Multivariate Ana

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MULTIVARIATE ANALYSIS Statistical techniques that simultaneously analyze

more than two variables

Multivariate techniques two categories1. Dependency techniques deal with one ormore dependent variables

One dependent variable - data metric Multiple Regression Analysis

Several dependent variable data metric Discriminant Analysis

2. Interdependency techniques More than two variables

Variables not segregated as dependent andindependent variables Interrelationships between the variables are

analyzed Data metric Factor Analysis, Cluster

Analysis


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MULTIVARIATE ANALYSIS

Multiple RegressionUsed to analyse quantitative data

To study cause and effect

relationship between a singledependent variable and two or morethan two independent variables

Used mainly for prediction/forecasting


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Greek letters are used for a (a) and b (b) when

denoting population parameters.

Y a b X b X b X k k

' ... 1 1 2 2

The general multiple regression with k

independent variables is given by:

X1 to Xkare the independent variables.

a is theY-intercept.

bj is called a partial regression coefficient. It is the

net change in Y for each unit change in Xj holding all

other variables constant, (where j=1 to k)

Multiple Regression Analysis


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Successive values of thedependent variable mustbe uncorrelated or notautocorrelated.

ASSUMPTIONS IN MULTIPLE REGRESSION

The independent variables

and the dependent variable

have a linear relationship.

The dependent

variable must becontinuous and atleast interval-scaled.

The variation in (Y-Y) orresidual must be the samefor all values ofY. Whenthis is the case, we say thedifference exhibitshomoscedasticity.

The residuals should

follow the normaldistributed with mean 0.


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Correlation Matrix

A correlation matrix is used to show all possiblesimple correlation coefficients among the variables.

The matrix is useful for locating correlatedindependent variables.

It shows how strongly each independent variable is

correlated with the dependent variable.

CorrelationCoefficients Cars Advertising Sales force

Cars 1.000

Advertising 0.808 1.000

Sales force 0.872 0.537 1.000


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Because determining b1, b2, etc. is

very tedious, a software package

such as Excel or MINITAB may be used.

The least squares criterion is used

to develop this equation.

Multiple Regression Analysis


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ANOVA TABLE

Source df SS MSRegression k-1 SSRS(YY)2 SSR/(k-1)Error n-k-1 SSE

S(Y-Y)2SSE/(n-k-1)

Total n-k-1 SS Total

S(Y-Y)Total Variation

ANOVA Explained Variation

Unexplained or Random Variation

Variation not accounted for by the

independent variables.

Variation

accounted

for by theset of

independent

variables.


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A market researcher for Super

Markets is studying the yearly

amount families of four or more

spend on food. Three

independent variables are

thought to be related to yearlyfood expenditures (Food). Those

variables are: total family

income (Income) in $00, size offamily (Size), and whether the

family has children in college

(College).

EXAMPLE 1


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The variable college is called a dummy orindicator variable. It can take only one ofthe two possible outcomes i.e. a child is a

college student or not.Examples of dummy variables: gender, thepart is acceptable or not, the voter will orwill not vote for the incumbent governor etc.

We usually code one value of the dummyvariable as 1 and the other 0.

Expenditure = a + b1*(Income) +b2(Size) + b3(College)


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Example 1 continued

Fam ily Food Incom e Size Student

1 3900 376 4 0

2 5300 515 5 13 4300 516 4 0

4 4900 468 5 0

5 6400 538 6 1

6 7300 626 7 17 4900 543 5 0

8 5300 437 4 0

9 6100 608 5 1

10 6400 513 6 111 7400 493 6 1

12 5800 563 5 0


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Example 1continued

From the analysis provided by MINITAB,

the estimated multiple regression equation is:

Y=954 +1.09X1 + 748X2 + 565X3

? What food expenditure would you estimate for afamily of 4, with no college student, and an income

of $50,000 (which is input as 500)?

Food Expenditure= 954+1.09*income+748*size

+565*college


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Example 1 continued

Each additional $100 dollars of income per year willincrease the amount spent on food by $109 per year.

An additional family member will increase the

amount spent per year on food by $748.A family with a college student will spend $565 moreper year on food than those without a college student.

Food Expend.=$954+$1.09*income+$748*size+$565*college

So a family of 4, with no college students, and an

income of $50,000 will spend an estimated $4,491.

Food Expend.=$954+$1.09*500+$748*4+$565*0


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13Example 1 continued

The regression equation is

Food = 954 + 1.09 Income + 748 Size + 565 Student

Predictor Coef SE Coef T P

Constant 954 1581 0.60 0.563

Income 1.092 3.153 0.35 0.738

Size 748.4 303.0 2.47 0.039

Student 564.5 495.1 1.14 0.287

S = 572.7 R-Sq = 80.4% R-Sq(adj) = 73.1%

Analysis of Variance

Source DF SS MS F P

Regression 3 10762903 3587634 10.94 0.003

Residual Error 8 2623764 327970

Total 11 13386667


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Correlation matrix

The coefficient ofdetermination is80.4percent. Thismeans that morethan 80 percent ofthe variation in the

amount spent onfood is accountedfor by the variablesincome, family

size, and student.

The strongest correlation between the dependentvariable and an independent variable is betweenfamily size and amount spent on food.

Food Income Size College

Food 1.000

Income 0.587 1.000

Size 0.876 0.609 1.000

College 0.773 0.491 0.743 1.000


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H H0 2 1 2

0 0: :b b

Conduct an individual test to determinewhich coefficients are not zero. This is the

hypotheses for the independent variablefamily size.

From the MINITABoutput, the onlysignificant variable is

FAMILY (family size)using the p-values.The other variables canbe omitted from the

model.

Thus, using the 5%level of

significance, rejectH0 if the p-value

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