21 Nov 2017
THE ANALYSIS OF VARIANCE (ANOVA), STUDENTS ‘T’ TESTS AND MATRIX ALGEBRA
By
ATUBI, A. 0. Ph.D.
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:
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 = x_{1}-x_{1}
vS_{1}^{2}/N_{1} + S_{2}^{2}/N_{2}
Where X_{1} and X_{2} are the means of the two sets of data; S_{1} and S_{2} their standard deviations; and N_{1} and N_{2} 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. 2x3 matrix can be multiplied by 3x2 matrix. But a 2x3 matrix cannot be multiplied by another 2x3 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 2x2 matrix
So far a 2x2 matrix, the inverse is the adjoint of that matrix over the determinant of the matrix.
The determinant of a 3x3 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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