ANOVA

Topics: Analysis of variance, Statistical hypothesis testing, Multiple comparisons Pages: 17 (1286 words) Published: December 19, 2013
Analysis of Variance (ANOVA)

Indian Institute of Public Health Delhi
MSc CR 2013-15

Outline of the session
• Need for Analysis of Variance
• Concept behind one way ANOVA

• Example
• Non-parametric alternative

When dependent variable is continuous
Type of
Dependent
variable

Type of
Independent
variable

Number
of
Groups

Continuous

Categorical

More
than
two

Non-parametric (Wilcoxon sign
rank)
Paired t – test

Not normal

Non-parametric (Wilcoxon sign
rank)
Independent z or t – test

Not normal

Non-parametric (Wilcoxon rank
sum or Mann-Whitney U )

Not normal

Unrelated or
independent

Not normal

Normal

Two

z or one sample t – test

Normal

Related

Choice of Significance test

Normal

NA

Distribution of
dependent
variable
Normal

One

Related/
Dependent

One way ANOVA/linear
regression
Non-parametric (Kruskal Wallis)

Normal

Repeated ANOVA

Not normal

Non-parametric (Friedmans test)

Unrelated

Related

Background
• When you have more than two groups to compare,
you can apply t-test multiple times
• But this is not done, why???
• Probability of type I error increases
• This increases as the number of comparison
increases
• Analysis of variance (ANOVA) is one way of dealing
with this problem which tests for overall significance

One way ANOVA
• Used to compare the mean of a numerical outcome
variable in the groups defined by an exposure level
with two or more categories

• Method is based on how much overall variation in the
outcome variables is attributable to differences
between the exposure group means
• This is equal to t-test for two sample with equal
variance

One-Way ANOVA
Partitions Total Variation

Total variation

Variation due to treatment

• Among Groups Variation

Variation due to random
sampling

• Within Groups Variation

One-Way ANOVA
• Difference between the means could be due to
variability between the groups and variability within
the groups
• Total variation= between group variation + within
group variation

• ANOVA, partitioned this sum of squares into two
– Sum of squares due to differences between the group means

– Sum of squares due to differences between the observations within each group, called as residuals

Test Statistic
• These sum of squares are divided by respective
degrees of freedom which is called as mean square
• The mean squares are then compared by using F test
• Hence, test statistic is given by

• With df = dfbetween groups, dfwithin groups = p-1, n-p
• Where n is total number of observations, p is
number of groups

Total Variation
SSTotal   X11  X   X 21  X     X ij  X  2

2

2

Response, X

X

Group 1

Group 2

Group 3

Treatment Variation
SST  n1 X1  X   n 2 X 2  X     n p X p  X  2

2

2

Response, X

X3
X

X2

X1
Group 1

Group 2

Group 3

Random (Error) Variation
SSE  X11  X1   X 21  X1     X pj  X p  2

2

2

Response, X

X3
X1
Group 1

Group 2

X2

Group 3

One-Way ANOVA: F-Test Statistic
• Test Statistic
– F = MST / MSE

SST /p  1

SSE /n  p

• MST is Mean Square for Treatment
• MSE is Mean Square for Error

• Degrees of Freedom
– 1 = p -1
– 2 = n - p

• p = # Populations, Groups, or Levels
• n = Total Sample Size

One-Way ANOVA
Summary Table
Source of
Variation

Degrees
of
Freedom

Sum of
Squares

Mean
Square
(Variance)

F

Treatment

p-1

SST

MST =
SST/(p - 1)

MST
MSE

Error

n-p

SSE

MSE =
SSE/(n - p)

Total

n-1

SS(Total) =
SST+SSE

F Distribution

Assumptions for ANOVA
• Independent Random Samples are Drawn
• Outcome variable should follow normal distribution:
each group should be approx normal
• Standard deviations of each group are approximately
same

Hypothesis test
• ANOVA tests following...
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