Analysis of Variance (ANOVA) King Saud University College of Pharmacy Department of Pharmaceutics PHT 463: Quality Control of Pharmaceutical Dosage Forms Summer Semester of 1423-1424H Analysis of Variance (ANOVA) Ibrahim A. Alsarra, Ph.D.

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.

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).

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.

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.

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.

Why not just use the Student's' t Test Each time two means are compared the probability (Type I error) =  (i.e. 0.05). M1 M2 Probability (Type I error) = 0.05 (Type I error) =  M1 M2 M3 Probability (Type I error) = 0.05+0.05

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

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) < 

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

Remember that… Total Variance Between Groups Within Groups

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

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

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

How to Setup the ANOVA Summary Table

How to Setup the ANOVA Summary Table Source

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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.

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