Lecture 7 - ANOVA 1 Flashcards
ANOVA associated with an __ test
F
We are doing _____ ______
multiple comparisons
Multiple Comparisons:
Suppose we wished to compare the means of several groups, K where K>__
2
Multiple Comparisons:
What is the Ho ?
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)
Multiple Comparisons:
What is the weakness of doing multiple tests?
Is that even if all the means were equal, we’re quite likely to get at least one significant result
Multiple Comparisons:
Where K = # of groups,
R = ?
2 (for pairwise comparisons)
Multiple Comparisons:
For a pairwise comparison amounts 5 means we have __ combinations
10
n over r = n! / r! (n-r)!
! = goes all the way down to zero
*see slide 5 if you’re confused girl
Multiple Comparisons:
What is the Pr [failing to reject ho in all 10 test ] for 10 combinations ?
0.95^10 = 0.6
Multiple Comparisons:
If Pr [ failing to reject Ho] = 0,6, what is Pr [ Rejecting Ho] ?
1 - 0.6 = 0.4
*Thus, there is a 40% chance at least one of the tests will detect a difference where none exists.
Statistical methods for dealing with multiple comparisons usually have 2 steps:
What are they?
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
Analysis of variance:
Don’t be mislead by the name, this test is about ____
means
What does ANOVA assess ?
assesses mean differences by comparing the variability explained by different sources
ANOVA:
What matters ?
What matters is not how far apart the sample means are, but how far apart they are relative to the variability of individual observations.
ANOVA:
What are we concerned with ?
- 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)
What is the No Treatment Effect ?
No treatment effect = between/within = random/random = approx 1
What is the Treatment Effect ?
Treatment effect = between/within = random + systematic / random > 1
ANOVA:
Ultimately our decision will be based on the F statistic in what form ?
F = Variance BETWEEN sample means/Variance among individuals WITHIN each sample
How do you find degrees of freedom for the F table?
df = vn, vd
vn = k - 1 vd = N - k
ANOVA:
What assumptions are we making?
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.
What is the null hypothesis of ANOVA (one-way model) ?
Mu(1) = Mu(2) = …. = Mu(k)
When does MSW (mean squared within) work ?
always works
When does MSB (mean squared between) work ?
only works if Ho is true
With what ratio do we judge Ho ?
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.
How do you read the F table ?
For y(1) part: k-1 degrees of freedom in the numerator
For y(2) part: N-k degrees of freedom in the denominator
When do you reject F value?
if it’s over the critical F value, you reject it
if it’s less than the critical F value, you accept it
What is SSW ?
sum of squares within
First we calculate the SSW (sum of squares within) variability within groups, then what do we calculate ?
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.
If Ho is true (i.e. the means are equal) then each x- is an estimate of ??
mu
How do you calculate the grand mean ?
grand mean = n1x1 + n2x2 + … + nkxk / N
SSB
sum of squares between
What is the formula for SSB (sum of squares between) ?
n1[x1-x(gm)]^2 + n2[x1-x(gm)] ^2
How do you calculate the variance by dividing the SS by the appropriate df ? (what is the formula?)
MSB = SSB / k - 1
If Ho is false, them MSB will _________ the variance.
overestimate
See the one way ANOVA table on slide 25
okay
Once you find the F table from the ANOVA table, what do you do?
calculate the critical F table using the df (k-1, N-k)
F < critical = accept
F > critical = reject
What is X(ij) ?
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
SST
Sum of squares total
What is the formula for SST ?
SST = SSW + SSB
Sum of squares are _____
additive
____ of square are not additive
Means
How do you calculate F?
It is the ratio of MSB/MSW
What is the P value ?
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
