Analysis of variance

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Anova, no fit.

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ANOVA fair fit

ANOVA very good fit

Fixed effects vs Random effects

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Example of ANOVA table

Analysis of Variance (ANOVA) is a collection of statistical models and their associated procedures used to analyze the differences among group means in a sample. The ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. This technique was developed by Ronald Fisher in the early 20th century and is a crucial tool in the field of statistics for comparing two or more means for statistical significance.

Overview

ANOVA tests the null hypothesis that samples in two or more groups are drawn from populations with the same mean values. To do this, it compares the ratio of systematic variance (variance between groups) to unsystematic variance (variance within groups). If the between-group variance is significantly larger than the within-group variance, it suggests that not all the group means are equal.

Types of ANOVA

There are several types of ANOVA, each suited for different experimental designs:

• One-way ANOVA: Used when comparing more than two groups based on one independent variable. It assesses the impact of a single factor on a single response variable.
• Two-way ANOVA: Allows for the investigation of the effects of two independent variables on a response variable. It can also evaluate the interaction between the two independent variables on the dependent variable.
• Multivariate analysis of variance (MANOVA): Extends ANOVA when there are two or more dependent variables that are correlated. MANOVA assesses the influence of independent variables on the combined dependent variables.

Assumptions

ANOVA makes several key assumptions:

• The responses for each group are normally distributed.
• Homogeneity of variances, which means that the variance among the groups should be approximately equal.
• Observations are sampled independently from each other.

Calculation

The calculation of ANOVA involves dividing the total variance observed in the data into the variance between groups and the variance within groups. The F-statistic is then calculated by dividing the mean square variance between the groups by the mean square variance within the groups. The resulting F-statistic is compared to a critical value from the F-distribution to determine the p-value.

Applications

ANOVA is widely used in many fields such as psychology, medicine, engineering, and agriculture to test the significance of differences among group means. It is particularly useful in experimental design and in situations where multiple comparisons are made.

Limitations

While ANOVA is a powerful and widely used method, it has limitations. It is sensitive to deviations from normality and homogeneity of variances. Additionally, ANOVA can identify that at least two groups are different but cannot specify which groups are different. Post-hoc tests are required to identify specific group differences after ANOVA indicates significant results.