ANOVA is a collection of statistical models used to analyze differences among means of more than two groups. Ronald Fisher introduced analysis of variance in the 1920s. ANOVA allows comparison of more than two groups' means and accounts for variation within and between groups. There are two main types of ANOVA - one-way and two-way. One-way ANOVA compares means of a single independent variable with more than two levels. Two-way ANOVA compares means between two independent variables and their interaction. Both require certain assumptions about the data and can be conducted in SPSS.

1. ANOVA
PRESENTED BY DR. HINA JALAL
DataAnalysis
2
EDUCATIONAL STATISTICS

2. ANOVA
Analysis of variance (ANOVA) is a
collection of statistical models used to
analyze the differences among means
of more than two groups and their
associated procedures (such as
"variation" among and between groups)

3. HISTORY OF ANOVA
• Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article The
Correlation Between Relatives on the Supposition of Mendelian Inheritance.
• His first application of the analysis of variance was published in 1921.
• Analysis of variance became widely known after being included in Fisher's 1925 book Statistical
Methods for Research Workers.
• Randomization models were developed by several researchers. The first was published in Polish
by Neyman in 1923.
• One of the major popularity of ANOVA was, it was upgraded version of t-tests with the range of more
then three groups’ average comparison. As it is impossible in t-tests.

4. WHY ANOVA?
 In real life things do not typically result in two groups being
compared. There are more than two also.
 Two-sample t-tests are problematic
 Increasing the risk of a Type I error
 So the more t-tests you run, the greater the risk of a type I error (rejecting
the null when there is no difference)
 ANOVA allows us to see if there are differences between means
with an OMNIBUS test

5. TYPES OF ANOVA
1. One Way ANOVA
2. Two Way ANOVA

6. ONE WAY ANOVA
 The one-way analysis of variance (ANOVA) is used to determine whether there
are any significant differences between the means of two or more
independent (unrelated) groups (although you tend to only see it used when
there are a minimum of three, rather than two groups).
For example, you could use a one-way ANOVA to understand whether exam performance differed based
on test anxiety levels amongst students, dividing students into three independent groups (Low, medium
and high-stressed students Since you may have three, four, five or more groups in your study design,
determining which of these groups differ from each other is important. You can use one way ANOVA to
findout significant sifference between the mean scores of these three groups.

7. ASSUMPTIONS OF ONE-WAY ANOVA
 Your dependent variable should be measured at the interval or ratio level (i.e., they are
continuous).
 Your independent variable should consist of two or more categorical, independent groups.
 You should have independence of observations.
 There should be no significant outliers.
 Your dependent variable should be approximately normally distributed for each category of
the independent variable.
 There needs to be homogeneity of variances.

8. TWO-WAY ANOVA
 The two-way ANOVA compares the mean
differences between groups that have been split on
two independent variables (called factors). The
primary purpose of a two-way ANOVA is to
understand if there is an interaction between the
two independent variables on the dependent
variable.
For example, you could use a two-way ANOVA to understand whether there is an interaction between gender
and educational level on test anxiety amongst university students, where gender (males/females) and education
level (undergraduate/postgraduate) are your independent variables, and test anxiety is your dependent variable.

9. ASSUMPTIONS OF TWO-WAY ANOVA
 Your dependent variable should be measured at the continuous level (i.e., they are interval or ratio variables).
 Your two independent variables should each consist of two or more categorical, independent groups.
 You should have independence of observations, which means that there is no relationship between the observations
in each group or between the groups themselves.
 There should be no significant outliers.
 Your dependent variable should be approximately normally distributed for each combination of the groups of
the two independent variables.
 There needs to be homogeneity of variances for each combination of the groups of the two independent
variables.

10. STEPS OF ONE-WAY ANOVA IS SPSS
 The following steps below show you how to analyze your data using
a one-way ANOVA in SPSS
 Step 1

11. STEP 2. TRANSFER THE DEPENDENT VARIABLES
AND FACTOR

12. STEP 3. CLICK THE BUTTON. TICK THE TUKEY CHECKBOX AS
SHOWN BELOW AND CLICK CONTINUE BUTTON.

13. STEP 4. CLICK OPTION BUTTON ON DESCRIPTIVE
SCREEN . THEN TICK THE DESCRIPTIVE CHECKBOX IN
THE –STATISTICS– AREA.
WHEN TESTING FOR SOME OF THE ASSUMPTIONS OF
THE ONE-WAY ANOVA, YOU WILL NEED TO TICK
MORE OF THESE CHECKBOXES.
CLICK THE CONTINUE BUTTON.
CLICK OK TO GENERATE THE OUTPUT.

14. SPSS OUTPUT OF ONE-WAY ANOVA
First Table
The descriptives table (see below) provides some very useful descriptive
statistics, including the mean, standard deviation and 95% confidence
intervals for the dependent variable for each separate group, as well as when
all groups are combined (Total).
These figures are useful when you need to describe your data.

15. SPSS OUTPUT OF ONE-WAY ANOVA
 Second Table
This is the table that shows the output of the ANOVA
analysis and whether we have a statistically significant
difference between our group means. Including
Significant Value

16. SPSS OUTPUT OF ONE-WAY ANOVA
•Third Table
The table below, Multiple Comparisons, shows which
groups differed from each other. The Tukey post-hoc test
is generally the preferred test for conducting post-hoc
tests on a one-way ANOVA, but there are many others
also.

17. ANOVA TABLE IN APA FORMAT

18. STEPS IN SPSS FOR TWO-WAY ANOVA
 The following steps
below show you how
to analyse your data
using a two-way
ANOVA in SPSS
Statistics, when the six
assumptions in the
previous section have
not been violated.
 Step 1. Click Analyze
> General Linear
Model> Univariate...
on the top menu, as
shown below:

19. STEP 2. YOU WILL BE
PRESENTED WITH
THE UNIVARIATE DIALOGU
E BOX, AS SHOWN
TRANSFER THE DEPENDENT
VARIABLE INTO
THE DEPENDENT
VARIABLE: BOX, AND
TRANSFER BOTH
INDEPENDENT INTO
THE FIXED FACTOR(S): BOX

20. STEP 3. CLICK ON PLOTS BUTTON AT
UNIVARIATE DAILOGUE BOX.
YOU WILL BE PRESENTED WITH
THE UNIVARIATE: PROFILE
PLOTS DIALOGUE BOX, AS SHOWN
TRANSFER THE INDEPENDENT VARIABLE
FROM THE FACTORS: BOX INTO
THE HORIZONTAL AXIS: BOX,
TRANSFER THE OTHER INDEPENDENT
VARIABLE INTO THE SEPARATE
LINES: BOX
CLICK THE ADD BUTTON. YOU WILL SEE
THAT ADDED TO THE PLOTS: BOX
CLICK CONTINUE

21. STEP 4. CLICK THE BUTTON. YOU WILL BE
PRESENTED WITH THE UNIVARIATE: POST HOC
MULTIPLE COMPARISONS FOR OBSERVED
MEANS DIALOGUE BOX, AS SHOWN
TRANSFER VARIABLES FROM
THE FACTOR(S): BOX TO THE POST HOC TESTS
FOR: BOX
YOU ONLY NEED TO TRANSFER INDEPENDENT
VARIABLES THAT HAVE MORE THAN TWO
GROUPS INTO THE POST HOC TESTS FOR: BOX.
WE ARE GOING TO SELECT TUKEY, FOR POST
HOC TEST
CLICK CONTINUE

22. STEP 5. CLICK THE OPTION BUTTON ON
MAIN DIALOGUE BOX, THIS WILL PRESENT
YOU WITH THE UNIVARIATE:
OPTIONS DIALOGUE BOX, AS SHOWN
TRANSFER VARIABLES FROM
THE FACTOR(S) AND FACTOR
INTERACTIONS: BOX INTO
THE DISPLAY MEANS FOR: BOX.
IN THE –DISPLAY– AREA, TICK
THE DESCRIPTIVE STATISTICS OPTION.
CLICK THE CONTINUE BUTTON TO
RETURN TO THE UNIVARIATE DIALOGUE
BOX.
CLICK THE OK BUTTON TO GENERATE
THE OUTPUT.

23. SPSS OUTPUT OF TWO-WAY ANOVA
Descriptive statistics Table
This table is very useful because it
provides the mean and standard deviation
for each combination of the groups of the
independent variables (what is sometimes
referred to as each "cell" of the design). In
addition, the table provides "Total" rows,
which allows means and standard
deviations for groups only split by one
independent variable, or none at all, to be
known. This might be more useful if you
do not have a statistically significant
interaction

24. SPSS OUTPUT OF TWO-WAY ANOVA
Plot of the results
Although this graph is probably not
of sufficient quality to present in your
reports (you can edit its look in SPSS
Statistics), it does tend to provide a
good graphical illustration of your
results. An interaction effect can
usually be seen as a set of non-
parallel lines. You can see from this
graph that the lines do not appear to
be parallel (with the lines actually
crossing).

25. SPSS OUTPUT OF TWO-WAY ANOVA
Statistical significance of the two-way ANOVA table
The actual result of the two-way ANOVA – namely, whether either of the two independent variables or
their interaction are statistically significant – is shown in the Tests of Between-Subjects Effects table,
as shown below:

26. SPSS OUTPUT OF TWO-WAY ANOVA
Post hoc tests – simple main effects in SPSS
Statistics
When you have a statistically significant interaction,
reporting the main effects can be misleading. Therefore,
you will need to report the simple main effects.

27. SPSS OUTPUT OF TWO-WAY ANOVA
Multiple Comparisons Table
If you do not have a statistically significant interaction, you might interpret the Tukey
post hoc test results for the different levels of education, which can be found in
the Multiple Comparisons table, as shown below:

28. TWO-WAY ANOVA IN APA FORMAT

29. Dr. Hina Jalal
@AksEAina (hinansari23@gmail.com)