Meaning of ANOVA.
ANOVA is a statistical technique used for portioning the
variation in an observed data into it’s different sources. In other words, it
is a technique employed to breakdown the total variation in an experiment to
it’s additive components. With this technique or method therefore, we can break
down the total variation, occurring in a dependent variable, into various
separate factors causing the variation.
ANOVA is a statistical method
developed by R.A Fisher in 1948 to handle research problems in agriculture and
biology. ANOVA can also be called F-ratio. It can be used for two comparative
samples but it is called student t-ratio, or t-test. Fisher defined ANOVA as
the separation of the variance due to other groups the overall approach in
analysis of variance is to partition the variation in the observed set of data
into a number of components. This technique does not stipulate any functional
difference between the dependent and the independent variables.
Terms used in ANOVA
i. Factor:
It is an independent variable to be
studied in an experiment. It is a variable whose effect on another variable is
to be studied. Example in the study of the effect of fertilizer on the
yield/growth rate of cassava, the factor on understudy is fertilizer.
ii. Factor
Level:
It is sources of a particular factor.
For instance, in the study of the effect of fertilizer on the growth rate/yield
of cassava, the different sources of fertilizer such as N.P.K, Nitrogen
phosphorus and potassium are called factor levels.
iii. Treatment:
A treatment is equivalent to a facto
in a single-factor analysis. In a multi-factor experiment or analysis, a
treatment is equivalent to a combination of factor levels.
iv. ANOVA
Table: It is a table that shows in
summary form, the computations for analysis of variance. This table enables us
to have a quick and convenient assessment of the sources of variation, their
respective sum of squares and degrees of freedom which are the results of
ANOVA. The ANOVA table.
1. The degree of freedom written as 9d.f) to
obtained the degree of freedom for variation within treatments is a-1 while the
d.f for total variation sis ab-1 and this is always equals to the sum of the
degrees of freedom for the between-treatments and within treatments.
2. Mean sum of squares. This is also called
the variance (S2). It is calculated as sum of squares divided by the
corresponding degrees of freedom i.e.
= summation or additive sign
:.
2 = sum of
squares (SS) of deviation from the mean.
N = 1
= Degrees of freedom (df).
Another
formular for calculating variance is
The
first formular is called deviation formular second is called the machine
formular.
3. F-test or F-calculated (fisher test).
This is obtained by dividing the mean square of treatment by the mean square of
the experimental or residual error.
In testing the significance of the F-test, if F-tabulated given the same degrees of freedom then the differences between the treatments are significant i.e the treatments given in the experiment have difference which must by carefully monitored. But if the F-Cal is smaller than the F-tab, the it means differences between the treatments given are similar.
THE ESSENCE OF ANOVA
Analysis of variance is a very
important analytical tool developed by R.A fisher for the analysis of
experimental data. Hence it is employed in such fields of study as agriculture,
medicine, biology, engineering, economics and other social researchers to
analyse data and achieve logical conclusions.
Besides determining the various
factors which cause variation of the dependent variable, ANOVA could also be
useful in following areas:
i. testing the overall significance of the
regression,
ii. testing the significance of the improvement in fit obtained by the
introduction of additional explanatory variables in the function.
iii. testing the equality of coefficient s
obtained from different samples.
iv. testing the stability of coefficients of
regression.
v. testing restrictions imposed on
coefficient of a function .
Basically, analysis of variance
technique is used to test hypothesis concerning population means; and it is
employed in regression analysis to conduct various tests of significance.
Hence, it is an essential component of any regression analysis.
Types of ANOVA.
ANOVA
is classified into two basic types: one factor classification and two-factor
classification the classification is based on the number of factors to be
studied.
A. One-factor classification
The analysis based on one factor
classification is used to find out the effect of a single independent variable (a
factor) on the dependent variable. For instance, in the study of the level of
cassava yield using different sources of nitrogen, we are actually considering
the effect of nitrogen on the yield of maize is dependent on it. To ascertain
if the variation in the cassava yield is due to the treatment (that is,
nitrogen application or due to natural) THE ONE-FACTOR ANALYSIS IS ADOPTED.
However, to adopt and apply this
one-factor analysis of variance technique, the following assumptions must be
made:-
1. The treatment effect is fixed
2. The total effect of the treatment is
equal to zero.
3. The expected value of the error effect is
equal to zero.
4. The error is normally and independently
distributed with mean zero and variance, 
2.
2.
b. Two-factor
ANOVA
Unlike the one-factor ANOVA, the two-factor
classification contains more than two variable. It is made up of one dependent
variable and two independent variables, variable, (factors) on the dependent
variable. This technique therefore, enables us to estimate not only the
separate effects of the factors (independent variables) but also the join
effect (Interaction effect) of these factors on the dependent variable.
Consider the study of the effects of
fertilizer and soil type on the yield of cassava. There are two independent
variables or factors namely; fertilizer and soil type, and one dependent
variable (cassava yield). Hence the effects of these two factors-fertilizer and
soil type- can be analyzed using the two factor analysis of variance technique.
By using this technique. The separate
effect of the two factors on the yield of cassava will be determined. Also, the
factors on the yield of cassava will be determined. Also, the combined effect
(interaction effect) of the two factors on cassava yield can be shown. However,
when the two factors do not have any interaction effect, we say that they have
additive effect is on the dependent variable (cassava yield).
In applying the two-factor analysis of
variance technique, the same assumptions holding in a one factor technique
still hold as follows;-
1. The treatment effect is fixed that is Tr
is fixed.
2. The total effect of the treatment is
equal to zero; that is
3. The sum
expected value of the error effect is equal to zero i.e
4. The error
is normally and independently distributed with mean zero and variance.