1. TWO WAY ANOVA
ANOVA IN LATIN SQUARE DESIGN
ANCOVA
2. DEFINITION
A two-way ANOVA test is a statistical analysis tool that
determines the effect of two independent variables on an
dependent variable( outcome), as well as testing how
altering the variables will affect the outcome
3. An ANOVA test determines whether a statistical
operation has outcomes that are useful or not.
In essence, it allows one to determine whether
to reject or accept a null hypothesis.
In a two-way ANOVA test, two variables are
used to determine this.
The two-way ANOVA test has the benefit of
having a lower chance of getting type 1
errors, which could corrupt the data collected.
4. TWO-WAY ANOVA TECHNIQUE IS USED WHEN THE
DATA ARE CLASSIFIED ON THE BASIS OF TWO
FACTORS
The agricultural output may be classified
on the basis of different varieties of seeds
on the basis of different varieties of fertilizers used.
A business firm may have its sales data classified
on the basis of different salesmen
on the basis of sales in different regions.
5. RESEARCHING TYPES OF FERTILIZERS AND
PLANTING DENSITY TO ACHIEVE THE
HIGHEST CROP YIELD PER ACRE
One could divide the land into portions and then assign
each portion a specific type of fertilizer and planting
density. Once the crop is mature, the yield in each plot
is measured.
Determines which combination of fertilizer and crop
density yields the best harvest
How the two variables affect the outcome
The two-way ANOVA test helps find this, along with
telling how the two factors influence the outcome.
6. TWO-WAY ANOVA
HYPOTHESES
NULL HYPOTHESIS (H0) ALTERNATE HYPOTHESIS (HA)
1.There is no difference in
average yield for any fertilizer
type.
1.There is a difference in average
yield by fertilizer type.
2.There is no difference in
average yield at either planting
density.
2.There is a difference in average
yield by planting density.
3.The effect of one independent
variable on average yield does
not depend on the effect of
the other independent
variable (a.k.a. no interaction
effect).
3.There is an interaction effect
between planting density and
fertilizer type on average yield.
7. You can use a two-way ANOVA when you have
collected data on a quantitative
dependent variable at multiple levels of two
categorical independent variables.
A quantitative variable represents amounts or
counts of things. It can be divided to find a group
mean.
Agricultural output is a quantitative variable because it
represents the amount of crop produced. It can be divided to find the
average agricultural output per acre.
A categorical variable represents types or categories
of things. A level is an individual category within the
categorical variable.
Fertilizer types 1, 2, and 3 are levels within the categorical
8. ASSUMPTIONS OF TWO WAY
ANOVA
1.Independence of variables
The two variable for testing should be independent to each other
2.Homoscedasticity
The variance around the mean for each set of data
should not vary significantly for all the groups
3.Normal distribution of variables
Two variables should have normal distribution.when
plotted individually each should have a bell curve
12. The sum of squares measures the deviation of data points away
from the mean value.A higher sum of squares indicates higher
variability while a lower result indicates low variability from the
mean.
To calculate the sum of squares, subtract the mean from the data
points, square the differences, and add them together.
There are three types of sum of squares: total, residual, and
regression.
The most widely used measurements of variation are the standard
deviation and variance. However, to calculate either of the two
metrics, the sum of squares must first be calculated. The variance is
the average of the sum of squares (i.e., the sum of squares
divided by the number of observations). The standard deviation
is the square root of the variance
The sum of squares is used to calculate whether a linear
relationship exists between two variables, and any unexplained
variability is referred to as the residual sum of squares.
13. A two-way design may have repeated measurements
of each factor or may not have repeated values.
The ANOVA technique is little different in case of
repeated measurements where we also compute the
interaction variation
16. ONE OBSERVATION FOR CELL
The various steps involved are
1) Take the total of the values of individual items (or their coded
values as the case may be) in all the samples and call it T.
2) Correction factor = (T)2
/n
3) Sum of squares of deviations for total variance or
total SS
∑X2
ij -(T)2
/n
Find out the square of all the item values (or their coded values as the
case may be) one by one and then take its total.
Subtract the correction factor from this total to obtain the sum of
squares of deviations for total variance
17. 4) The sum of squares of deviations for variance
between columns or (SS between columns)
Take the total of different columns and then obtain the square of each column
total and divide such squared values of each column by the number of items in
the concerning column and take the total of the result thus obtained.
Finally, subtract the correction factor from this total to obtain the sum of
squares of deviations for variance between columns or (SS between columns).
5)The sum of squares of deviations for variance
between rows (or SS between rows).
Take the total of different rows and then obtain the square of each row total
and divide such squared values of each row by the number of items in the
corresponding row and take the total of the result thus obtained.
Finally, subtract the correction factor from this total to obtain the sum of
squares of deviations for variance between rows (or SS between rows).
18. 6)Sum of squares of deviations for residual or error
variance
[Sum of squares for residual or error variance=
Total SS – (SS between columns + SS between rows)]
MS residual or the residual variance provides the basis for the F-
ratios concerning variation between columns treatment and
between rows treatment. MS residual is always due to the
fluctuations of sampling, and hence serves as the basis for the
significance test.
19. 7) Degrees of freedom (d.f.) can be worked out as
under:
d.f. for total variance = (c . r – 1)
d.f. for variance between columns = (c – 1)
d.f. for variance between rows = (r – 1)
d.f. for residual variance = (c – 1) (r – 1)
where c = number of columns
r = number of rows
21. Both the F-ratios are compared with their corresponding
table values, for given degrees of freedom at a specified
level of significance, as usual and if it is found that the
calculated F-ratio concerning variation between columns
is equal to or greater than its table value, then the
difference among columns means is considered
significant. Similarly, the F-ratio concerning variation
between rows can be interpreted.
24. ANOVA TABLE
5% level as the calculated F-ratio of 4 is less than the table value of 5.14,
but the variety differences concerning fertilizers are significant as the
calculated F-ratio of 6 is more than its table value of 4.76.
26. MORE THAN ONE OBSERVATION PER CELL
The ANOVA technique is little different in case
of repeated measurements where we compute
the interaction variation.
Interaction variation.=Total ss-total ss of colums
+rows+within samples
27. EXAMPLE
Set up ANOVA table for the following
information relating to three drugs testing to
judge the effectiveness in reducing blood pressure
for three different groups of people:
28. Do the drugs act differently?
Are the different groups of people affected
differently?
Is the interaction term significant?
Answer the above questions taking a
significant level of 5%.
30. d.f. for interaction = d.f. for colums × d.f. for rows
d.f. for error = d.f. for total-
-d.f. for column
-d.f. for rows
-d.f. for interactions
31. ANOVA TABLE
The above table shows that all the three F-ratios are significant of 5% level which
means that the drugs act differently, different groups of people are affected
differently and the interaction term is significant.
=18.7
32. "The only difference between one-way and two-way
ANOVA is the number of independent variables.
A one-way ANOVA has one independent variable, while
a two-way ANOVA has two.
One-way ANOVA:
Testing the relationship between shoe brand (Nike,
Adidas, Saucony, Hoka) and race finish times in a
marathon.
Two-way ANOVA:
Testing the relationship between shoe brand (Nike,
Adidas, Saucony, Hoka), runner age group (junior, senior,
master’s), and race finishing times in a marathon."
33. ANOVA IN LATIN-SQUARE DESIGN
The treatments in a L.S. design are so allocated among
the plots that no treatment occurs more than once in any
one row or any one column. The two blocking factors may
be represented through rows and columns (one through
rows and the other through columns) An experiment has
to be made through which the effects of five different
varieties of fertilizers on the yield of a certain crop,
34. In such a case the varying fertility of the soil in different
blocks in which the experiment has to be performed must
be taken into consideration; otherwise the results
obtained may not be very dependable because the
output happens to be the effect not only of
fertilizers, but it may also be the effect of fertility
of soil. Similarly, there may be impact of varying
seeds on the yield.
To overcome such difficulties, the L.S. design is used
when there are two major extraneous factors such as the
varying soil fertility and varying seeds.
The Latin-square design is one wherein each fertilizer, in
our example, appears five times but is used only once in
each row and in each column of the design
35. The ANOVA technique in case of Latin-square design
remains more or less the same as in case of a two-way
design, excepting the fact that the variance is
splitted into four parts as under:
(i) variance between columns;
(ii) variance between rows;
(iii) variance between varieties;
(iv) residual variance.
40. The above table shows that variance between rows and variance between
varieties are significant and not due to chance factor at 5% level of
significance as the calculated values of the said two variances are 8.85 and
9.24 respectively which are greater than the table value of 4.76. But variance
between columns is insignificant and is due to chance because the calculated
value of 1.43 is less than the table value of 4.76.
42. ANALYSIS OF CO -VARIANCE (ANCOVA)
A variable that is not among the main research
variables may affect the dependent variable and its
relation with the independent variable, which, if
identified, can be involved in the modeling and its linear
effect are controlled. This is not a dependent or independent
variable, this type of variable is known as covariate
43. The statistical
method that can
combine ANOVA
and regression
for adjusting
linear effect of
covariate and
make a clearer
picture is called the
analysis of
covariance
(ANCOVA).
44. Regression analysis primarily uses data to establish a
relationship between two or more variables. It assumes
that past relationships will also reflect in the present or
future.
We are taking the relationship between the prices of an
antique collection for auction and its age duration.
The older an antique gets, the more the price it fetches.
Assuming that we have set data for the last 50 items
auctioned, we can predict the future auction prices based
on the item’s age.
46. The effectiveness of different treatment methods on
patient recovery times, while adjusting for variables
like age or baseline health status.
In social science, it might be used to examine the
impact of an educational program on student
performance, taking into account factors such as
socioeconomic status or prior knowledge.
47. How are ANCOVA and ANOVA different?
While ANOVA can compare the means of three or more
groups, it cannot control for covariates.
ANCOVA builds on ANOVA by introducing one or more
covariates into the model.
ANCOVA discovers the variance changes of the dependent
variable due to change in covariate variable and
discriminates it from the variance changes due to changes
in the levels of the qualitative variable; so it reduces the
uncertain changes of the variance of dependent variable
(error) and make pure results as well as increases the
analytical power.
48. WHAT IS TWO WAY ANOVA TEST?
A two-way ANOVA is a statistical test that is used to find the effect of multiple
levels of two independent variables on a response variable(Dependent) and the
interaction effect, if any, between the two. The two-way ANOVA tests and
compares the differences between the mean variables and uses this information to
check the variance. It also measures the degree of interaction between the variables
and its effect on the response
WHEN TO USE TWO WAY ANOVA TEST?
One can use a two-way ANOVA test to check the effect of two factors on a dependent
variable and if there is any interaction effect between the two
WHAT IS ANCOVA?
To control the effect of covariate variable, not only the changes in variance of the
dependent variable are examined (ANOVA), but also the relationship between the
dependent variable and covariate in different levels of a qualitative variable is
analyzed
51. Y is the covariate (or concomitant) variable. Calculate the adjusted total,
within groups and between groups, sums of squares on X and test the
significance of differences between the adjusted means on X by using the
appropriate F-ratio. Also calculate the adjusted means on X.
![6)Sum of squares of deviations for residual or error
variance
[Sum of squares for residual or error variance=
Total SS – (SS between columns + SS between rows)]
MS residual or the residual variance provides the basis for the F-
ratios concerning variation between columns treatment and
between rows treatment. MS residual is always due to the
fluctuations of sampling, and hence serves as the basis for the
significance test.](https://image.slidesharecdn.com/twowayanova-250415050626-0ac9f52e/85/TWO-WAY-ANOVA-pptx-biostatistics-and-reasearch-18-320.jpg)
