Name: Jenifer L. Acido
MAFSTA 620 (ISTADISTIKA SA FILIPINO)
Professor: Elmer G. De Jose, PhD
Analysis of Variance (Two-Way Anova)
Sir Ronald Aylmer Fisher, byname R.A. Fisher, (born
February 17, 1890, London, England—died July 29,1962,
Adelaide, Australia), British statistician and geneticist who
pioneered the application of statistical procedures to the
design of scientific experiments. He was the one who
developed ANOVA is also called the Fisher analysis of
variance, and it is the extension of the t- and z-tests.
What is Two-way Analyis of Variance?
ANOVA (Analysis of Variance) is a statistical test used to analyze the difference between the
means of more than two groups.
When to use a two-way ANOVA
Use a two-way ANOVA when you have collected data on a quantitative dependent
variable at multiple levels of two categorical independent variables. A quantitative
variable represents amounts or counts of things. It can be divided to find a group mean.
Why does need to use two way ANOVA?
Use a two-way ANOVA when you want to know how two independent variables, in
combination, affect a dependent variable. To tests three null hypotheses at the same time and to
provide all the information needed to interpret the model.
Three Null Hypotheses
There is no difference in group means at any
level of the first independent variable.
There is no difference in group means at any
level of the second independent variable.
The effect of one independent variable does not
depend on the effect of the other variable (a.k.a
no interaction effect)
Three Alternative Hypotheses
There is a difference in group means at any level
of the first independent variable.
There is a difference in group means at any level
of the second independent variable.
The effect of one independent variable does
depend on the effect of the other variable (a.k.a
no interaction effect)
Assumptions of the two-way ANOVA
To use a two-way ANOVA the data should meet certain assumptions. The two-way ANOVA
makes all of the normal assumptions of a parametric test of difference:
Homogeneity of variance (a.k.a. homoscedasticity)
The variation around the mean for each group being compared should be similar among all groups.
Independence of observations
The independent variables should not be dependent on one another (i.e. one should not cause the
other). And also the dependent variable should represent unique observations.
Normally-distributed dependent variable
The values of the dependent variable should follow a bell curve. If your data don’t meet this
assumption, you can try a data transformation.
Model
Degrees of Freedom (Df) shows the degree of freedom for each variable (number of levels in the
variable minus 1)
Sum of Squares (Sum sq) is the sum of squares (a.k.a the variation between the group means created
by the levels of the independent variable and the overall mean).
Mean Square (Mean sq) shows the mean sum of squares (sum of squares divided by the degrees of
freedom.
F Value is the test statistic from the F-test (the mean square of the variable divided by the mean square
of each parameter).
Pr(>F) is the p-values of the F-Statistic, and shows how likely it is that the F-Value calculated from
the F-Test would have occurred if the null hypothesis of no difference was true.
Here is the example of computing Two-Way Anova using
 Manual Computation
 Excel
 Jamovi Software
Manual Computation
Students
Low Noise
Male
Students
10
12
11
9
12
13
10
13
C1=90
Female
Students
Medium
Noise
7
9
8
12
13
15
12
12
C2= 88
Loud Noise
Column
Total
Does Noise has an effect on students scores?
Does Gender has an effect on students scores?
Does Gender effect how students react to noise?
4
5
6
5
6
6
4
4
C3=40
1st Step: State a Null Hypothesis
Null Hypothesis
There is no effect of one variable on the other.
2nd Step: Calculate the Row Total and Column Total
Row
Total
R1=98
R2=120
Step 2
3rd Step: Calculate the Correction Factor
4th Step: Sum of Squares of Total
5th Step: Sum of Squares of Column
6th Step: Sum of Squares of Row
7th Step: Sum of Squares within groups
4
17
8th Step: Residual Sum of Square
9th Step: Make a table for different models
10th Step: Get the F Crtitical and make a conclusion
Using Excel
1ST Step: Make a Table
2nd Step: Find Data Tool
3rd Step: Click Two-Factor with Replication
4th Step: Input Range and Select Range in the sheet
5th Step: Output Range and Click OK
6th Step: Results
Interpret the
Results
Using Jamovi Software
Step 1: Input all the Data
Step 2 and 3 : Analyze if the variable is Independent or Independent
Interpret the Data Results
Bevans, R. (2002) Two-Way ANOVA | Examples & When To Use
https://www.scribbr.com/author/beccabevans/
Introduction to Fisher https://link.springer.com/chapter/10.1007/978-1-4612-4380-9_7
Vectors Academy https://youtu.be/0K-bfzLTRiY