Abstract
Independence of observations is one of the standard assumption in analysis of Variance (ANOVA) table. Where the error terms in the model are independent, identically distributed normal variables with null means and homogeneous variances. In this paper investigate the effect of dependence of observations in ANOVA for unbalanced 2-way nested fixed model and developing a method for adjusting it. When the error terms are correlated and focus on the effects of departures from independence assumptions on hypothesis testing by determining the expect mean squares for errors as well as treatments for this model and correcting the F statistics for testing the factor effect. The model considered is one in which all measurements have same variance 2 σ , and the covariance matrix enjoy a structure defined as follows: every pair of measurements comes from: i) The same experimental observation and the same experimental unit; ii) Different experimental observation, but in the same experimental unit; ii) Different experimental unit; has covariance 2 2
Highlights
The assumption of independence of observation in ANOVA table may seem like a reasonable assumption in examining data using experimental designs .Rarely is the independent assumption verified in an ANOVA the analysis of data from experimental design is often hampered by lack of technique to correct the usual F-test for the effect of correlation
The aim of this work explains a method for adjusting ANOVA table when observation are correlated, that is, when the error terms are correlated and focus on the effects of departures from independence assumption on hypothesis by determining the expected mean squares for errors as well as treatments for un balanced two way nested fixed effects modeland correcting the F statistics for testing the factor effect
Correlation's constant C1, C2 small correlation can be amplified by the number of treatment nij,i = 1,2,.., m j = 1,2,..., p
Summary
The assumption of independence of observation in ANOVA table may seem like a reasonable assumption in examining data using experimental designs .Rarely is the independent assumption verified in an ANOVA the analysis of data from experimental design is often hampered by lack of technique to correct the usual F-test for the effect of correlation. The aim of this work explains a method for adjusting ANOVA table when observation are correlated, that is, when the error terms are correlated and focus on the effects of departures from independence assumption on hypothesis by determining the expected mean squares for errors as well as treatments for un balanced two way nested fixed effects model (balanced model is special case of un balanced model)and correcting the F statistics for testing the factor effect. Let Yijk be the observation of the kth treatment of the jth experimental observation from the ith experimental unit with i =1,2,...,m , j =1,2, ..., p ,k=1,2,..., nij. We assume that the Yijk distributed normally with mean ijk and that the measurements have the same variance 2 ,and every pair of measurements comes from i) The same experimental observation and the same experimental unit; has covariance 2 1, 2 2 and 2 3 respectively. Analysis of variance table for the model of study, which is given in equation (5), is given in table 1
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