ANOVA Model

Examples of ANOVA Model in the following topics:

• Qualitative Variable Models

  • Dummy variables are "proxy" variables, or numeric stand-ins for qualitative facts in a regression model.
  • Analysis of variance (ANOVA) models are a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups).
  • One type of ANOVA model, applicable when dealing with qualitative variables, is a regression model in which the dependent variable is quantitative in nature but all the explanatory variables are dummies (qualitative in nature).
  • This type of ANOVA modelcan have differing numbers of qualitative variables.
  • Graph showing the regression results of the ANOVA model example: Average annual salaries of public school teachers in 3 regions of a country.

• ANOVA Design

  • It is also common to apply ANOVA to observational data using an appropriate statistical model.
  • In a 3-way ANOVA with factors $x$, $y$, and $z$, the ANOVA model includes terms for the main effects ($x$, $y$, $z$) and terms for interactions ($xy$, $xz$, $yz$, $xyz$).
  • Random effects models are used when the treatments are not fixed.
  • The fixed-effects model would compare a list of candidate texts.
  • Differentiate one-way, factorial, repeated measures, and multivariate ANOVA experimental designs; single and multiple factor ANOVA tests; fixed-effect, random-effect and mixed-effect models

• Two-Way ANOVA

  • The two-way analysis of variance (ANOVA) test is an extension of the one-way ANOVA test that examines the influence of different categorical independent variables on one dependent variable.
  • Another term for the two-way ANOVA is a factorial ANOVA.
  • In a 3-way ANOVA with factors $x$, $y$, and $z$, the ANOVA model includes terms for the main effects ($x$, $y$, $z$) and terms for interactions ( $xy$, $xz$, $yz$, $xyz$).
  • Caution is advised when encountering interactions in a two-way ANOVA.
  • Distinguish the two-way ANOVA from the one-way ANOVA and point out the assumptions necessary to perform the test.

• ANOVA Assumptions

  • The results of a one-way ANOVA can be considered reliable as long as certain assumptions are met.
  • The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:
  • Kempthorne uses the randomization-distribution and the assumption of unit-treatment additivity to produce a derived linear model, very similar to the one-way ANOVA discussed previously.
  • The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies.
  • In summary, the normal model based ANOVA analysis assumes the independence, normality and homogeneity of the variances of the residuals.

• ANOVA

  • A consumer looking for a new car might compare the average gas mileage of several models.
  • ANOVA is a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups).
  • ANOVA is the synthesis of several ideas and it is used for multiple purposes.
  • ANOVA with a very good fit and ANOVA with no fit are shown, respectively, in and .
  • Recognize how ANOVA allows us to test variables in three or more groups.

• An alternative test statistic

  • The analysis of variance (ANOVA) technique introduced in Section 5.5 uses this general principle.
  • ANOVA can be further employed in advanced regression modeling to evaluate the inclusion of explanatory variables, though we leave these details to a later course.

• Introduction

  • Discuss two uses for the F distribution: One-Way ANOVA and the test of two variances.
  • A consumer looking for a new car might compare the average gas mileage of several models.
  • In this chapter, you will study the simplest form of ANOVA called single factor or One-Way ANOVA.
  • This is just a very brief overview of One-Way ANOVA.
  • For further information about One-Way ANOVA, use the online link ANOVA2 .

• Further Discussion of ANOVA

  • Due to the iterative nature of experimentation, preparatory and follow-up analyses are often necessary in ANOVA.
  • Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level.
  • Eta-squared is a biased estimator of the variance explained by the model in the population (it estimates only the effect size in the sample).
  • It is prudent to verify that the assumptions of ANOVA have been met.
  • Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and modeled data values.

• F Distribution and One-Way ANOVA: Purpose and Basic Assumptions of One-Way ANOVA

  • The purpose of a One-Way ANOVA test is to determine the existence of a statistically significant difference among several group means.
  • In order to perform a One-Way ANOVA test, there are five basic assumptions to be fulfilled:

• Reading an ANOVA table from software

  • The calculations required to perform an ANOVA by hand are tedious and prone to human error.
  • An ANOVA can be summarized in a table very similar to that of a regression summary, which we will see in Chapters 7 and 8.
  • Table 5.30 shows an ANOVA summary to test whether the mean of on-base percentage varies by player positions in the MLB.
  • ANOVA summary for testing whether the average on-base percentage differs across player positions.