One way ANOVA-post hoc t-tests and Factorial ANOVA

What I'm going to do in this blog is talking about one way ANOVA-post hoc t-tests and factorial ANOVA.

 

1. One way ANOVA-post hoc t-tests : it just means how to follow up a significant F-test in one way ANOVA with pairwise comparisons of group means using pairwise t-tests or confidence intervals. They tell us which group means differ significantly and in what direction.

 1-1) Follow up comparisons : It's also referred to as post-hoc comparisons. The post-hoc indicates that we're making comparisons after the fact. This implies using two-sided tests or confidence intervals. To find out how we should interepret a significant overall effect, we'll determin post-hoc confidence intervals.

 1-2) The number of comparisons to do : If we have g grops, there are g times g-1 divided by 2 comparisons to be made. These comparisons should only be performed if the overall test is significant.

** when we perform the comparisons, whether using pairwise t tests or confidence intervals, the same assumption should hold as for the f test, which is independence, normality, homogeneity.

 1-3) The formulas for the T test and confidence interval : these are almost the same as for the regular T test and confidence interval for two independen groups assuming equal population variances.

In post-hoc comparisons, we use Fisher's least significant differnece method.

 1-4) The ways to correct for the inflated family wise error rate : Since it makes multiple comparisons, we should correct for the family wise error rate, the probability that at least one of the comparisons will result in a false rejection of the null hypothesis. Here, I'll introdue two of these.

•  The Bonferroni method : The Bonferroni method involves dividing the desired overall alpha by the number of comparisons and using the result corrected alpha for the individual comparisons. With this correction, the actual probability of falsely rejecting the null hypothesis will be smaller than or equal to the desired overall alpha. In many cases, the correction is overly conservative resulting in a smaller alpha and less power.

  

• Tukey's honestly significant difference method : This way is less conservative. The actual probability of falsely rejecting the null is closer to the desired overall alpha. This results in more powerful tests and narrower confidence interval methods than the Bonferroni method.

 

2. Factorial ANOVA : Factorial ANOVA with one dependent variable combines the levels of all factors, allowing us to test whether the means differ between the factor levels for each factor separately. The factors are categorical independent variables each with two or more categories.