Going to be present by: Payman Going to be present by: Payman EHSASEHSASUnder Supervision of : Prof.Dr. Under Supervision of : Prof.Dr. Cuma AKBAYCuma AKBAY November November

20152015

ContentsContents::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.

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 test paired 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.

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 Assumptions for the Two Factor ANOVAFactor ANOVA

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-way The null hypotheses in a Two-way ANOVAANOVA

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-way Advantages of Two-way ANOVAANOVA

Source of variation

SS Df MS[SS/df]

F Pvalue Fcrit

Main effect A

Main effect B

Interactive effect

Within

Total

Basic Two-way ANOVA Basic Two-way ANOVA tabletable

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

ss

F

Main Effect 2

2

2

w

factorB

ss

F

Interaction

2

2

w

ninteractio

ssF

Logic of Two-way ANOVA Logic of Two-way ANOVA (TABLE)(TABLE)

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 Usage side of Two-way ANOVA ANOVA

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

D 25 , 26 , 24 29 , 29 , 25 29 , 30 , 31

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.

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. ErrorNumber 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. ErrorNumber of bacteria: 21.5 a 22.8 b 25.0 c 0.47

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:

MANOVA

MANOVA

Multivariate analysis o

f

Multivariate analysis of

variance

variance

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

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?

Multivariate normality.

Homogeneity of covariance matrices.

Independence of observations.

Linearity.

Assumption of MANOVAAssumption of MANOVA

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

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

Advantages of MANOVA over Advantages 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.

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 of Theoretical Considerations of MANOVAMANOVA

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:

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 ?

Weight Type of weightType of weight Sickness Sickness Fat normal Fat normal thin thin 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 normal Fat normal thin thin 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

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.

1.Table: Impact of sickness to ill and tiredness. Sort of sicknessSort of sickness

AA B C St. Error B 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.

Weightness Weightness

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

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.

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.

ReferencesReferences: :

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.net2.www.academia.org

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