How do you choose between a one-way and two-way ANOVA?

One-way ANOVA is suitable when you have one categorical independent variable (factor) with three or more levels and a continuous dependent variable. It assesses whether there are statistically significant differences in the means of the dependent variable across the different levels of the independent variable.

One-way ANOVA is used whenever a single factor (at 2 or more levels) is to be investigated. However, in practice it is pretty rare that a single source of variation is to be accounted for.

It is important to note that prior to conducting a One-Way ANOVA, the researchers must ensure that the tests assumptions are met. They are: (1) Normality. Scores must be normally distributed and follow a bell-shaped curve. (2) Homogeneity of Variance. The variance between groups must not be significantly different. (3) Independent Observations. Scores obtained from each subject cannot be influenced by other subjects. (4) Random Sampling. Subjects must be assigned to groups randomly.

-> One way Analysis of Variance -> Considers only one categorical variable -> It is possible to compare means of three or more samples. -> Null Hypothesis: All population means are equal -> Alternate Hypothesis: At least one of the means is different -> Alternate Hypothesis: At least one of the means is diffrent.

To clarify, the Bonferroni correction itself is not a posthoc test, but rather an adjustment you make to simultaneous confidence intervals. This is to ensure that when you are looking for significant differences between two of the multiple groups you just compared with ANOVA, you are not increasing your chance of making a Type I error.

Two-way ANOVA is applicable when you have two independent variables (factors) and one continuous dependent variable. It assesses the influence of each independent variable individually, as well as their interaction effect on the dependent variable. Two-way ANOVA is more complex but provides insights into how factors may interact to impact the dependent variable.

Two-way ANOVA is used whenever a two factors capturing a known source of variation are to be investigated. Depending on whether replication was used, one may explore the main effects of the factors and their interaction (whenever there are replications).

The assumptions for a Two-way ANOVA are very similar to One-Way ANOVA (i.e., data independence, random sampling, normality, and homogeneity of variance). Prior to conducting the ANOVA test, these assumptions must be met. It is generally good practice to perform the following assumptions checks: (1) Normality. This assumption can be checked using visualization (e.g., density plot, histogram, or scatter-plot) or statistical test (e.g., Shapiro-Wilk). A non-significant result (p >.05) indicates that data is normally distributed. (2) Homogeneity of Variance. A Levene's Test should be used. A non-significant result (p >.05) indicates that the variance between both groups are equal.

-> Two-way analysis of variance. -> We investigate the influence of two independent variable on a dependent variable. -> Also finds if there exist any inter-relationship exist between the independent variable.

Choose between one-way and two-way ANOVA based on the complexity of your experimental design. If you have only one categorical independent variable, use one-way ANOVA. If you have two independent variables and want to examine their individual and interactive effects, use two-way ANOVA. Consider the research question and design of your study to determine the most appropriate analysis.

The choice between one way or two way anova choice is self explanotary. If the test is about measuring a single feature like age for different groups, regular one way anova is used. The idea is to perform a hypothesis testing of mean differences.

If you're just collecting data relevant to a single independent variable, a one-way ANOVA is sufficient. If, however, your study has multiple independent variables, a two-way or MANOVA approach is best.

The choice of a one-way or a two-way ANOVA depends on the structure of the data. This is an experimental design question. The correct ANOVA model must reflect the underlying data structure.

Prior to pick which statistical test to perform, I must ensure that my (1) Independent variable(s) (IV) are categorical and (2) Dependent variable (DV) is continuous. If I only have one IV with two or more levels (i.e., two or more category), I will perform a one-way ANOVA. For example, I am investigating whether the following treatments (medication-only, therapy-only, combination, placebo) improve cognitive functions in dementia patients using the Mini Mental State Exam (MMSE). In this scenario, my IV is treatment with 4 levels, all categorical, and my DV is MMSE scores.

The ANOVA model must be correctly specified and must reflect pairing, repeated measures, different sizes of experimental units, constraints to randomization, type of factor (crossed, nested), (fixed vs. random), data imbalance.

Although ANOVA is somewhat robust to assumptions with large and equal sample sizes for each group, it is a good idea to test for these assumptions, consider the impact if violated and then perform appropriate remedial (e.g., Welch test for not equal variances or notable differences in sample sizes for each group; or the nonparametric test like Kruskal-Wallis). You should also consider using multiple comparisons if you do reject the null hypothesis. The ANOVA will let you know at least one of the groups are different, and the multiple comparison will enable you to determine which one is different.

Performing ANOVA involves calculating the F-statistic, which compares the variance between group means to the variance within groups. For one-way ANOVA, use software or statistical packages to conduct the analysis. For two-way ANOVA, the procedure is more involved, including assessing main effects and interaction effects. Utilize statistical software like R, Python, or dedicated statistical packages (e.g., SPSS, SAS) for accurate implementation.

In Python, you'd use scipy.stats.f_oneway() for ANOVA. Import it from scipy.stats, then input your data arrays for each group. In R, the aov() function conducts ANOVA. Use a formula like aov(response ~ y, data=df) where response is the numeric variable, y is the categorical variable, and df is your dataset. Both methods help analyze if there are significant differences between groups' means given their data. For one-way ANOVA, you're comparing the means of three or more groups to determine if at least one of the groups is significantly different from the others. The functions mentioned For two-way ANOVA, which involves analyzing the influence of two categorical independent variables on one continuous dependent variable

I tend to use programming languages (Python & R) to perform statistical test. In Python, the package scipy.stat has the package "f_oneway" to perform the oneway ANOVA and "tukey_hsd" for any post-hoc tests that we may need.

Interpret ANOVA results by examining the F-statistic, associated p-value, and effect sizes. In one-way ANOVA, a significant result indicates at least one group mean differs from others. In two-way ANOVA, consider main effects and interaction effects. Interpretation involves assessing whether differences are statistically significant and understanding the practical significance of those differences.

To understand a 2-way ANOVA, you need to observe your main effects (the effect of factor A and factor B). It should be noted that these main effects exist when controlling for the additional factor (i.e. Factor A = effect of Factor A when Factor B is fixed). Then you can look at the interaction effect. Two way interactions can be simple but can get complicated pretty quickly. Likely scenarios: 1. multiplicative effects - the whole is greater than the sum of its parts. 2. detractive effects - one factor reduces the effect of the other when interacting with each other. 3. additive - one factor increases the effects of the other in a linear fashion 4. subtractive - one factor reduces the effects of the other in a linear fashion.

When interpreting scores from ANOVA, one should look at the following: (1) p-value. Depending on the pre-determined alpha level, if p-value < alpha, we can conclude that a significant difference in DV exists between groups. (NOTE: at this point, we cannot say which group has the highest scores yet). (2) TukeyHSD should be performed next. These are essentially multiple t-tests to identify whether the mean difference between any two groups are significant. (3) visualization. A bar-plot should be constructed to compare the mean differences between groups. Only when the above steps are completed, one can make inferences about the relationship between IV and DV.

Sample Size: Larger samples improve the reliability of both tests. Consider power analysis to ensure enough data. Post-hoc Tests: If main effects are significant, use post-hoc tests to identify specific group differences in one-way ANOVA. In two-way ANOVA, explore specific interactions between factors. Visualizations: Graphs and plots can bring your analysis to life. Use them to illustrate main effects, interactions, and group differences.

Try to get acquinted with more sophisticated methods and algorithms for hypothesis testing and bayesian modeling : multiarm bandits, belief networks, causal inference.

Whenever two factors are considered, one must also determine whether there are different experimental unit sizes (such as repeated measures designs or constraints to randomisation - split-plot, split-block designs) to correctly specify the ANOVA model. Failure to do so may result in incorrect inferences).

Once you choose which ANOVA to leverage, consider how best to visualize the results. For a one-way ANOVA, which tests for differences among group means based on a single factor, the following visualizations are useful: boxplots, bar charts, histograms, or violin plots. For a two-way ANOVA, which tests for differences among group means based on two factors, these visualizations are appropriate: interaction plot, grouped bar chart, heatmap, or multiple boxplots/violin plots.

While robust, ANOVA depends on assumptions that can easily be violated (normality and homoscedasticity especially). Be sure to check these assumptions and verify that they are met. If they are not, you have better-suited, non-parametric options such as the Kruskal-Wallis and/or Friedman test.