Analysis of variance (ANOVA)
It is a collection of statistical model and their associated estimation procedures (such as the "variation" among and between groups) used to analyse the differences among group means in a Sample. ANOVA was developed by the Statistician Ronald Fisher. The ANOVA is based on the law of total variance , where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means.
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
Basically, we are testing groups to see if there’s a difference between them. Examples of when you might want to test different groups:
- A group of psychiatric patients are trying three
different therapies: counselling, medication and biofeedback. You want to
see if one therapy is better than the others.
- A manufacturer has two different processes to make
light bulbs. They want to know if one process is better than the other.
- Students from different colleges take the same exam.
You want to see if one college outperforms the other.
Replication: It’s whether you are replicating (i.e. duplicating) your
test(s) with multiple groups. With a two way ANOVA with replication,
you have two groups and individuals within that group are doing more than one
thing (i.e. two groups of students from two colleges taking two tests). If you
only have one group taking two tests, you would use without replication.
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.
·
One Way
ANOVA
·
A one way ANOVA is used to
compare two means from two independent (unrelated) groups using the F-distribution . The null hypothesis for the test is that the two means are equal. Therefore, a Significant result means that the two means are unequal.
·
Examples
of when to use a one way ANOVA
·
Situation
1: 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.
Situation 2: Similar to situation 1, but in this case the individuals
are split into groups based on an attribute they possess. For example, you
might be studying leg strength of people according to weight. You could split
participants into weight categories (obese, overweight and normal) and measure
their leg strength on a weight machine.
·
Limitations
of the One Way ANOVA
·
A one way ANOVA will tell you
that at least two groups were different from each other. But it won’t tell
you which groups were different. If your test returns a significant
f-statistic, you may need to run an ad hoc test (like the Least significant difference to
tell you exactly which groups had a difference in means. Back to Top.
·
Two Way
ANOVA
· A Two Way ANOVA is an extension of the One Way ANOVA. With a One Way, you have one independent variable affecting a dependent variable With a Two Way ANOVA, there are two independents. Use a two way ANOVA when you have one measurement variable and two nominal variable . In other words, if your experiment has a quantitative outcome and you have two categorical explanatory variables a two way ANOVA is appropriate.
·
For example, you might want to
find out if there is an interaction between income and gender for anxiety level
at job interviews. The anxiety level is the outcome, or the variable that can
be measured. Gender and Income are the two categorical variables. These categorical variables
are also the independent variables, which are called factors in a Two
Way ANOVA.
·
The factors can be split into levels.
In the above example, income level could be split into three levels: low,
middle and high income. Gender could be split into three levels: male, female,
and transgender. Treatment groups are all possible combinations of the factors.
In this example there would be 3 x 3 = 9 treatment groups.
·
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. For the above example, multiple stress scores could be entered into
cells. If you do enter multiple observations into cells, the number in each
cell must be equal.
·
Two null hypothesis 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.
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.
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