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.

          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.
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.
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