1. By: Mona Tarek Assistant Researcher Department of Pharmaceutics Statistics Supervised by Dr. Amal Fatany King Saud University

2.  It compares variance (or means) of test groups in relation to the associated error.  Useful in: Designs for tests comparing more than 2 groups. Separation of variation due the treatment from variation due to experimental error. (within group variation)

3.  This is the ANOVA form used for: Comparing means from two or more groups. Parallel groups design  It is the multiple analogue of the two independent t test (unpaired data)

4.  But is more accurate than t-test as: T-testANOVA It doesn’t differentiate between difference in means due to treatment and due to error. It gives the difference due to treatment only; it removes any errors ( as variability among experimental units & other experiment error sources among a single group.) E.g. manufacture, personal error, and time factor. A t-value is calculated and compared to a tabulated one. More complex calculations that lead to F-value that is then compared to a tabulated F one.

5.  The design of ANOVA ( randomized block design): The experimental units are divided into “t” no. of groups that equals no, of applied treatments. Total no. of observations (experiment units N) is conveniently chosen to be divisible by t i.e. N/t = integral no. that is the no. of units in each group.( ie the number of units in each group is the same)

6.  F-ratio relates (variance due treatment /variance due error)  To calculate the F-ratio, we calculate:  BSS: the between sum of squares ----  represent the actual difference among the tested treatment ---  the lager value numerator  WSS: the within sum of squares ------  represent the difference within a single treatment group. i.e represent error due to variability among experiment units denominator  As the value of WSS  F meaning: more significant difference is declared with more confidence due to the treatments not error.

7.  Example: Groups of three subjects were given 1 of 10 food regimen & showed the weight gain in kgs in the following table. These are unpaired data & it’s a completely randomized experiment. There are only two sources of variation the variation between the regimens & the variation within regimens. Are all the food regimens the same?

8. No. of treatments ( groups) (t=10) No. Of Units in each group n Total Number of observations (N)= 30 N.B. n= N/t sum of observations in each group separately sum of all observations Square each observation and add them up in each gp. Sum of squares of all observations

9.  First construct the null hypothesis: (at p=0.05) There is no difference between the two regimens  Then construct another table: Source of variation DFSum of squares (SS) Mean Square (SS/ DF) F-ratio (BSS/WSS) Between regimens (BSS) t-1= 9160.54 Within regimens (WSS) N-t=20 totalN-1=29203.87 ∑X² - ( ∑X )²/ N (∑x₁ )²/n₁ +(∑x₂)²/n₂+(∑x₃)²/n₃ + ∙ ∙ ∙ +(∑x͵)²/n͵ - (∑X)²/N (∑X)²/N is called correction term 43.33 Total-BSS 160.54/9 = 17.81 43.33/20 =2.17 17.81/ 2.17= 8.22

10.  Calculated F-value= 8.22  Then we look at a tabulated F-value at DF 1 = 9 (between regimens) DF 2 = 20 (within regimens) Tabulated F-value = 2.39 at p-value= 0.05 ( There are different f-tables depending on the p-value we are working at) Calculated value is larger than the tabulated value at p=0.05 :. Null hypothesis is rejected i.e. there is a significant difference between the regimens.