Transcript Basic principles of probability theory

Design of experiment: ANOVA and testing
hypotheses
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ANOVA
Interactions
Contrasts
Multiple hypothesis tests
ANOVA tables
Standard ANOVA tables look like
effect
df
SSh
MS
F
prob
v1
d1
SS1
MS1=SS1/d1
MS1/MSe
pr1
...
...
...
...
…
…
vp
dp
SSp
MSp=SSp/dp
MSp/MSe
prp
error
de
SSe
MSe=SSe/de
total
N
SSt
Where v1,,,vp are values we want to test if they are 0. df-s is degrees of freedom corresponding to
these values. SSh-s are corresponding sum of the squares (h denotes hypothesis). F is Fvalue used for F distribution. Its degrees of freedom are (di,de). Prob-s are corresponding
probability. If a particular probability is very low (less than 0.05) then we reject hypothesis
that this value is 0. These values are calculated using likelihood ratio test. Let us say we
want to test hypothesis:
H0: vi=0 vs H1:vi0
Then we maximise likelihood under null hypothesis to find corresponding variance and then we
maximise the likelihood under alternative hypothesis and find corresponding variance.
Then we calculate sum of the squares for null and alternative hypotheses and find Fstatistics
LR test for ANOVA
Suppose variances are:
ˆˆ 2 for null hypothesis ˆ 2 for thealternative hypothesis
Then mean sum of the squares for the null and alternative hypotheses as:
SSh 
SSe 
ˆˆ 2  
ˆ2

dfh
and for the alternative hypothesis
ˆ2

dfe
Since the first sum of the squares is 2 with degrees of freedom dfh and the second sum of squares is 2 with
degrees of freedom dfe and they are independent then their ratio has F-distribution with degrees of
freedom (dfh,dfe). Degrees of freedom of hypothesis is found using the number of levels of the factor
minus 1in the simplest case.
Using this type of ANOVA tables we can only tell if there is significant differences between means. It does
not tell which one is significantly different.
Degrees of freedom of hypothesis is defined by the number of constraints it implies. Degrees of freedom of
error is as usual the number of observations minus the number of (estimated) parameters
Example: Two way ANOVA
Let us consider an example taken from Box, Hunter and Hunter. Experiment was done on
animals. Survival times of the animals for various poisons and treatment was tested.
Table is:
treatment
A
poisons
I
B
C
D
0.31 0.82 0.43
0.45 1.10 0.45
0.46 0.88 0.63
0.43 0.72 0.76
0.45
0.71
0.66
0.62
II
0.36
0.29
0.40
0.23
0.92
0.61
0.49
1.24
0.56
1.02
0.71
0.38
III
0.22
0.21
0.18
0.23
0.30
0.37
0.38
0.29
0.44
0.35
0.31
0.40
0.23
0.25
0.24
0.22
0.30
0.36
0.31
0.33
ANOVA table
ANOVA table produced by R:
Df
Sum Sq
pois
2
1.03828
treat
3
0.92569
pois:treat 6
0.25580
Residuals 36 0.83013
Mean Sq
0.51914
0.30856
0.04263
0.02306
F value
22.5135
13.3814
1.8489
Pr(>F)
4.551e-07 ***
5.057e-06 ***
0.1170
Most important values are F and Pr(>F).
In this table we have tests for main effects - pois. and treat. Moreover we have “interactions”
between these two factors. If there are significant interactions then it would be difficult to
separate effects of these two factors. They should be considered simultaneously. In this
case pr. for interaction is not very small and it is not large enough to discard interaction
effects with confidence. In these situations transformation of the observations might help.
Let us consider ANOVA table for the transformed observations. Let us use
transformation 1/y. Now ANOVA table looks like:
Df
Sum Sq Mean Sq
F value
Pr(>F)
pois
2
34.903
17.452
72.2347
2.501e-13 ***
treat
3
20.449
6.816
28.2131
1.457e-09 ***
pois:treat 6
1.579
0.263
1.0892
0.3874
Residuals 36
8.697
0.242
ANOVA table
According to this table pr. corresponding to the interaction terms is high. It means that
interaction for the transformed observations is not significant. We could remove
interaction terms. We can build ANOVA table without the interactions. It will look like:
Df Sum Sq
pois
2 34.903
treat
3 20.449
Residuals 42 10.276
Mean Sq
17.452
6.816
0.245
F value
71.326
27.858
Pr(>F)
3.124e-14 ***
4.456e-10 ***
Now we can say that there is significant differences between poisons as well as treatments.
Sometimes it is useful to use transformation to reduce the effect of interactions. For this
several different transformations (inverse, inverse square, log) could be used.
Interactions
For one way ANOVA we need 1) to find out if there is significant effect
(difference between means); 2) to analyse and find out where this difference
is located
For two way or higher levels we need first to consider interactions. We would like
to remove interactions if we can. Otherwise tests for the main effects
(differences between means of observations corresponding to different levels
of factors) may not be reliable. If we see significant interactions then we can
try transformations of the data. Box and Cox family of variance stabilisiting
transformations is one way of removing interactions. If we find necessary
transformation then we can carry out further analysis using the transformed
data or use GLM with corresponding link function.
If interactions can not be removed then we may need to combine all pairs together
and work with them as if we have one way anova with IJ levels, where I is
the number of levels for the first factor and J is the number of levels for the
second factor.
What to do if there is an effect
ANOVA tables will tell us if there is an effect somewhere. But it does not say where is this
effect. If we can make conclusions that there is significant differences then we
should carry on and find out where are these differences. One way (not
recommended) is to look at the confidence intervals using confint. For example for
poison data:
2.5 %
97.5 %
(Intercept) 0.22201644 0.40631690
poison1
-0.09299276 0.01986776
poison2
-0.13414253 -0.06898247
treatB
0.23217989 0.49282011
treatC
-0.05198677 0.20865344
treatD
0.08967989 0.35032011
Since these intervals were calculated after making decision that there are significant main
effects they meant to be more reliable (we know that there are effects so effects we
see on this table may correspond to actual effects that exist). It is called Fisher least
significant differences - Fisher’s LSD. Confidence intervals based directly on t
distribution are very optimistic when many tests are performed simultaneously.
Contrasts
Let us say we have m parameters -  estimate and we have a vector c with m elements and with
the property: ici=0. If we are dealing with unbalanced anova then inici=0 should be
considered. Then we can build hypothesis H0: i ici=0.
i ici is called contrast. If we have a mxk matrix C then we can form k contrasts. Matrix C is
called contrasts matrix. Usually this matrix is orthogonal. I.e.
jcij =0
jcij ckj =0
In anova we are interested in effects, i.e. differences between between effects of different
levels of some factors. They can be written using contrasts matrix. Default contrasts
matrix used in R - contr.treatment uses this type of contrast matrix.
Note that R uses contrasts by default and during the estimation usually not the parameters
themselves but those corresponding to contrasts are estimated.
Finding where the effect is becomes multiple hypotheses testing problem.
Multiple comparison
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One comparison - use t-test or equivalent
Few comparisons - use Bonferroni
Many comparisons - Tukey’s honest significant differences, Holm, Scheffe or
others
Bonferroni correction
If there is only one comparison then we can use t-test or intervals based on t
distribution. However if the number of tests increases then probability that
significant effect will be observed when there is no significant effect becomes
very large. It can be calculated using 1-(1-)n, where  is significance level and
n is the number of comparisons. For example if the significance level is 0.05
and the number of comparisons (tests) is 10 then the probability that at least one
significant effect will be detected by chance is 1-(1-0.05)10=0.40. Bonferroni
suggested using /n instead of  for designing simultaneous confidence
intervals. It means that the intervals will be calculated using
i-j  t/(2n)(se of comparison)0.5
Clearly when n becomes very large these intervals will become very conservative.
Bonferroni correction is recommended when only few effects are compared.
pairwise.t.test in R can do Bonferroni correction.
When finding p-values then p values are multiplied by the number of comparisons
(Note that of we are testing effects of I levels of factor then the number of
comparisons is I(I-1)/2
Bonferroni correction: Example
Let us take the example dataset - poisons and try Bonferroni correction for each factor:
pairwise.t.test(poison,treat,”none”,data=poisons)
1
2
2 0.32844 3 3.3e-05 0.00074
pairwise.t.test(poison,treat,”bonferroni”,data=poisons)
1
2
2 0.9853 3 1e-04 0.0022
As it is seen each p-value is multiplied by the number of comparisons 3*2/2 = 3. If the
corresponding adjusted p-value becomes more than one then it is truncated to one.
It says that there is significant differences between effects of poisons 1 and 3 and between 2
and 3. Difference between effects of poisons 1 and 2 is not significant.
Note: Command in R - pairwise.t.test can be used for one way
anova only.
Holm correction
Another correction for multiple tests - Holmes correction is less conservation than
Bonferroni correction. It is a modification of Bonferroni correction. It works in
a sequential manner.
Let us say we need to make n comparisons an significant level is . Then we
calculate p values for all of them and sort them and apply the following
procedure:
1)
set i = 1
2)
If pi< /(n-i+1) then it is significant, otherwise it is not.
3)
If a comparison number i is significant then increment i by one and if i  n go
to the step 2
The number of significant effects will be equal to i where the procedure stops.
When reporting p-values Holm correction works similar to Bonferroni but in a
sequential manner. If we have m comparisons then the smallest p value is
multiplied by m, the second smallest is multiplied by m-1, j-th comparison is
multiplied by m-j+1
Holm correction: example
Let us take the example - the data set poisons and try Holm correction for each factor:
pairwise.t.test(poison,treat,”none”,data=poisons)
1
2
2 0.32844 3 3.3e-05 0.00074
pairwise.t.test(poison,treat,”holm”,data=poisons) # this correction is the default in R
1
2
2 0.3284 3 1e-04 0.0015
The smallest is multiplied by 3 the second by 2 and the largest by 1
It says that there is significant differences between effects of poisons 1 and 3 and between 2
and 3. Difference between effects of poisons 1 and 2 is not significant.
Tukey’s honest significant difference
This test is used to calculate simultaneous confidence intervals for differences of all
effects.
Tukey’s range distribution. If we have a random sample of size N from normal
distribution then the distribution of stundetised range - (maxi(xi)-mini(xi))/sd is
called Tukey’s distribution.
Let us say we want to test if i- j is 0. For simultaneous 100% confidence intervals
we need to calculate for all pairs lower and upper limits of the interval using:
difference ± ql, sd (1/Ji+1/Jj)0.5 /2
Where q is the -quantile of Tukey’s distribution, Ji and Jj are the numbers of
observations used to calculate I and j, sd is the standard deviation, l is the
number of levels to be compared and  is the degree of freedom used to
calculate sd.
Tukey’s honest significant difference
R command to perform this test is TukeyHSD. It takes an object derived using aov as an input
and gives confidence intervals for all possible differences. For example for poison data
(if you want to use this command you should use aov for analysis)
lm1 = aov(time~poison+treat,data=poisons)
TukeyHSD(lm1)
$poison
diff
lwr
upr
p adj
2-1 -0.073125 -0.2089936 0.0627436 0.3989657 # insignifacnt
3-1 -0.341250 -0.4771186 -0.2053814 0.0000008 # significant
3-2 -0.268125 -0.4039936 -0.1322564 0.0000606 # significant
$treat
B-A
C-A
D-A
C-B
D-B
D-C
diff
0.36250000
0.07833333
0.22000000
-0.28416667
-0.14250000
0.14166667
lwr
upr
p adj
0.18976135 0.53523865 0.0000083 #sginificant
-0.09440532 0.25107198 0.6221729 #insgnific
0.04726135 0.39273865 0.0076661 #significant
-0.45690532 -0.11142802 0.0004090 #significant
-0.31523865 0.03023865 0.1380432 #insignific
-0.03107198 0.31440532 0.1416151 #insignific
Scheffe’s simultaneous confidence intervals
If we have a parameter vector  then a linear combination cT is estimatable if there
exists a vector a so that E(aTy) = cT.
Scheffe’s theorem states that simultaneous 100(1-)% confidence interval for all
estimatable  is:
 + (q Fq,n-r() )1/2 (var()1/2
q is the dimension of the dimension of space of all possible contrasts, r is the rank of
X (design matrix), n is the number of observations. It can also be applied for
regression surface confidence intervals
xT + (q Fq,n-r() )1/2 (var(xT(XTX)-1x))1/2
Testing for homogeneity of variances
One of the simplest tests for homogeneity is using Levene’s test. It can be done using
the algorithm: 1) fit linear model; 2) calculate residuals; 3) fit linear model
using absolute values of residuals as a response vector and 4) test for
differences. If p-values are very small then variances are not equal (homogenic)
otherwise variances are not homogenic. There is also an R command levene.test
Another test for homogeneity of variances is Bartlett test. It can be done using
bartlett.test in R. Bartlett test is sensitive to violation normality assumption.
Levene’s test is less sensitive to such violation.
Testing for homogeneity of variances
Example:
lm2 = lm(time~poison+treat)
summary(lm(abs(lm2$res)~poison+treat))
Residual standard error: 0.09448 on 42 degrees of freedom
Multiple R-squared: 0.2533,
Adjusted R-squared: 0.1644
F-statistic: 2.849 on 5 and 42 DF, p-value: 0.02649
P value is 0.026 small enough (usually you would look around 0.01).
lm2 = lm(1/time~poison+treat)
summary(lm(abs(lm2$res)~poison+treat))
Residual standard error: 0.2634 on 42 degrees of freedom
Multiple R-squared: 0.1494,
Adjusted R-squared: 0.04812
F-statistic: 1.475 on 5 and 42 DF, p-value: 0.2183
Now p-value is sufficiently big. So we can say that variances are homogenic.
References
1.
2.
3.
4.
5.
Stuart, A., Ord, KJ, Arnold, S (1999) Kendall’s advanced theory of
statistics, Volume 2A
Box, GEP, Hunter, WG and Hunter, JS (1978) Statistics for
experimenters
Berthold, MJ and Hand, DJ. Intelligent Data Analysis
Cox, GM and Cochran WG. Experimental design
Dalgaard, P. Introductory Statistics with R