ANOVA theory qadm pptx in qualitative decision making

ANALYSIS OF VARIANCE
(ANOVA)
Course: QADM
One-Way ANOVA

Introduction
• ANOVA is a tool to test equality of more than two means simultaneously.
OR
• It is a tool for analyzing how the mean value of a quantitative response
variable is related to one or more categorical explanatory factors.
• F-test is used to determine the significance difference among three or
means.
• It was developed by Sir R. A. Fisher, an English Statistician.
One-Way ANOVA

Examples
• To determine significant differences for mean time of solving a computer problem
by four groups of students, using C, C #, C++ and Python.
• To determine significant differences for Software Effort among different phases of
SDLC.
• To determine significant differences for software metrics such as: Defect metric,
process metric, KSLOC, FPs, etc.
• To determine interaction effect of testing technique, software type, expertise level
etc.
One-Way ANOVA

F-test
• Consider two independent normal populations with common variance σ2
.
If random samples of sizes n1 & n2 are drawn from these populations then:
where are two sample variances.
(Note: the larger of the variances is placed in the numerator of the F formula)
One-Way ANOVA

Assumptions & Conditions for F-test
• The samples are independent random samples.
• The distribution of the response variable is a normal curve within
each population.
• The different populations may have different means.
• All populations have the same standard deviation, σ.
One-Way ANOVA

• Normality: Normal probability plots of the sample data are effective
in detecting gross violations of normality.
• A rule of thumb for equality of variances: one can consider that
condition met if the ratio of the largest to the smallest sample standard
deviation is less than 2

Sensitivity of F-statistic
• The F-statistic is sensitive to differences among a set of sample
means.
• The greater the variation among the sample means, the
larger is the value of the test statistic.
• The smaller the variation among the observed means, the smaller the value
of the test statistic.
One-Way ANOVA

ANOVA theory qadm pptx in qualitative decision making

F-distribution
• If x is an F random variable with u numerator and υ denominator
degrees of freedom, then the PDF of x is:
One-Way ANOVA

Why not t-test ?
Why t-test should not be done while comparing several means taking
two at a time?
• when one is comparing two means at a time, the rest of the means
under study are ignored.
• the more means there are to compare, the more t tests are needed.
• For the comparison of 5 means two at a time, 10 tests are required.
• for the comparison of 10 means two at a time, 45 tests are required.
• the more t tests that are conducted, the greater is the likelihood of
getting significant differences by chance alone.
One-Way ANOVA

Why the procedure is called ANOVA?
• The name Analysis of Variance is derived from a partitioning of total
variability into its component parts.
One-Way ANOVA

One-Way ANOVA
• Random samples of size n are selected from each of k populations.
The k different populations are classified on the basis of a single
criterion such as different treatments or groups.
• It is assumed that the k populations are independent and normally
distributed with means µ1, µ2, … µk and common variance σ2
.
One-Way ANOVA

Model for One-way ANOVA
• µ is the Grand Mean of all µi,
• The represents random error (within group variation).
• is the effect of ith
treatment with constraint
One-Way ANOVA

Summary procedure of ANOVA
α: 0.05, 0.01, or 0.10
One-Way ANOVA

Example # 1
• A researcher wishes to try three different techniques to lower the blood pressure
of individuals diagnosed with high blood pressure. The subjects are randomly
assigned to three groups; the first group takes medication, the second group
exercises, and the third group follows a special diet. After four weeks, the
reduction in each person’s blood pressure is recorded. At α = 0.05, test the claim
that there is no difference among the means. The data follow.
One-Way ANOVA

Solution (Example 01)
• Find Grand Mean (GM) as:
• Find b/w group variance as:
• Find within group variance as:
• F-test as:
One-Way ANOVA

Solution (Example 01, Contd.)
• Decision: the decision is to reject Ho as :
• There is enough evidence to reject the claim and conclude that at least one
mean is different from the others.
One-Way ANOVA

Why the word Variance in ANOVA?
• The procedure for comparing the means
analyzes the variation in the sample data.

Example # 02 [Plasma Etching Experiment]
Recall that the engineer is interested in determining if the RF power setting affects
the etch rate, and she has run a completely randomized experiment with four
levels of RF power and ﬁve replicates.

One-Way ANOVA

Solution (Example # 05)
SSTR = SS Treatments

Two-way ANOVA
• The two-way ANOVA is an extension of the one way analysis of
variance; it involves two independent variables. The independent
variables are also called factors.

Example # 06 (Gasoline Consumption)
• A researcher wishes to see whether the type of gasoline used and the
type of automobile driven have any effect on gasoline consumption.
Two types of gasoline, regular and high-octane, will be used, and two
types of automobiles, two-wheel- and four-wheel drive, will be used
in each group. There will be two automobiles in each group, for a
total of eight automobiles used.

Solution (Example 06, Hypotheses)

Solution (Example 06, ANOVA table)
• the null hypotheses concerning the type of automobile driven and the
interaction effect should be rejected.

Practice Questions
Q1)
One-Way ANOVA

ANOVA theory qadm pptx in qualitative decision making

More Related Content

Similar to ANOVA theory qadm pptx in qualitative decision making (20)

Recently uploaded (20)

ANOVA theory qadm pptx in qualitative decision making

Editor's Notes