How do you choose the right ANOVA test?

You can use 1-way ANOVA if you have 2 or more categories of an independent variable but when you get the real data you might not understand which one is an independent variable and which one is the dependent variable. The reason is that we don't get structured data. For example; if you have the first column in Excel with three levels of irrigation say low, medium, high, and the values of the dependent variable in the second column and you want to apply one-way ANOVA in Excel using Data Analysis plug-in in Data Tab of your Excel then you need to select Grouped By Rows instead of Grouped By Columns which is the default. So, make sure to understand your independent variable carefully as different software tools are used differently.

Choose the right ANOVA test by considering the number of independent variables, nature of variables (categorical or repeated measures), assumptions of normality and homogeneity, sample size, and available statistical software. Use one-way ANOVA for one factor, two-way for two factors, and repeated measures for repeated observations. Check assumptions via visual inspection and tests. Transform data or use non-parametric tests if assumptions are violated. Post hoc tests and robust procedures enhance reliability. Always report any deviations for transparency.

The underlying data structure or experimental design (or lack of) must be accounted for. Failure to do so is such a common mistake. Incorrect ANOVA model specification may lead to inaccurate F statistics and corresponding p-values, therefore incorrect inferences. One must determine whether factors are fully crossed, nested? Are there blocking structures, covariates to account for? Was the experiment fully randomized or were there constraints resulting in split-plot, split-block designs with several error terms. Are there repeated measures? One must also worry about data imbalance. Is the ANOVA balanced, if not are some effects/interactions non-estimable? Running an ANOVA is easy, making sure it’s correctly specified is something else.

Two-way is only appropriate when there are two independent variables affecting the dependent variable. It can determine the main effects of each independent variable and whether there is an interaction effect between them.

Choosing between factorial and nested designs in ANOVA requires understanding the structure of your independent variables. Factorial design fits when variables apply to all groups, like testing fertilizer and irrigation on crops. In contrast, nested design suits situations where variables apply to specific subsets, such as shampoo and hair type affecting hair quality. My experience underscores the significance of this distinction; using the appropriate ANOVA variant ensures accurate analysis, especially in complex experimental setups. It's like having the right tool for the job, optimizing the insights gained from diverse experimental designs.

Choosing the right ANOVA involves ensuring two key assumptions are met: Normality: Residuals should follow a normal distribution. Check via visual tools or statistical tests. Transform data or consider non-parametric tests if normality is not met. Homogeneity of Variances: Variability within groups should be roughly equal. Test with Levene's or Bartlett's. Address by transforming data or using Welch's ANOVA for unequal variances. Consider data transformation, non-parametric alternatives, bootstrapping, and robust procedures. Larger sample sizes enhance ANOVA robustness. Post hoc tests should be robust to normality violations. Always report any deviations from assumptions for transparency and rigor.

First of all, DO NEVER replace your ANOVA with non-parametric methods or transformation without considering the consequences! You will entirely change the null hypothesis to something you might never want! Change of medians, it not median change is not mean change. Moreover, neither of the mentioned ones compare medians unless very strong assumptions are met. They assess stochastic superiority. Failed assumptions DO NOT mean you need non-param methods! Just run ANOVA over a more general parametric model, like the GLM (logistic, Poisson, gamma etc), use GEE estimation followed by Wald's analysis permutation ANOVA. Remember about modern ART ANOVA, ATS, WTS and quantile regression followed by Wald's. Don't stick to dinosaurs, limited and weak.

Choosing the right ANOVA test involves considering the assumptions of normality and homogeneity. Normality: ANOVA makes the assumption that every sample came from a population that was distributed normally. Histograms and Q-Q plots are good visual aids for verifying this assumption. Formal statistical tests such as Shapiro-Wilk, Kolmogorov-Smironov, Jarque-Barre, and D'Agostino-Pearson can also be used. Homogeneity of Variance: The ANOVA makes the assumption that the population variances from which the samples are drawn are equal. Should this presumption be broken, you can apply the Brown and Forsythe or Welch tests. You might also perform a nonparametric Kruskal-Wallis H Test.

Specifying the appropriate ANOVA test depends on several elements. 1) The structure of the empirical data. It requires the identification of the experimental design (if any). 2) What are experimental units? Are there experimental units of different sizes (split-plots, split-blocks, repeated measures...), will this lead to a single or several error terms? 3) The correct specification of the ANOVA model depends on the type & nature of the factors: are they crossed, nested, fixed or random. 4) Model specification must also account for blocking and covariates. 5) One must also account for data imbalance and potential confounding in main effects and interactions.

The choice between these ANOVA tests depends on the specific research questions. One-way ANOVA assesses differences between groups, two-way ANOVA explores main effects and interactions, while repeated measures ANOVA examines changes within the same subjects across multiple measurements. Each serves a different purpose based on the experimental design and hypotheses being tested.