This document provides information about analysis of variance (ANOVA). It discusses different types of ANOVA based on the number of independent variables and samples. A one-way ANOVA has one independent variable and can be independent or repeated measures. Independent ANOVA involves different samples for each group, while repeated measures ANOVA involves measuring the same sample under different conditions. The document provides an example of a one-way repeated measures ANOVA comparing student test scores on different days to determine if their knowledge differed significantly between days. It outlines transferring the data to SPSS and running appropriate tests such as Mauchly's test of sphericity and Bonferroni post hoc test for repeated measures ANOVA.

1. ANOVA
Analysis of Variance
Ibrahim bin Abdullah
ibrahim.lecturer@gmail.com
www.facebook.com/ibrahim.abdullah

2. Types of samples and
appropriate testing:
1 sample
•Use 1-sample t-test
2 samples
•Use 2 samples t-test
3 samples
•Use ANOVA

3. ANOVA can be:
• 1-way
 1 independent variable
• 2-way
 2 independent variable
• 3,4,etc-way
 3,4,etc independent variable

4. In ANOVA, whatever the
type, there is always only 1
Dependent Variable
ANOVA is UNIVARIATE (1 Dependent Variable).
If there are more than 1 Dependent
Variables, use MANOVA

5. It can be further classified:
INDEPENDENT
ANOVA
1-WAY ANOVA
REPEATED
MEASURE
ANOVA

6. INDEPENDENT
ANOVA
REPEATED
2-WAY ANOVA MEASURE
ANOVA
MIXED ANOVA
So, they are called 2-way independent Anova, 2-way mixed Anova, etc

7. Specky
Students taking
blueberry
We are testing the Non-specky
effect of blueberry
on the eye sight. Specky
Students NOT
taking blueberry
Non-specky
We can do t-test TWICE to test the samples. However, doing that
will increase α (type 1 error ie. we tend to reject Ho when Ho should
not be rejected). Instead of doing t-test repeatedly, we must do
ANOVA

8. One-way
Independent
ANOVA
INDEPENDENT
ANOVA First part of this chapter deals with 1-way
1-WAY ANOVA Independent Anova
REPEATED
MEASURE ANOVA Later we will look at 1-way Repeated

9. One-way Independent ANOVA
Assumptions that MUST be fulfilled:
1. Normality (any one of three)
W-S
1.
2.
3.
W-S or K-S (p ≥ 0.05)
Analyze
Descriptive
Explore
Skewness test (within S 2SE)
4. Plot
5.  Normality
s
 Coefficient of variation: 100 30%
x
2. Homogeneity of variance
Levene’s test (p ≥ 0.05)

10. One-way Independent ANOVA
Hypotheses:
1. Ho
μ1 = μ2 = μ3 ….. μi
2. HA
At least one pair of means is not equal
(it can be μ1≠μ2 = μ3 etc)

11. One-way Independent ANOVA
If p < 0.05 (significant, ie Ho rejected), then
must do Post Hoc test (multiple pairwise
comparison test)
Post Hoc Tests
Tukey Dunnette Bonferroni
Test Test Test
If homogenous, No If not homogenous, For repeated measure
control Has control
On the other hand if not significant, test stops

12. One-way Independent ANOVA
A study is carried out to determine if there is difference in the
knowledge of Vision and Mission of the university among students
of first year, second year and third year of The Management and
Science University (MSU)
Knowledge Score on Vision and Mission of MSU
Student Yr1 60 55 45 50 55 60 70 45 35 35
Student Yr2 65 60 70 75 70 78 79 80 81 82 85
Student Yr3 60 60 60 60 70 70 70 70 75 70
Hypotheses:
Ho: μ1 = μ2 = μ3
HA: At least one pair of means is not equal (it can be μ1≠μ2 = μ3 etc)
Transfer the data into PASW.
Remember, since this is an independent test, all samples are recorded in similar column.

13. One-way Independent ANOVA
Variable view

14. One-way Independent ANOVA
Transfer the data from the test conducted in
“Data View”
Since this is an independent test, same column (in
this example labeled “year”) used for all samples
In repeated measure test we use different column
for every variable

15. ANALYSIS
1
Normality test

16. One-way Independent ANOVA
Descriptives
MSU Year
MENU Knowledge Year 1 Mean
Statistic
51.000
Std. Error
3.5590
95% Confidence Interval Lower 42.949
1. Analyze for Mean Bound
2. Descriptive Statistics Upper 59.051
3. Explore
Bound
4. Dependent = score 5% Trimmed Mean 50.833
5. Factor = year Median 52.500
6. Plots Variance 126.667
7.  Normality plot Std. Deviation 11.2546
Minimum 35.0
Maximum 70.0
Case Processing Summary Range 35.0
MSU Year Cases Interquartile Range 17.5
Skewness -.018 .687
Valid Missing Total Kurtosis -.563 1.334
Year 2 Mean 75.000 2.3549
N Percent N Percent N Percent 95% Confidence Interval Lower 69.753
for Mean Bound
Knowledge Year 1 10 100.0% 0 .0% 10 100.0% Upper 80.247
dimension1
Year 2 11 100.0% 0 .0% 11 100.0% Bound
5% Trimmed Mean 75.278
Year 3 10 100.0% 0 .0% 10 100.0%
Median 78.000
Variance 61.000
Tests of Normality
Std. Deviation 7.8102
MSU Year Kolmogorov-Smirnova Shapiro-Wilk Minimum 60.0
Maximum 85.0
Statistic df Sig. Statistic df Sig.
Range 25.0
Knowledge Year 1 .139 10 .200* .952 10 .695
Interquartile Range 11.0
Year 2
dimension1
.195 11 .200* .931 11 .424
Skewness -.731 .661
Year 3 .327 10 .003 .770 10 .006 Kurtosis -.396 1.279
a. Lilliefors Significance Correction Year 3 Mean 66.500 1.8333
95% Confidence Interval Lower 62.353
*. This is a lower bound of the true significance. for Mean Bound
Upper 70.647
Bound
5% Trimmed Mean 66.389
S-W test showed that Year 1 and Year 2
Median 70.000
were normal but Year 3 was not Variance 33.611
Std. Deviation 5.7975
So, check Year 3 skewness: Minimum 60.0
•Skewness -.192 .687
Maximum 75.0
It showed normal. So we can use ANOVA Range 15.0
Interquartile Range 10.0
Skewness -.192 .687
Kurtosis -1.806 1.334

17. ANALYSIS
2
The ANOVA test

18. One-way Independent ANOVA
MENU
1. Analyze
2. Compare means
3. One-way ANOVA If we have control, under Post Hoc choose
4. Dependent = score Dunnette only
5. Factor = year
6. Post Hoc If p > 0.05, use Tukey
7.  Tukey If p < 0.05, use Dunnette’s T3
8.  Dunnette’s T3
9. Option Remember, we look at Post Hoc only if we
10.  Descriptive reject Ho (ie there is at least a pair of means
11.  Homogeneity not equal)
Test of Homogeneity of Variances
Knowledge
Levene Statistic df1 df2 Sig.
2.022 2 28 .151 P > 0.05, so homogeneity is assumed
Knowledge
MSU Year Subset for alpha = 0.05
Homogenous subset
N 1 2 See there are group 1 (Year 1) and group 2
(Year 3 and Year 2)
Tukey HSDa,b Year 1 10 51.000
So, Year 2 and Year 3 are not significant,
Year 3 10 66.500 but both are significant when compared to
Year 1
dimension1
Year 2 11 75.000
Sig. 1.000 .079 μ1 ≠ μ2 = μ3 (ie at least one pair of means
is not equal)
Means for groups in homogeneous subsets are displayed.
a. Uses Harmonic Mean Sample Size = 10.313.
b. The group sizes are unequal. The harmonic mean of the group sizes
is used. Type I error levels are not guaranteed.

19. ANALYSIS
3
GLM (General Linear Model) test
The general linear model incorporates a number of different statistical models: ANOVA, ANCOVA, MANOVA, MANCOVA, ordinary linear regression, t-test and F-test.
GLM is therefore a more general concept, compared to ANOVA.

20. One-way Independent ANOVA
MENU
1
1. Analyze
2. General Linear Model
3. Univariate
2
Plots
Click year to Horizontal Axis first, then click Add

21. One-way Independent ANOVA
3 5 Save
Post Hoc
Cook’s distance shows the outliers. The value should be less than 1.
Value of more than 1 means outlier (that can be removed). See Cook’s
distance at DATA VIEW under COO_1
4
Options
Estimate of effect size will returns “Partial ETA Squared”. Value of 0.14
or more means high. Effect size is NOT influenced by sample number
(as opposed to p value, which can be influenced by sample size)
If p is high (not significant, ie rejecting Ho), look at Observed Power (B).
If B is high (0.8 ie. 80% or more), then confirm to reject Ho. If B is low,
probably means that the low sample size used in the test results in
rejection of Ho. Ho can still be accepted, instead of rejected – refer to
type II error

22. One-way Independent ANOVA
Estimated Marginal Means
This refers to unweighted means. This is MSU Year
important when comparing the means of Dependent Variable:Knowledge
unequal sample sizes (as in ANOVA), where MSU Year 95% Confidence Interval
you take into consideration each mean in
porportion to its sample size. Unequal Lower Upper
sample size can occur eg. due to drop-out Mean Std. Error Bound Bound
of participants which can destroy the Year 1 51.000 2.707 45.454 56.546
random assignment of subjects to
Year 2 75.000 2.581 69.712 80.288
conditions, a critical feature of the
dimension1
experimental design Year 3 66.500 2.707 60.954 72.046
Estimates of Effect Size (Partial ETA Squared) and Observed Power
Tests of Between-Subjects Effects
Dependent Variable:Knowledge
Source Type III Sum of Partial Eta Noncent. Observed
Squares df Mean Square F Sig. Squared Parameter Powerb
Corrected Model 3075.242a 2 1537.621 20.976 .000 .600 41.952 1.000
Intercept 127380.859 1 127380.859 1737.717 .000 .984 1737.717 1.000
year 3075.242 2 1537.621 20.976 .000 .600 41.952 1.000
Error 2052.500 28 73.304
Total 134160.000 31
Corrected Total 5127.742 30
a. R Squared = .600 (Adjusted R Squared = .571)
b. Computed using alpha = .05

23. One-way Independent ANOVA
Cook’s Distance

24. One-way Independent ANOVA
Profile Plot
The profile plot can be included in the thesis result

25. One-way
Repeat Measure
ANOVA
INDEPENDENT
ANOVA We have looked at 1-way
1-WAY ANOVA Independent Anova
REPEATED
MEASURE ANOVA Now, we look at 1-way Repeated

26. One-way Repeat Measure ANOVA
In Repeat Measure, we repeat the test on the SAME
sample but at DIFFERENT time intervals.
The data for different time or day must be put in
DIFFERENT COLUMNS of PASW Variable View.
In this test, we are not concerned about homogeneity.
Rather we are concerned about sphericity (Maunchly’s
Sphericity Test). The value, W>0.05 showed
sphericity.
[If W>0.05, read Sphericity row. If W<0.05, read Greenhouse row]
For pairwise comparison (Post Hoc), we do not use
Tukey or Dunnette but Bonferroni Test.

27. One-way Repeat Measure ANOVA
A study is carried out to determine if there is difference in the
knowledge of Vision and Mission of the university on different days
among students of first year of The Management and Science
University (MSU)
Knowledge Score on Vision and Mission of MSU
Sunday 60 55 45 50 55 60 70 45 35 35 65
Monday 60 55 45 50 55 82 85 60 60 60 60
Friday 85 60 60 60 60 70 70 70 70 75 70
Hypotheses:
Ho: μ1 = μ2 = μ3
HA: At least one pair of means is not equal (it can be μ1≠μ2 = μ3 etc)
Transfer the data into PASW.
Remember, since this is repeated measure test, all samples are recorded in different columns.

28. One-way Repeat Measure ANOVA
Variable view

29. One-way Repeat Measure ANOVA
Transfer the data from the test conducted in “Data View”
In repeated measure test we use different column for every variable

30. One-way Repeat Measure ANOVA
MENU
1. Analyze
2. General Linear Model
3. Repeated Measures
4. Factor
5. Number of levels = 3
6. Define
7. (Move all knowledge to right)
8. Option
9.  Compare main effects
10. (See picture on right, Select Bonferroni)
11.  Descriptive
12.  Estimates
13.  Observed power
14. Save
15.  Cook’s distance
16. Plots
17. Move factor1 to Horizontal Axis
18. Add
19. Continue

31. One-way Repeat Measure ANOVA
Mauchly's Test of Sphericityb
Measure:MEASURE_1
Within Subjects Effect Epsilona
Approx. Chi- Greenhouse-
Look at Mauchly’s W Mauchly's W Square df Sig. Geisser Huynh-Feldt Lower-bound
dimension1 factor1 .961 .359 2 .835 .962 1.000 .500
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to
In this example, W = 0.961
an identity matrix.
a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of
Since W > 0.05, we will read Within-Subjects Effects table.
b. Design: Intercept
Sphericity, not Greenhouse
Within Subjects Design: factor1
Tests of Within-Subjects Effects
Measure:MEASURE_1
Source Type III Sum Partial Eta Noncent. Observed
of Squares df Mean Square F Sig. Squared Parameter Powera
W > 0.05 factor1 Sphericity Assumed 1397.515 2 698.758 9.347 .001 .483 18.694 .957
W < 0.05 Greenhouse-Geisser 1397.515 1.925 726.116 9.347 .002 .483 17.990 .951
Huynh-Feldt 1397.515 2.000 698.758 9.347 .001 .483 18.694 .957
Lower-bound 1397.515 1.000 1397.515 9.347 .012 .483 9.347 .787
Error(factor1) Sphericity Assumed 1495.152 20 74.758
Greenhouse-Geisser 1495.152 19.246 77.685 See that the Observed Power is high
Huynh-Feldt 1495.152 20.000 74.758
Lower-bound 1495.152 10.000 149.515
a. Computed using alpha = .05
[If W>0.05, read Sphericity row. If W<0.05, read Greenhouse row]

32. One-way Repeat Measure ANOVA
Pairwise Comparisons
Measure:MEASURE_1
(I) factor1 (J) factor1 95% Confidence Interval
Pairwise Comparisons here Mean
Difference Std.
for Differencea
Lower Upper
is Bonferroni test (I-J) Error Sig.a Bound Bound
1 2 -8.818 3.508 .092 -18.886 1.250
1 and 2 are not significant (p=0.092)
dimension2
3 -15.909* 4.035 .008 -27.490 -4.328
2 1 8.818 3.508 .092 -1.250 18.886
1 and 3 are significant (p=0.08) dimension1 dimension2
3 -7.091 3.491 .209 -17.112 2.930
3 1 15.909* 4.035 .008 4.328 27.490
So we reject Ho because at least one pair of dimension2
2 7.091 3.491 .209 -2.930 17.112
means is not equal
Based on estimated marginal means
a. Adjustment for multiple comparisons: Bonferroni.
*. The mean difference is significant at the .05 level.