By: Mona TarekAssistant Researcher

Department of PharmaceuticsStatistics

Supervised by Dr. Amal FatanyKing Saud University

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)

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)

But is more accurate than t-test as:

T-test ANOVAIt 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.

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)

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.

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?

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

First construct the null hypothesis: (at p=0.05)

There is no difference between the two regimens Then construct another table:

Source of variation

DF Sum of squares (SS)

Mean Square(SS/ DF)

F-ratio(BSS/WSS)

Between regimens(BSS)

t-1= 9 160.54

Within regimens(WSS)

N-t=20

total N-1=29 203.87

∑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

Calculated F-value= 8.22

Then we look at a tabulated F-value at DF1 = 9 (between regimens)DF2 = 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 rejectedi.e. there is a significant difference between the regimens.