each of the treatment categories (within the group)

ANOVA quantifies both of these levels of variability

Question 3

Why is pairwise comparisons (t-tests) for multiple groups unfavorable?

Answer 3

• It’s extra work (need to do for every pair combination)
• It can lead to lots of false positives. With every test there is a chance of incorrectly rejecting the null. Therefore, risk increases the more tests we do.

Question 4

How is the notation set up for ANOVA?

Answer 4

Y(ij) means the jth response in the ith group

note: ij is in lower case

The number of different groups is denoted K, and the number of
responses in the ith group is denoted ni

Question 5

What is the model for ANOVA?

Answer 5

Y(ij)= ui +eij

µi is the true mean response for the ith group at the population level.
eij is the error term for the jth response in the ith group

Question 6

What is ‘special’ about the error term in the ANOVA model?

Answer 6

The error terms are assumed to be independent, and to follow a
N(0, σ2) with constant variance.

Question 7

What is RSS?

Answer 7

A measure of the variation in the data that is not explained by
differences between groups.

Question 8

What is TSS?

Answer 8

TSS is a measure of the total amount of variation in the data.

Question 9

What is GSS?

Answer 9

can be interpreted as a measure of the variation that
is explained by differences between groups.

This is the same as ESS

Question 10

What is the null hypothesis for ANOVA? What is the alternative hypothesis?

Answer 10

Null means there is no difference between the groups
H0 : µ1 = µ2 = · · · = µK

Alternative means there is a difference between the groups (note: not all the means have to be different it just implies the null is not true)
HA : µ1, µ2, . . . , µK not all equal

Question 11

What do we expect of the GSS when group means are very different as opposed to when group means are very similar?

Answer 11

From the previous discussion, we expect GSS to be relatively large
when the group means are very different.

We expect GSS to be relatively small when then group means are
similar.

Question 12

Graphically what would a significant difference in group means look like?

Answer 12

GSS (green) would be large compared to RSS (red)

If there was not a significant difference the lines would be equal in length

Question 13

How do you calculate the F statistic for the hypothesis test invovled in an ANOVA?

Answer 13

Refer to slide 798 but basically…

the group mean square/ residual mean square

Question 14

How are the various sum of squares and related quantities frequently displayed?

Answer 14

An ANOVA table

Question 15

Using an ANOVA table what gives you the F statistic required for a hypothesis test?

Answer 15

GMS/ RMS

Question 16

What would result in the failure of the null hypothesis?

Answer 16

⇒ Large differences between group means
⇒ Relatively large value of GSS (top of equation for F stat large)
⇒ Large value of F

Question 17

If the null hypothesis is true what does the F-statistic follow? How many degrees of freedom is associated with this distribution?

Answer 17

• F distribution.
• The F distribution is specified in terms of two degrees of freedom which are the numerator and denominator of the F stat. They are (K − 1) and (n − K)

Question 18

Where do P values come from?

Answer 18

Computing the F stat, same rules as always (less then 0.05= significant)