Factorial Experiments
Factorial experiments
In CRD, RBD or LSD we consider
the effects of only one factor. Such experiments with one factor are known as
simple experiments, whereas the effects of more than one factors are considered
together are known as factorial experiments. In other words, the experiment
where the effects of several factors of variations are studied and investigated
simultaneously, the treatments being all the combinations of different factors
under study is known as factorial experiment.
Consider
a simple case of a factorial experiment. The yield of a crop depends on the
variety of crop and also on manure. We may carry out two simple experiments,
one for variety and other for manures. The first experiment will give
information on whether the different varieties of crop are equally effective or
not. The second experiment will give information on whether the
different manures are equally effective or not. But none of these two
experiments will give us, any information about the dependence or independence
of the effects of the varieties on those of the manures. The only way to know
about the behavior of the different varieties in the presence of different manures
is to have all possible combinations of the varieties and manures in the same
experiment. That is to conduct a
factorial experiment with two factors variety and manure.
22
factorial experiment
Let us consider two
factors A and B, each at two levels. Let a0 and a1 denote the first and second
levels of factor A and b0 and
b1 denote the first and
second levels of factor B. Then there will 22 = 4 treatment combinations namely a0b0, a0b1,
a1b0, and a1b1.
According to Yates’s notations the small letters a and b denote the second
levels of factors A and B respectively. The first level of the factor is
expressed by the absence of the corresponding letter in the treatment
combinations. The four treatment combinations can be written as follows:
a0b0 or ‘1’
: Factor A at first level and B at first level.
a1b0 or ‘a’
: Factor A at second level and B at first level.
a0b1
or
‘b’ : Factor A at first level
and B at second level.
a1b1
or ‘ab’
: Factor A at second level and B at second level.
Main effects and interactions
Suppose
the 22 factorial experiment is conducted in r
replicates. The symbols [1], [a], [b] and [ab] will be used to denote the total
of all the observations receiving the treatment combinations 1, a, b and ab respectively. The symbols (1), (a), (b) and (ab) will be
used to denote the mean of all observations receiving the corresponding
treatment combinations.
Consider the effects of factor A:
The effect of changing factor A from its first level a0 to second level a1 in the presence of the first level of b0 of factor B is given by,
(a1b0) - (a0b0) = (a) - (1)
Similarly the
effect of A in presence of level b1 of B is given by
(a1b1) - (a0b1) = (ab) - (b)
The
effects (a) - (1) and (ab) - (b) are called the simple effects of factor A.
The
average of two simple effects of factor A is a defined as the main effect due
to factor A. Therefore the main effect of factor A is,
A =
The simplified
form of this is,
A =
where the R.H.S.
is to be expanded algebraically and then the treatment combinations are to be
replaced by corresponding treatment means.
Consider
the effects of factor B:
The simple
effect of factor B at the level a0 of A is,
(a0b1)
- (a0b0)
= (b) - (1)
Similarly the
simple effect of B at level a1 of A is,
(a1b1)
- (a1b0)
= (ab) - (a)
Therefore
the main effect of factor B is,
B =
The simplified
form of this is,
B =
Interaction
If
the factors A and B are independent in their effects, then we expect the two
simple effects of factor A to be equal. If the two factors are not independent,
then the simple effects of factor A are not same. Half (one by two) of the
difference of the two simple effects of A is a defined as a measure of
dependence or interaction.
Therefore
interaction of factor A with B is,
AB =
=
Similarly the
interaction of factor B with A is,
BA =
=
Form (1) and (2) we observe that interaction AB is same as interaction BA. That
is interaction does not depend on the order of factors.
Note:
1. The main effects and interactions can
be easily obtained from the following table which gives the signs with which to
combine the treatment mean and also the divisor. The first line gives the
general mean.
Statistical analysis of 22 factorial experiment
Factorial experiments are conducted in CRD, RBD or LSD. Hence analysis of factorial experiment is same as usual analysis of the design except that in this case the treatment sum of squares is splitted into three orthogonal components each with one degree of freedom. The sum of squares due to factorial effects are computed by using factorial effects totals which are given by,
[A] = [ab] + [a] - [b] - [1], [B] = [ab] - [a] + [b] - [1] and
[AB] = [ab] - [a] - [b] + [1]
where [ab], [a], [b] and [1] denote the treatment totals for all the
replicates, for corresponding treatment combinations .
If each treatment
combination is replicated r times, then
the sum of squares due to factorial effects are obtained by dividing the square
of the effect total by 22
× r = 4.r . Thus,
where G is total of all the observations and β
Null hypothesis
H0 = The factorial effects are insignificant.
We can write the
separate hypothesis for each factorial effect as,
H0A = Main effect A is insignificant.
H0B = Main effect B is insignificant.
H0AB = Interaction AB is insignificant.
ANOVA table for 22 factorial experiment in RBD with r blocks
Test procedure
For testing the null hypothesis, we compare the calculated values of F- ratio for different main effects and interactions with the tabulated value of F with [1, 3(r -1)] degrees of freedom at specified level of significance. If calculated value of F for some factorial effect is larger than the tabulated value of F, we conclude that the corresponding factorial effect is significant, otherwise it is insignificant.
Yate’s method for computing factorial effect totals
Yates’s developed a systematic
method of obtaining the various effect totals for 2n factorial experiment,
for n =2, 3, 4,….. without writing the algebraic
expressions for factorial effect totals. The computational procedure involves
following steps.
1. In the first column of Yates’s table write the treatment combinations systematically starting with treatment combination 1 and then introducing letters a, b,…. in turn. After introducing a letter write down its combinations with all the previous treatment combinations and then introduce new letter.
For example : In 23 factorial experiment the order will be 1,a,b,ab,c,ac,bc,abc.
2. Write down the treatment total from all the replicates in the second column, against the appropriate treatment combination.
3. For obtaining third column, break the values in the second column in two consecutive pairs. Then the first half of the third column is obtained by writing down in order the sum of the pairs and second half is obtained by writing the differences of the pairs of the second column.( the first value being subtracted from second member of a pair).
4. To obtain fourth column, the whole
procedure explained in step 3 is repeated on column 3
and the fifth
column is derived from fourth column in similar manner and so on.
For a 2n experiment, we have to
repeat the procedure of sum and difference n times and then the values in (n +2)th
column give the factorial effect totals. The first
entry in the last column is always the grand total.
Example: Yates’s table for 23 factorial experiment.
|
Treatment Combination (1) |
Treatment Total (2) |
(3) |
(4) |
(5) |
|
1 |
[1]
|
[1]+[a] |
[1]+[a]+ [b]+
[ab] = U1 |
G = U1+
U2 |
|
a |
[a]
|
[b]+ [ab] |
[c]+[ac]+
[bc]+[abc] = U2 |
[A]= U3+
U4 |
|
b |
[b]
|
[c]+[ac] |
[a]-[1]+ [ab]-[b]
= U3 |
[B]= U5+
U6 |
|
ab |
[ab]
|
[bc]+[abc] |
[ac]-[c] +[abc]-[bc]
= U4 |
[AB]= U7+
U8 |
|
c |
[c]
|
[a]-[1] |
[b]+ [ab]-
[1]-[a] = U5 |
[C]= U2-
U1 |
|
ac |
[ac]
|
[ab]-[b] |
[bc]+[abc]-
[c]-[ac] = U6 |
[AC]= U4 -
U3 |
|
bc |
[bc]
|
[ac]-[c] |
[ab]-[b]- [a]+[1]
= U7 |
[BC]= U6 –
U5 |
|
abc |
[abc]
|
[abc]-[bc] |
[abc]-[bc]-
[ac]+[c] = U8 |
[ABC]= U8 –
U7 |
Yates’s table for 22 factorial experiment.
|
Treatment Combination (1) |
Treatment total(2) |
(3) |
(4) |
|
1 |
[1]
|
[1]+[a] |
[1]+[a]+ [b]+ [ab] |
|
a |
[a]
|
[b]+ [ab] |
[ab]+[a]- [b]-[1] |
|
b |
[b]
|
[a]- [1]
|
[ab]+[b]- [a]-[1] |
|
ab |
[ab]
|
[ab]-[b] |
[ab]-[b] -[a]+[1] |
23 Factorial Experiment
Consider an experiment with three
factors A, B and C, each at two levels. Let a0, b0 and c0
be the first levels and a1, b1 and c1 be
the second levels of factor A, B and C respectively. Then there will 23 =
8 treatment combinations namely a0b0c0, a1b0c0,
a0b1c0, a0b0c1, a1b1c0,
a1b0c1, a0b1c1 and a1b1c1 According to Yates’s notations the
small letters a, b and c denote the second levels of factors A, B and C
respectively. The first level of the factor is expressed by the absence of the
corresponding letter in the treatment combinations. The eight treatment
combinations can be written as follows:
a0b0c0 or ‘1’
: Factor A at first level , B at first level and C at first level
a1b0c0 or ‘a’
: Factor A at second level, B at first level and C at first level.
a0b1c0
or
‘b’ : Factor A at first level, B
at second level and C at first level.
a0b0c1 or ‘c’ : Factor A at first level, B at first level and C at second level.
a1b1c0 or
‘ab’ : Factor A at second level , B at second level
. and C at first level
a1b0c1
or
‘ac’ : Factor A at second level,
B at first level and C at second level.
a0b1c1 or ‘bc’ : Factor A at first level, B at second
level and C at second level.
a1b1c1
or ‘abc’ : Factor A at second level, B at second
level and C at second level.
Main effects and interactions
Suppose, the 2^3 factorial
experiment is conducted in r replicates. Let the symbols [1],[a],[b],….,[abc]
denote the total yields from all the replicates and (1),(a),(b),…., (abc) denote the treatment means of the
corresponding treatment combinations.
Consider
the simple effects of factor A, with different levels of factor b and c.
|
Level
of b |
Level
of c |
Simple effect of A |
|
b0 |
c0 |
(a1b0c0)
-(a0b0c0) = (a)-(1) |
|
b1 |
c0 |
(a1b1c0)
-(a0b1c0) = (ab)-(b) |
|
b0 |
c1 |
(a1b0c1)
-(a0b0c1) = (ac)-(c) |
|
b1 |
c1 |
(a1b1c1)
-(a0b1c1) = (abc)-(bc) |
A = 1/4 { (abc) - (bc)+ (ac) - (c) +(ab)- (b)+ (a)-(1)} = 1/4 { (a-1) (b+1) (c+1)}
where the R.H.S.
is to be expanded algebraically and then the treatment combinations are to be
replaced by corresponding treatment means.
Similarly,
we can write the main effects of factor B and C as,
B = 1/4 {
(abc) + (bc)- (ac) - (c) +(ab) +(b)- (a)-(1)}
= 1/4 {
(a+1) (b-1) (c+1)}
and C = 1/4 {
(abc) + (bc)+ (ac) + (c) -(ab)-(b)- (a)-(1)}
= 1/4 {
(a+1) (b+1) (c-1)}
First order Interactions
(Two factor interactions)
Consider
the average effects of factor A, with different levels of factor B.
Average effect
of A with b0 of B = 1/4 {(a)
- (1) + (ac) - (c)}
Average effect
of A with b1 of B = 1/4 {
(ab) - (b)+ (abc) - (bc) }
The interaction
of factor A with B is given by, half of the difference between average effect
of A at level b1 and average effect of A with level b0 of B. Therefore
interaction of A with B is,
AB = 1/4 {
(abc) +(ab) - (bc)- (ac)- (a)-(b)+ (c) +(1)}
= 1/4 (a-1) (b-1) (c+1)
Similarly, we
have
AC = 1/4 {(abc)
+ (ac) - (ab) - (bc) - (a) + (b) - (c) + (1)}
= 1/4 (a-1) (b+1) (c-1) and
BC = 1/4 {(abc)
- (ac) - (ab) + (bc) + (a) - (b) - (c) + (1)}
= 1/4 (a+1) (b-1) (c-1)
Second order Interactions
(Three factor interactions)
Consider the interaction between A and B at each
level of factor C.
Interaction AB at level c0 of C = 1/4 {(ab)-(b) – (a) + (1)} and
Interaction AB at level c1 of C = 1/4 {(abc)-(bc) – (ac) +(c)}
The second order interaction ABC is given by half of
the difference between interaction AB at level c1 of C and
interaction AB at level c0 of C. Therefore,
ABC = 1/4 {(abc) - (ac) - (ab) - (bc) + (a) + (b) + (c)
- (1)}
= 1/4 (a-1)
(b-1) (c-1)
Note:
1. All the seven factorial
effects A, B, C, AB, BC, AC and ABC are mutually orthogonal contrasts.
2. We have AB = BA,
AC = CA, BC = CB and ABC = ACB = CAB etc.
3. Let M denotes
the mean of all the observations then,
M = 1/4 {(abc) + (ac) + (ab) + (bc) + (a) + (b) + (c)
+ (1)}
= 1/4 (a+1) (b+1) (c+1)
4. The following
table gives the signs and divisors with which the various treatment means are
to be combined to get the factorial effects.
|
Factorial effect |
Treatment means |
Divisors |
|
(abc) (ab)
(ac) (bc) (a)
(b) (c) (1) |
||
|
M |
+ + + + +
+ + + |
8 |
|
A |
+ + + - +
- -
- |
4 |
|
B |
+ + - +
-
+ -
- |
4 |
|
C |
+ - + +
-
- + - |
4 |
|
AB |
+ + - -
-
- + + |
4 |
|
AC |
+ - + - -
+ -
+ |
4 |
|
BC |
+ - - + + - -
+ |
4 |
|
ABC |
+ - - -
+
+
+ + |
4 |
Rules to write
signs for treatment combinations
i) For main
effects give a positive (+) sign where the corresponding factor is at second
level and negative (-) sign where it is at first level.
ii) The signs of
the two factors interaction are obtained by combining the corresponding signs
of the two main effects.
Statistical
analysis of 23 factorial experiment
Factorial
experiments are conducted in CRD, RBD or LSD. Hence analysis of factorial
experiment is same as usual analysis of the design except that in this case the
treatment sum of squares is split into seven orthogonal components each with
one degree of freedom. The sum of squares due to factorial effects are computed
by using factorial effect totals which are given by,
[B] = [abc] +[bc] - [ac] + [ab]
-[a]+[b]-[c]- [1]
[C] = [abc] +[bc] + [ac] - [ab] -[a]-[b]+[c]+
[1]
[AB] = [abc] -[bc] - [ac] + [ab] +[a]+[b]-[c]-
[1]
[BC] = [abc] +[bc] - [ac] - [ab] +[a]-[b]-[c]- [1]
[AC] = [abc] -[bc] + [ac] - [ab] - [a]+[b]-[c]+ [1] and
[ABC] = [abc] -[bc] - [ac] + [ab] +[a]+[b]+[c]- [1]
where [abc], [bc], -------, [1] denote the treatment totals
from all the replicates for the corresponding treatment combination. These
factorial effect totals can easily be obtained by using Yates’s method for
computing factorial effect totals. If each treatment combination is replicated
r times the sums of squares due to factorial effects are obtained by dividing
the square of effect total by 23. r= 8r.
Each of these sums of squares carries one degree of freedom.
The remaining sums
of squares that is TSS, SSB and SSE are calculated as usual. That is,
Also, SSE = TSS – SSB – SSA – SSB - SSAB – SSC – SSAC - SSBC - SSABC
where G is total of all the observations and βj is
the total of observations in the jth
block ( j = 1,2,..,r)
Null hypothesis
H0 = All the factorial effects are insignificant.
We can write the
separate hypothesis for each factorial effect as,
H0A = Main effect A is insignificant.
H0B = Main effect B is insignificant.
H0AB = Interaction AB is insignificant.
H0C = Main effect C is insignificant.
H0AC = Main effect AC is insignificant.
H0BC = Interaction BC is insignificant.
H0ABC = Interaction ABC is insignificant.
Confounding in Factorial experiments
In factorial experiments, as the number of
factors and the levels of the factors increases the number of treatment
combinations to be compared also increases. This in turn requires the blocks of
large size to accommodate all the treatment combinations. Example: In a 25
experiment there should be 32 plots in a block, but it has been found that the
experimental error increases with an increase in the size of a block, as the
blocks of large size are heterogeneous. In order to maintain the homogeneity
within the blocks replicate is divided into number of equal blocks (incomplete
blocks) and then the treatment combinations are allocated to these blocks so
that only the unimportant treatment comparisons (contrasts) get mixed up or entangled
with block contrasts. These treatment combinations are then said to be mixed up
or confounded with block effects.
Confounding can be defined as, ‘the
process by which unimportant comparisons are purposively mixed up or entangled
with the block comparisons for the purpose of assessing more important comparisons
with greater precision.
It may also be defined as, ‘the
technique of reducing the size of replication over a number of blocks at the
cost of losing some information on some effect which is not of much practical
importance.
The confounded effects
cannot be separately estimated or tested, but the remaining effects are still
capable of separate estimation and testing. The device of confounding consists
of subdividing the replicate into two or more equal blocks and then various
treatment combinations into the same number of groups of equal size following
certain rule by which we lose some information on certain higher order
interactions and allocating the treatment combinations in any group to any
block at random.
Total
(complete) confounding
In complete confounding,
we confound the same interaction in all the replicates, therefore we loose information
on that interaction from all the replicates and hence we cannot estimate and
test the confounding effect, but the unconfounded effects are orthogonal to the
blocks of the replicates and can be estimated and tested as in a complete block
design.
Complete confounding in
a 23 factorial
experiment
In
a 23 experiment i.e. an experiment with three factors each at two
levels, there are 8 treatment combinations.
Let
A, B and C be the three factors. Let a, b and c denote the second levels of the
factors and first levels be denoted by absence of the letter. Then the
treatment combinations will be 1, a ,b, c, ab, bc, ac and abc. Since there are 8 treatment combinations,
replicate requires 8 plots. If we decide to use blocks of four plots each then
replicate will contain only two blocks. In this case, 8 treatment combinations
are divided into two groups of four treatments each, so as to confound an unimportant
interaction with blocks and these groups are allocated at random in the two
blocks.
Let us consider to confound highest order interaction ABC, which is given by, ABC = 1/4 {(abc) + (ac) + (ab) + (bc) - (a) - (b) - (c) - (1)}
To
confound interaction ABC with blocks all the four treatment combinations with
positive sign are allocated at random in one block and those with negative sign
in the other block. The following arrangement gives ABC confounded with the
blocks.
Replication I
Block 1 Block 2
|
a |
ab |
|
abc |
1 |
|
c |
ac |
|
b |
bc |
From above arrangement, we observe that the contrast estimating abc also contains block effect. All the other factorial effects are orthogonal with block effects and hence can be estimated and tested as usual without any difficulty.
Analysis
of 23 factorial experiment in which abc is completely confounded.
Let us consider a 23 factorial
experiment conducted in r replicates with interaction abc is confounded in every
replicate. The effect totals for all the
factorial effects are obtained as usual by using Yates’s method. The sum of
squares due to the factorial effects except the confounded factorial effect abc
are obtained as,
Null hypothesis
H0: The factorial effects except confounded effect are insignificant.
Test procedure
To test for the
significance of the factorial effects (except confounded effect) we compare the
calculated value of F with the tabulated value of F with (1, 6(r -1)) degrees
of freedom at specified level of significance. If calculated value of F, is
larger than the tabulated value of F, we conclude that the corresponding
factorial effect is significant otherwise it is in significant.
Partial confounding
Sometimes we may wish to test all
the factorial effects. In
complete confounding, entire information on confounded effect is loosed and
hence we cannot estimate the confounded factorial effect. To avoid this we shall
distribute the loss of information among more than one interaction and shall
get some information on each of them.
A confounded factorial design where the same
factorial effect is not confounded in all the replicates is known as partially compounded
design and the confounding is known as partial confounding. In partial
confounding the sum of squares due to unconfounded effects is obtained in usual
way by using all the replicates while the sum of squares due to confounded
effects are obtained from only those replicates in which the corresponding
effects are not confounded. Since the partially confounded interactions are
estimated from only a portion of observations they are estimated with lower
degree of precision than the other factorial effects.
Partial confounding in 23
factorial experiments
Let us consider an experiment with three factors A, B and C each at two
levels. The treatment combinations will be 1, a, b, ab, c, ac, bc, abc.
Consider
a plan with four replicates where each replicate is divided into two blocks of
four plots each. The eight treatment combinations are allocated to the blocks
of the replicates at random so that the interactions AB, AC, BC and ABC are
confounded with blocks of the replicates I, II, III and IV respectively.
Rep. I Rep. II Rep. III Rep. IV
In above plan main effects A, B and C are not confounded in any of the replicates and therefore they are estimated from all the four replicates. On the other hand, each interaction is confounded in one of the replicates and left unconfounded in remaining three replicates. Hence we can estimate them from only three replicates in which they are not confounded. For example interaction AB will be estimated from replicates II, III and IV.
Analysis:
Suppose that above plan is replicated r times,
then there will be 4.r replicates and each interaction will be confounded in r
of the replicates.
The
block sum of squares will be computed from 8.r blocks as,
Sum of squares due to
unconfounded factorial effects
Sum of squares due to confounded interactions
Each of these sums of squares carries 1 d. f.
Null Hypothesis
H0: All the
factorial effects are insignificant.
Test procedure
To test for the
significance of the factorial effects we compare the calculated value of F with
the tabulated value of F with [ 1, 24r- 7] degrees of freedom at specified
level of significance. If calculated value of F, is larger than the tabulated
value of F, we conclude that the corresponding factorial effect is significant
otherwise it is insignificant.
To calculate the sum of
squares for confounded factorial effects (in partial confounding) (For practical)
The
sum of squares due to confounded effects is obtained from those replicates in
which it is not confounded. From practical point of view, the effect totals for
confounded factorial effects can be obtained from Yates’s table, for all the
replicates by applying the adjustment factor (A.F.).
The A.F. for confounded effects is computed as follows:
1.
Note, the replicate in which given effect is confounded.
2.
Note the sign of treatment combination (1) in the algebraic expression of the
corresponding effect.
3.
Let T1 be the total of observations in the block containing
treatment combination (1) of the replicate in which the effect is confounded and
T2 be the total of other block of the same replicate.
4.
a) if sign of (1) is positive, then A.F.
= T1 – T2
b) if sign of (1) is negative, then A.F.
= T2 – T1
5.
To obtain correct effect totals for the
confounded effects, the A.F. is a subtracted from the effect total obtained from Yates’s table.
Example:
Correct effect total for AB = [AB] * = [AB] – A.F.
6.
The sum of squares for confounded effect is the ratio of square of correct
effect total to the number of observations from all the replicates in which the
factorial effect is not confounded.
Advantages and disadvantages of confounding
The greatest advantage
of confounding is that, it reduces the experimental error considerably by
dividing the experimental material into homogeneous block and hence results in
increase in precision of experiment. In
total compounding entire information on the confounded effect is lost and hence
it is not possible to estimate the confounded effect. In partial confounding
the confounded effects are estimated only from a part of observation and hence
there is loss of information on confounded effects and they are estimated with
lower degree of precision. The algebraic
calculations are usually more difficult and the statistical analysis is
complicated specifically when some of the observations missing.
***
Model adequacy
checking
using residual analysis
The major assumptions concerning the error term that we have made in our study of regression analysis are as follows:
1. The relationship between the
response y and the regressors is linear, at least approximately.
2. The error term ε has zero mean.
3. The error term ε has constant variance σ2.
4. The errors are uncorrelated.
5. The errors are normally
distributed.
Taken
together, assumptions 4 and 5 imply that the errors are independent random
variables. Assumption 5 is required for hypothesis testing and interval
estimation.
We
should always consider the validity of these assumptions to be doubtful and
conduct analyses to examine the adequacy of the model we have tentatively
entertained. The types of model inadequacies discussed here have potentially
serious consequences. Gross violations of the assumptions may yield an unstable
model in the sense that a different sample could lead to a totally different
model with opposite conclusions. We usually cannot detect departures from the
underlying assumptions by examination of the standard summary statistics, such
as the t or F statistics,
or R2. These are
“global” model properties, and as such they do not ensure model adequacy.
We
can study several methods useful for diagnosing violations of the basic
regression assumptions. These diagnostic methods are primarily based on study
of the model residuals. Methods for
dealing with model inadequacies, as well as additional more sophisticated diagnostics.
Residual Analysis
The analysis of residuals plays an important role in validating the regression model. If the error term in the regression model satisfies the assumptions noted earlier, then the model is considered valid. Since the statistical tests for significance are also based on these assumptions, the conclusions resulting from these significance tests are called into question if the assumptions regarding ε are not satisfied.
The ith residual is the difference between the observed value of the dependent variable, yi, and the value predicted by the estimated regression equation, ŷi. These residuals, computed from the available data, are treated as estimates of the model error, ε. As such, they are used by statisticians to validate the assumptions concerning ε. Good judgment and experience play key roles in residual analysis.
A residual is the vertical
distance between a data point and the regression line. Each
data point has one residual. They are positive if they are above
the regression line and negative if they are below the regression line.
If the regression line actually passes through the point,
the residual at that point is zero.
Residual Plots:
A residual plot is
a graph that shows the residuals on the vertical axis and the independent
variable on the horizontal axis. If the points in a residual plot are randomly
dispersed around the horizontal axis, a linear regression model is appropriate
for the data; otherwise, a nonlinear model is more appropriate.
The
table below shows inputs and outputs from a simple linear regression analysis.
|
x |
y |
ŷ |
e |
|
60 |
70 |
65.411 |
4.589 |
|
70 |
65 |
71.849 |
-6.849 |
|
80 |
70 |
78.288 |
-8.288 |
|
85 |
95 |
81.507 |
13.493 |
|
95 |
85 |
87.945 |
-2.945 |
And the chart below displays the residual (e)
and independent variable (X) as a residual plot.
The residual plot shows a fairly random
pattern - the first residual is positive, the next two are negative, the fourth
is positive, and the last residual is negative. This random pattern indicates
that a linear model provides a decent fit to the data.
Below, the residual plots show three typical
patterns. The first plot shows a random pattern, indicating a good fit for a
linear model.
The other plot patterns are non-random
(U-shaped and inverted U), suggesting a better fit for a nonlinear model.
Mostly researchers do not
considered the ANOVA assumption of normality,
they directly use
regular Parametric test
such as One-way
ANOVA or CRD and
Two-way ANOVA or
RBD for non-normal data. So
before use this parametric
test, data pass through test
of normality such as Anderson- Darling Test.
This test is used to check the data follow normal distribution or does
not follow. Anderson-Darling Test compares the data nature with normal
distribution i.e. this test is based on normal distribution. In this test, if
p<0.05 then data is significant (Reject Null Hypothesis) i.e. data does not
follow normal distribution and if p > 0.05 then non- significant (Accept
Null Hypothesis) i.e. data follow normal distribution. The graph of normal and
non-normal distribution as below,
Anderson-Darling Test shows non-
normal data pattern. (p < 0.05)
i) Square root transformation for
counts.
Suppose data items are
integer valued (count data). For example, Number of patients recovered due to
certain medicines, Number of defects / accidents. In this situation yij’s
are discrete hence normality assumption is not valid. In this case yij’s
are assumed to be Poisson random variables. To conduct ANOVA, Square root
transformation is suggested.
ii) Sin-1
(.) transformation for proportions.
Suppose the data items
are percentages and proportions. For example, Treatments are same pesticides
and we find the percentage of plants cured.
yij =
proportions (0 <yij< 1), yij’s are Non-Normal. We
use Sin-1(yij) the arcsine transformation and conduct
ANOVA as usual.
iii) Kruskal Wallis
test.
Suppose for some reasons we cannot
conduct ANOVA
In situations where the normality
assumption is unjustified, the experimenter may wish to use an alternative
procedure to the F test analysis of variance that does not depend on this assumption.
Such a procedure has been developed by Kruskal and Wallis (1952). The Kruskal-Wallis test
is used to test the null hypothesis that the a treatments are identical
against the alternative hypothesis that some of
the treatments generate observations that are larger than others. Because
the procedure is designed to be sensitive for testing
differences in means, it is sometimes convenient to think of the Kruskal-Wallis test
as a test for equality of treatment means. The Kruskal-Wallis test is a nonparametric alternative
to the usual analysis of variance.
To perform a Kruskal-Wallis test,
first rank the observations yij
in ascending order and replace each observation by its rank,
say Rij ,with the smallest observation having rank 1. In the case of ties (observations having
the same value), assign the average rank to each of the tied observations. Let Ri be the sum of the ranks in the ith treatment. The test statistic is;
Where n, is the number of observations in the i th treatment, N is the total number of observations, and
When the number of ties is moderate, there will be little difference between equations (1) and (3), and the simpler form (Equation3) may be used. If they are reasonably large, say ni≥ T5, H is distributed approximately as χ2a-1 under the null hypothesis.
Therefore, if H > χ2a-1the
null hypothesis is rejected. The P-value
approach could also be used.
***
MCQ Test
Que: Choose the most correct alternative.
1. A graph of three quartiles, mimnium and maximum
values in data is known as ---
(a) histogram
(b) residual
(c) box
plot (d) normal plot
2. A missing value in an experiment is estimated by
the method of_____-
(a)
minimizing the error mean square (b) minimizing the total mean square
(c) minimizing
the block mean square (d) all (a), (b)and (c)
3.
In a randomized block design with 4 blocks and 5 treatments having one missing
value, the error degrees of freedom will be_____
(a)
12 (b) 11 (c) 10 (d) 9
4. A Latin square design controls_____
(a) two way variation (b)
three way variation
(c)
multiway variation (d)
no variation
5. Which of the following is a contrast?
(a)
3T1+T2-3T3+T4 (b)
T1+3T2-3T3+T4
(c) -3T1-T2+T3+3T4 (d)
T1+T2+T3-T4
6. Missing observation in a completely randomized
block design is to be______
(a)
estimated (b)
deleted
(c)
guessed (d)
none of the above
7. A randomized block design has_____
(a)
one way classification (b)
two way classification
(c)
three way classification (d)
no classification
8. In Latin
square design, number of rows, columns and treatments are_____
(a)
all different (b)
always equal
(c)
not necessarily equal (d)
none of the above
9. While analyzing the data of a k*k Latin
square, the error d. f. in analysis of variance is equal to_____
(a)
(k-1) (k-2) (b) k (k-1) (k-2)
(c) k2-2 (d) k2-k-2
10. The method of confounding is a device to reduce
the size of ______
(a)
experiments (b) replications (c) blocks (d)
all the above
11. The principle of design of experiments is
proposed by___
(a)
F. Yates (b) R.A. Fisher (c) C.R. Rao (d)
Karl Pearson
12. Local control helps to ______
(a)
reduce the number of treatments (b)
increase the number of plots
(c) reduce the error variance (d)
increase the error d.f.
13.
The additional effect gained due to combined effect of two or more factors is
known as______
(a)
main effect (b)
interaction effect
(c)
either of (a) or (b) (d)
neither of (a) or (b)
14. If different effects are confounded in different
blocks, it is said to be_____
(a)
complete confounding (b) partial confounding
(c)
balanced confounding (d)
none of the above
15.
The effect, which is confounded in all the blocks in an experimental
design_____
(a) is estimated more
precisely (b) is estimated less
precisely
(c) cannot be estimated (d) none of the above
16.
The analysis of variance of an experimental data is based on the assumptions
that______
(a)
the response variable is distributed normally (b) the errors are independent
(c)
the errors are homoscedastic
(d) all
the above
17.
In one way classification, with more than two treatments, the equality of
treatment means is tested by_____
(a)
t-test (b) chi-square test (c) F- test (d) none of these
18. For a 6x6 Latin Square design there will be
observations----
(a) 6 (b) 12 (c) 24 (d)
36
19. In a CRD with t treatments and n experimental
units, error d .f. is equal to---
(a) n-t (b)
n-t-1 (c) n-t+1 (d) t-n
20. In which of the following design all the three
principles of design of experiments are used_______
(a) CRD (b)
RBD (c) Both a and b (d) none of these
21.
Randomization is a process in which the treatments are allocated to the
experimental units_____
(a)
in a sequence (b)
with equal probability
(c)
both (a) and (b) (d)
none of the above
22.
If E is the error variance of the design then amount of information or
efficiency of design is given by ----
(a) E (b) E2 (c)
23.
Let the relative efficiency of design D1 whose error variance is E1,
w. r. t D2 whose error variance is E2, the relative
efficiency of D1 with respect to D2 is-
(a)
E1/ E2 (b) E2/ E1
(c) E1*E2
(d) E1-E2
24. CRD are most suitable in the situations
when_____
(a)
all experimental units are homogeneous
(b)
the units are likely to be destroyed during experimentation
(c)
some units are likely to fail to response
(d) all the above
25.
Analysis of non-normal data is done
using _____
(a)
square root transformation (b) Sin
inverse transformation
(c) KrusKal Wallis test (d)
all of these
Q 1.The factors like spacing, date of sowing and breeds are often used as:
a) Experimental unit b) Treatment c) Replicate d) None of the above
Q 2.Randomization is a process in which the treatments are allocated to the experimental units:
a) At the will of the investigator
b) In a sequence
c)
With the probability
d) None of the above
Q 3. Randomization is the process which enables the experimenter to:
a) Apply mathematical theories
b) Make probability statements
c) Treat error independent
d) All the above
Q 4. Replication in the experiment means
a) The number of blocks
b) Total number of treatments
c)
The number
of times a treatment occurs in a experiment
d) None of the above
Q 5. The decision about the number of replication is taken in view of:
a) Size of experimental units
b) Competition among experimental units
c) Fraction to be sampled
d) All the above
Q 7.Expermental error is due to:
a) Experimenter’s mistakes
b)
Extraneous factors
c) Variation in treatment effects
d) None of the above
Q 8.Statistical model for a completely randomized design under model I and model II is
a) Unspecified
b)
Completely specified
c) Incompletely specified
d) None of the above
Q 9.A randomized block design has
a) Two way classification
b) One way classification
c) Three way classification
d) No classification
Q 10.In the analysis of data of a randomized block design with b block and v treatments, the error degrees of freedom are
a) b(v-1)
b) v(b-1)
c) (b-1)(v-1)
d) none of the above
Q 11.The formula for estimating one missing value in a randomized block design having b blocks and k treatments with usual notation is:
a) bT’+kB’-G/(b-1)(k-1)
b) bB’+ bT’-G/(b-1)(k-1)
c) kT’+ bB’-G’/(b-1)(k-1)
d) bT’+kB’-G/(b-1)(k-1)
Q 12.A missing value in an experiment is estimated by the method of :
a) Minimizing the error mean square
b) Analysis of covariance
c) Both a) and b)
d) Neither a) and b)
Q 13. A Latin square design controls:
a) Two way variation
b) Three way variation
c) Mulyway variation
d) No variation
Q 15.While analyzing the
data of k x k Latin square, the error d.f in analysis of variance is equal to a) (k-1)(k-2)
b) k(k-1)(k-2)
c) k2-2
d) k2-k-2
Q 16.All contrast, representing the effects of a 2n factorial can easily be estimated with the help of
a) Simple effects
b) Contrast
c) both a) and b)
d) Neither a) and b)
Q 17.The effect, which is confounded in all the blocks in an experimental design
a) Is estimated more precisely
b) Is estimated less precisely
c) Cannot be estimated
d) None of the above
Q 18. If different effects are confounded in different blocks it is said to be
a) Complete confounding
b)
Partial confounding
c) Balanced confounding
d) None of the above
Q 19. The statistical model used for CRD RBD and LSD is
a)
Linear and additive
b) Non linear and additive
c) Linear and multiplicative
d) Non linear
Q 20. Error sum of square in LSD as compared to CRD using the same experimental material will a) Same b) Greater c) Less d) No conclusion be drawn
Q 21.The principle of controlling heterogeneity in designing an experiment is
a) Randomization b) Replication c) Both
a) and b) d) Neither a) nor b)
Q-23.Confounding in large size factorial experiment is implemented for not to defeat the following principle of design of experiments
a) Randomization
b)
Local control
c) Replication
d) Total control
Q-24.CRD is suitable when
a) Heterogeneity of experimental material is along two direction
b) Heterogeneity is in an erratic fashion
c) Homogeneous experimental units
d) None of the above
Q -25. The number of replication in an experiment is based on
a) The precision required
b) Experimental material available
c)
Heterogeneity
of experimental material
d) Al the above
Q-26.The process of pair wise addition and subtraction in Yates’s table of factorial effect total is carried out in as many column as the number of
a) Replication
b) Treatment combination
c)
Factors
d) Interactions
Q -27. The allocation of treatment to the experimental units with equal probability is known
a) Replication
b) Randomization
c) Local control
d) None of the above
Q-28. In RBD, blocks are formed in --- direction to the fertility gradient
a)
Perpendicular
b) Horizontal
c) Parallel
d) None of these
Q-29.The analysis of CRD is analogous to ANOVA for ---
a)
One way classification
b) Two way classification
c) Both a) and b)
d) None
Q-30. In 23 factorial experiment with 4 blocks the degrees of freedom for error are ---
a) 32 b) 31 c) 21 d) None of these
Q-31.When the interaction effect is confounded in all the replicates, then it is called--- confounding.
a) Partially b ) Complete c) Incomplete d) None of these
Q-34.Two types of effects measured in factorial experiments are
a)
Main and interaction effect
b) Simple and complex effects
c) Simple and factorial effects
d) None of these
Q-35.In Latin square design with 5 treatments having one missing value, the errors degrees of freedom in ANOVA table will be
a) 12 b) 10 c)11 d) 9
Q-36.Statistical error of the difference two treatment means in case of m x m Latin square design with mean error sum square S2E will be
a)
 |
S2E
b)
√2 ∗ SE2/m
c) 2*S2E /m
d) None of these
Q-37.In experimental designs, randomization is necessary to make the estimates:
a) Valid b) accurate c) precise d) biased
Q-38.For two treatments there can be in all:
a) One contrast
b)
Two contrast
c) Three contrast
d) No contrast
Q-40.In one way classification with more than two treatments, the equality of treatment means is tested by:
a) t-test b) χ2 test c) F-test d) None of the above
Q-41.If in a randomized block design having 5 treatments and 4 replications, a treatment is added, the increase in error degrees of freedom will be:
a) 1 b) 2 c) 3 d) 4
Q-42.The additional effect gained due to combined effect of two or more factors is known as:
a) Main effect
b) Interaction effect
c) Either of (a) or (b)
d) Neither of (a) or (b)
Q-44. In a completely randomized design with t treatments and n experimental units, error degrees of freedom is equal to:
a) n-t b) n-y-1 c) n-t+1 d) t-n
Q-45 Each contrast among k treatment has:
a) (k-1) b) One d.f. c) k d.f. d) None of the above
Q-46.Aalysis of experimental data means:
a) Estimation of treatment effects
b) Dividing the total variance into component variances
c) Testing of hypothesis about the parameters involved in the experimental model
d) All the above
Q-47.Local control in experimental design is mean to:
a) Increase the efficiency of the design
b) Reduce experimental error
c) To form homogeneous blocks
d)
All the above
Q-48.Experimental error is necessarily required for:
a) Testing the significance of treatments effects
b) Comparing treatment effects calculating the information released from an experiment
c) Calculating the information released from an experiment
d)
All the above
Q-50. Which of the following is not a contrast among three treatments?
a) T1+2T2-T3
b) T1- T3
c) T1-2T2+T3
d) -T1+2T2-T3
***
Design
of experiments (MCQ)
1. A completely randomized design is also known
as______
(a)
unsystematic design (b)
non-restrictional design
(c)
single block design (d)
all the above
2. A missing value in an experiment is estimated by
the method of_____-
(a)
minimizing the error mean square (b)
analysis of covariance
(c) both (a) and (b) (d)
neither (a) and (b)
3. In a randomized block design with 4 blocks and 5
treatments having one missing value, the error degrees of freedom will be_____
(a)
12 (b) 11 (c) 10 (d) 9
4. A Latin square design controls_____
(a) two way variation (b) three way variation
(c)multiway
variation (d) no
variation
5. Which of the following is a contrast?
(a)
3T1+T2-3T3+T4 (b)
T1+3T2-3T3+T4
(c) -3T1-T2+T3+3T4 (d) T1+T2+T3-T4
6. Missing observation in a completely randomized
block design is to be______
(a)
estimated (b)
deleted
(c)
gussed (d)
none of the above
7. A randomized block design has_____
(a)
one way classification (b)two way classification
(c)
three way classification (d)
no classification
8. Error sum of squares in RBD as compared to CRD
using the same material is_____
(a)
more (b) less (c)
equal (d) not comparable
9. In Latin square design, number of rows, columns
and treatments are_____
(a)
all different (b)
always equal
(c)
not necessarily equal (d)
none of the above
10. While analyzing the data of a k*k Latin square,
the error d.f. in analysis of variance is equal to_____
(a) (k-1) (k-2) (b) k (k-1) (k-2)
(c) k2-2
(d)
k2-k-2
11. The method of confounding is a device to reduce
the size of ______
(a)
experiments (b) replications (c) blocks
(d) all the above
12. The precision of whole-plot treatment can be
increased by assigning the treatments to whole plots_____
(a)
randomly (b)
in randomized block arrangement
(c) in a Latin square arrangement (d) all the above
13. The concept of fractional factorial design was
first expounded by______
(a)
F. Yates (b) D.J. Finney (c) C.R. Rao (d)
G.E.P. Box
14. Local control helps to ______
(a)
reduce the number of treatments (b)
increase the number of plots
(c) reduce the error variance (d) increase the error d.f.
15. The additional effect gained due to combined
effect of two or more factors is known as______
(a)
main effect (b)
interaction effect
(c)
either of (a) or (b) (d)
neither of (a) or (b)
16. If different effects are confounded in different
blocks, it is said to be_____
(a)
complete confounding (b) partial confounding
(c)
balanced confounding (d) none of the
above
17. The effect, which is confounded in all the
blocks in an experimental design_____
(a) is estimated more
precisely (b) is estimated less
precisely
(c) cannot be estimated (d) none of the above
18. If the interactions AB and BC are confounded
with incomplete blocks in a 2n factorial experiment, then
automatically confounded effect is_____
(a)
ABC (b) AC (c)
A (d) C
19. The analysis of variance of an experimental data
is based on the assumptions that______
(a)
the response variable is distributed normally
(b)
the errors are independent
(c)
the errors are homoscedastic
(d) all the above
20. In one way classification, with more than two
treatments, the equality of treatment means is tested by_____
(a)
t-test (b)
chi-square test
(c) F- test (d) none of the above
21. The total sum of squares due to all orthogonal
contrasts in 2n factorial experiment is equal to_____
(a)
replication S.S. (b)
treatment S.S.
(c)
total S.S. (d)
error S.S.
22. The effect which is utilized to divide a
replicate into a fraction is called_____
(a) defining contrast (b) alias
(c)
confounded effect (d)
all the above
23. In a CRD with t treatments and n experimental
units, error d .f. is equal to_____
(a) n-t (b)
n-t-1 (c) n-t+1 (d) t-n
24. Randomized block design is a_______
(a)
three restrictional design (b)
two restrictional design
(c) one restrictional design (d) no restrictional design
25. Randomization is a process in which the
treatments are allocated to the experimental units_____
(a)
in a sequence (b)
with equal probability
(c)
both (a) and (b) (d)
none of the above
26. When there occurs a missing value in an
experiment, treatment sum of square has______
(a) an upward bias (b)
a downward bias
(c)
no bias (d)
none of the above
27. Two types of effects measured in a factorial
experiment are_____
(a) main and interaction effects (b) simple and complex effects
(c)
both (a) and (b) (d)
neither (a) nor (b)
28. The contrast representing the quadratic effect
among four treatments is_____
(a)
3T1+T2-3T3+T4 (b)
-T1+3T2-3T3+T4
(c) -3T1-T2+T3+3T4 (d) T1-T2-T3+T4
29. CRD are most suitable in the situations
when_____
(a)
all experimental units are homogeneous
(b)
the units are likely to be destroyed during experimentation
(c)
some units are likely to fail to response
(d) all the above
30. In the analysis of data of a RBD with b blocks
and v treatments, the error degrees of freedom are_____
(a)
b(v-1) (b) v(b-1) (c) (b-1) (v-1) (d)
none of the above
***
Question Bank
B.Sc.
III (Semester- V) Examination Oct 2023
Paper
No. XI: Design of experiments
Q 1) Choose the most correct
alternative:
1. A completely randomized design is also known
as______
(a) unsystematic
design (b) non-restrictional design
(c)
single block design (d)
all the above
2. A missing value in an experiment is estimated by
the method of_____-
(a)
minimizing the error mean square (b)
analysis of covariance
(c)
both (a) and (b) (d) neither (a) and (b)
3. In a randomized block design with 4 blocks and 5
treatments having one missing value, the error degrees of freedom will be_____
(a)
12 (b) 11 (c) 10 (d) 9
4. A Latin square design controls_____
(a) two way variation (b) three way variation
(c) multiway
variation (d) no
variation
5. Which of the following is a contrast?
(a)
3T1+T2-3T3+T4 (b)
T1+3T2-3T3+T4
(c) -3T1-T2+T3+3T4 (d) T1+T2+T3-T4
6. Missing observation in a completely randomized
block design is to be______
(a)
estimated (b)
deleted
(c)
gussed (d)
none of the above
7. A randomized block design has_____
(a)
one way classification (b)two way classification
(c)
three way classification (d)
no classification
8.
Error sum of squares in RBD as compared to CRD using the same material is_____
(a)
more (b) less (c)
equal (d) not comparable
9. In Latin square design, number of rows, columns
and treatments are_____
(a)
all different (b)
always equal
(c)
not necessarily equal (d)
none of the above
10. While analyzing the data of a k*k Latin square,
the error d.f. in analysis of variance is equal to_____
(a) (k-1) (k-2) (b) k (k-1) (k-2)
(c) k2-2
(d)
k2-k-2
11. The method of confounding is a device to reduce
the size of ______
(a)
experiments (b) replications (c) blocks
(d) all the above
12.
The precision of whole-plot treatment can be increased by assigning the
treatments to whole plots_____
(a)
randomly (b)
in randomized block arrangement
(c) in a Latin square arrangement (d)
all the above
13. The concept of fractional factorial design was
first expounded by______
(a)
F. Yates (b) D.J. Finney (c) C.R. Rao (d)
G.E.P. Box
14. Local control helps to ______
(a)
reduce the number of treatments (b)
increase the number of plots
(c) reduce the error variance (d) increase the
error d.f.
15.
The additional effect gained due to combined effect of two or more factors is
known as______
(a)
main effect (b)
interaction effect
(c)
either of (a) or (b) (d)
neither of (a) or (b)
16. If different effects are confounded in different
blocks, it is said to be_____
(a)
complete confounding (b) partial confounding
(c)
balanced confounding (d) none of the
above
17.
The effect, which is confounded in all the blocks in an experimental
design_____
(a) is estimated more
precisely (b) is estimated less
precisely
(c) cannot be estimated (d)
none of the above
18.
If the interactions AB and BC are confounded with incomplete blocks in a 2n
factorial experiment, then automatically confounded effect is_____
(a)
ABC (b) AC (c)
A (d) C
19.
The analysis of variance of an experimental data is based on the assumptions
that______
(a)
the response variable is distributed normally
(b)
the errors are independent
(c)
the errors are homoscedastic
(d) all the above
20.
In one way classification, with more than two treatments, the equality of
treatment means is tested by_____
(a)
t-test (b)
chi-square test
(c) F- test (d) none
of the above
21.
The total sum of squares due to all orthogonal contrasts in 2n
factorial experiment is equal to_____
(a)
replication S.S. (b)
treatment S.S.
(c)
total S.S. (d)
error S.S.
22. The effect which is utilized to divide a
replicate into a fraction is called_____
(a) defining contrast (b)
alias
(c)
confounded effect (d)
all the above
23.
In a CRD with t treatments and n experimental units, error d .f. is equal to_____
(a) n-t (b)
n-t-1 (c) n-t+1 (d) t-n
24. Randomized block design is a_______
(a)
three restrictional design (b)
two restrictional design
(c) one restrictional design (d) no restrictional design
25.
Randomization is a process in which the treatments are allocated to the
experimental units_____
(a)
in a sequence (b)
with equal probability
(c)
both (a) and (b) (d)
none of the above
26.
When there occurs a missing value in an experiment, treatment sum of square
has______
(a) an upward bias (b)
a downward bias
(c)
no bias (d)
none of the above
27. Two types of effects measured in a factorial
experiment are_____
(a) main and interaction effects (b) simple and complex effects
(c)
both (a) and (b) (d)
neither (a) nor (b)
28. The contrast representing the quadratic effect
among four treatments is_____
(a)
3T1+T2-3T3+T4 (b)
-T1+3T2-3T3+T4
(c) -3T1-T2+T3+3T4 (d) T1-T2-T3+T4
29. CRD are most suitable in the situations
when_____
(a)
all experimental units are homogeneous
(b)
the units are likely to be destroyed during experimentation
(c)
some units are likely to fail to response
(d) all the above
30.
In the analysis of data of a RBD with b blocks and v treatments, the error
degrees of freedom are_____
(a)
b(v-1) (b) v(b-1) (c) (b-1)
(v-1) d) none of the above
31. Each contrast among k treatment has ---
(a)
one (b) k (c) k-1 (d) none of these
32. Efficiency of experimental design D1,
over design D2 is denoted by
E. If E > 1
Then design D1 is ---- efficient than
design D2 .
(a) less (b) more (c) equally (d) none of these
Q
2) Attempt any two of the following:
1)
What are the three basic principles of design of experiments? Explain each
in brief.
2)
Give the concept and definition of efficiency of a design. Derive the
expression for efficiency of RBD over CRD.
3)
What is confounding in factorial experiment? Explain partial confounding in 23
factorial experiment with ANOVA table and test statistic.
4)
What is LSD? Give mathematical model,
split the total sum of squares, hypothesis to be tested and ANOVA table.
5)
What is confounding in factorial experiment? Explain total confounding in 23
factorial experiment with ANOVA table and test statistic.
6)
Give the concept and definition of efficiency of a design. Derive the
expression for efficiency of LSD over CRD.
7)
Give the concept and definition of efficiency of a design. Derive the
expression for efficiency of LSD over RBD when rows are taken as blocks.
8)
Give the concept and definition of efficiency of a design. Derive the
expression for efficiency of LSD over RBD when columns are taken as blocks.
9)
What is CRD? Give mathematical model,
split the total sum of squares, hypothesis to be tested and ANOVA table.
10)
What is RBD? Give mathematical model,
split the total sum of squares, hypothesis to be tested and ANOVA table.
11)
Derive the formula for estimating one missing observation in case of LSD.
12)
Derive the formula for estimating one missing observation in case of RBD.
Q 3)
Attempt
any four of the following:
1) Give the test for equality of two specified
treatment effects in CRD.
2) Give the test for equality of two specified
treatment effects in RBD.
3) Give the test for equality of two specified
treatment effects in LSD.
4) Explain in brief replication, the one of the
basic principles of design.
5) Explain in brief randomization, the one of the
basic principles of design.
6) Explain in brief local control, the one of the
basic principles of design.
7) Explain main effects and interaction effects in
factorial experiments.
8) What is confounding? Explain total confounding in
brief.
9) What is confounding? Explain partial confounding
in brief.
10) State the advantages and disadvantages of
confounding
11) Explain: i) Treatment ii) Experimental error
12) Explain: i) Block ii) Experimental unit 13) Explain: i) Absolute experiments ii) Comparative experiments
14) Write a short note on choice of size and shape
of plots.
15) Write a short note on precision of the
experiment
16)
State the advantages and disadvantages of CRD.
17)
State the advantages and disadvantages of RBD.
18)
State the advantages and disadvantages of LSD.
19)
Write a note on Yates table for computing the effect totals.
20)
Give the mathematical model and assumptions for the 23 factorial
experiment.
***
Comments
Post a Comment