THE ANALYSIS OF VARIANCE (ANOVA), STUDENTS ‘T’ TESTS AND MATRIX ALGEBRA

Introduction

The ANOVA sometimes referred to as the F test (named after the statistician Sir Roland Fisher, the author of this test) is a set of procedures for testing the significance of differences among more than two independent means. This procedure determines the extent to which there are significant differences between the means of three or more samples with a single level of significance. Because this procedure and others like it focus on variances, they are referred to as the analysis of variance or ANOVA

One Way Analysis of Variance

The step in ANOVA can be illustrated by an example

Example 5.1.1

The number of crimes committed in 4 months in various parts of a town are list below:

Month

Artisan quarters

Slums

CDB

GRA

January

February

March

April

20

16

32

16

52

44

56

36

36

40

36

40

16

20

32

20

At 0.05 level of confidence, are crime frequencies related to urban morphology or month of the year?

First set up a null hypothesis of equality of the means Let Ho be: “There is no significant difference between the crimes frequencies and urban morphology or month of the year.

Next, find the total of the four areas as listed below then their means

Total =Artesan SlumsCBDGRA

8418815288

Mean

Next, calculate the Grand mean

Next, calculate the sum of squares for each population. Note that in ANOVA the number of samples do not have to be same

Table 4.1.1: Means and sums of squares of crimes committed

In ANOVA, we determine differences between means by calculating their variability. Three types of variability are estimated:

The variation within each sample result

The variation between the sample results

The total variation of the samples, regardless of the sample grouping to which they belong (Anyadike, 2009)

Next, calculate the sums of the sum of squares for each column (i.e Variable):

SSW = ∑∑(Xij – X)2 = 172+236+16+144 =568

Calculate sum of square between = ? N (X —X)2

=4(21-3 2)2+4(473 2)2+4 (3832)2 + 4(2232)2

= 484 +900+144+400

=1928

Calculate the within group sum of square. MSW = SSW

n-rn

Where n=no of observations

M=no of groups

Similarly, the mean s square between will be

MSb =SSb

M-l

Therefore

MSW =SSb=568=568

n-m 16-4 12

= 47.33

The ratio between the variance estimate is known as the Snedecor’s variance ratio test or Snedecor’s F

Now set up an ANOVA table:

Finally, test for significance:

The degrees of freedom, V are:

V- numerator = M-1=4-1=3

V- denominator = n-m= 16—4=12

From the Table of the F-Distribution, critical value of F at 0.05 3 and 12 =3.49

Calculated value is 13.18

Since the calculated F of 13.58 is greater (>) 3.49, Ho is rejected. Therefore, “There is a significant difference between crime frequencies with regards to urban morphology or month of the year.

THE STUDENT ‘T TEST

The most powerful test for the comparison of sample means is the student ‘t’ test. It is a parametric test and is used to determine whether or not the differences between two sample means are sufficiently great as to justify a conclusion that the means of their populations also differ significantly. It is also used for small samples

The student ‘t’ is expressed as:

T = x1-x1

vS12/N1 + S22/N2

Where X1 and X2 are the means of the two sets of data; S1 and S2 their standard deviations; and N1 and N2 the number of observations.

The degree of freedom, V is expressed as follows:

V = N+N,-2

Degree of Freedom

There is often confusion among students about the concept of “degree of freedom”. Basically, if the sum of a set of elements and the sum of all but one is of its elements are both known, then the value of the last element must also be known, i.e. it is not, unlike the others, free to vary (Ayandike, 2009). For example, if the sum of 8 elements is 30 and the values of 7 of the elements variously add up to 28, then the value ofthe final (i.e. the 8”) element must be (30-28) = 2, i.e. if is not free to adopt any other value. The degree of freedom in this case is thus (8-1) = 7, i.e. in the set of 8 element, 7 of them are free to take on any values to sum up to 28, leaving value inviolate

ELEMENTS OF MATRIX ALGEBRA

Introduction

The matrix is a rectangular array of number arranged in rows n and columns, m i.e

Each of the numbers is called an elements. The position of each element is determined by its position in the row as well as in the

The size of the matrix is given by the number of rows (n) and number of column (m) for example.

A matrix which has the same number of rows and columns is called a square matrix. In the example above, a and c are square matrices. A matrix with a single row is called a row vector, while a matrix of a single column is called a column vector. Example of a row vector is

{3 5 7 8}

example of column vector is

Matrix can be added, subtracted, multiplied and inverted but cannot be divided. However, they can only be divided by a scalar (i.e an ordinary number).

Addition of Matrix

Matrices to be added must be of the same size as one another. That is they must have the same number of columns and row s. This is because each element of one matrix must be added to the same element of the other matrix e.g. supposing we are adding two matrix [A] ± [B]

[A] + [BI =[C]

i.e you add element by element

N.B:The number of columns and rows must be the same before it can be added

Matrix Subtraction

The same rule as addition is applied e.g to subtract [BI from [A]

[A] [B]

Matrix Multiplication

There are two aspect of matrix multiplication namely:

a.Multiplication of matrix by a scaler

b.Multiplication of matrix by two matrices

By Scaler

Supposing we are multiplying the matrix below by 4

Multiplication by Two Matrices

When two matrices are to be multiplied, the number of columns in the first matrix must be equal the number of rows in the second matrix e.g. 2×3 matrix can be multiplied by 3×2 matrix. But a 2×3 matrix cannot be multiplied by another 2×3 matrix because the number of rows there is not equal to the number of columns in the second e.g.

The result will have as many rows as the first and as may columns as the second. Multiplication of a matrix by vector, example

An identity or unity matrix 1, is a matrix where the diagonal consist of l’s and the rest of the elements are zero e.g.:

Matrix Inversion

In matrix, algebra function of division is changed to that of inversion. The inverse of the matrix is it’s reciprocal i.e.

Only square matrices have inverses. A matrix that cannot be inverted is called a singular matrix. Several methods exists for finding the inverse of a matrix. They includes

1. The classical method

This is to set the matrix beside an identity matrix, and to perform all operations simultaneously in both matrices, for example, if you are to invert this matrix [A], you first place it beside an identity matrix

You have

Step 1

Subtract row 2 from 3, multiply row 1

Row 2(3xrow 1)

Step 3

2. By Determinants

This is the more modern one. A determinant is a single number extracted from a square matrix by series of operations. It is represented by either det A or /A/

The process of obtaining a determinant from a matrix is called evaluating the determinant. Using determinant, the inverse of matrix [A] becomes

/A/ = ad—bc

The adjoint of a matrix is the transposed matrix of co -factors with the signs taken into consideration. The signs are alternating +, -, across and down the element of the matrix e.g. in a 2×2 matrix

So far a 2×2 matrix, the inverse is the adjoint of that matrix over the determinant of the matrix.

The determinant of a 3×3 matrix

The minus for each leading element becomes

Uptill i.

The Solution of Simultaneous Equation

The major interest in matrices (and its greatest strength) is their use in the resolution of the unknowns in simultaneous equations (Anya dike, 2009)

SIMULTANEOUS EQUATIONS WITH TWO UNKNOWNS

3x+4y 10

2x +‘7y = 11

Set in matrix form

1.Using the classical method, our equation in matrix form is [A] [x] [B]

The matrix of the unknown

[x]=[A]=[B]

For a 2 x 2 matrix the adjoint of it is

Since our matrix of unknown is

Check with original equation

3x+4y =10

2x +7y1l

Check

3(2)+4(1)= 10 =6+4+10

2(2) +7 (1) =1

4+7=11

1. By determinant method

The matrix in our example is [A] [x] [B]

This 13 is called the common denominator. Then we find the numerator of x, which is the determinant of the main matrix.

:. X2/A1.B/-÷/A!13113 =1 =y

Therefore,Y = 1

X=2

REFERENCES

Anyadike, R.N.C (2009) Statistical methods for social and Environmental Sciences. Spectrum Books Limited Ibadan.

Anyadike, R.N.C (2009): Statistical methods for social and Environmental sciences. Spectrum Books limited Ibadan.

Atubi, A.O. (2010d) Road Traffic Accident variations in Lagos State, Nigeria: A Synopsis of variance Spectra. African research Review, Vol 4(2) pp. 197-218. Ethiopia.

Ewhmdjakpor C, Atubi, A.O. and Odemerho F. (2006): Statistics for social investigations. Delsu Investment Nigeria, Limited, Delta State University, Abraka.

Ewhrudjakpor. C, Atubi, A.O, and Odermerho F (2006). Statistics for social Investigations. Delsu Investment Nigeria, Limited. Delta State University, Abraka.

Spiegel, M.R (1972): Theory and problems of statistics. McGraw-Hill, New York.

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