ANOVA is a method of analyzing data fromdesigned experiments whose objective is tocompare two or more group means.

Factorial ANOVA has two independent variables which are crossed with each other.That means each value of one variable ispaired with every value of other variable.

TYPES OF ANOVATYPES OF ANOVA

ONE WAY ANOVA

TWO WAY ANOVA

ANOVA

One-Way ANOVA Two way ANOVA

•One independent variable

•One dependent variable For Example, Only temperature as independent variable

•Two or more independent variables

•Two dependent variables

For Example, Both Temperature and Concentration asindependent variable

Concentration

Pure Methanol

50% Methanol

Room Temperature

108 120

98 130

600

Temperature194 144

202 140

            Main Effect A

  Pure Methanol50%

Methanol Sum Square Sum of Squares SS/4 (SS/4)-CT

Room 108 120 456 207936      

Temperature 98 130     670336 167584 6272

60° C 194 144 680 462400      

Temperature 202 140          

Sum 602 534 1136        

Square 362404 285156 1290496        

SS 647560 161312 CT      

SS/4 161890            

Main Effect B 578            

concentration

It is square of sum of grand total of all the observations divided by number of observations

It is sum of squares of grand total of the observations (in column) divided by product of number of rows and replication which is subtracted by CT

       140202

   80656156816284396144194

3042171204342408    13098

   6250042436250206120108

MS-A-B-CTSS/2Sum of Squares

Square of sum

Square of sumSumSum

50% Methanol

Pure Methanol 

Interaction term AB         

Room

temp.

60 °C

concentration

            TOTAL

  Pure Methanol 50% Methanol Square SquareSum of

squares SS-CT

108 120 11664 14400    

98 130 9604 16900 171344 10032

194 144 37636 20736    

202 140 40804 19600    

Room

temp.

60 °C

Residual = Total (10032) – Main effect A (578) – Main effect B (6272) –Interaction term AB (3042)

= 140

Main effect A= (∑ Cj2)/Rr - CT Main effect B= (∑ Ri

2 ) /Rr – CTCj= Sum of observations in column jRi= Sum of observations in row iR= number of rows r= number of replicates per cell CT= Correction term

Interaction term AB= ∑ Cij2/2 – A – B - CT

Total= ∑ X2 - CTX = All Observations in each column

and row

Source of Variation DF SS

MS F P

Main Effect A 1 6272 6272 179.2 <0.001

Main Effect B 1 578 578 16.514 0.015

Interaction Term AB 1 3042 3042 86.914 <0.001

Residual 4 140 35   

Total 7 10032 1433.143   

ANOVA          

 Df SS MS F Significance F

Regression 3 9892 3297.333 94.20952 0.000363456

Residual 4 140 35   

Total 7 10032     

Sigma stat plot Multiple linear regression

 

While in both the case RESIDUAL will be same

Source of Variation DF SS

Main Effect A 1 6272

Main Effect B 1 578

Interaction Term AB 1 3042

Residual 4 140

  DF SS

Regression 3 9892

Residual 4 140

Sigma stat plot is one of the software to carry out ANOVA.

Over here we get individual sum of squares which is an advantage over multiple linear regression analysis.

While any of the independent variable has a significant

effect on the dependent variable can be easily known from multiple linear regression analysis.