which analysis of variance should be applied when an experiment has more than one independent variable. Discuss the assumptions and limitations of the analysis of variance in comparison with the t test including a discussion of post hoc tests

When an experiment has more than one independent variable, the appropriate analysis of variance (ANOVA) to be applied is called a factorial ANOVA.

In a factorial ANOVA, each independent variable is called a factor, and their interaction is examined to determine if it has a significant effect on the dependent variable. This allows researchers to investigate not only the main effects of each independent variable but also their combined effects.

Assumptions of Analysis of Variance:
1. Independence: The observations should be independent of each other.
2. Normality: The distribution of the dependent variable in each group should be approximately normally distributed.
3. Homogeneity of Variance: The variances of the dependent variable should be equal across all groups.
4. Linearity: The relationship between the independent and dependent variables should be linear.

Limitations of Analysis of Variance compared to t-test:
1. Factorial ANOVA can become complex and difficult to interpret as the number of independent variables and levels increase.
2. It assumes that there are no interactions between independent variables, which may not always be the case.
3. Violations of the assumptions can impact the validity of the results.
4. It can only determine if there are significant differences among groups but does not identify which specific groups differ from each other.

Post hoc tests are used after a significant factorial ANOVA result to identify which specific groups differ significantly from each other. These tests allow for multiple comparisons, taking into account the increased chances of type I errors. Common post hoc tests include Tukey's Honestly Significant Difference (HSD), Fisher's Least Significant Difference (LSD), and Bonferroni correction.

To perform a factorial ANOVA with post hoc tests, you will need statistical software like SPSS, R, or Excel with an ANOVA function. First, collect and organize your data, ensuring you have a sufficient sample size. Then, input the data into your chosen software, designating the independent variables as factors. Run the factorial ANOVA analysis, and if it yields significant results, proceed to conduct post hoc tests to determine specific group differences.