Statistical test

1. ANOVA TEST

2. Analysis of Variance (ANOVA) is a parametric statistical technique used to compare datasets.
This technique was invented by R.A. Fisher, and is thus often referred to as Fisher’s ANOVA, as
well.
It is similar in application to techniques such as t-test and z-test, in that it is used to compare
means and the relative variance between them.
However, analysis of variance (ANOVA) is best applied where more than 2 populations or
samples are meant to be compared.
An ANOVA test is a way to find out if survey or experiment results are significant.
In other words, they help you to figure out if you need to reject the null hypothesis or accept
the alternative hypothesis

3. The basic principle of ANOVA is to test for differences among the means of the populations by
examining the amount of variation within each of these samples, relative to the amount of
variation between the samples.
In terms of variation within the given population, it is assumed that the values of (Xij) differ from
the mean of this population only because of random effects i.e., there are influences on (Xij)
which are unexplainable, whereas in examining differences between populations we assume that
the difference between the mean of the jth population and the grand mean is attributable to what
is called a ‘specific factor’ or what is technically described as treatment effect.
Thus while using ANOVA, we assume that each of the samples is drawn from a normal
population and that each of these populations has the same variance.
We also assume that all factors other than the one or more being tested are effectively controlled.
We further assume the absence of many factors that might affect our conclusions concerning the
factor(s) to be studied.
In short, we have to make two estimates of population variance viz., one based on between
samples variance and the other based on within samples variance.

4. Then the said two estimates of population variance are compared with F-test, wherein we work
out.
F = Estimate of population variance between samples variance DIVIDED by Estimate of
population variance within samples variance
This value of F is to be compared to the F-limit for given degrees of freedom.

5. TYPES OF TESTS
There are two main types:
one-way and two-way.
Two-way tests can be with or without replication.
One-way ANOVA between groups: used when you want to test two groups to see if there’s a
difference between them.
Two way ANOVA without replication: used when you have one group and you’re double-
testing that same group. For example, you’re testing one set of individuals before and after they
take a medication to see if it works or not.
Two way ANOVA with replication: Two groups, and the members of those groups are doing
more than one thing. For example, two groups of patients from different hospitals trying two
different therapies.

6. HOW TO RUN AN ANOVA
These tests are very time-consuming by
hand.
In nearly every case you’ll want to use
software.
For example, several options are available
in Excel:
Two way ANOVA in Excel with replication
and without replication.
One way ANOVA in Excel 2013.

7. STEPS
Find the mean for each of the groups.
Find the overall mean (the mean of the groups combined).
Find the Within Group Variation; the total deviation of each member’s score from the Group
Mean.
Find the Between Group Variation: the deviation of each Group Mean from the Overall Mean.
Find the F statistic: the ratio of Between Group Variation to Within Group Variation.

8. MAIN EFFECT AND INTERACTION EFFECT
 The results from a Two Way ANOVA will calculate a main effect and an interaction effect.
 The main effect is similar to a One Way ANOVA: each factor’s effect is considered separately.
 With the interaction effect, all factors are considered at the same time. Interaction effects between factors
are easier to test if there is more than one observation in each cell.
 Two null hypotheses are tested if you are placing one observation in each cell. For this example, those
hypotheses would be:
H01: All the income groups have equal mean stress.
H02: All the gender groups have equal mean stress.
 For multiple observations in cells, you would also be testing a third hypothesis:
H03: The factors are independent or the interaction effect does not exist.
 An F-statistic is computed for each hypothesis you are testing.

9. ASSUMPTIONS FOR TWO WAY ANOVA
The population must be close to a normal distribution.
Samples must be independent.
Population variances must be equal.
Groups must have equal sample sizes.