rm anova.pptx

7/30/2019 RM ANOVA.pptx

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Analysis of Variance (ANOVA):

The essence is that the total amount of variation in aset of data is broken down into two types (1) amountattributed to chance (2) amount attributed to specifiedcauses.

Through ANOVA we can investigate any number offactors which are hypothesized or said to influence thedependent variable.

We make 2 estimates of population variance

onebased on variance between samples and one based onvariance within samples.

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Then the 2 estimates are compared with the F-test:

F = (Estimate of the population variance based on

variance between samples) (Estimate of

population variance based on variance within

samples)

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One way ANOVA:

1. Obtain mean of each sample i.e. X1 bar.Xk bar

2. Find X double bar

3. SS between = n1(X1 bar X dbl bar)2 + n2(X2 bar X

dbl bar)2 +.+

nk(Xk bar

X dbl bar)2

4. Mean Square (MS) between = SS between / (k-1)

5. SS within = (X1i X1 bar)2 + (X2i X2 bar)

2 + ..... +(Xki Xk bar)

2

6. Mean Square (MS) within = SS within / (n k)where n = total number of items in all samples

k = number of samples

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7. To check we can find the SS for the total variance by:

SS (for total variance) = (Xij X dbl bar)2

SS (for total variance) = SS between + SS within

The d.f. is (n-1) = (k -1) + (n k)

8. F-ratio = MS between / MS within



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a. This ratio is used to judge whether the difference

among several sample means is significant or is just amatter of sampling fluctuations.

b. If the calculated value of F < table value of F, thenthe difference is insignificant and the null hypothesis

of no difference between the means stands.c. If calculated F value > table F value then the

difference is significant and the samples did notcome from the same universe.

d. The higher the calculated F value above the table, themore definite one can be about the conclusion.



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Example 1:

Set up an analysis of variance table for the following per

acre production data for 3 varieties of wheat, each

grown on 4 plots and state if the variety differences

are significant:



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Plot OfLand

Per Acre Production Data

Variety of Wheat

A B C

1 6 5 52 7 5 43 3 3 34 8 7 4



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2 way ANOVA :

Used when data are classified on the basis of 2 factors

For example:

Agricultural output may depend on seeds and fertilizers

Sales data may be classified on the basis of salesperson

and geographic region



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Steps Involved in 2 way Anova:

1. Total all values in all samples i.e. find T

2. Calculate the correction factor T2/n

3. Calculate Total SS = X2ij (T2/n)

4. SS between columns5. SS between rows

6. SS for residual or error variance = Total SS (SSbetween columns + SS between rows)



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7. Degrees of Freedom:

For total variance = (c.r-1)

Variance between columns = (c-1)

Variance between rows = (r-1)

Residual variance = (c-1)(r-1)

Where:

c = number of columnsr= number of rows



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Source of

Variation

Sum of

squares (SS)

Degrees ofFreedom

(d.f.)

Mean Square

(MS)F-ratio

Betweencolumns

treatment

(T2j/nj)-T2/n

c-1SS between

columns /(c-1)

MS betweencolumns/MS

residual

Between rowstreatment

(T2i/ni)-T2/n

r-1SS betweenrows/(r-1)

MS betweenrows/MSresidual

Residual or

error

Total SS-(SS

col+SS row)

(c-1)(r-1)SS residual/(c-

1)(r-1)

Total X2ij-T2/n (c.r-1)



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Example 2: Per Acre Production Data of Wheat

Varieties ofseeds

A B C

Varieties ofFertilizers

W 6 5 5

X 7 5 4

Y 3 3 3

Z 8 7 4


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Source of

Variation

Sum of

squares (SS)

Degrees of

Freedom(d.f.)

Mean Square

(MS) F-ratio

Betweencolumns

treatment8 3-1=2 8/2=4 4/1=4

Between rowstreatment 18 4-1=3 18/3=6 6/1=6

Residual orerror 6

(3-1)(4-1)=6 6/6=1

Total 32(3x4)-

1=11http://sites.google.com/site/hvimcost 13


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5% F-Limit (or the table values)

F (2,6) = 5.14

F (3,6) = 4.76



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The previous example was a two-way design of

experiment without repeated values.

The next example we will see ANOVA of repeated

values.



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Example 3: Amount of blood pressure reduction in mmHg


Drug

X Y Z

Group ofPeople -A

14 10 1115 9 11

B12 7 10

11 8 11

C10 11 8

11 11 7


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Example 3 continued:

Test the 3 drugs to judge the effectiveness in reducingblood pressure by answering the following:

1. Do the drugs act differently?

2. Are the different groups of people affecteddifferently?

3. Is the interaction term significant?

Answer the above questions taking a significant level of5%.


rm anova.pptx

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