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Variance
Simple analysis of variance
Confidence Interval of the Mean
t values
Do two means differ
F test
Simple analysis of variance
So far, we’ve assumed that all observed
variance comes from a single, random source.
not likely
there can be many sources of variance
We’ll now introduce a way to analyze
the variance in sample sets.
Analysis of variance
In general, when sources of variance are linearly
related (independent and uncorrelated), the variances
are additive.
We often need to do experiments to evaluate the
magnitude and sources of variance.
s
total
2
= s
1
2
+ s
2
2
+.... + s
k
2
Determining sources of variance
Let’s start with a simple example where there
should only be two potential sources of variance.
In this example, a series of four samples
are obtained and each is analyzed in
triplicate.
Sample Replicates Mean
Determining sources of variance
Since there is error in any measurement, it’s not
surprising that the means are different.
We want to know if the difference is due to variance in
the method or real sample differences.
Simple two level model
S
between = variance of sample material
S
within = variance of analytical method
S
total
= S
between
+ S
within
Simple two level model
We have two potential sources of variance.
Samples may actually be different.
Run to run errors.
A simple set of calculations can be used to sort
out the sources of variance.
The F test can then be used to determine
if the variance values are significant.
Simple two level model
Level 1
Level 2
Level 1 gives us an idea as to sample variability
Level 2 tells use about the method variability
X
1
X
2
X
3
X
4
x
1
’ x
1
’ x
1
’ x
2
’ x
2
’ x
2
’ x
3
’ x
3
’ x
3
’ x
4
’ x
4
’ x
4
Simple analysis of variance
First, calculate sum of
squares for all values
in your sample :
The variation of the
total mean can the be
calculated as:
ssT = x i - x
T
! ` j
2
MST =
dfT
x i - x
T
! ` j
2
df
T
= total - 1
x T
= Grand Mean
mean of all the points
ss between = nr x
s
- x
T
` j
2
MS between =
df (^) s
ss between
Next, calculate the between sample variance
Then the mean square for the samples
x s
= mean of each sample
df (^) s =# samples - 1
nr =# replicates per sample
ssT = ss between + sswithin
Since you know SST and SSbetween, you can find the
within sample variance by:
The mean sum of squares for our replicates is then:
MSwithin =
dfT - # samples
^ ss T - ss between h
Simple
analysis of
variance
Back to the
example.
ReplicatesReplicatesReplicates
1 15.9 16.1 16.
2 14.9 15.2 15.
3 14.8 15.8 15.
4 16.2 16.0 15.
Source df SS MS
Total
SS
T
Sample
SS
between
Replicate
SS
within
F test
OK. We’ve done several
calculations. Now what?
We can now use the F test to
determine if there is a significant
difference between the two
sources of variance.
F is then compared to F c to see if
the difference is significant. This
will be covered in a bit.
F =
s
small
2
s
big
2
Using XLStat
Note: XLstat does not report Fc values - just the P value
- the probability that your values are NOT different.
Data must be ordered in a single column.
Two methods are available for calculating sum of
squares for your groups - Type I and III. These are only
useful for more complex multivariable ANOVA
Might as well review them at this point.
Sum of squares analysis.
Type I (Sequential)
The Sums of Squares obtained by fitting effects in the order
specified in the model. Type I SS for each effect will change
if the order of the effects in the model is changed.
Type III (Marginal)
The Sums of Squares obtained by fitting each effect after all
the other terms in the model. The Type III SS do not depend
upon the order in which effects are specified in the model.
Sum of squares analysis.
Type I SS - Useful to explore unbalanced experimental data - where
some effects are measured more than others. Can also show flaws in
an experimental design (next chapter)
Type III Sums of Squares are preferable in most cases since they
correspond to the variation attributable to an effect after correcting for
any other effects in the model. They are unaffected by the frequency
of observations.
With a balanced experiment (all combinations measured with equal
frequency), Type I and III give the same results.
Analysis of variance:Analysis of variance:Analysis of variance:Analysis of variance:Analysis of variance:Analysis of variance:
Source DF
Sum of
squares
Mean
squares
F Pr > F
Model 11 438.943 39.904 40.264 < 0.
Error 36 35.678 0.
Corrected
Total
Fcrit =
There is a difference. Can we tell what it is?
In this example, there is <0.01% chance of there NOT
being a difference.
XLStat results.
Lead / Standardized
coefficients
Chemist-A1 Chemist-A
Chemist-A
Chemist-A4 Chemist-A
Chemist-A
Chemist-A
Chemist-A
Chemist-A
Chemist-A
Chemist-A
Chemist-A
-0.
0
1
Variable
Standardized coefficients
This plot shows
how each chemist
performed.
While results have
been normalized,
you’d get the
same basic plot
with the raw data.
Using XLStat
The Dunnett test is used to
compare samples (your chemists)
to a control.
There actually is no control but
the test provides a useful way of
comparing results.
In this case, choose Chemist A
because his/her results were the
lowest, causing the results to be
positive for the others.
Dunnett test
Compares group means.
Each is pitted against one control or
reference group.
Calculate a t test values for each group
comparison.
Test typically can only be used when all
groups are of equal size.
Dunnett test
Category Difference
Standardize
d difference
Critical
value
Critical
difference
Pr > Diff Significant
A1 vs A12 -10.798 -15.339 2.890 2.034 0.000 Yes
A1 vs A9 -8.078 -11.475 2.890 2.034 0.000 Yes
A1 vs A5 -6.345 -9.014 2.890 2.034 0.000 Yes
A1 vs A4 -6.328 -8.989 2.890 2.034 0.000 Yes
A1 vs A7 -5.838 -8.293 2.890 2.034 0.000 Yes
A1 vs A10 -5.227 -7.426 2.890 2.034 0.000 Yes
A1 vs A8 -4.903 -6.964 2.890 2.034 0.000 Yes
A1 vs A6 -4.475 -6.357 2.890 2.034 0.000 Yes
A1 vs A3 -2.575 -3.658 2.890 2.034 0.007 Yes
A1 vs A11 -2.165 -3.076 2.890 2.034 0.032 Yes
A1 vs A2 -0.105 -0.149 2.890 2.034 1.000 No
Tukey Test
“Honestly Significantly Different (HSD) test.
Based on pairwise comparison among means.
Mi - Mj = difference between pair means
MSE = mean square error
nh = the harmonized mean
Harmonized mean is the weighted
arithmetic mean, with each value's weight
being the reciprocal of the value.
ts =
nh
MSE
Mi - Mj
Harmonized
mean
nh =
xi
i = 1
n /
x
Tukey test
Contrast Difference
Standardized difference
Critical value Pr > Diff Significant
A12 vs A1 10.798 15.339 3.490 < 0.0001 Yes
A12 vs A2 10.693 15.190 3.490 < 0.0001 Yes
A12 vs A11 8.633 12.263 3.490 < 0.0001 Yes
A12 vs A3 8.223 11.681 3.490 < 0.0001 Yes
A12 vs A6 6.323 8.982 3.490 < 0.0001 Yes
A12 vs A8 5.895 8.374 3.490 < 0.0001 Yes
A12 vs A10 5.570 7.913 3.490 < 0.0001 Yes
A12 vs A7 4.960 7.046 3.490 < 0.0001 Yes
Compares each
chemist’s results
to see if there is a
significant
difference.
Tukey test
Provides grouping
of chemists with
statistically similar
results (95%
confidence.)
Chemist Means GroupsGroupsGroupsGroupsGroupsGroups
A12 45.170 A
A9 42.450 B
A5 40.718 B C
A4 40.700 B C
A7 40.210 B C
A10 39.600 C
A8 39.275 C D
A6 38.848 C D E
A3 36.948 D E
A11 36.538 E F
A2 34.478 F
A1 34.373 F
t values
t values account for error introduced based on
sample size, degrees of freedom and potential
sample skew. Actually use χ
distribution - chi
squared.
n-
= (n-1) s
This allows us to estimate population
variance from sample variance. All of
this is tied together into the t values.
t
values
Confidence level
Degrees of 90% 95% 99%
freedom t . t . t .
Example
Data: 1.01, 1.02, 1.10, 0.95, 1.
mean = 1.
s
x
s
x
t values for 4 degrees of freedom
90% confidence = 2.
95% confidence = 2.
Example
CI
CI
2.13 x 0.
5
1/
2.78 x 0.
5
1/
t test example
Beyond the mean
You can have samples that are considered significantly
different and still have the same mean.
In both examples, the populations would be considered to
be different - even though the means, medians and modes
are identical in example on the right.
The F test
This test can be used to tell if two populations
are different based on changes in variance.
Examples
Has the measurement precision changed?
Has the method been altered?
Were there any significant changes due to
the lab or analyst?
The F test
Calculation of F
F is always 1 or greater and depends on the
confidence level and degrees of freedom for
both data sets
You can look up the F
c
value for the
desired levels or use a spreadsheet.
F =
S
2 larger
S
2 smaller
Example
A - mean = 50 mg/l, s = 2.0 mg/l, n = 5, df = 4
B - mean = 45 mg/l, s = 1.5 mg/l, n = 6, df = 5
F = 2
F
c is 5.19 at 95% confidence
The variance values are essentially the same so the
means must really differ.
You need to be concerned with differences
in both the mean and sample variance.
For this example, F
would not exceed
F
c but the means
are significantly
different.
Comparison of the methods
The difference in the means is
smaller than the sample
variance.
Comparison of the methods
Here, the means are identical but
the distributions look different.
However, the lower curve is for a
much smaller data set.
The F test would show them to be
the same.
It accounts for the variations in
sample size - using df.
