Which analysis of variance should be applied when an experiment has more than one independent variable.
What are the assumptions and limitations of the analysis of variance in comparison with the t test including post hoc tests.
When an experiment has more than one independent variable, the appropriate analysis of variance (ANOVA) to use is called a factorial ANOVA.
To conduct a factorial ANOVA, you need to follow these steps:
1. Determine the number and levels of the independent variables: Identify the independent variables and their levels in your experiment. For example, if you are studying the effects of both gender (male vs. female) and age group (young vs. old) on a dependent variable, you have two independent variables with two levels each.
2. Collect and organize the data: Gather data for your dependent variable from each combination of the independent variables. For example, if you have 10 males and 10 females in each age group, you will have a total of 40 participants and 40 data points.
3. Calculate the sum of squares: Use the data to calculate the sum of squares between and within groups. The sum of squares between groups measures the variability between different combinations of independent variables, while the sum of squares within groups measures the variability within each combination of independent variables.
4. Test the significance: Perform the calculations to compare the between-groups variability to the within-groups variability using appropriate statistical tests, such as F-tests. This will determine if there are significant differences in the dependent variable across the different combinations of independent variables.
Assumptions and limitations of ANOVA compared to t-tests with post hoc tests:
Assumptions of ANOVA:
1. Independence: The observations should be independent of each other.
2. Normality: The dependent variable should be normally distributed within each combination of independent variables.
3. Homogeneity of variances: The variances of the dependent variable should be equal across the different combinations of independent variables.
Limitations of ANOVA compared to t-tests with post hoc tests:
1. Factorial ANOVA cannot determine which specific combinations of independent variables are significantly different from each other. It only provides an overall test for differences across the combinations.
2. ANOVA assumes that there are no interactions between the independent variables, meaning that the effect of one independent variable does not depend on the levels of another independent variable.
3. ANOVA can be more complex to interpret and may require additional statistical knowledge compared to t-tests with post hoc tests, especially when multiple independent variables are involved.
4. ANOVA assumes that the samples are representative of the population, which may not always be the case.
In summary, factorial ANOVA is the appropriate analysis to use when an experiment has more than one independent variable. It has assumptions related to independence, normality, and homogeneity of variances. It provides an overall test for differences but does not directly point out specific significant differences between combinations of independent variables. It also assumes no interactions between the independent variables and requires careful interpretation.