Two and more factors in analysis of variance

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Two and more factors in analysis of variance

• Factorial and nested designs

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Factorial design

• Each level of the first factor is combined with
  each level of the second one. By two levels in
  each factor
• 2 factors -gt 4 combinations
• 3 factors -gt 8 combinations
• Generally Number of combinations is product of
  number of levels for each factor

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Mowing, fertilization, removing of dominant
Usually each combination in several replications
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Factorial designs in terrain - factors shape and
pattern
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Another possibility - nested design
factor A (local)
factor C (plant)
sing. observ.
Plant 1 from the first locality has nothing
common with plant 1 from any other locality.
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Factorial design
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Proportional design

• The same proportion of replications of each
  factor at each level of other factor contingency
  table of no. of replications ?2 equals zero -
  i.e. factors are absolutely independent
• In ideal case is the same number of observations
  in all combinations, but proportional design is
  enough

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formula for expected frequency in contingency
table
So, for example for non-fertilized non-mowed
I.e. the same proportional representation of the
first factors level by each level of the second
factor then we consider the factors independent
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When factors are independent, and design is
balanced
Balanced design Weights of rats
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When factors are independent, and design is
proportional
Proportion design Weights of rats
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When factors are dependent, i.e. design isnt
balanced nor proportional
Non-proportional design Weights of rats
According to marginal means it seems as listening
of music can affect weight of rats. (There are
methods, which can partly cope with it LS
means, but power of test is lowered for both
factors).
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Statistica can compute anything, but

• If I have proportion design, the result should be
  always the same.
• Two-way ANOVA can be computed even in
  non-proportion design default there (Type III
  sum of squares - orthogonal) is alright, but I
  can, according to the experiment situation,
  decide myself for other type (perhaps Type I -
  sequential), and I should know, what means what
  (and why are results different).

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Model of two-way ANOVA
Two factors (mown and fertilized) - index i is
level of the first factor (non-mown, mown), index
j is level of the second one, k replication in
within group response is e.g. number of species.
Grand mean
Effect of fertilization
Effect of mowing
Error variability
Interaction
Parameterisation is usually such, that a, ß, and
? would be balanced around zero (then µ is really
mean of everything).
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Three null hypothesis

• ai0 for all i mowing has no effect
• ßi0 for all j fertilization has no effect
• ?ij0 for all combinations of ij - there is no
  interaction between mowing and fertilization
• Null interaction means, that main effects are
  purely additive

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Null interaction
Effect of every factor is independent of the
level of other factor ATTENTION it means
additivity
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Interaction is deviation from additivity
e.g.
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Can be seen well in graphs (interaction plot)
Do not forget to stress, that connection of means
isnt an interpolation here we just want to
visualize interaction with help of (non)
parallelism of lines
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Can be seen well in diagram (interaction plot)
When I refer about result, it isnt enough to
write that interaction is significant, but one
need to say why (where is the deviation from
additivity).
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Null hypothesis of main effects - averaged over
all levels of the second effect

• ai0 for all i mowing has no effect
  (at mean over all levels of fertilization)
• ßi0 for all j fertilization has no effect
  (at mean over all levels of mowing)

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You have to use head when interpreting results!!!
(and look at diagram)
Administrate two medicines separately and
together (factorial design) - main effects are
insignificant it doesnt mean the medicines are
ineffective though. Just their effects cancel
when applied together.
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Holds again grand/overall variability expressed
with help of SSTOT can be divided
SSA SSB SSAB(interaction)
SSTOT sum of deviations from grand mean SSA
sum of deviations of marginal means of factor A
groups from grand mean, weigh by number of
observations (similar to SSB) SSAB weigh sum of
squares of deviations of means combination from
means if there is pure additivity
Explained by model
Error (Residual)
SSerror(resid)
Expected without interaction
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Example mown, fertilized, number of species as
response
Test of null hypothesis, that mean number of
species is zero everywhere
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a, b are sums of levels for factors A and B, n is
number of observations in all groups Holds DFA
a-1, DFBb-1, DFAB(a-1)(b-1), DFTOTn-1 DFerror
DFTOT - DFA - DFB - DFAB Holds again, that
fraction MS SS/DF is estimation of grand
variance, if null hypothesis is true
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If all the effects are fixed
Test Feffect MSeffect / MSerror
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Problem what is in denomination depends on
which factor is with fixed effect and which
factor with random effect (especially important
if one of the factors is experimental (and thus
of our major interest), and the other is
locality. Important for experimental design
planning!
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I, the experimenter, am the one deciding, which
model I will use
classic ANOVA factorial
ANOVA without interactions (also Main effects
ANOVA) - non-additivity is part of random
variability it makes possible to work with data
with one observation for each factor combination
(better avoid it though)
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Experimental design
C RANDOMIZED BLOCKS
WRONG
LATIN SQUARE
Pseudoreplications
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Completely randomized blocks

• I test by two-way analysis of variance without
  repetition (error variability is deviations from
  additivity, i.e. interaction between block and
  treatment)
• It can give more powerful test, if blocks explain
  something, i.e. help to control variability.

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Multiple comparison
Similar to one-way analysis of variance if I do
it on interaction I compare all
factorially-made groups with each other if I do
it on main effect, I compare additive effects of
single levels. I am the one deciding what will be
compared.
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Friedman test - nonparametric ANOVA for
completely randomized blocks
Based on sequencing values inside block
where a is number of levels of factor studied, b
is number of blocks and Ri is sum of ranks for
level i of factor studied.
31
Two-factorial experiment I compare daisy and
sunflower and their response to level of
nutrients (response is height of plant)
Three null hypothesis 1. Height of daisies and
sunflowers isnt different (it can sometimes
happen, we are testing totally unrealistic null
hypothesis, we didnt need to test this one
obviously) 2. Height of plants is independent of
level of nutrients 3. Effect of level of
nutrients is the same for both species
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We have a problem

• Data are positively skewed (the least important
  problem)
• There is distinctive inhomogeneity of variances
  (CV could be constant, i.e. SD linearly depends
  on mean)
• Classic interaction tests additivity thus if
  fertilization elongates daisies from 10 to 20 cm,
  sunflowers should be elongated from 100 to 110
  cm. From biological point of view this isnt
  absolutely the same effect to both species.

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Additive effect
Multiplicative effect
with every value we multiply error thus SD is
linearly dependent on mean. eijk has lognormal
distribution centered around 1.
After log-transformation
is multiplicative effect changed to additive
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Logarithmic transformation

• Changes lognormal distribution to normal one
• If SD was linearly dependent on mean, it leads to
  homogeneity of variances
• Changes multiplicative effects to additive ones
• ATTENTION it makes everything simultaneously
  I cannot want just one of those

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Many biological data contain zeroes

• Transformation often used X log(X1) has
  similar quality, but not exactly the same,
  especially if there are low X values.
  Particularly inaccurate can be the change from
  multiplicativity to additivity!!!
• Sometimes is used X log(bXa), where a and b
  are constants. (but the change to additivity from
  multiplicativity is never achieved)

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Other transformations used

• For Poisson distribution (numbers of randomly
  placed individuals)
• For percentages (p as a number between 0 and 1)

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Nested design
We measure length of corollas tubes
factor A (local)
factor C (plant)
sing. observ.
Plant 1 from the first locality has nothing
common with plant 1 from any other locality.
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The top factor in hierarchy can be either with
fixed effect or with random one

• Factors in lower position of hierarchy are almost
  always with random effect (it is possible to
  compute it also with fixed one, but it is very
  unusual case)
• In analysis of sum of squares we count squares of
  differences of each observation (or mean) and its
  hierarchically nearest upper relevant mean.
• If hierarchically lower effects are random, then
  we test every effect against nearest
  hierarchically lower effect

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Test of null hypothesis, that mean tube length is
zero
Null hypothesis on lower hierarchical levels
plants do not differ in mean length of their
tubes in scope of locality
Flocality MSlocality/MSplant
Fplant MSplant/MSerror 2,15/2,240,96
Ideal, when model is balanced - Statistika
compute it even if it isnt, but they are various
approximations.
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Most frequent use

• Analysis of variability among single hierarchical
  levels, e.g. in taxonomy
• Often I am interested mainly (only) in
  hierarchically higher factor, everything else is
  just for increasing test power.
• I.e. I can have just 6 pounds, three pastured and
  three non-pastured (I am not able to have more).
  In each of them I lay out 10 squares for biomass
  sampling, and I do three analytic determinations
  from every square. Analysis of variability can
  help me to plan optimal sampling design.

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Mind mixed samples

• I can spare my work, but they must be
  independently replicated!

These arent independent observations
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More complicated models of ANOVA

• Factorial and nested designs can be combined in
  different ways, whereas some of them will be with
  fixed effect and some with random one

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Split plot (main plots and split plots - two
error levels)
6 plots (3 calcite, 3 granite), 3 types of
impacts in each plot
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ANOVA - Repeated measures

• I have some experimental design and I follow the
  state of individual objects in time, e.g. growing
  plants, etc.

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Replicated BACI - repeated measures