BY VISWANTH REDDY.S DEPARTMENT OF PHARMACOLOGY GOKARAJU RANGARAJU COLLEGE OF PHARMACY

Analysis of variance(ANOVA) Experimental designs

CRD RCBD LSD

Applications of biostatistics

Its mainly employed for comparison of means of three or more samples including the variations in each sample.

this statistical technique first devoloped by R.A.Fisher and was extensively used for agricultural experiments.

The analyis of variance is a method to estimate the contribution made by each factor to the total variation.the total variation splits in to the following two components .

1.variation with in the samples 2.variation between the samples

There are two classifications for the analysis of variance when we classify data based on one factor analysis it is known as

one way ANOVA When we classify data on the basis of two factors which is known

as two way ANOVA

The technique of analysing variance in case of one factor and two factors is similar.however , incase of onefactor analysis the total variance is divided in to twoparts only

1. Variance between samples2. Variance with in the samples.the variance with in the samples is residual variance.

In case of two factor analysis ,the total variance is divided in to 3parts viz.,,

variance due to factor number one Variance due to factor number two Residual variance

PROCEDURE FOR CALCULATING F-STATISTIC:

T-test employed for two mean samples

F-test is employed for comparison means of three or more samples. in this case , the variation between the treatments and the replicates are shown in columns and rows, respectively. Now we have to find out whether these variations are significant and if so what level of significance, for this purpose calculate the F-statistic which is the ratio of variances. The detailed procedure as follows:

TREATMENTS

1 2 3

1 X11 X21 X31---------------∑XR1 R E

P 2 X12 X22 X32----------------∑XR2 L I

C 3 X13 X23 X33-----------------∑XR3 A T E S

∑X= ∑XC1 ∑XC2 ∑XC3= GRAND TOTAL(G)

∑X2= ∑ XC21+ ∑ XC2

2+ ∑XC32---------------------------------------------------------------A

(∑X)2/nc= (∑ XC1)2/nc1+ (∑ XC2)2/nc2+ (∑XC3)2/nc3-----------------B (∑X)2/nr= (∑ XR1)2/nr1+ (∑ XR2)2/nr2+ (∑XR3)2/nr3-------------------C

C.F = (∑X)2/n= G2/n---------------------------------------------------------------------D

Now total sum of squares=A-D between treatments sum of squares=B-D between rows sum of square= C-D residual sum of squares= (A-D)-[(B-D)+(C-D)]

SOURCE OF VARIATION

DEGREES OF FREEDOM(d.f)

SUM OF SQUARES(SS)

MEANS OF SQUARES(MS)

BETWEEN TREATMENTS

c-1 B-D B-D/c-1

BETWEEN ROWS r-1 C-D C-D/r-1

RESIDUAL (C-1)(r-1) (A-B-[(B-D)+(C-D)]

(A-B-[(B-D)+(C-D)]/(C-1)(r-1)

TOTAL Cr-1 A-D

TREATMENTS

1 2 3

1 X11 X21 X31 R E

P 2 X12 X22 X32 L I

C 3 X13 X23 X33 A T E S

∑X= ∑XC1 ∑XC2 ∑XC3= GRAND TOTAL(G)1. Find the total sum of squares ∑X2= ∑ XC2

1+ ∑ XC22+

∑XC32--------A

2. Square the coloumn total and divide separately each total by number of observations inn each coloumn denoted by C1,C2,C3------etc

(∑X)2/nc= (∑ XC1)2/nc1+ (∑ XC2)2/nc2+ (∑XC3)2/nc3-----------------B

3.Find the grand total∑X= ∑XC1 + ∑XC2 + ∑XC3= GRAND TOTAL(G)4.Square the grand total and divide it by the number of observations(n). correction factor, C.F.=( ∑X)2/n or GT2/n---------------------------------D5. Calculate the F value F=BETWEEN TREATMENT MEAN SQUARE/RESIDUAL MEAN SQUARE

SOURCE OF VARIATION

DEGREES OF FREEDOM(d.f)

SUM OF SQUARES(SS)

MEANS OF SQUARES(MS)

F VALUE

BETWEEN TREATMENTS

c-1 B-D B-D/c-1B-D/c-1/A-B/C(r-1)

RESIDUAL C(r-1) A-B A-B/C(r-1)

TOTAL Cr-1 A-D

In one way classification we have studied influence of one factor.however , in two way classification we will study the influence of two factors.

In such cases , data are classified based on two criteria..for example , the yield of different varieties of wheat may be affected by the application of different fertilizers.

Therefore analysis of variance can be used to test the effects of these two factors simultaneosly.

The calculation in two factors analysis is more or less the same In addition to the calculation based on rows.

In one way classification columns are taken into consideration . However in two way analysis both coloumns and rows are considered.

TREATMENTS

1 2 3

1 X11 X21 X31--------------- ∑ XR1 R E

P 2 X12 X22 X32---------------- ∑XR2 L I

C 3 X13 X23 X33----------------- ∑XR3 A T E S

∑X= ∑XC1 ∑XC2 ∑XC3= GRAND TOTAL(G)

∑X2= ∑ XC21+ ∑ XC2

2+ ∑XC32---------------------------------------------------------------A

(∑X)2/nc= (∑ XC1)2/nc1+ (∑ XC2)2/nc2+ (∑XC3)2/nc3-----------------B (∑X)2/nr= (∑ XR1)2/nr1+ (∑ XR2)2/nr2+ (∑XR3)2/nr3-------------------C

C.F = (∑X)2/n= G2/n---------------------------------------------------------------------D

Now total sum of squares=A-D between treatments sum of squares=B-D between rows sum of square= C-D residual sum of squares= (A-D)-[(B-D)+(C-D)]

SOURCE OF VARIATION

DEGREES OF FREEDOM(d.f)

SUM OF SQUARES(SS)

MEANS OF SQUARES(MS)

F VALUE

BETWEEN TREATMENTS

c-1 B-D B-D/c-1B-D/c-1/(A-B-[(B-D)+(C-D)]/(C-1)(r-1)

BETWEEN ROWS

r-1 C-D C-D/r-1 C-D/r-1/(A-B-[(B-D)+(C-D)]/(C-1)(r-1)

RESIDUAL (C-1)(r-1) (A-B-[(B-D)+(C-D)]

(A-B-[(B-D)+(C-D)]/(C-1)(r-1)

TOTAL Cr-1 A-D

A statistical design is a plan for the collection and analysis of data.

It mainly deals with the following parameters..

However the selection of an efficient design requires careful planning in advance of data collection and also analysis

A B

C

D A

A

B

B

C

C

D

D

CDA B

A

A

B

B

C

C

D

D

To eliminate bias To ensure independence among observations Required for valid significance tests and interval

estimates

Old New Old New Old New Old New

In each pair of plots, although replicated, the new variety is consistently assigned to the plot with the higher fertility level.

Low High

The repetition of a treatment in an experiment

A A

A

B

B

B

CC

C

D

D

D

Ex: If physicians wants to know whether a

particular drug which has been invented will be benificial in the treatment of particular disease

A farmer wants to know whether new type of fertilizer will give him better yields..he will frane his investigation interms of some suitable hypothesis.

There are many types of experimental designs… in which the most imp are as follows….

Complete randomized design(CRD)

Randomized complete block design(RCBD)

Latin square design(LSD)

DEPT OF PHARMACOLOGY

Where the treatments are assigned completetly at random so that each treatment unit has the same chance of receiving any one treatment.

This is suitable for only the expriment material is homogenous.(ex:laboratory experiments, green house studies etc.)

Not suitable for heterogenous study.(ex: field experiments)

DEPT OF PHARMACOLOGY

Advantages : Simple and easy Provides maximum number of degrees of freedom

Disadvantages: Only suitable for small number of treatments and for homogenous experimental material. Low precision if the plots are not uniform

A B

C

D A

A

B

B

C

C

D

D

Simplest and least restrictive Every plot is equally likely to be

assigned to any treatment

A A

A

B

B

B

CC

C

D

D

D

We have an experiment to test three varieties: the top line from Oregon, Washington, and Idaho to find which grows best in our area ----- t=3, r=4

1 2 3 4

5 6 7 8

9 10 11 12A

A

A

A

12156

Layout of CRD: The step by step procedures for randamization and

layout of a CRD are given for a field experiment with four treatments with five replications.

Determine the total number of experimental units (n) as the number of treatments and number of replications.

n=r×t→5×4=20 The entire experimental material is divided in to “n”

number of experiments. ex: five treatments with four replicatons . We

need 20 experimental units.the 20 units are numberd as follows……

DEPT OF PHARMACOLOGY

1 2 3 4 5

6 7 8 9 10

11 12 13. 14 15

16 17 18 19 20

Assign the treatments to the experimental units by 3 digit random numbers , selected from random number table.

The random numbers written in order and are ranked , however the lowest random number gives rank1, the highest rank allotted to large number. These ranks corresponds to unit number

Then the first set of r units are alloted to treatment T1 Then the next set of r units are alloted to treatment T2

Then the other set of r units T3 & so on…

random number rank treatment

937 17 149 02 908 15

T1

361 07 953 19 749 13 180 04

T2

951 18 953 19 749 13 180 04

T3

951 18 957 20 157 03 571 11

T4

226 05

Final layout:

DEPT OF PHARMACOLOGY

1T3

2T1

3T5

4T2

5T5

6T4

7T1

8T3

9T4

10T4

11T5

12T4

13T2

14T3

15T1

16T3

17T1

18T2

19T2

20T5

Analysis of variance: There are two sources of variation among

these observations obtained from a CRD trial.

1. Treatment variation 2. Experimental error The relative size of the two is used to

indicate whether the observed difference among the treatment is real or due to chance.

Calculations: 1. Correction factor(C.F)= (GT)2/n2. Total sum of squares(total ss)=total ss-c.f3. Treatment sum of squares(TSS)=TSS-cf4. Error sum of squares(ESS)=total ss – TSS

These results are summarized in the ANOVA table & the mean squares and F are calculated.

ANOVA table:

DEPT OF PHARMACOLOGY

Source of variation

df ss ms F

treatments t-1 TSS TMS=TSS/t-1

TMS/EMS

Error n-t ESS EMS=ESS/n-t

Total n-1 Total SS

Most widely used experimental designs in agricultural research.

The design also extensively used in the fields of biology, medical, social sciences and also business research.

Experimental material is grouped in to homogenous sub groups… the sub group is commonly termed as block.since each block will consists the entire set of treatments , a block is equivalent to a replication.

Ex: in field experiments , the soil fertility is an important character that influences crop responses.

Hence the treatments applied at random to relatively homogenous units with in each block and replicated over all the blocks, the design is known as a RBD.

divides the group of experimental units into n homogeneous groups of size t.

These homogeneous groups are called blocks. The treatments are then randomly assigned to the

experimental units in each block - one treatment to a unit in each block.

Advantages of RCBD: this design has been shown to be more efficient or accurate

than CRD for most of types of experimental work . The elimination of between SS from residual SS , usually results in a decrease of error of mean SS.

Flexibility is another advantage of RCBD. Large number of treatments can be included in this design.

Dis advantages of RCBD: not suitable for large number of treatments … because if the

block size is large it may be difficult to maintain homogenicity with in blocks. Consequently error will be increased.

Advantages& Disadvantages of RCBD:

Layout of RCBD:

let us consider that the experiment is to be conducted on 4 blocks of land, each having 5 plots. Now we take in to consideration five treatments , each replicated 4 times, we divide the whole experimental area in to 4 relatively homogenous blocks and each block into five plots or units. Treatments allocated at random to the units of a block .

A E B D C

E D C B A

C B A E D

A D E C B

PLOTS

BLOCKS

1 2 3 4 5

1 2 3 4

The Anova Table for a randomized Block Experiment

Source of variation

d.f S.S. M.S.S F

Treatments t-1 SST SST/t-1 SST/t-1/SSE/(t-1)(r-1)

Blocks r-1 SSB SSB/r-1 SSB/r-1/SSE/(t-1)(r-1)

Error (t-1)(r-1) SSE SSE/(t-1)(r-1)

Total rt-1 total SS

By comparing the variance ratio of treatments with the critical value of F we can find out if the different treatments are significantly differe

The conclusion will be irrespective of the difference on account of blocks.

Ex:

A Latin Square experiment is assumed to be a three-factor experiment.

The factors are rows, columns and treatments.

It is assumed that there is no interaction between rows, columns and treatments.

The degrees of freedom for the interactions is used to estimate error

differ from randomized complete block designs in that the experimental units are grouped in blocks in two different ways, that is, by rows and columns.

A requirement of the latin square is that the number of treatments, rows, and number of replications, columns, must be equal; therefore, the total number of experimental units must be a perfect square. For example, if there are 4 treatments, there must be 4 replicates, or 4 rows and 4 columns.

• .

Latin Square Designs Selected Latin Squares

3 x 3 4 x 4A B C A B C D A B C D A B C D A B C DB C A B A D C B C D A B D A C B A D CC A B C D B A C D A B C A D B C D A B

D C A B D A B C D C B A D C B A 

5 x 5 6 x 6A B C D E A B C D E FB A E C D B F D C A EC D A E B C D E F B AD E B A C D A F E C BE C D B A E C A B F D

F E B A D C

The layout LSD is shown below for an experiment with five treatments A,B.C,D,E . The 5×5 LSD plan given as follows.

Later on the process of randomization is done with the help of table of random numbers method. for this select 5 three digit random numbers.

A B C D E

B A E C D

C D A E B

D E B A C

E C D B A

Random numbers

sequence rank

628846475902452

1 2. 3 4 5

34251

Now use the rank to represent the existing row number of the selected plan and sequence to represents the row number of new plan.

However the third row of the selected plan (rank=3) becomes the firstrow(sequence=1)then so on.....

The column should be randomized in the same way by using the same procedure used for rearrangement… the five random numbers selected are as follows:

C D A E B

D E B A C

B A E C D

E C D B A

A B C D E

Random numbers sequence rank

792032947293196

1 2. 3 4 5

41532

However , the rank will now used to represent the column number of the plan obtained above and the sequence will be used to represent the column number of the final plan.

In this way ,the fourth column of the above plan becomes the first column of the final plan. In addition to this , the fifth column becomes third: third becomes fourth and seconds becomes fifth.the final plan which becomes the layout of the design , is as follows:

Row number

1 2 3 4 5

12345

EACBD

CDBEA

BCDAE

ABEDC

DEACB

ANALYSIS OF VARIANCE FOR LSD: C.F=(GT)2/n Total SS=∑X2-CF Row SS=1/n ∑R2-CF Column SS=1/n ∑C2-CF Treatment SS=1/n ∑T2-CF Error SS=Total SS-Row SS-ColumnSS-

Treatment SS

The Anova Table for a Latin Square Experiment

Source d.f. SS M.S. F

Treat n-1 TSS TMS TMS/EMS

Rows n-1 RSS RMS RMS/EMS

Cols n-1 CSS CMS CMS/EMS

Error (n-1)(n-2) ESS EMS

Total n2 - 1 Total SS

Controls more variation than CR or RCB designs because of 2-way stratification. Results in a smaller mean square for error.

Simple analysis of data Analysis is simple even with missing plots.

Advantages

Disadvantages

Number of treatments is limited to the number of replicates which seldom exceeds 10.If have less than 5 treatments, the df for controlling random variation is relatively large and the df for error is small.

Applications of biostatistics in pharmacy:  Public health, including epidemiology, health services research, nutrition,

environmental health and healthcare policy & management. Design and analysis of clinical trials in medicine Population genetics, and statistical genetics in order to link variation in

genotype with a variation in phenotype. This has been used in agriculture to improve crops and farm animals (animal breeding). In biomedical research, this work can assist in finding candidates for gene alleles that can cause or influence predisposition to disease in human genetics

Analysis of genomics data, for example from microarray or proteomics experiments.Often concerning diseases or disease stages.

Ecology, ecological forecasting Biological sequence analysis Systems biology for gene network inference or pathways analysis Statistical methods are beginning to be integrated into medical informatics,

public health informatics, bioinformatics and computational biology.

Applications of biostatistics in pharmacy:

Test whether the new treatments / new diagnostics / new vaccine works or not?

Ideally clinical trial should include all patients. Is it practically possible? No We test the new treatments / new diagnostics / new vaccine on a representative sample of the population

Statistics allows us to draw conclusions about the likely effect on the population using data from the sample

BUT ALWAYS REMEMBER…

Statistics can never PROVE or DISPROVE a hypothesis, it only suggests to accept or reject the hypothesis based on the available evidences

REFERENCESREFERENCES

 Hinkelmann and Kempthorne (2008, Volume 1, Section 6.6: Completely randomized design; Approximating the randomization test)

http://en.wikipedia.org/wiki/Analysis_of_variance

Montgomery (2001, Section 5-2: Introduction to factorial designs; The advantages of factorials)

http://www.slideshare.net/Medresearch/analysis-of-variance-ppt-powerpoint-presentation

http://www.synchronresearch.com/pdf_files/Application-Biostatistics-in-Trials.pdf

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