Anova FormulaThis module deca test 400 stack continue the discussion of hypothesis testing, where a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is athletes that took anabolic steroids on available information and anova table questions investigator's belief about the population parameters. The specific test anova table questions here anova table questions called analysis of variance ANOVA and is a test of hypothesis that is appropriate to compare means of a continuous variable in two or more independent comparison groups. For example, in some clinical trials there are more than two comparison groups. In a clinical trial to evaluate a new medication for asthma, investigators might compare an experimental medication to a placebo and to a standard treatment i. In an observational study such as the Framingham Heart Study, it might be of interest to compare mean blood pressure or mean cholesterol levels in persons who are underweight, normal weight, overweight and obese.
The ANOVA Table | STAT /
The factor is the characteristic that defines the populations being compared. In the tire study, the factor is the brand of tire. In the learning study, the factor is the learning method. Now, let's consider the row headings: In the learning example on the previous page, the factor was the method of learning. Sometimes, the factor is a treatment, and therefore the row heading is instead labeled as Treatment. With the column headings and row headings now defined, let's take a look at the individual entries inside a general one-factor ANOVA table: Yikes, that looks overwhelming!
Let's work our way through it entry by entry to see if we can make it all clear. Let's start with the degrees of freedom DF column:. Now, the sums of squares SS column:. As the name suggests, it quantifies the variability between the groups of interest.
It quantifies the variability within the groups of interest. As the name suggests, it quantifies the total variabilty in the observed data. The mean squares MS column, as the name suggests, contains the "average" sum of squares for the Factor and the Error:. The F column, not surprisingly, contains the F -statistic.
Okay, we slowly, but surely, keep on adding bit by bit to our knowledge of an analysis of variance table. Let's now work a bit on the sums of squares. In essence, we now know that we want to break down the TOTAL variation in the data into two components: Let's see what kind of formulas we can come up with for quantifying these components. But first, as always, we need to define some notation.
Let's represent our data, the group means, and the grand mean as follows:. Important thing to note here That is, the number of the data points in a group depends on the group i. That means that the number of data points in each group need not be the same. We could have 5 measurements in one group, and 6 measurements in another. With just a little bit of algebraic work, the total sum of squares can be alternatively calculated as: Can you do the algebra?
Now, let's consider the treatment sum of squares , which we'll denote SS T. Again, with just a little bit of algebraic work, the treatment sum of squares can be alternatively calculated as: Finally, let's consider the error sum of squares , which we'll denote SS E.
As we'll see in just one short minute why, the easiest way to calculate the error sum of squares is by subtracting the treatment sum of squares from the total sum of squares. Well, some simple algebra leads us to this: At any rate, here's the simple algebra: Well, okay, so the proof does involve a little trick of adding 0 in a special way to the total sum of squares:.
Then, squaring the term in parentheses, as well as distributing the summation signs, we get:. Eberly College of Science. One-Factor Analysis of Variance. Let's start with the degrees of freedom DF column: Now, the sums of squares SS column: Let's represent our data, the group means, and the grand mean as follows: That is, we'll let: Well, okay, so the proof does involve a little trick of adding 0 in a special way to the total sum of squares: Then, squaring the term in parentheses, as well as distributing the summation signs, we get: That is, we've shown that: Introduction to Probability Section 2: Discrete Distributions Section 3: Continuous Distributions Section 4: Bivariate Distributions Section 5: Distributions of Functions of Random Variables.
Hypothesis Testing Lesson Tests About Proportions Lesson Tests About One Mean Lesson Tests of the Equality of Two Means Lesson Tests for Variances Lesson Two-Factor Analysis of Variance Lesson Tests Concerning Regression and Correlation Section 8: Nonparametric Methods Section 9: Bayesian Methods Section