This document provides an overview of two-way ANOVA and MANOVA. It defines two-way ANOVA as an analysis method used for studies with two or more categorical explanatory variables and a quantitative outcome. Two-way ANOVA allows investigating the main effects of two factors and their interaction. The document also describes MANOVA, which assesses the effect of one or more independent variables on two or more dependent variables. It provides the assumptions, advantages, and an example of two-way MANOVA. In conclusion, two-way ANOVA and MANOVA are appropriate for analyzing studies with multiple explanatory variables and outcomes.

1.  

2. Going to be present by: Payman EHSASGoing to be present by: Payman EHSAS
Under Supervision of : Prof.Dr. Cuma AKBAYUnder Supervision of : Prof.Dr. Cuma AKBAY
NovemberNovember
20152015
NovemberNovember
20152015

3. Contents:Contents:
i. Introduction of Two-way ANOVA.
ii. Assumption for two factor ANOVA.
iii. The null hypotheses for Two-way ANOVA.
iv. Advantages of Two-way ANOVA.
v. Logic of Two-way ANOVA table.
vi. Example of two-way ANOVA.
vii.MANOVA with details.
viii.Assumption of MANOVA.
ix. Advantage of MANOVA.
x. Example of MANOVA.

4. Two- way (or multi-way) ANOVA is an appropriate analysis method
for a study with a quantitative outcome with two or more categorical
explanatory variables .
On the other hands:On the other hands:
Tow-way ANOVA means groups are defined by two independent
variables.
With Two-way ANOVA there are two main effects and one
interaction so these main effects are typically called factors.
Two-way ANOVA is an extension of the paired t testpaired t test to more than
two treatments.
Introduction:Introduction:
Two-way ANOVATwo-way ANOVA
In Minitab, use the Two-way ANOVA when you have exactly two factors. If you have three or more factors,
use the Balanced ANOVA. Both the Two-way and Balanced ANOVA require that all combinations of factor
levels (cells) have an equal number of observations, i.e., the data must be balanced.
In Minitab, use the Two-way ANOVA when you have exactly two factors. If you have three or more factors,
use the Balanced ANOVA. Both the Two-way and Balanced ANOVA require that all combinations of factor
levels (cells) have an equal number of observations, i.e., the data must be balanced.

5.  Observations within each sample are independent.
 Populations are normally or approximately normally
distributed.
 Populations from which the samples are selected
must have equal variances(homogeneity of variance)(homogeneity of variance).
 The groups must have the same sample size.
Assumptions for the Two FactorAssumptions for the Two Factor
ANOVAANOVA

6.  The null hypotheses in a two-way ANOVA are
these:
 The population means for the DV are equal across
levels of the first factor.
 The population means for the DV are equal across
levels of the second factor.
 The effects of the first and second factors on the DV
are independent of one another.
The null hypotheses in a Two-wayThe null hypotheses in a Two-way
ANOVAANOVA

7.  More efficient to study two factors (A and B)
simultaneously rather than separately.
 We can investigate interactions between factors.
 In a two way ANOVA (A×B design) there are four
sources of variations
 Variation due to factor A
 Variation due to factor B
 Variation due to the interactive effect of A & B
 Within cell or error variation
Advantages of Two-wayAdvantages of Two-way
ANOVAANOVA

8. Source of
variation
SS Df MS
[SS/df]
F Pvalue Fcrit
Main effect A
Main effect B
Interactive
effect
Within
Total
Basic Two-way ANOVABasic Two-way ANOVA
tabletable

9. The variance(total sum of squares) is first partitioned into WITHIN and
BETWEEN sum of squares , sum of squares BETWEEN is next
partitioned by intervention, blocking and interaction.
SS TOTAL
SS WITHIN
SS BETWEEN
SS INTERVENTION SS BLOCKING SS INTERACTION
Main Effect 1
2
2
w
factorA
s
s
F =
Main Effect 2
2
2
w
factorB
s
s
F =
Interaction
2
2
w
ninteractio
s
s
F =
Logic of Two-way ANOVALogic of Two-way ANOVA
(TABLE)(TABLE)

10.  Two-way ANOVA applications:
 In agriculture you might be interested in the effects of both
potassiumpotassium and nitrogennitrogen (factors) on the growthgrowth of potatoes
(response).
 In medicine you might want to study the effects of
medicationmedication and dosedose (factors) on the duration of
headachesheadaches (response).
 In education you might want to study the effects of grade
levellevel and gendergender (factors) on the time required to learn a
skillskill (response).
 A marketing experiment might consider the effects of
advertising eurosadvertising euros and advertising mediumadvertising medium (factors) on
salessales (response).
Usage side of Two-way ANOVAUsage side of Two-way ANOVA

11. Example:
There is four different type of bactericidesbactericides (A.B.C &D) and three
different percent of dosagedosage (10%.20%.30%) to confirm on
bacteriabacteria. The rate of usage is given below , so find the group of
bactericides and impact of dosage of use on bacteria?
30% 20%30% 20%
10%10%
A 21 , 20 , 22 23 , 25 , 21
26, 25 , 24
B 19 , 22 , 25 19 , 18 , 20
20 , 21 , 22
C 18 , 20 , 16 20 , 22 , 24
24 , 25 , 23

15. In first step of spss outcome:
As we see in table, in front of bactericides sig. (p) value is (p=0.000)(p=0.000) (lower than
alpha ) therefor it shows significant , it means impact of bactericides was significant
on the bacteria .
Same as upside the p value of use dosage is (p= 0.00)(p= 0.00) lower than alpha so the
impact of bactericide dosage is also significant on bacteria reproductive system.
On other side complex usage of (bactrecides+dosage) p value is (p=0.012)(a<0.05)(p=0.012)(a<0.05)
So it shows both them had significant impact on bacteria reproduction system.

18. If we gather these information in same text so:
1.Table: Impact of bactericides on the bacteria.
Sort of bactericidesSort of bactericides
A B C D St. Error
Number of bacteria: 23.0 a 20.67 b 21.33 c 27.33 0.54
2. Table: Impact of different amount of bactericides usage on bacteria.
DosageDosage
30% 20% 10% St. Error
Number of bacteria: 21.5 a 22.8 b 25.0 c 0.47

19.  In influence of bactericides and dosage for this analysis (Two-factor analysis): both
bactericides and dosage had good impress.
 As it showed in Duncan multi option analysis result in this analysis from type of
bactericides, D had highest average , B and Cs average was approximately same ,
so comparatively average of A and D bactericides was the same after each other.
 In dosage of bactericides amount as it shows in 2nd table: (10%) dosage was had
high average and 20% , 30% dosage of bactericide had estimated same average.
 If the purpose of this research would be average of bactericides dosage, therefor
we could recommend D bactericides for 10% of dosage.
 If the aim would be on the lowest point of bactericides so we could advice B or C
bactericides with 20% or 30% of dosage.
 Two-way ANOVAs analysis has different standard error in each different group so
for this analysis also it had different standard errors as it shows in table.
Result of example:Result of example:

20. M
ANOVA
M
ANOVA
M
ultivariate analysis of
M
ultivariate analysis of
variance
variance

21.  MANOVA is variation of ANOVA… Which you already know!
 MANOVA assesses the statistical significance of the effect of one or
more IV’s on a set of two or more dependent variables.
 A MANOVA or multivariate analysis of variance is a way to test the
hypothesis that one or more independent variables, or factors, have an
effect on a set of two or more dependent variables.
 The purpose of MANOVA is to test whether the vectors of means for the
two or more groups are sampled from the same sampling distribution.
 MANOVA tests whether mean differences among groups on a
combination of DVs is likely to occur by chance.
 The more important purpose is to explore how independent Variables
influence some patterning of response on the dependent variables.
What is MANOVA?
IV – Independent Variable, DV – Dependent Variable

22. BecauseBecause :
MANOVA has the ability to examine more than one DV
at once or simultaneous effect of IV’s on multiple DV’s.
The major benefit of MANOVA over multiple ANOVAs is
equal to controlling Type I Error rate.
IV – Independent Variable, DV – Dependent Variable
WHYMANOVA?

23.  Multivariate normality.
 Homogeneity of covariance matrices.
 Independence of observations.
 Linearity.
Assumption of MANOVAAssumption of MANOVA

24. Multivariate normality:
• DV should be normally distributed within groups.
• Linear combinations of DV must be distributed.
• All subjects of variables must have multivariate normal
distribution.
Homogeneity of covariance matrices:
• The inter correlations (co variances) of the multiple DV across the
cells of design.
• BOX test is used for this assumption.
Assumption of MANOVAAssumption of MANOVA

25. Outliers :
 As in ANOVA, MANOVA sensitive to outliers.
 It may produce Type 1 error.
 But no indication which type of error it is.
 Several programs available to test for univariate & multivariate
analysis.
Multicollineararity & Singularity:
 High correlation between dependent variables.
 One DV becomes linear combination of other DV.
 So it becomes statistically redundant & suspect to include both
combinations.
Assumption of MANOVAAssumption of MANOVA

26. Advantages of MANOVA overAdvantages of MANOVA over
ANOVAANOVA
 MANOVA produces significant main effect on the DV but
ANOVA do not.
 Because variables are more significant together than
considered separately.
 It considers inter correlations between DV’s.
 It controls the inflation of Type I error*.
 With a single DV you can put all of your eggs in one basket.
 Under certain (rare) conditions MANOVA may find
differences that do not show up under ANOVA.

27. The interpretation of MANOVA results are always taken
in the context of the research design.
Once again, fancy statistics do not make up for poor
design.
Use of IVs change the interpretation of other IVs, so
choice of IVs to include needs to be thought about
carefully.
Choice of DVs also needs to be carefully considered,
highly correlated DVs severely weaken the power of the
analysis.
Generalizability is limited to the population studied.
Theoretical Considerations ofTheoretical Considerations of
MANOVAMANOVA

28. Two-way MANOVATwo-way MANOVA
Two-way MANOVA is also same as one-way ANOVA but it
has some differences in IVs and DVs.
Basic factors for Two-way MANOVA:Basic factors for Two-way MANOVA:
Two independent variables.
One or more than one dependent variables.
For more detail of this analysis we toughly pass on example:

29. ExampleExample::
In three different type of sicknesssickness (A,B and C)
To three different weightweight groups ( fat, Normal and
thin) it seems illill and tirednesstiredness symptoms, Therefor
find the type of sicknesses, differences of weight, ill
and tiredness symptoms impress ?

30. Weight
Type of weightType of weight
SicknessSickness
Fat normalFat normal
thinthin
AA 2 , 4 , 2 , 2 2 , 2 , 2 , 3 3 , 3 , 2 , 2
BB 4 , 4 , 3 , 4 4 , 4 , 3 , 3 4 , 5 , 4 , 3
CC 5 , 4 , 3 , 4 5 , 4 , 3 , 3 5 , 4 , 4 , 4
TirednessTiredness
Groups of weightGroups of weight
SicknessSickness
Fat normalFat normal
thinthin
AA 3 , 3 , 5 , 4 3 , 3 , 3 , 3 4 , 4 , 3 , 3
BB 5 , 4 , 4 , 4 5 , 4 , 4 , 5 5 , 5 , 6 , 6
CC 5 , 4 , 4 , 5 5 , 5, 4 , 4 6 , 6 , 5 , 4

34. In first step of spss outcome:
 The p value of ill is (p=0.00)(p=0.00) and the p value of Tiredness is (p=0.00).(p=0.00). it means the p
value of both of them is lower than (0.0.5).(0.0.5). so the effect of ill on the tiredness is
significant.
The p value of weight factor on the ill (p=0.375)(p=0.375) doesn't had significant effect But
tiredness p value (p=0.021)(p=0.021) is lower then alpha (p<0.05) so it shows effectives or
significant.
If we obtain the impact of ill and type of weight in two bilateral (illness*weight factor). It
requires in ill factor (p=0.994),(p=0.994), and tiredness requires (p=0.240)(p=0.240) ,so the average of this
two factors doesn’t had a effective or significant so those are acceptable.

37. 1.Table: Impact of sickness to ill and tiredness.
Sort of sicknessSort of sickness
AA B C St. ErrorB C St. Error
Ill:Ill: 2.42 a 3.75 b 4.00 b 0.21
Tiredness:Tiredness: 3.42 a 4.75 b 4.75 b 0.19
2. Table: Impact of weight on ill and tirdness.
WeightnessWeightness
Fat normal Thin St. ErrorFat normal Thin St. Error
ill :ill : 3.42 a 3.17 a 3.58 a 0.47
Tiredness:Tiredness: 4.17 a 4.00 a 4.75 b 0.19

38. If we review the 1.table: in the A group of sicknesses,
average of ill and tiredness is higher than the B and C
groups in the other words the significant of B and C
groups of sickness is lower than A group of sickness.
The B and C groups of sicknesses averages are similar
in tiredness and illness.
The 2.tablo shows that the factor of weight above the ill
was too weak but the significant of tiredness for thins
patients are observe higher than fats and normal
sickness.

39. Conclusion:Conclusion:
TWO-WAY ANOVA:TWO-WAY ANOVA:
Two- way (or multi-way) ANOVA is an appropriate analysis method for a study
with a quantitative outcome with two or more categorical explanatory variables.
More efficient to study two factors (A and B) simultaneously rather than
separately and we can investigate interactions between factors.
In a two way ANOVA (A×B design) there are four sources of variations.
MANOVA:MANOVA:
MANOVA assesses the statistical significance of the effect of one or more IV’s
on a set of two or more dependent variables.
MANOVA has the ability to examine more than one DV at once or simultaneous
effect of IV’s on multiple DV’s.
 Two-way MANOVA has two IV’s and one or more than one DV’s.

40. References:References:
1. ÇİMEN,Murat,2015,SPSS uygulamalı veri Analyzi (first.ed.)
ISBN:978-605-355-366-3, H.Ibrahim somyürek. ANKARA-
TURKEY.
2. KALAYCI, Şeref,2014, SPSS uygulamali çok değişkenli
istatistik teknikleri,ÖZ BARAN,Ofset.(6th
.ed), ANKARA-
TURKEY.
3. Aaron French, Marcelo Macedo, John Poulsen. Multivariate
Analysis of Variance (MANOVA).
4. Cooley, W.W. and P.R.Lohness. 1971. Multivariate Data
Analysis. John Wiley & Sons, Inc.
Internet:
1.www.slideshare.net
2.www.academia.org

42. Thanks from you’reThanks from you’re
patience!patience!