1. Analysis of Variance (ANOVA)
King Saud University
College of Pharmacy
Department of Pharmaceutics
PHT 463: Quality Control of Pharmaceutical Dosage Forms
Summer Semester of H
Analysis of Variance (ANOVA)
Ibrahim A. Alsarra, Ph.D.

2. Introduction
T-test is the statistical technique we use when we want to see if there is a significant difference in the mans of two groups on the performance of some measures.
The variable forms the groups is called the independent variable and the variable that is measured is called the dependent variable.

3. Introduction…cont.
The analysis of variance, or ANOVA, is an extension of the t-test.
It is used when you have more the TWO groups (so if you have THREE or MORE variables, you would use the ANOVA rater that the t-test).

4. ANOVA
It is a useful tool, which helps the user to identify sources of variability from one or more potential sources, sometimes referred to as “treatments” or “factors”.
The main purpose of analysis of variance is to test for significant differences between groups of data.

5. One-Way ANOVA
The one-way ANOVA is a method of analysis that requires multiple experiments or readings to be taken from a source that can take on two or more different inputs or settings.
The one-way ANOVA performs a comparison of the means of a number of replications of experiments performed where a single input factor is varied at different settings or levels.

6. One-Way ANOVA…cont.
The aim of this comparison is to determine the proportion of the variability of the data that is due to the different treatment levels or factors as opposed to variability due to random error.
The model deals with specific treatment levels and is involved with testing the null hypothesis.
It deals with one independent variable and one dependent variable.

7. Why not just use the Student's' t Test
Each time two means are compared the probability
(Type I error) =  (i.e. 0.05). M M2
Probability (Type I error) = 0.05
(Type I error) =  M M M3
Probability (Type I error) =

8. Why not just use the Student's' t Test …cont.
Multiple t-tests are not the answer because as the number of groups grows, the number of needed pairs comparisons grows quickly

9. Why not just use the Student's' t Test …cont.
Analysis of Variance protects from inflating Type I error by making the
Experiment-Wise
Probability (Type I Error) < 

10. Key Terms in ANOVA Between and within groups Treatments Sum of squares
Degree of freedom
F-test

11. Remember that…
Total Variance
Between Groups
Within Groups

12. Keep in Mind…
ANOVA
Mathematically, ANOVA is the ratio of the variation between groups to the variation within groups = F

13. Between Groups Variance
The variance in the data that can be attributed to the independent variable
The variance among the means

14. Within Groups Variance
The variance due to all other sources:
Subject factors
Error Variance
Residual variance
Variance among data and group means

15. How to Setup the ANOVA Summary Table

16. How to Setup the ANOVA Summary Table
Source

17. How to Setup the ANOVA Summary Table …cont.
Source
Total

18. How to Setup the ANOVA Summary Table …cont.
Source
Between
Within
Total

19. How to Setup the ANOVA Summary Table …cont.
Source
Sum of Squares
Between
SSB
Within
SSW
Total
SST

20. How to Setup the ANOVA Summary Table …cont.
Source
Sum of
Squares
Degree of Freedom (df)
Between
SSB
dfB
Within
SSW
dfW
Total
SST
dfT

21. How to Setup the ANOVA Summary Table …cont.
Source
Sum of
Squares
Degree of Freedom (df)
Mean Squares
Between
SSB
dfB
MSB
Within
SSW
dfW
MSW
Total
SST
dfT

22. How to Setup the ANOVA Summary Table …cont.
Source
Sum of
Squares
Degree of Freedom (df)
Mean Squares
Between
SSB
dfB
MSB
Within
SSW
dfW
MSW
Total
SST
dfT F

23. Calculating F Statistic
Calculate sums of squares
Calculate df
Calculate mean squares
Calculate F

24. Determining the Critical F
Alpha = 0.05
Find Column for df between
Find Row for df within
Compare Critical F (Table F) to Obtained (our) F

25. Statistical Decision Making
If Critical F > Obtained F
Failed to reject null hypothesis
If Critical F < Obtained F
Reject null hypothesis

26. Multiple Comparison Tests
All a significant ANOVA tells us is that there is a significant difference between two or more of the mean, but not which means
To find out where the significant differences lie, we need to carry out a POST-HOC ANALYSIS

27. Multiple Comparison Tests…cont.
POST-HOC ANALYSIS: means analysis after the initial analysis, so though the ANOVA gave us some information.
There are a number of post-hoc analyses possible. Two are used often and will suit most situations: Tukey’s Test and Scheffe’s Test.

28. Tukey’s Tests
It is often called the Honestly Significant Difference (HSD) Test.
It is best used when the groups have the same number of subjects.
It is a fairly easy to calculate
It is a more conservative test that is usually used to compare the significant differences between each set of pairs

29. Scheffe’s Tests
It is best used when there are different numbers of individuals in the various groups, however it can also be used when there are equal sample sizes.

30. Two-Way ANOVA
Two or three independent variables and one dependent variable
The one-way ANOVA performs a comparison of the means of a number of replications of experiments performed where a single input factor is varied at different settings or levels.

31. Two-Way ANOVA…cont.
The key is:
INTERACTION