Lecture 7 - ANOVA 1

Question 1

ANOVA associated with an __ test

Answer 1

F

Question 2

We are doing _____ ______

Answer 2

multiple comparisons

Question 3

Multiple Comparisons:

Suppose we wished to compare the means of several groups, K where K>__

Answer 3

2

Question 4

Multiple Comparisons:

What is the Ho ?

Answer 4

When comparing means of several populations, the temptation is to test the Hypothesis Ho: Mu(i) = Mu(j) for all possible pairs Mu(i), Mu(j)

Question 5

Multiple Comparisons:

What is the weakness of doing multiple tests?

Answer 5

Is that even if all the means were equal, we’re quite likely to get at least one significant result

Question 6

Multiple Comparisons:
Where K = # of groups,
R = ?

Answer 6

2 (for pairwise comparisons)

Question 7

Multiple Comparisons:

For a pairwise comparison amounts 5 means we have __ combinations

Answer 7

10

n over r = n! / r! (n-r)!

! = goes all the way down to zero

*see slide 5 if you’re confused girl

Question 8

Multiple Comparisons:

What is the Pr [failing to reject ho in all 10 test ] for 10 combinations ?

Answer 8

0.95^10 = 0.6

Question 9

Multiple Comparisons:

If Pr [ failing to reject Ho] = 0,6, what is Pr [ Rejecting Ho] ?

Answer 9

1 - 0.6 = 0.4

*Thus, there is a 40% chance at least one of the tests will detect a difference where none exists.

Question 10

Statistical methods for dealing with multiple comparisons usually have 2 steps:

What are they?

Answer 10

1 - An OVERALL test to see if there is good evidence that any of the parameters differed from its hypothesized value

2 - A detailed FOLLOW-UP analysis to decided which of the parameters differs from their hypothesized value and to estimate the size of the difference

Question 11

Analysis of variance:

Don’t be mislead by the name, this test is about ____

Answer 11

means

Question 12

What does ANOVA assess ?

Answer 12

assesses mean differences by comparing the variability explained by different sources

Question 13

ANOVA:

What matters ?

Answer 13

What matters is not how far apart the sample means are, but how far apart they are relative to the variability of individual observations.

Question 14

ANOVA:

What are we concerned with ?

Answer 14

• Variability WITHIN each group (which is always random as each x is treated in the same way)
• Variability BETWEEN the groups (which is due to both random variability within the groups, as well as any systematic differences between the groups due to an experimental treatment)

Question 15

What is the No Treatment Effect ?

Answer 15

No treatment effect = between/within = random/random = approx 1

Question 16

What is the Treatment Effect ?

Answer 16

Treatment effect = between/within = random + systematic / random > 1

Question 17

ANOVA:

Ultimately our decision will be based on the F statistic in what form ?

Answer 17

F = Variance BETWEEN sample means/Variance among individuals WITHIN each sample

Question 18

How do you find degrees of freedom for the F table?

Answer 18

df = vn, vd

vn = k - 1
vd = N - k

Question 19

ANOVA:

What assumptions are we making?

Answer 19

1) Random samples - simple random samples eliminate bias
2) Independence - Critical assumption. We have independent samples, one from each of k populations. For dependent measures (repeated measures on the same subject) a repeated measures ANOVA model is available.
3) Independent observations within each sample - critical
4) Normal population - The population from which the simple random samples are drawn must be normally distributed. ANOVA is somewhat robust to this assumption for larger samples
5) Homogeneity of Variance: The variance must be equal (even when the group means are different). There is no simple rule of assessing this, as a rule of thumb the F test will be approximately correct when the largest sample variance is no more than twice as large as the smallest sample variance.

Question 20

What is the null hypothesis of ANOVA (one-way model) ?

Answer 20

Mu(1) = Mu(2) = …. = Mu(k)

Question 21

When does MSW (mean squared within) work ?

Answer 21

always works

Question 22

When does MSB (mean squared between) work ?

Answer 22

only works if Ho is true

Question 23

With what ratio do we judge Ho ?

Answer 23

Using the ratio statistic:

F = MSB / MSW

*Once the F statistic is calculated (and a level of significance, alpha = 0.05) is chosen, we find the critical value for the F statistic from the table.

Question 24

How do you read the F table ?

Answer 24

For y(1) part:
k-1 degrees of freedom in the numerator 

For y(2) part:
N-k degrees of freedom in the denominator

Question 25

When do you reject F value?

Answer 25

if it’s over the critical F value, you reject it

if it’s less than the critical F value, you accept it

Question 26

What is SSW ?

Answer 26

sum of squares within

Question 27

First we calculate the SSW (sum of squares within) variability within groups, then what do we calculate ?

Answer 27

The variance (mean square in ANOVA) by dividing the SSW by the appropriate degrees of freedom

MSW = SSW/N - k

Where N equals the total number of observations.

Question 28

If Ho is true (i.e. the means are equal) then each x- is an estimate of ??

Answer 28

mu

Question 29

How do you calculate the grand mean ?

Answer 29

grand mean = n1x1 + n2x2 + … + nkxk / N

Question 30

SSB

Answer 30

sum of squares between

Question 31

What is the formula for SSB (sum of squares between) ?

Answer 31

n1[x1-x(gm)]^2 + n2[x1-x(gm)] ^2

Question 32

How do you calculate the variance by dividing the SS by the appropriate df ? (what is the formula?)

Answer 32

MSB = SSB / k - 1

Question 33

If Ho is false, them MSB will _________ the variance.

Answer 33

overestimate

Question 34

See the one way ANOVA table on slide 25

Answer 34

okay

Question 35

Once you find the F table from the ANOVA table, what do you do?

Answer 35

calculate the critical F table using the df (k-1, N-k)

F < critical = accept
F > critical = reject

Question 36

What is X(ij) ?

Answer 36

The total variability of the observations X(ij) - ignoring groups is measured in terms of the deviation of each observation X(ij) around the overall mean:
_
X(ij) - X

Question 37

SST

Answer 37

Sum of squares total

Question 38

What is the formula for SST ?

Answer 38

SST = SSW + SSB

Question 39

Sum of squares are _____

Answer 39

additive

Question 40

____ of square are not additive

Answer 40

Means

Question 41

How do you calculate F?

Answer 41

It is the ratio of MSB/MSW

Question 42

What is the P value ?

Answer 42

The probability of this occurring due to chance.

So if you’re rejecting Ho, P value < 0.05

If you’re accepting Ho, P value is > 0.05