Multiway, Multivariate,
Covariate, ANOVA
MARE 250
Dr. Jason Turner
One-way, Two-way…
For Example…
One-Way ANOVA – means of urchin #’s from
each location (shallow, middle, deep) are equal
Response – urchin #, Factor – location
Two-Way ANOVA – means of urchin’s from each
location collected with each quadrat (0.25, 0.5)
are equal
Response – urchin #, Factors – location, quadrat
If our data was balanced – it is not!
Two-Way – 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 location (shallow, middle, deep)
are equal
Responses – urchin #, Factor – location, quadrat
Unbalanced…No Problem!
Multi-Way – ANOVA
Multi-way ANOVA:
Response:1
Factors: >2
Multi-Way ANOVA – means of urchin’s from each
location (shallow, middle, deep) collected with
each quadrat (¼m, ½m), across different years
(2009, 2010, 2011, 2012) are equal
Response – urchin #, Factors – location,
quadrat, year
Multi-Way – ANOVA
Multi-way ANOVA:
Run using a General Linear Model (GLM) – very
similar to the way a 2-way test is run
Multivariate – ANOVA (MANOVA)
Multivariate ANOVA (MANOVA):
Response:>1
Factors:1-?
MANOVA– means of urchin’s and species from
each location (shallow, middle, deep) are equal
Responses – urchin #, species #
Factor – location
MANOVA
Multivariate Analysis of Variance - compare means
of multiple responses at multiple factors
Run using General MANOVA program – like GLM
MANOVA
Multivariate Analysis of Variance - compare means
of multiple responses at multiple factors – results to 4
different types of MANOVAs given as output
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 Hotelling’s Trace
MANOVA
Pillai's - will give similar results to the Wilks' and LawleyHotelling's tests
Roy's - use only when the mean vectors are collinear;
does not have a satisfactory F approximation
For this class we will use Wilks’
MANOVA
Multivariate Analysis of Variance – compare means
of multiple responses at multiple factors
Responses: #Urchins, #Species
Factors: Distance
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.
Error! Error!
The probability of making a TYPE I Error
(rejection of a true null hypothesis) is called
the significance level (α) of a hypothesis test
TYPE II Error Probability (β) – nonrejection of
a false null hypothesis
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
MANOVA
H0: (μUrchS = μUrchM = μUrchD)(μSpecS = μSpecM = μSpecD)
Ha: All means not equal
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
Analysis of Covariance
We can assess Covariance in 2 ways:
1. Run a covariance test
2. Run a correlation
Both help us to determine whether (or not)
there is a linear relationship between
two variables (our responses)
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.
Assessing Covariance using Correlation
Relationship between covariance and correlation
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
Co-whattheheckareyoutalkingabout?
Pearson correlation (just like our RJ test)
(greater than 0 – linear relationship; H0: r=0)
Co-whattheheckareyoutalkingabout?
Pearson correlation (just like our RJ test)
(greater than 0 – linear relationship; H0: r=0)
Co-whattheheckareyoutalkingabout?
Covariances: #Urchins, #Species
(positive # = relationship; negative = negative
Co-whattheheckareyoutalkingabout?
Covariances: #Urchins, #Species
(positive # = relationship; negative = negative
Co-whichoneshouldIuse?
It is important to note that covariance does
not imply causality (relationship between
cause & effect)
Can determine that using Correlation
SO…run a Correlation between responses to
determine if there is Covariance
If Covariance than run MANOVA with other
Response as a Covariate
Co-whichoneshouldIuse?
If you run a MANOVA, and fail to accept the
null hypothesis (H0: means are equal)
Then need to run ANOVA w/ Tukeys on each
individual Response Variable
MANOVA
H0: (μUrchS = μUrchM = μUrchD)
Ha: All means not equal
H0: (μSpecS = μSpecM = μSpecD)
Ha: All means not equal