MARE 250

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Multiway, Multivariate, Covariate, ANOVA
MARE 250 Dr. Jason Turner
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One-way, Two-way
For Example One-Way ANOVA means of urchin s
from each distance (shallow, middle, deep) are
equal Response urchin , Factor
distance Two-Way ANOVA means of urchins from
each distance collected with each quadrat (0.25,
0.5) are equal Response urchin , Factors
distance, quadrat
If our data was balanced it is not!
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Two-Way Multiway ANOVA
The two-way ANOVA procedure does not support
multiple comparisons To compare means using
multiple comparisons, or if your data are
unbalanced use a General Linear Model General
Linear Model - means of urchin s and species
s from each distance (shallow, middle, deep)
are equal Responses urchin , Factor
distance, quadrat UnbalancedNo Problem! Or
multiple factors General Linear Model - means of
urchin s and species s from each distance
(shallow, middle, deep) are equal Responses
urchin , Factor distance, quadrat, transect
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Analysis of Covariance
Interaction relationship between two factors
when the effect of one factor is not independent
of the effect of another e.g. of urchins at
each distance is effected by quadrat
size Covariance relationship between two
responses when two responses are not
independent e.g. - of urchins and species
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Analysis of Covariance

• We can assess Covariance in 2 ways
• 1. Run a covariance test
• Run a correlation
• Both help us to determine whether (or not) there
  is a linear relationship between two variables
  (our responses)

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Assessing Covariance using Correlation
Relationship between covariance and
correlation Although both the correlation
coefficient and the covariance are measure of
linear association, they differ in the following
ways correlations coefficients are
standardized, thus a perfect linear relationship
will result in a coefficient of 1. covariance
values are not standardized, thus the value for a
perfect linear relationship will depend on the
data. The correlation coefficient is a function
of the covariance. The correlation coefficient is
equal to the covariance divided by the product of
the standard deviations of the variables. Thus, a
positive covariance will always result in a
positive correlation and similarly, a negative
covariance will always result in a negative
correlation.
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Co-whattheheckareyoutalkingabout?
Pearson correlation (just like our RJ
test) Urchins and Species 0.642 P-Value
0.000 (greater than 0 linear relationship) Covar
iances Urchins, Species Urchins
11.991055 Species 1.582084
0.506967 (positive relationship negative
negative
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Co-whichoneshouldIuse?
It is important to note that covariance does not
imply causality (relationship between cause
effect) Can determine that using
Correlation SOrun a Correlation between
responses to determine if there is Covariance If
Covariance than run MANOVA with other Response as
a Covariate
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Assessing Covariance using Correlation
Scatterplot graph of one response (x-axis)
plotted versus another (y-axis)
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Assessing Covariance using Correlation
Correlation coefficient (Pearson) measures the
extent of a linear relationship between two
continuous variables (responses)
Null Correlation 0 Alternative Correlation ?
0 Pearson correlation of Urchins and Species
0.642 P-Value 0.000 (Correlation is
Significantly different than Zero) IF p lt 0.05
THEN the linear correlation between the two
variables is significantly different than 0 IF p
gt 0.05 THEN you cannot assume a linear
relationship between the two variables Conclusion
there IS a linear relationship
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Assessing Covariance using Correlation
Null Correlation 0 Alternative Correlation ?
0 Pearson correlation of Urchins and Depth
0.109 P-Value 0.071 IF p lt 0.05 THEN the
linear correlation between the two variables is
significantly different than 0 IF p gt 0.05 THEN
you cannot assume a linear relationship between
the two variables Conclusion there IS NO
linear relationship
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MANOVA
Multivariate Analysis of Variance - compare means
of multiple responses at multiple factors
MANOVA for Method s 1 m 0.0 n 26.5
Test DF
Criterion Statistic F Num
Denom P Wilks' 0.63099
16.082 2 55 0.000 Lawley-Hotelling
0.58482 16.082 2 55
0.000 Pillai's 0.36901 16.082 2
55 0.000 Roy's
0.58482 16.082 2 55 0.000
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MANOVA
By default, MINITAB displays a table of the four
multivariate tests for each term in the
model Wilks' test - the most commonly used test
because it was the first derived and has a
well-known F approximation Lawley-Hotelling -
also known as Hotelling's generalized T statistic
or Hotellings Trace Pillai's - will give
similar results to the Wilks' and
Lawley-Hotelling's tests Roy's - use only when
the mean vectors are collinear does not have a
satisfactory F approximation Though Wilks' test
is the most widely used method, Pillai's is often
considered the best test to use in most situations
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MANOVA
Multivariate Analysis of Variance compare means
of multiple responses at multiple factors
MANOVA for Method s 1 m 0.0 n 26.5
Test DF
Criterion Statistic F Num
Denom P Wilks' 0.63099
16.082 2 55 0.000 Lawley-Hotelling
0.58482 16.082 2 55
0.000 Pillai's 0.36901 16.082
2 55 0.000 Roy's
0.58482 16.082 2 55 0.000
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MANOVA
Multivariate Analysis of Variance compare means
of multiple responses at multiple
factors Responses Urchins, Species Factors
Distance, Quadrat Q - Why not just run multiple
one-way ANOVAs????? A - When you use multiple
one-way ANOVAs to analyze data, you increase the
probability of a Type I error. MANOVA controls
the family error rate, thereby minimizing the
probability of making one or more type I errors
for the entire set of comparisons.
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Error! Error!
The probability of making a TYPE I Error
(rejection of a true null hypothesis) is called
the significance level (a) of a hypothesis
test TYPE II Error Probability (ß)
nonrejection of a false null hypothesis
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MANOVA
In Conclusion We run ANOVA instead of multiple
t-tests to investigate 1 response versus multiple
factors We run MANOVA instead of multiple
one-way ANOVAs to investigate multiple responses
versus multiple factors