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Analysis of Variance, Logic of ANOVA

The analysis of variance is a collection of the statistical models and their associated procedures, where the observed variance of the particular variable is partitioned in the components which are attributable to various sources of variations. ANOVA generalizes the test in more than two groups and provides the statistical testing for determining whether the means of several groups are equal or not. When doing the multiple two sample tests, there are increased chances of committing a type 1 error. Hence ANOVA is found useful when comparing two three or more means.

The three classes of models used for analysis of variance are-

1. The fixed effects models- this model of analysis is applied to the situations where one or more treatments to the subjects are applied in order to find out the value changes in the response variables. Therefore the experimenter can stimulate the ranges of the response values of the variables which the treatment would generate in the population as a whole.

2. Random effects models- these models are used when the treatments are not fixed in cases where the factor levels are taken in the form of samples form a larger population. These levels themselves random variables and therefore some of the assumptions and the method of contrasting their treatments differ from the ANOVA models.

3. Mixed effect model- this analysis method contains different interpretations and analysis of both the fixed and random effect variables.

Linear model is the most common approach that relates to the response and blocks, among the several variance analysis models approaches.

Logic of ANOVA

Partitioning of the sum of squares-the fundamental techniques used is by portioning the total sum of all squares S into various components which are related to various effects being used in the model. As an example, a model for a simplified ANOVA is shown with a similar type of treatments at different levels.

S_Total = S_Error+S_Treatments

Hence, we can partition the number of degrees of freedom f in a similar way and can also specify the chi-square distribution for describing the associated sum of squares by

f_Total = f_Error+f_Treatments

The f test- for comparing the components of total deviation we use the f test. The statistical significance in a one way or single factor ANOVA is tested by the comparison of the f test statistic.

f = variance between items/variance within items

f^*=MSTR/MSE

MSTR = SSTR/I-1

MSE= SSE / n_T - 1 Here,

n_T = total number of cases and I = number of treatments

There are various kinds of ANOVAs. Some statisticians give design of experiment as the basis of ANOVA, especially when the protocol specifies the random assignment of treatment of subjects. The specification of the structure of treatments must be included in the specification of structure of the treatments. The ANOVA is also applied to the observational data by using an appropriate statistical model.

The popular designs of ANOVA are-

The One-Way ANOVA- this is used for testing the differences between two or more independent groups for example for different levels of urea applied in a crop. Most commonly, the one way ANOVA is used for testing the differences in at least three groups, as the two groups cases can be covered by the t test.

The Factorial ANOVA- for studying the interaction between effects of treatments, factorial INPVA is used.

The repeated measures ANOVA- these are used when the same subjects are used for each individual treatment.

Multivariate analysis of variants-when there is more than one response variable, the MANOVA is used.

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