A ppt about testing hypothesis using one way ANOVA

2. Q1. Why is it called One-Way ANOVA ?
Q2. What does Two-Way ANOVA mean ?
Q3. What does MANOVA means ?

3. When to use?
CHARACTERISTICS
No. of groups 3 or more
No. of data sets 3 or more
No. of variables One
Data used in hypothesis Sample variance or the
groups
Purpose To test the significance of
the difference of the means
of three or more groups
Statistical tool to use F test

4. Remember:
We can use t-test to compare the
means of two independent groups.
NOTE: Comparing the mean of each groups.
When getting the difference of two independent groups → T test.

5. Reasons why we can’t use t-test
1. When you are comparing two means
at a time, the rest of the means under
study are ignored. With the F test, all
the means are compared
simultaneously.

6. Reasons why we can’t use t-test
2. When you are comparing two means at a time and making
all pairwise comparisons, the probability of rejecting the null
hypothesis when it is true is increased, since the more t tests
that are conducted, the greater is the likelihood of getting
significant differences by chance alone.
Recall: Type 1 Error → rejecting the true H₀.
Type 2 Error → accepting the false H₀.
3. The more means there are to compare, the more tests are
needed.

7. Assumptions for the F-test for comparing three or more
means:
1. The population from which the samples
were obtained must be normally of
approximately normally distributed
2. The samples must be independent of
one another.
3. The variances of the populations must
be equal.

8. F-test Estimates two Population variance:
1. Between
• Group variance - it finds the
variance of the means.
2. Within
• Group variance - it computes the
variance using all the data and is not
affected by the differences in the
means
Formula for variance:

9. 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 persons blood pressure is recorded.
At 0.05, test the claim that there is difference in the
means. The data are shown.

10. Hypothesis testing:
Problem: Is there a significant mean difference among the three groups on
the reduction of their blood pressure.
Critical test: F test using one-way ANOVA
Step 1:
• H₀ : there is no significant mean difference
among the three groups on the reduction
of their blood pressure
• In symbols, µ₁ =µ₂ =µ₃ ( claim)

11. Medication Exercise Diet
10 6 5
12 8 9
9 3 12
15 0 8
13 2 4
x
̅ ₁= 11.8 x
̅ ₁= 3.8 x
̅ ₁= 7.6
S²= 5.7 S²= 10.2 S²= 10.3
Example 1:

12. Hypothesis testing:
Alternative
hypothesis:
Step 1:
There is a significant mean difference among the three groups on the reduction
of their blood pressure. In symbols, µ₁ ≠µ₂≠ µ₃ , or at least one mean is different
from the others.
(This means that the average of the reduction of blood pressure
among the three groups are not the same or at least one mean is
different from the others.)

13. Hypothesis testing:
Step 2: Find the critical value.
Since k= 3 and N= 15
d.f.N = k-1 = 3-1 = 2
d.f.D = N-k = 15-3 = 12
The critical value is 3.89, obtained from the f distribution table with ∝ = 0.05 .
Note:
The degrees of freedom for this F test are d.f.N = k-1, where k is the number of
groups, and d.f.D = N-k, where N is the number of the sample sizes of the groups
N = n₁ + n₂ +… + nk . The sample sizes need not be equal. The F test to compare
means is always right-tailed.

14. Hypothesis testing:
Step 3:
Compute the test value, using the procedure outlined here.
a.) Find the mean and variance of each sample ( these are already given).
x
̅ ₁= 11.8 x
̅ ₁= 3.8 x
̅ ₁= 7.6
S²= 5.7 S²= 10.2 S²= 10.3

15. Hypothesis testing:
Step 3:
b.) Find the grand mean. The grand mean, denoted by x̅gm , is the mean of all the values in
the samples.
x̅gm = ZX / N = 10 +12+9+…+4 ₌ 7.73
15

16. Hypothesis testing:
Step 3:
c.) Find the between-group variance, denoted by S²B
SB² = ∑ni (x̅i – x̅gm)²
k – 1
SB² = 5(11.8 – 7.73)² + 5(3.8 – 7.73)² + 5(7.6 – 7.73)²
3-1
= 104.8 = 80.07
12
Note: This formula finds the variance among the means by using the
sample sizes as weights and considers the differences in the means.

17. Hypothesis testing:
Step 3:
d.) find the within-group variance, denoted by S²w .
Sw² = ∑ (Ni – 1)Si² = (5-1)(5.7)+(5-1)(10.2)+(5-1)(10.3)
∑( Ni – 1) (5-1)+(5-1)+(5-1)
= 104.8 = 8.73
2
Note:
This formula finds an overall variance by calculating a weighted average of the
individual variances. It does not involve using differences of the means.

18. Hypothesis testing:
Step 3:
e.) find the F test value.
F = SB² = 80.07
Sw² 8.73
F = 9.17

19. Hypothesis testing:
Step 4:
Using traditional method:
If the computed F – value is greater than the critical/tabled value, reject H0
Otherwise, do not reject H0 .
Solution:
Computed value Critical/tabled value
9.17 > 3.89
Since the computed F – value of 9.17 is greater than the
critical F – value of 3.89, the null hypothesis is rejected.

20. Hypothesis testing:
Step 5:
We are 95% confident that there is enough evidence to reject the claim and
conclude that at least one mean is different from the others.

21. How to make a summary of the
computed results for ANOVA :
Table 11-1 Analysis of Variance Summary Table
Source Sum of
squares
d.f. Mean
square
F
Between SSB k-1 MSB
Within
(error)
SSW N-k MSw
Total
In the table,
SSB = sum of squares between groups
SSw = sum of squares within groups
k= number of groups
N= n1 + n2 + … + nk = sum of sample sizes for groups
MSB = SSB ; MSW = SSW ; F = MSB
k-1 N-k MSW

22. How to make a summary of the
computed results for ANOVA :
Using the example we have:
Analysis of variance summary table
Source Sum of
squares
d.f. mean
square
F
Between 160.13 80.07 9.17
Within
(error)
104.80 8.73
Total 264.93

23. Steps in finding the F-test value for the
Analysis of Variance:
Step 1:
Find the mean variance of each sample,
( x̅1 , S1² ) , ( x̅2 , S2² ) , … , ( x̅k , SI² )
Step 2:
Find the grand mean.
x̅gm = Zx
N
Step 3:
Find the between-group variance.
SB² = ∑ni(x̅i – x̅gm)²
k – 1

24. Steps in finding the F-test value for the
Analysis of Variance:
Step 4:
Find the within group variance,
Sw² = ∑ (Ni – 1)Si²
∑( Ni – 1)
Step 5:
Find the F value
F = SB²
Sw²
The degrees of freedom are d.f.N. = k-1, where k is the number of groups, and d.f.D. =
N-k, where N is the sum of the sample sizes of the groups N = n1 + n2

25. Example 2:
Example 2:
A state employee wishes to see if there is a significant difference in the number of
employees at the interchanges of three state toll roads. The data are shown. At ∝ =
0.05, can it be concluded that there is a significant difference in the average number of
employees at each interchange?
Problem:
Is there a significant difference
Statistical tool:
F test using one-way ANOVA

26. Example 2:
Step 2: find the critical value since
k = 3 and N = d.f.N = k-1 = 2
d.f.D = N-k = 18-3 = 15
At ∝ = 0.05, the critical value is 3.68 .
Step 1:
H0:µ1=µ2=µ3
H1: µ1≠µ2≠µ3
(Since the mean of the three groups are not equal)

27. Example 2:
Pennsylvania
Turnpike
Greensburg
Bypass/Mon-
Fayette
Expressway
Beaver Valley
Expressway
7 10 1
14 1 12
32 1 1
19 0 9
10 11 1
11 1 11
x
̅ 1 = 15.5 x
̅ 2 = 4.0 x
̅ 3 = 5.8
S1² = 81.9 S2² = 25.6 S3² = 29.0

28. Example 2:
b.) Find the grand mean.
x̅gm = ∑X =
N
7+14+32+19+10+11+10+1+1+0+11+1+1+12+9+1+1+11
18
= 8.44
Step 3: compute test value, using the procedure
outline here.
a.) Find the mean and variance of each sample
x̅ = 15.5 x̅ = 4.0 x̅ = 5.8
S²1 = 81.9 S²2 = 25.6 S²3 = 29.0

29. c.) Find the between-group variance, S²B
SB² = ∑ni(x̅i – x̅gm)²
k – 1
= 6(15.5-8.44)+6(4.0-8.44)+6(5.8-8.44)
3-1
SB² = 6(49.8436)+6(19.7136)+6(6.9696) SB² = 299.58
2

30. d.) Find the within-group variance, S²w
Sw² = ∑(ni - 1)Si²
∑(ni - 1)
= (6-1)(81.9)+(6-1)(25.6)+(6-1)(29.0)
(6-1)+(6-1)+(6-1)
Sw² = 45.5
e.) find the F test value
F= SB² /Sw² = 229.58/45.5 = 5.05

31. Step 5:
Conclusion: we are 95% confident that there is enough
evidence to reject the claim and conclude that at least
one mean is different from the others.
Step 4: Decision
Computed value > critical value
5.05 > 3..68
Reject H0

32. What is the general purpose of ANOVA? :
One-way
ANOVA
1. Two-way
ANOVA
2. N-way
multivariate
ANOVA
3.
Researchers and students use ANOVA in many ways. The use
of ANOVA depends on the research design. Commonly,
ANOVAs are used in three ways.

33. What is one-way ANOVA?
A one-way ANOVA refers to the number of independent
variables—not the number of categories in each variable. A
one-way ANOVA has just one independent variable. For
example, difference in IQ can be assessed by Country, and
Country can have 2,20, or more different countries in that
variable.

34. What is two-way ANOVA?
A two-way ANOVA refers to an ANOVA using 2 independent
variables. Expanding the examples above, a 2-way ANOVA
can examine differences in IQ scores ( the dependent variable)
by Country ( independent variable1) and gender ( independent
variable 2).

35. What is two-way ANOVA?
Two-way ANOVA’s can be used to examine the
INTERACTION between the two independent variables.
Interactions indicate that differences are not uniform across all
categories of the independent variables. For example, females
may have higher IQ scores overall compared to males, and are
much greater in European Countries compared to north
American Countries.

36. What is N-way ANOVA?
A researchers can use many independent variables and this is
an n-way ANOVA. For example, potential differences in IQ
scores can be examined by Country, Gender, Age group,
Ethnicity, etc, simultaneously.

37. What are some examples of Research Questions
ANOVA can examine?
1. One-way
ANOVA
• Are there differences in GPA by
grade level (freshmen vs.
sophomores vs. juniors)?
2. Two-way
ANOVA
• Are there differences in GPA by
grade level (freshmen vs.
sophomores vs. juniors) and
gender ( male or female)?