ANalysis Of VAriance

1
ANalysis Of VAriance (ANOVA)
Comparing gt 2 means Frequently applied to
experimental data Why not do multiple t-tests?
If you want to test H0 m1 m2 m3 Why not
test m1 m2 m1 m3 m2 m3
For each test 95 probability to correctly fail
to reject (accept?) null, when null is really true
0.953 probability of correctly failing to
reject all 3 0.86
2
Probability of if incorrectly rejecting at least
one of the (true) null hypotheses 1 - 0.86
0.14 As you increase the number of means
compared, the probability of incorrectly
rejecting a true null (type I error) increases
towards one Side note possible to correct
(lower) ? if you need to do multiple tests
(Bonferroni correction)- unusual
3
ANOVA calculate ratios of different portions of
variance of total dataset to determine if group
means differ significantly from each
other Calculate F ratio, named after R.A.
Fisher
1) Visualize data sets 2) Partition variance (SS
df) 3) Calculate F (tomorrow)
4
Pictures first
3 fertilizers applied to 10 plots each (N30),
yield measured How much variability comes from
fertilizers, how much from other factors?
8
7
6
5
Overall mean
Yield (tonnes)
4
3
2
Fert 1
Fert 3
Fert 2
1
0
0
10
20
30
Plot number
5
Thought Questions
What factors other than fertilizer (uncontrolled)
may contribute to the variance in crop
yield? How do you minimize uncontrolled factors
contribution to variance when designing an
experiment or survey study? If one wants to
measure the effect of a factor in nature (most of
ecology/geology), how can or should you minimize
background variability between experimental
units?
6
Fertilizer (in this case) is termed the
independent or predictor variable or explanatory
variable Can have any number of levels, we have
3 Can have more than one independent variable.
We have 1, one way ANOVA Crop yield (in this
case) is termed the dependent or response
variable Can have more than one response
variable. multivariate analysis (ex MANOVA).
Class taught by J. Harrell
7
Pictures first
-calculate deviation of each point from
mean -some and some - -sum to zero
(remember definition of mean)
8
7
6
5
Overall mean
Yield (tonnes)
4
3
2
Fert 1
Fert 3
Fert 2
1
0
0
10
20
30
Plot number
8
Sum the squared values
Square all values
9
why (n-1)?? Because all deviations must sum
to zero, therefore if you calculate n-1
deviations, you know what the final one must be.
You do not actually have n independent pieces of
information about the variance.

n-1
SS not useful for comparing between groups, it is
always big when n is big. Using the mean SS
(variance) allows you to compare among groups
calculate mean SS a.k.a. variance
10
Partitioning Variability
Back to the question How much variability in
crop yield comes from fertilizers (what you
manipulated), how much from other factors (that
you cannot control)?
Calculate mean for each group, ie plots with
fert1, fert2, and fert3 (3 group means)
But first imagine a data set where
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-Imagine case were the group (treatment) means
differ a lot, with little variation within a
group -Group means explain most of the
variability
Group means
8
7
6
5
Overall mean
Yield (tonnes)
4
3
2
Fert 1
Fert 3
Fert 2
1
0
0
10
20
30
Plot number
12
Now. imagine case were the group (treatment)
means are not distinct, with much variation
within a group -Group means explain little of
the variability -3 fertilizers did not affect
yield differently
Group means
8
7
6
5
Overall mean
Yield (tonnes)
4
3
2
Fert 1
Fert 3
Fert 2
1
0
0
10
20
30
Plot number
13
H0 mean yield fert1 mean yield fert2 mean
yield fert3 Or Fertilizer type has no effect
on crop yield
-calculating 3 measures of variability, start
by partitioning SS
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Sum of squares of deviations of data around the
grand (overall) mean (measure of total
variability)
Total SS
Within group SS (Error SS)
Sum of squares of deviations of data around the
separate group means (measure of variability
among units given same treatment)
Unfortunate word usage
Sum of squares of deviations of group means
around the grand mean (measure of variability
among units given different treatments)
Among groups SS
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k number experimental groups Xij datum j in
experimental group I Xbari mean of group I Xbar
grand mean
Sum of squares of deviations of data around the
grand (overall) mean (measure of total
variability)
Total SS
2
ni
k
Total SS
?
?
Xij - X
j1
i1
Sum of deviations of each datum from the grand
mean, squared, summed across all k groups
Total SS
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k number experimental groups Xij datum j in
experimental group I Xbari mean of group I Xbar
grand mean
Sum of squares of deviations of data around the
separate group means (measure of variability
among units given same treatment)
Within group SS
2
ni
k
?
?
Xij - Xi
Within group SS
i1
j1
Sum of deviations of each datum from its group
mean, squared, summed across all k groups
Within group SS
17
k number experimental groups Xij datum j in
experimental group I Xbari mean of group I Xbar
grand mean
Sum of squares of deviations of group means
around the grand mean (measure of variability
among units given different treatments)
Among groups SS
2
k
?
Among groups SS
Xi - X
ni
i1
Sum of deviations of each group mean from the
grand mean, squared
Among groups SS
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partitioning DF
Total number experimental units -1 In fertilizer
experiment, n-1 29
Total df
units in each group -1, summed for all groups In
fertilizer experiment, (10-1)3 9327
Within group df (Error df)
Unfortunate word usage
Number group means -1 In fertilizer experiment,
3-12
Among groups df
19
SS and df sum
Total SS within groups SS among groups
SS Total df within groups df among groups df
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Mean squares
Combine information on SS and df
Total mean squares total SS/ total
df total variance of data set
Within group mean squares within SS/ within
df variance (per df) among units given
same treatment
Error MS
Unfortunate word usage
Among groups mean squares among SS / among
df variance (per df) among units given
different treatments
21
Tomorrow the big F example calculations
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Mean squares
Combine information on SS and df
Total mean squares total SS/ total
df total variance of data set
Within group mean squares within SS/ within
df variance (per df) among units given
same treatment
Error MS
Unfortunate word usage
Among groups mean squares among SS / among
df variance (per df) among units given
different treatments
23
? Back to the question Does fitting the
treatment mean explain a significant amount of
variance? In our example. if fertilizer doesnt
influence yield, then variation between plots
with the same fertilizer will be about the same
as variation between plots given different
fertilizers
Among groups mean squares
F
Within group mean squares
Compare calculated F to critical value from table
(B4)
24
If calculated F as big or bigger than critical
value, then reject H0 But remember. H0 m1
m2 m3 Need separate test (multiple
comparison test) to tell which means differ from
which
25
Remember Shape of t-distribution approaches
normal curve as sample size gets very
large But. F distribution is
different always positive skew shape differs
with df
See handout
26
Two types of ANOVA fixed and random effects
models
Calculation of F as
Among groups mean squares
F
Within group mean squares
Assumes that the levels of the independent
variable have been specifically chosen, as
opposed to being randomly selected from a larger
population of possible levels
27
Exs Fixed Test for differences in growth rates
of three cultivars of roses. You want to decide
which of the three to plant. Random Randomly
select three cultivars of roses from a seed
catalogue in order to test whether, in general,
rose cultivars differ in growth rate Fixed
Test for differences in numbers of fast food
meals consumed each month by students at UT, BG,
and Ohio State in order to determine which campus
has healthier eating habits Random Randomly
select 3 college campuses and test whether the
number of fast food meals per month differs among
college campuses in general
28
In random effects ANOVA the denominator is not
the within groups mean squares Proper
denominator depends on nature of the
question Be aware that default output from
most stats packages (eg, Excel, SAS) is fixed
effect model
29
Assumptions of ANOVA Assumes that the variances
of the k samples are similar (homogeneity of
variance of homoscedastic) robust to
violations of this assumption, especially when
all ni are equal Assumes that the underlying
populations are normally distributed also
robust to violations of this assumption
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Model Formulae
Expression of the questions being asked
Does fertilizer affect yield?
(word equation)
yield fertilizer
response var
explanatory var
Right side can get more complicated
31
General Linear Models
Linear models relating response and explanatory
variables and encompassing ANOVA ( related
tests) which have categorical explanatory
variables and regression ( related tests) which
have categorical explanatory variables
In SAS proc glm executes ANOVA, regression and
other similar linear models Other procedures can
also be used, glm is most general
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data start infile 'C\Documents and
Settings\cmayer3\My Documents\teaching\Biostatisti
cs\Lectures\ANOVA demo.csv' dlm',' DSD
input plot fertilizer yield options ls80
proc print data one set start proc glm
class fertilizer model yieldfertilizer run
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The SAS System 5

1253 Thursday, September 22, 2005
The GLM Procedure
Class Level Information
Class Levels Values
fertilizer 3 1 2
3 Number of Observations
Read 30 Number of
Observations Used 30
The SAS System
6
1253 Thursday, September 22, 2005
The GLM
Procedure Dependent Variable yield
Sum of Source
DF Squares Mean Square F
Value Pr gt F Model 2
10.82274667 5.41137333 5.70 0.0086
Error 27 25.62215000
0.94896852 Corrected Total 29
36.44489667 R-Square Coeff
Var Root MSE yeild Mean
0.296962 20.97804 0.974150
4.643667 Source DF Type
I SS Mean Square F Value Pr gt F
fertilizer 2 10.82274667
5.41137333 5.70 0.0086 Source
DF Type III SS Mean Square F Value
Pr gt F fertilizer 2
10.82274667 5.41137333 5.70 0.0086