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1. (One way analysis of variance)
and
(Multivariate Analysis of variance)
Prepared by
Ammara Aftab
ammara.aftab63@gmail.com
Applied Statistics (MSc) 2015
(University of Karachi)
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2. (one way analysis of variance)
Independent
Dependent
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3. Is to test differences in means (for groups or variables) for
statistical significance..
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4. ONE-WAY ANOVA
H0: There are no differences among the mean values of the groups being compared
(i.e., the group means are all equal)– H0: µ1 = µ2 = µ3 = …= µk
Ha (Conclusion if H0 rejected)?
Not all group means are equal
(i.e., at least one group mean is different from the rest).
H0 in ANOVA?
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5. Number of steps involved in ANOVA
2 Groups: A B
one-step test :
Step 1: Check weather the mean of two groups are different or not
Scenario 2: If we are comparing 3 or more groups :
>3 Groups: A B C
It is a two-step test:
Step 1: Overall test that examines if all groups are equal or not.
And, if not all are equal (H0 rejected), then:
Step 2: Pair-wise (post-hoc) comparison tests to see where (i.e.,
among which groups) the differences exit, and how.
Scenario 1 If we are comparing 2 groups :
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7. MANOVA
(multivariate analysis of variance)
Independent
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8. Is an extension of ANOVA methods to cover
cases where there is more than one dependent
variable.
Is to tests for the difference in two or
more vectors of means.
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9. What are the Basic
requirements?
2 or more continuous DVs
1 or more categorical IVs
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10. Examples
 With a single DV you “put all of your eggs in one basket”
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11. Assumptions
Normal distribution.
Linearity.
Homogeneity of variance:
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12. WHY WE NEED MANOVA
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13. It measure >1 dependent variable
–Multiple correlated responses
It provides a joint test for any significant
effects.
It is use to tests for patterns.
Its…
–Power can be reduced by irrelevant variables
–Tests linear combinations of variables
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14. by Ammara aftab

15. Pillai - This is Pillai's Trace, one of the four multivariate criteria test statistics used in
manova. We can calculate Pillai's trace using the generated eigenvalues Divide each
eigenvalue by (1 + the eigenvalue).
Hotellings - This is Lawley-Hotelling's Trace. It is very similar to Pillai's Trace. It is the
sum of the eigenvalues and is a direct generalization of the F statistic in ANOVA.
Wilks - This is Wilk's Lambda. This can be interpreted as the proportion of the variance
in the outcomes that is not explained by an effect. To calculate Wilks' Lambda, for each
eigenvalue, calculate 1/(1 + the eigenvalue), then find the product of these ratios.
Roys - This is Roy's Largest Root. We can calculate this value by dividing the largest
eigenvalue by (1+largest eigenvalue).
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16. Theoretical and practical issues in
MANOVA
• The interpretation of MANOVA results are always taken in the context
of the research design.
• Choice of IVs and DVs takes time and a thorough research of the
relevant literature
• As with any analysis, theory and hypotheses come first, and these
dictate the analysis that will be most appropriate to your situation.
• Choice of DVs also needs to be carefully considered, and very highly
correlated DVs weaken the power of the analysis
• Missing data, unequal samples, number of subjects and power are also
issue.
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17. Anova vs. Manova
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18. Anova vs. Manova
One way anova just have 1 dependent and 1 independent variable while
manova have more than dependent variable.
Measuring several dependent variables in a single experiment, there is a
better chance of discovering which factor is truly important. While in one
way anova we have to perform every Dv’s indivually again and again.
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19. One-way ANOVA
(one way analysis of variance)
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22. In “factor” box you have to put
the predictor(IV)
In Dependent list
box you have to
put the (DV’s)
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23. While none of the three ANOVAs were statistically significant at the
alpha = .05 level, in particular
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24. Post Hoc tests is used to conduct a
separate comparison between DV’s
and IV’s
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25. Post hoc
Homogeneous Subsets
In this result We can see the
see separate comparison
between all DV’s and IV’s
All the red
circle shows
That with
different
method we get
same result of
significance
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26. Homogeneity of variance test
Click on the Options button in the One-Way ANOVA
dialog box. The One-Way ANOVA Options dialog box
appears:
Click on Homogeneity of Variance
(to get a test of the assumption of
homogeneity of varianceby Ammara aftab

27. If the p value is less than or equal to your α level for this test, then you can
reject the H0 and say that the variances are equal. If the p value is greater
than α level for this test, then we fail to reject H0 which increases our
confidence that the variances are not equal and the homogeneity of
variance assumption may be reasonably satisfied.
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28. For each dependent variable the descriptive output gives the sample
size, mean, standard deviation, minimum, maximum, standard error,
and confidence interval for each level of the independent variable.
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29. MANOVA
(multivariate analysis of variance)
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31. In manova you have
more than one DV,
so you can select
many DVs and put
them into this box
together.
In the “fixed factor”
box you have to put
the predictor(IV’s)
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32. We want to
test for an
effect of the
IV on all DV’s
at once.
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33. Click on the
“observed power”
button
to get the statistical
power of the
multivariate tests.
h. Pillais - This is Pillai's Trace, one of the four multivariate criteria test statistics used in manova. We can calculate Pillai's trace using the generated eigenvalues (see superscript m
i. Hotellings - This is Lawley-Hotelling's Trace. It is very similar to Pillai's Trace. It is the sum of the eigenvalues (see superscript m) and is a direct generalization of the F statistic in
j. Wilks - This is Wilk's Lambda. This can be interpreted as the proportion of the variance in the outcomes that is not explained by an effect. To calculate Wilks' Lambda, for each eig
k. Roys - This is Roy's Largest Root. We can calculate this value by dividing the largest eigenvalue by (1+largest eigenvalue).
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34. The output has two segments. The first part of the results section.
We can see the interaction effects b/w DV’s AND IV’s
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35. The 2nd part gives you the results of the
multivariate tests.
The red circle four numbers are the p-values of the four different multivariate tests. These
results tells there is a significant effect of the IV on all of the DVs, considered as a
group.
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36. by Ammara aftab