Multivariate analysis allows you to determine the effect of more than 2 Independent variables on several dependent variables

1. Multivariate analysis of
variance

2. Life is Not That Simple
• The world is complex and multivariate in nature, and
instances when a single variable completely explains a
phenomenon are rare.
• For example, when trying to explore how to grow a
bigger tomato, we would need to consider factors that
have to do with the
• plants' genetic makeup,
• soil conditions,
• lighting,
• temperature, etc..
• So analyze all those variables Multivariate analysis is
used

3. Multivariate Analysis
• Many statistical techniques focus on just one
or two variables
• Multivariate analysis (MVA) techniques allow
more than two variables to be analysed at once

4. MANOVA
• 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
• An extension of univariate ANOVA procedures to situations in
which there are two or more related dependent variables
(ANOVA analyses only a single DV at a time)
• The more important purpose is to explore how independent
Variables influence some patterning of response on the
dependent variables.

5. Basic Requirements
• 2 or more DVs (I, R)
• 1 or more categorical IVs (N, O)

6. Proper Usage
•MANOVA is appropriate when we have several
DVs which all measure different aspects of
some cohesive theme, e.g., several different
types of academic achievement (e.g., Maths,
English, Science).
•MANOVA works well in situations where there
are moderate correlations between DVs.

7. MANOVA vs ANOVA
• Because variables are more significant
together than considered separately.
• It considers inter correlations between DV’s.
• It controls the inflation of Type I error*.

8. ADVANTAGES
•It tests the effects of several independent
variables and several outcome (dependent)
variables within a single analysis
•It can provide a more powerful test of
significance than available when using
univariate tests
•It reduces error rate compared with performing
a series of univariate tests

9. Assumptions of MANOVA
Multivariate normality:
• DV should be normally distributed within groups..
Homogeneity of covariance matrices:
• The inter correlations (co variances) of the multiple DV across
the cells of design.

10. Assumptions of MANOVA
Independence of observations:
• Subject score on DV are not influenced or related to other subject
scores.
Linearity
• Linear relationship against
▫ All pairs of dependent variables,
▫ All pairs of covariates,
▫ All dependent variable – covariate pairs in each cell.

11. THEORY
• Mathematical Vector Model
• Where overall mean
ith treatment effect with
jth error for the ith group
jth observation of the ith group
• Hypothesis
against HA = at least one inequality

12. Theory
Observational data
• No of groups (i)= k
• No of observations In each group= n i
• Then MANOVA equation is

13. Theory
Sources of
variations
Degree of
freedom
Treatment effect
(B)
(variability
between the
different groups)
k-1
Residual effect
(W)
(variability within
the group)
n-k
Total SSCP
(W+B)
n-1

14. Multivariate Test Statistics
• Wilks' lambda (λ)
The smaller the value of Wilks' lambda, the
larger the between-groups dispersion
λ = IWI
IW+BI
Ho rejects if λ is small

15. Hypothesis Testing

16. Example
• In certain district of Punjab hospitals are
classified on the bases of ownership (private,
government, non- profit). The study was made
to investigate effect of ownership on the costs
to hospitals:
• X 1: cost of nursing
• X 2: maintenance cost
• Does the type of ownership effects costs to
hospital?

17. Data Given
• p=2 k=3 n=8
• np= 3 ng = 2 nnp = 3
Type of ownership obs.no. Cost of nursing (X1)
(in million)
Maintenance cost (X2)
(in millions)
Private 1 9 3
2 6 2
3 9 7
Government 1 2 2
2 2 2
Non - profit 1 3 8
2 1 9
3 2 7

18. Computed Means
Type of ownership X1 (mean) X2 (mean)
Private 8 4
Government 2 2
Non - profit 2 8
Grand mean 4 5

19. Calculations
X1
Calculations Result
Total SSQ 52+22+52+(-2) 2+(-2) 2+(-1) 2+(-3) 2+(-2) 2 76
Treatment SSQ 3*42+2*(-2) 2+3*(-2) 2 68
Residual SSQ 12+(-2)2+12+0+0+12+(-1) 2+0 8
X2
Total SSQ (-2) 2+(-3) 2+22+(-3) 2+(-3) 2+32+42+22 64
Treatment SSQ 3*(-1) 2+2*(-3) 2+3*32 48
Residual SSQ (-1) 2+(-2) 2+32+0+0+0+1+(-1) 2 16
Cross Products
Total SCP 5*(-2)+2*(-3)+5*2+(-3)*(-2)+(-2)*(-3)+(-1)*3+(-3)*4+(-2)*2 -13
Treatment SCP 3*4*(-1)+2*(-2)*(-2)+3*(-2)*3 -18
Residual SCP 1*(-1)+(-2)*(-2)+1*3+0+0+0+(-1)*1+0 5

20. Calculations
Variance co-variance matrix
X1 X2
X1 a c
X2 c b
Sources of variation degree of freedom Matrix Determinant
Treatment effect (B) 2 68 -18 2940
-18 48
Residual effect (w) 5 8 5 103
5 16
Total effect 7 76 -13 4695
-13 64

21. Calculations
 λ = IWI
IW+BI
= 103/(4695) = .0219
As p=2 k=3
Test statistics =
= 11.514

22. Result
• Significance level = .01
• Tabulated F4,8 (.01) =7.01
• As Test statistics (11.514) > tabulated F4,8 (.01) =7.01
• Hypothesis H0 =
is rejected.
• That is to say that average costs (D.V.) differ with
different type of ownership (I.V.).