the document describes about the analysis of ANOVA

1. 1/26/2024 Research Methods 1
ANOVA (Analysis of Variance)
Research Methods

2. 1/26/2024 Research Methods 2
Introduction
 The use of t-test is made to determine whether there is any
significant difference b/n the means of two random samples.
 But if we have five randomly drawn samples from a population and
we want to determine whether there are any significant differences
among their means, this requires 10 t-tests.
 Therefore, a single F-test (ANOVA) makes it possible to determine
the differences among these sample means.
 ANOVA is an extension of t-test.

3. 1/26/2024 Research Methods 3
Introduction
 ANOVA (analysis of variance) is a mechanism to
compare means of greater than two groups where as
the t-test is appropriate to compare means of two
groups.
 In an ANOVA the null hypothesis is tested using some
procedures such as computing group means (X-bar- X),
grand mean (XG), F- ratio (F) and degrees of freedom
(df) which will be discussed in detail with examples.

4. 1/26/2024 Research Methods 4
When to Use ANOVA
 The purpose of ANOVA is to draw inferences about population
means.
 In testing the null hypothesis the researcher follows certain
procedures.
 The researcher begins by using sample data to compute a test
statistic.
 The statistic is treated as a “score” in a sampling distribution that
assumes the null hypothesis is true.
 If the statistics falls within the rejection region of the sampling
distribution (i.e, the “Score” is improbable if the null is true), then we
reject the null hypothesis and conclude that the group means are not
equal.

5. 1/26/2024 Research Methods 5
Null and Alternative Hypothesis
 The null hypothesis (Ho) assumes that
 µ1= µ2= µ3= µ4=,
 where µ1= population mean of Group 1
 µ2= population mean of Group 2
 µ3= population mean of Group 3
 µ4= population mean of Group 4
 µn = population mean of Group n
 On the other hand, the basic alternative hypothesis (H1) assumes that: H1
not H0, that is µ1 ≠ µ2, µ3 ≠ µ4.
 There might be some other ways in which the null hypothesis (Ho)
becomes false.

6. 1/26/2024 Research Methods 6
One way ANOVA
 One way ANOVA involves the comparison of group means for three
or more independent groups in which the researcher manipulates
the effect of a single independent variable upon the dependent
variables under study.
 When the independent variable may have two levels, we need to
apply a two way ANOVA.

7. 1/26/2024 Research Methods 7
Steps in Computing F-Statistics
 Computing F- statistic involves calculating three Sums of Squared
deviations:
 SSTotal,
 SSBetween, and
 SSwithin.
 Initially, we need to to calculate X bar (Mean for each group) and XG
(Grand Mean) from the given set of data.
 SST= Sum of Square Total (deviation of each individual score from the
grand mean)
 SSB = Sum of Squares Between (deviation of each group score from
the grand mean)
 SSW = Sum of Square Within (deviation of each individual score from
their respective group mean)

8. 1/26/2024 Research Methods 8
Steps in Computing F-Statistics
 Then, we calculate
 dfB = degree of freedom between
 dfW = degree of freedom within
 Finally, we calculate F-ratio using the formula
 F= (SSB/dfB)/ (SSW/dfW)
 Now let’s see the example.

9. 1/26/2024 Research Methods 9
A. SSW
Deviations from Group means (within Group Variation)
Group 1 Group 2 Group 3
X X- X1 (X- X1)2 X X- X2 (X- X2)2 X X- X3 (X- X3)2
0 -3 9 1 -1 1 5 -2 4
6 3 9 4 2 4 6 -1 1
2 -1 1 3 1 1 10 3 9
4 1 1 2 0 0 8 1 1
3 0 0 0 -2 4 6 -1 1
3 2 7
Σ=20 Σ= 10 Σ= 16
Grand Mean= 4 SSW = 46

10. 1/26/2024 Research Methods 10
B. SST
Deviations from Grand means (Total Variation)
X X- XG (X –XG)2
A1 0 -4 16
A2 6 2 4
A3 2 -2 4
A4 4 0 0
A5 3 -1 1
B1 1 -3 9
B2 4 0 0
B3 3 -1 1
B4 2 -2 4
B5 0 -4 16
C1 5 1 1
C2 6 2 4
C3 10 6 36
C4 8 4 16
C5 6 2 4
M=3 SST= 116

11. 1/26/2024 Research Methods 11
C. SSB
Deviation of Group means from Grand mean (Between group Variation)
X X - XG (X - XG)2 N (XG2) N
3.0 -1 1 5 5
2.0 -2 4 5 20
7.0 3 9 5 45
M=4.0 SSB 70.0

12. 1/26/2024 Research Methods 12
D. F-Ratio
 SSB (Sum of squares between) = 70.0
 SSW (Sum of squares within) = 46.0
 SST= 116.0
 Next using the above, compute F- statistic which is the result of:
 F= (SSB/dfB)/ (SSW/dfW)
 dfB = degree of freedom between (dfB = K-1) = 2
dfW = degree of freedom within (dfW = n-K) = 12
 where K= number of groups and
 n = total number of subjects in the study.
 Therefore, MSB = (SSB/dfB)= (70/ (3-1)) =35.0
 MSW= (SSW/dfW) = (46/ (15 -3)) = 3.83
 F = (70/ (3-1)) / (46/ (15-3))
 = 35/3.83
 = 9.13

13. 1/26/2024 Research Methods 13
Interpreting F-value
 The computed F (9.13) is greater than 1.0 which is the value
expected if the null hypothesis were true.
 Because F distribution for various degrees of freedom shows three
values of alpha (α) equals 0.05, 0.01 and 0.001, we need to
determine significance level.
 By consulting the table of significance level and the degree of
freedom, we can read 3.88 for α equals 0.05 with F (2,12).
 Hence, we can reject the null hypothesis that three group means are
equal.

14. 1/26/2024 Research Methods 14
Post hoc comparisons
 One important thing to note about the F-test is that it is a
global test.
 What that means is that if we find a significant difference
(p-value <0.05) all we know is that overall there is a
significant difference somewhere in the comparisons
between the three groups.
 We don’t know where the significance lies. It could be ….
 Therefore, we may need to find a way of telling us which
comparisons are significantly different.
 A number of tests, called post hoc comparisons, have
been developed that allow us to do just that.

15. 1/26/2024 Research Methods 15
Interaction Effects
 Another useful thing that we can do in ANOVA is look at so-called
‘interaction effects’.
 What is an interaction effect?
 When we have a significant interaction effect, it means that the effect
of one variable on another is different for different conditions of a third
variable.
 For example, if we were looking at the effect of assessment methods,
it may be that boys do better using exams while girls do better when
essay-style assessment is used.
 In order to capture this kind of effect in ANOVA, we can introduce
interaction effects which allow us to see whether the relationship
between independent and dependent variables is mediated in any way
by third variables.
 All additional variables have to be nominal or ordinal with a limited
number of categories.