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 Introduction

 Different types of ANOVA

 Assumption of ANOVA

 Hypothesis of ANOVA

 F-statistics and F-test

 Techniques of analyzing variance

 Application of ANOVA

 References

 

 ANOVA is the abbreviation for the full name of the method:
Analysis of variance. Invented by Ronald Fischer.

 Analysis of variance (ANOVA) is a collection of statistical
models and their associated estimation procedures (such as the
“variation” among and between groups) used to analyze the
differences among group means in a sample.

 Example:
 A group of psychiatric patients are trying three different

therapies: counseling, medication and biofeedback. You want
to see if one therapy is better than the others.

 

 

A one way ANOVA is used to compare two means from two
independent (unrelated) groups using the F-distribution.

When to use a one way ANOVA

Situation : You have a group of individuals randomly split into
smaller groups and completing different tasks. For example, you
might be studying the effects of tea on weight loss and form
three groups: green tea, black tea, and no tea.

 

 For example, adventurous researchers studying a
population of walruses might ask “Do the walruses
weigh more in early or late mating season?

 

 Here, the independent variable or factor (the two
terms mean the same thing) is “month of mating
season”.

 In an ANOVA, our independent variables are
organised in categorical groups.

 For example, if the researchers looked at walrus
weight in December, January, February and March,
there would be four months analyzed, and therefore
four groups to the analysis.

 

 In a one-way ANOVA there are two possible
hypotheses.

 The null hypothesis (H0) is that there is no
difference between the groups and equality
between means. (Walruses weigh the same in
different months)

 The alternative hypothesis (H1) is that there is a
difference between the means and groups.
(Walruses have different weights in different
months)

 

 Normality – That each sample is taken from a
normally distributed population

 Sample independence – that each sample has been
drawn independently of the other samples

 Variance Equality – That the variance of data in the
different groups should be the same

 Your dependent variable – here, “weight”, should be
continuous – that is, measured on a scale which can
be subdivided using increments (i.e. grams,
milligrams)

 

A one way ANOVA will tell you that at least two
groups were different from each other. But it
won’t tell you what groups were different.

 

 A two-way ANOVA is, like a one-way ANOVA, a
hypothesis-based test.

 It examines the influence of two different categorical
independent variables on one continuous dependent
variable.

 The two-way ANOVA not only aims at assessing the
main effect of each independent variable but also if
there is any interaction between them.

 

 Thinking again of our walruses, researchers might use a
two-way ANOVA if their question is: “Are walruses
heavier in early or late mating season and does that
depend on the gender of the walrus?”

 

 In this example, both “month in mating season” and
“gender of walrus” are factors –i.e, there are two
factors. Further, each factor’s number of groups must
be considered – for “gender” there will only two
groups “male” and “female”.

 The two-way ANOVA therefore examines the effect of
two factors (month and gender) on a dependent variable
– in this case weight, and also examines whether the
two factors affect each other to influence the
continuous variable.

 

 Your dependent variable – here, “weight”, should
be continuous – that is, measured on a scale which
can be subdivided using increments (i.e. grams,
milligrams)

 Your two independent variables – here, “month”
and “gender”, should be in categorical,
independent groups.

 

 Sample independence – that each sample has
been drawn independently of the other
samples

 Variance Equality – That the variance of data
in the different groups should be the same

 Normality – That each sample is taken from a
normally distributed population

 

 There are three pairs of null or alternative hypotheses for
the two-way ANOVA. Here, for walrus experiment, where
month of mating season and gender are the two
independent variables.

 H0: The means of all month groups are equal
 H1: The mean of at least one month group is different

 H0: The means of the gender groups are equal
 H1: The means of the gender groups are different

 H0: There is no interaction between the month and gender
 H1: There is interaction between the month and gender

 

 The key differences between one-way and two-way
ANOVA are summarized clearly below.

 1. A one-way ANOVA is primarily designed to enable the
equality testing between three or more means. A two-way
ANOVA is designed to assess the interrelationship of two
independent variables on a dependent variable.

 2. A one-way ANOVA only involves one factor or
independent variable, whereas there are two independent
variables in a two-way ANOVA.

 

 3. In a one-way ANOVA, the one factor or
independent variable analyzed has three or more
categorical groups. A two-way ANOVA instead
compares multiple groups of two factors.

 4. One-way ANOVA need to satisfy only two
principles of design of experiments, i.e. replication and
randomization. As opposed to Two-way ANOVA,
which meets all three principles of design of
experiments which are replication, randomization, and
local control.

 

 If you are comparing means between more than two groups,
why not just do several two sample t-test to compare their
mean from one group with the mean from each of the other
group?

 Before ANOVA, this was the only option available to
compare means between more than two groups.

 The problem with the multi t-test approach is that as the
number of groups increases, the number of two sample t-
test also increases.

 As the number of test increases the probablity of making a
type1 error also increases.

 

 The F-test can assess the equality of variances. F-tests are
named after its test statistic, F, which was named in honor
of Sir Ronald Fisher.

 The F-statistic is simply a ratio of two variances. Variance
is the square of the standard deviation.

 To use the F-test to determine whether group means are
equal, it’s just a matter of including the correct variances in
the ratio. In one-way ANOVA, the F-statistic is this ratio:

 F = variation between sample means / variation within
the samples

 

 The technique of analysing variance in case of single
variable and in case of two variable is similar.

 In both cases a comparision is made between the variance
of sample means with the residual variance.

 However, in case of single variable, the total variance is
divided into two two parts only,viz..,

 Variance between the samples and the variance within the
samples

 

 The later variance is the residual variance. In case of
two variables the total variance is divided in three
parts viz..,

1. Variance due to variable no.1
2. Variance due to variable no.2
3. Residual variance

 

 This is particularly applicable to experiment otherwise
difficult to implement such as is the case in clinical trails.

 In the bioequivalence studies the similarities between the
samples will be analyzed with ANOVA only.

 Pharmacovigillance data can also be evaluated using
ANOVA.

 Pharmacodynamic data can also be evaluated with ANOVA
only.

 That means we can analyze our drug is showing
pharmacological action or not’

 

 Pharmacy

 Biology

 Microbiology

 Agriculture

 Statistics

 Marketing

 Business research

 Finance

 Mechanical calculation