OneWay Analysis of Variance

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One-Way Analysis of Variance
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The salmon feeding experiment
The numbers are the final weights of salmon after
being fed different formulations of food
WHAT IS THE MAIN DIFFERENCE IN THESE DATA?
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Difference is in how they are distributed (the
coloured numbers represent the different
formulations of feed)
TRIAL 1
2
3
1
1
3
2
3
Within group variability is small while between
group variability is large. (i.e. groups from
distinct clusters)
2
1
1
1
2
2
3
3
15 20 25 30
35 40
TRIAL 2
1
3
3
Within group variability is large and while group
variability is small (i.e. its difficult to pick
out distinct clusters)
1
1
1
1
1
2
2
2
2
3
3
3
2
15 20 25 30
35 40
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• If you want to compare the results of the two
  trials
• or ask the question -
• Is there a difference in the results of the two
  trials?
• You would frame an hypothesis (for each trial)

H0 m1 m2 m3 There is no difference
between the feeds
H1 m1 ? m2 ? m3 There is a difference
between the feeds
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Experimental Design and ANOVA
This experiment is a completely randomized
design because 1) took 15 young salmon 2)
randomly assigned a salmon to a feed regime 3)
each feed formulation is a level of the factor
This kind of design is analysed with a one-way
(or single factor or single classification) ANOVA
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Our hypotheses are
H0 m1 m2 m3 There is no difference
between the feeds
H1 m1 ? m2 ? m3 There is a difference
between the feeds
Why not just analyse these hypotheses with a
series of t-tests?
H0 m1 m2 There is no difference between
the feeds 1 and 2 H1 m1 ? m2 There is a
difference between the feeds 1 and 2
H0 m1 m3 There is no difference between
the feeds 1 and 3 H1 m1 ? m3 There is a
difference between the feeds 1 and 3
H0 m2 m3 There is no difference between
the feeds 2 and 3 H1 m2 ? m3 There is a
difference between the feeds 2 and 3
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As you increase the number of means analysed
pairwise, you increase the chances of a Type I
error (rejecting a valid hypothesis).
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Only formula needed for a one-way ANOVA
Sum of squared differences from the mean Degrees
of freedom
Variance
S(Xi - X)2 n-1

The variance in an ANOVA analysis is called the
Sum of Squares or SS
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There are several sources of variation (as
measured by the sum of squaresSS) in these
data. The goal of an ANOVA analysis is to measure
this variability and decide where it comes from.
The first is Total Variability - measured as
Total Sum of Squares or SST
S(X)2 N
SST SX2 -
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There are several sources of variation (as
measured by the sum of squaresSS) in these
data. The goal of an ANOVA analysis is to measure
this variability and decide where it comes from.
Total Variability can be broken down into two
components
Total Variability

• Variability between treatments
• Which comes from
• The effect of the treatment itself
• Differences between the subjects in the
  experiment
• Experimental error

• Variability within treatments
• Which comes from
• Differences between the subjects in the
  experiment
• Experimental error

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Total Variability

• Variability between treatments
• Which comes from
• The effect of the treatment itself
• Differences between the subjects in the
  experiment
• Experimental error

• Variability within treatments
• Which comes from
• Differences between the subjects in the
  experiment
• Experimental error

We compare these by computing an F statistic
Variability between treatments Variability within
treatments
F
treatment effect individual differences
experimental error individual differences
experimental error

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Back to our data set and consider Trial 1
Overall mean 25
How do Formulae 1, 2, and 3 differ from one
another? i.e. What is the source of any variation
between them?
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COMPUTATION

• Total variability
• Total sum of squares SST

S(X)2 N
SST SX2 -
9513 - 140,625 15 138
Overall Mean - 25
Note in class, we did this as SST (X1 - X)2
(X2 - X)2 (X3 - X)2 (Xn - X)2 (20 -
25)2 (22 - 25)2 (21 - 25)2 (28 - 25)2
138
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COMPUTATION
2) Within treatment variability sum of the sum
of squares SSSWITHIN EACH TREATMENT
S(X)2 N
SSwithin F1 SX2 -
2209 - 11025 5 4
Overall Mean - 25
SSS SSwithin F1 SSwithin F2 SSwithin F3
4 2 2 8 Therefore SSwithin 8
SSwithin F2
3382 - 16900 5 2
SSwithin F2
3922 - 19600 5 2
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COMPUTATION

• 3) Between treatment variability
• - can be computed in two ways

SX2 - S(X)2 n N
SSbetween
Overall Mean - 25
OR Just subtract the SSwithin from the
SStotal Since SStotal SSwithin
Ssbetween So SSbetween SStotal - Sswithin
138 - 8 130
SSbetween 130
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COMPUTATION
4) Calculate Mean Square
Remember formula for variance (slightly
modified!) s2 SS/df
In ANOVA - substitute Mean Square (MS) for s2
SSwithin dfwithin
MSwithin
130/2 65
SSbetween dfbetween
MSbetween
8/12 0.66
MSbetween
And F (the calculated statistic for ANOVA)
65/.66 97.59
MSwithin
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For a simple one-way ANOVA The degrees of
freedom (df) are Dfwithin N - K Dfbetween
K - 1 Dftotal N - 1 Where N total number
of values in the entire experiment K the
number of treatment conditions
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The results of the ANOVA are presented in an
ANOVA table
Source of SS df MS (SS/df) variation
Between 130 2 66 (treatment) (K-1) Within 8
12 0.65 (error, residual) (N-K) Total 138 14
(N-1)
F 97.59
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The Final Step !!!
Look up the critical value for p .05 for 2 and
12 degrees of freedom in any F-table F(.05, 2,
12) 5.10 Since our F-value of 97.59 is much
larger than 5.10 p ltltlt .05 And to go back to
our original hypotheses
H0 m1 m2 m3 There is no difference
between the feeds
H1 m1 ? m2 ? m3 There is a difference
between the feeds
Reject H0 and accept H1 that there is a
difference between the feeds
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Summary of ANOVA logic and procedure
Total variability
S(X)2 N
SST SX2 -

• Between treatments
• Treatment effect
• Individual differences
• Experimental error

• Within treatments
• Individual differences
• Experimental error

SSwithin SSSWITHIN EACH TREATMENT
SX2 - S(X)2 n N
SSbetween
SSbetween dfbetween
MSbetween
SSwithin dfwithin
MSwithin
MSbetween
F
MSwithin