Understanding Variance Analysis

2010

Analysis of variance (ANOVA) was initially developed by RA Fisher, beginning around 1918, and had early applications in agriculture. It is now a dominant and powerful statistical technique used extensively in psychology. In ANOVA, a dependent variable is predicted by a mathematical model comprising one or more predictor variables, which may be categorical (factors) or quantitative and continuous (covariates; regressors).

Analysis of variance (ANOVA) is the method used to compare continuous measurements to determine if the measurements are sampled from the same or different distributions. It is an analytical tool used to determine the significance of factors on measurements by looking at the relationship between a quantitative "response variable" and a proposed explanatory "factor." This method is similar to the process of comparing the statistical difference between two samples, in that it invokes the concept of hypothesis testing. Instead of comparing two samples, however, a variable is correlated with one or more explanatory factors, typically using the F-statistic. From this F-statistic, the P-value can be calculated to see if the difference is significant. For example, if the P-value is low (P-value<0.05 or P-value<0.01-this depends on desired level of significance), then there is a low probability that the two groups are the same. The method is highly versatile in that it can be used to analyze complicated systems, with numerous variables and factors. INTRODUCTION ANOVA is a quantitative research method that tests hypotheses that are made about differences between two or more means. If independent estimates of variance can be obtained from the data, ANOVA compares the means of different groups by analyzing comparisons of variance estimates. There are two models for ANOVA, the fixed effects model, and the random effects model (in the latter, the treatments are not fixed). ANOVA is a very useful technique for testing the equality of more than two means of population. The word analysis of variance is used because the technique involves first finding out the total variation among the observation in the collection data, then assigning causes of components of variation to various factors and finally drawing conclusion about the equality of means. It also used to test the significance of a regression equation as a whole i.e., whether all the equation are equal to zero. Factor analysis is the process by which a complicated system of many variables is simplified by completely defining it with a smaller number of "factors." If these factors can be studied and determined, they can be used to predict the value of the variables in a system.

This article demonstrates that the variance of three or four observations can be expressed in terms of the range and the first order differences of the observations. A more general result, which holds for any number of observations, is also stated.

2017

In this paper the simplest variant of analysis of variance is under consideration. Three examples from textbooks by Lakin (1990) and Rokitsky (1973) were re-considered. It was obtained that traditional one-way ANOVA and Kruskal – Wallis criterion can lead to unreal results about factor’s influence on value of characteristics. Alternative way to solution of the same problem is under consideration too.

Physical therapy, 1985

In summary, one-way ANOVA for independent groups is used to test whether the group means for a specific dependent variable differ significantly after exposing each group to a unique level of a single factor or independent variable. You may recognize the preceding sentence. We hope it makes more sense to you now than when you first read it at the beginning of this article.

2008

Naive Statistics: Intuitive Analysis of Variance David L. Trumpower (david.trumpower@uottawa.ca) Olga Fellus (ofell070@uottawa.ca) University of Ottawa, Faculty of Education, 145 Jean-Jacques-Lussier Street Ottawa, ON K1N 6N5 Canada example, subjects’ estimates of the mean of skewed distributions are typically biased toward the median. Similarly, Kareev, Arnon, and Horwiz-Zeliger (2002) have shown that estimates of variance tend to be low. It is well known that we are often biased toward making use of the most salient data when making judgements about averages (i.e., the availability heuristic; Kahneman & Tversky, 1973) and correlations (Chapman & Chapman, 1969). It is also well known that we tend to make use of data which confirms our preconceived hypotheses about such things as correlations between variables (i.e., confirmation bias, Wason & Johnson-Laird, 1972; for a review, see Klayman & Ha, 1987). If students do hold certain intuitive biases or misconceptions which affect stati...

Wiley StatsRef: Statistics Reference Online, 2014

Analysis of Variance (ANOVA) ANOVA investigates special linear models, used for planning experiments or quality control. Here the matrix of the deterministic predictors is a so-called design-matrix with 0-1 entries indicating that which predictors influence the response at all. For testing hypotheses, we will intensively use the following theorem and its corollaries.

2003

One-way analysis of variance Analysis of variance (ANOVA) involves comparing random samples from several populations (groups). Often the samples arise from observing experimental units with different treatments applied to them and we refer to the populations as treatment groups. The sample sizes for the groups are possibly different, say, N i and we assume that the samples are all independent. Moreover, we assume that each population has the same variance and is normally distributed. Assuming different means for each group we have a model y i j = µ i + ε i j , ε i j s independent N(0, σ 2) or, equivalently, y i j s independent N(µ i , σ 2),

Teaching Statistics, 2009

This article demonstrates that the variance of three or four observations can be expressed in terms of the range and the first order differences of the observations. A more general result, which holds for any number of observations, is also stated.

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