[L9] Analysis of Variance & [L10] The Kruskal-Wallis Test Flashcards
- Very flexible and general technique, and the principles
can be applied to a wide range of statistical tests.
ANOVA
ANOVA is a ___ test
parametric
Has a wide range of applications.
ANOVA
Many of applications make some tricky assumptions
about the data.
ANOVA
In ANOVA we measure an ___ variable (also called
a ___ variable).
* This outcome must be measured on a ___ scale.
outcome; dependent - continuous
It is called dependent because it depends on one or more
__ variables
predictor
__
_ variables can be Manipulated (Treatment) or
variables we simply measure (Sex).
Predictor
In ANOVA, predictor variables are mostly ___,
although continuous variables can also be used in the
same framework
categorical
When predictor variables are categorical, they are also
called “__“_
FACTORS or INDEPENDENT VARIABLES.
___ – measurement of differences
ANOVA
Differences happen for two reasons: ___
(a) because of the
effect of predictor variables (b) because of other reasons
In ANOVA, we want to know two things:
___
- How much of the variance (difference) between the
two groups is due to the predictor variable
- How much of the variance (difference) between the
- Whether this proportion of variance is statistically
significant, that is, it is larger than we would expect by
chance if the null hypothesis were true?
- Whether this proportion of variance is statistically
We can divide (statisticians sometimes say partition)
variance into three different types:
___
- The Total Variance
- Variance due to treatment, (Differences between Group)
- Variance due to Error (Differences within Group)
In ANOVA, the variance is conceptualized as sums of
_
__
squared deviations from the mean
In ANOVA, the variance is conceptualized as sums of
squared deviations from the mean.
* It is usually shortened to___ and denoted by
__
sum of squares; SS.
The 3 Sum of Squares
- Total Sum of Squares, SS total
- Between-groups Sum of Squares, SS between
- Error Sum of Squares, SS within
___– this is the variance
that represents the difference between the groups, and this
is called _
_
. Sometimes it refers to the betweengroups
sum of squares for one predictor, in which case it
is called SS predictor. Sometimes it is called___.
Between-groups Sum of Squares; SSbetween; SStreatment
The ___-groups variance is the variance that we are
actually interested in.
between
We are asking whether the difference between the groups
(or the effect of the predictor) is big enough that we could
say it is ___
not due to chance
_
_
_– also called within-groups sum
of squares.
Error Sum of Squares
It’s within the groups, because different people, who
have had the same treatment, have different scores.
Error Sum of Squares
They have different scores because of error. So this is
called either ___
SSwithin, or SSerror.
We need to calculate the three kinds of Sum of Squares,
___
TOTAL, WITHIN GROUPS, and BETWEEN
GROUPS.
_
_
_sum of squared differences between the mean
and each score.
SStotal –
___
* To know how large the effect of the treatment has been
Calculating the Effect Size:
The same as asking what
___ the treatment effect
has been responsible for.
proportion of the Total
Variance (or Total Sum of Squares)
Effect Size goes under two different names: these are ___.
RSquared
or eta-Squared
Mean Squares. Often written as MS.
* These are ___
MSbetween, MSwithin, MStotal
Three sets of degrees of freedom
- df total, df between, and df within
Finally we calculate the relative size of the two values,
by dividing MS between by MS within.
* This gives us the statistic for ANOVA, which is called F,
or sometimes the
__
__
F-ratio.
To find the probability value associated with F we need
to have two sets of degrees of freedom, the ___
between and
within.
__
_are exactly the same test.
* It is just a different way of thinking about the result (when we have two groups).
ANOVA and t-test
In fact, if we take the ___and square it. We get the
value of F.
value of t
This is a general rule when there are 2 groups:
___
F = t-squared.
Question: If we covered t-tests, why are we doing it
again?
_
- t-test – restricted to comparing 2 groups.
- ANOVA extends in a number of useful directions.
___ extends in a number of useful directions.
* Can be used to compare 3 groups or more, to calculate
the p-value associated with the Regression Line, and ina
wide range of ___ situations
ANOVA, other
When there are 2 groups, ANOVA is equivalent to a
___ and it therefore makes the same assumptions as
the t-test.
t-test,
same assumptions as
the t-test. and it makes these assumptions regardless of the number
of
_
__ that are being compared
groups
Assumptions in ANOVA
- Normal distribution within each group
- Homogeneity of Variance
We do not assume that the outcome variable is normally
distributed.
* What we do assume is that __ each group are
normally distributed.
data within; Normal distribution within each group
_
* As with the t-test, we assume that the standard deviation
within each group is approximately equal.
Homogeneity of Variance
the variance being the square of the
.
SD
- However, as with the t-test, we don’t need to worry
about this assumption, if we have approximately equal
__
_ in each group.
numbers of people
ANOVA comparing Three Groups
- Formulae are all the same.
__most elementary analysis of
variance
One way ANOVA –
One way ANOVA – Also called as __
simple-randomized groups design,
independent groups design, or the single factor
experiment, independent groups design.
_
__that is being investigated, there
are two or more levels or conditions of the IV, and
subjects are randomly assigned to each condition.
Only one IV (one factor); one-way anova
ANOVA – not limited to ___experiments.
single factor
The effect of many different __ may be investigated
at the same time in one experiment
anova; factors
_ – one in which the effects of two
or more factors or IVs are assessed in one experiment
Factorial experiment
Conditions or treatments used are combinations of the
__
levels of factors.
_more complicated, However, we get a lot more information
Two way ANOVA –
Two way ANOVA –It allows in one experiment to evaluate the __ of two
IVs and the __ between them.
effect; interaction
__– the levels of each factor were
systematically chosen by the experimenter rather than
being randomly chosen
Fixed effects design
We want to determine whether factor A has a
significant effect, disregarding the effect of factor B.
This is called the __
main effect of factor A.
We want to determine whether factor B has a
significant effect, without considering the effect of factor
A. This is called the ___
main effect of factor B.
finally, we want to determine whether there is an
interaction between factors A and B. This is called the
__
interaction effect of factors A and B.
Three analyses in fixed effects design:
- main effect of factor A.
- main effect of factor B.
3.interaction effect of factors A and B.
In analyzing data from a two-way ANOVA, we determine
four variance estimates:
- MS within cells
- MS rows
- MS columns
- MS interaction
The estimate ___ is the within cells variance
estimate and corresponds to the within groups variance
estimate used in the one-way ANOVA
MS within cells
It becomes the
__ against which the other
estimates, MS rows, MS columns, and MS interactions,
are compared.
standard
The other estimates are sensitive to the__
effects of the IVs.
The estimate MS rows is called the_
row variance
estimate.
row variance
estimate is based on the variability of the row means and,
hence, is sensitive to the_
effects of variable A.
The estimate MS columns is called the _
column variance
estimate.
column variance
estimate is based on the variability of the column means and,
hence, is sensitive to the_
effects of variable B.
The estimate MS interaction is the __
interaction variance
estimate
interaction variance
estimate is based on the variability of the cell means and,
hence, is sensitive to the _
interaction effects of variables
A and B.
If variable A has no effect, MS rows is an
__of the __
independent
estimate; σ-squared.
Finally, if there is no interaction between variables A and
B, MS interaction is also an _
independent estimate of σ –
squared.
Thus, the estimates MS rows, MS columns, and MS
interaction are analogous to the ___
_
of the one-way ANOVA design
between-groups variance estimate
Each F (or F obtained) value is evaluated against
__
(critical value) as in the one way analysis
F crit
In a two-way ANOVA, we can essentially two one-way
experiments, plus we are able to evaluate the interaction
between the __
two independent variables.
In a 2-way ANOVA, we partition the total sum of
squares (SS total), into four components:
- the withincells sum of squares,
- the row sum of squares,
- the column sum of squares,
- and the interaction sum of
squares.
When these Sum of Squares (SS) are divided by the
appropriate degrees of freedom, they form four variance
estimates..
(MS within-cells, MS rows, MS columns and
MS interaction
Only difference is that with the row sum of squares we
use the __ means, whereas the between-groups sum of
squares used the __ means.
row; group
In ANOVA we aim to find out if there are differences
between the groups, but not _
_
what those differences are.
Usually, we test the hypothesis that: μ1 = μ2 = μ3
* In the case of two groups, this is not a problem, because
if the mean of group 1 is different from the mean of
group 2, that can only happen in __
one way.
However, when we reject a null hypothesis when there
are three or more groups, we aren’t really saying
enough.
* We are just saying that group 1, group 2, and group 3
(and so on, up to group k) are ___
not the same.
Unlike the two-group solution, this can happen in __
lots of
ways.
_
to answer the question of where the
differences come from.
Post Hoc tests –
Post hoc” is Latin, and means “
after this”.
Post hoc tests are tests done after
__.
* They are based on –
ANOVA; t-tests.
It is possible to just do t-tests to compare groups, but this
would cause a problem called -
alpha inflation
Alpha is the Type__error rate.
I
A _
_is where we reject a null hypothesis that is
true.
Type I error
The probability value that we choose as a cut-off
(usually 0.05) is the __.
Type I error rate
That is, it is the probability of getting a __ result
in our test, if the population effect is actually zero.
significant
When 3 tests are done, a cut-off of 0.05 is used, and most
think that the probability of a Type I error is __.
still 0.05
We call 0.05 our _
_error rate, because that
is the Type I error rate we have named.
nominal Type I
The problem is that the Type I error rate has __, and it is no longer our true type I error rate.
risen above
0.05
When we do multiple t-tests, following an ANOVA, we
are at risk of
__.
capitalizing on chance
The probability that one of those tests will be statistically
significant is not 0.05, but is actually closer to __
three
times 0.05 or 0.15, about 1 in 7.
So our actual type I error rate is much – than our
nominal rate.
higher
we need to perform some sort of __
and we can’t use our plain ordinary t-test.
modified test
-
* Assumption of homogeneity of variance
Bonferroni Correction
Bonferroni Correction
What is done here is to calculate the pooled standard
error, and then calculate three t-tests using this
__
pooled
standard error.
However, there are 2 reasons, why we are not going to do this. (Bonferroni Correction)
1st: it is tricky
2nd: It is so unintuitive.
__
* When there are two groups, we calculate the standard
error, and then calculate the confidence interval, based
on multiplying the SE by the critical value for t. at 0.05
level.
Bonferroni Corrected Confidence Intervals
Bonferroni Corrected Confidence Intervals: We carry out the same procedure, except we are no
longer using the
__
95% level.
We have to adjust alpha by dividing by __, to give 0.0166.
3
We then calculate the critical value for _
_, using the new
value for alpha.
t
To calculate the confidence intervals, we need to know
the _
critical value of t.
Since we are now using the value of alpha corrected for
the number of tests (say 3), we are now going to be
doing __, so we need to use 0.05/3 = 0.0166.
three tests
Before we can determine the critical value, we need to
know the
__
* The df are calculated in the same way as the t-test. That
is, df = N-2, where N is the total sample size for the two
groups we are comparing.
degrees of freedom (df).
- Calculation of statistical significance is also
straightforward once we have the standard errors of the
differences-
Bonferroni Corrected Statistical Significance
The value for t is equal to the __
difference divided by the
standard error of difference.
Bonferroni Correction
__ – to find probability value.
Computer
Bonferroni Correction
Two advantages:
- it controls our type I error rates,
which means that we don’t go making spurious
statements. 2. it is easy to understand.
Whenever we do ___, we can Bonferroni
correct by multiplying the probability value by the
number of tests, and treating this as the probability
value.
multiple tests
Or equivalently, dividing our cut-off by the ___, and rejecting only null hypotheses that have
significance values lower than that cut-off.
number of
tests
Problem: Bonferroni Correction
it is a very unwieldy and very blunt tool.
* Not that precise.
* The p-values required for statistical significance rapidly
become very small.
- Non-parametric test used with independent groups design.
The Kruskal-Wallis Test
Substitute for one-Way ANOVA if assumptions are
violated.
The Kruskal-Wallis Test
The Kruskal-Wallis Test Does not assume population __
normality or homogeneity of
variance.
The Kruskal-Wallis Test: Requires only __ scaling of
__ variable
ordinal; dependent
Kruskal-Wallis Test: The statistic we compute is .
H
Kruskal-Wallis Test
Step 1:
All of the scores are grouped together and rank-ordered, assigning the rank
Kruskal-Wallis Test
Step 2:
When this is done, the ranks for each condition or sample are
summed- evaluate stats
To use the Kruskal-Wallis test, the data must be of at least
__ scaling.
ordinal
there must be at least __ scores in each sample to
use the probabilities given in the table for Chi-square.
five
