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Lecture - 8Hypothesis formulation and testing

Contd..

Test for normalityNormal distribution

not normal distribution

- Non-parametric tests
- Distribution free tests
- Data are far from normal or data do not follow

any distribution pattern such as normal, linear,

binomial, exponential etc. e.g. no. of insects,

bacterial count, disease incidence, salary of

staff etc. - Few samples/replications can cause non-normality
- Problems in measurement e.g. GPA which do not

exactly measure the intelligence of students - Means, SD, SE, or variance do not represent the

data

- Why non-parametric tests?

- Observations are independent of each other
- Scale of measurement is rank
- Have low power than the parametric tests if

parametric tests are not applicable only, these

methods should be applied - These methods are recently becoming popular as

distribution free data are quite common

- Steps in non-parametric tests - Ranking
- Example 1
- Female heights (cm)193, 170, 188, 178, 183, 180,

185, - Male heights 175, 173, 163, 168, 165
- Methods
- Step 1
- Sorting by ascending
- or Descending order
- Step 2
- Ranking of the
- data from all
- the groups (it is the
- basic principle)

- Ranking
- Example 2
- Tied ranks
- There are two 32
- they get
- 3 and 4 ranks
- therefore,
- averaged rank
- is 3.5
- Similarly, three 44
- with ranks 8, 9 10,
- therefore they all
- get the mean rank i.e. 9

- Mann-Whitney test (U-test)
- Two groups (k2) i.e. similar to t-test for

non-normal distribution i.e. non-parametric - U n1n2 n1(n11)/2 R1
- Where,
- n1 is the number of samples in the first group
- n2 is the number of samples in the second group
- R1 is the sum of the ranks of the first group
- R2 is the sum of the ranks of the second group
- Here, assumption is n1 gt n2, but if n2 gt n1 then,

the equation should be - U n1n2 n2(n21)/2 R2

- Mann-Whitney test
- Example 1
- H0 Males females
- n1 7 and n2 5
- U 7578/2 30
- 33
- U 0.05, 5, 7
- 30 (From table)
- Reject H0

- Mann-Whitney test
- Example 2 Ordinal data
- H0 Males females
- n1 9 and n2 8
- U 98910/2 69.5
- 47.5
- U 0.05, 8, 9
- 57 (From table)
- Accept H0
- There is no difference
- between grades obtained
- by male female students

- Wilcoxons test (paired samples)
- Also called Rank Sum, Matched pair and

Signed Rank tests - Analogous to paired t-test (but low power)
- Example test whether the new breed of goat has

longer hind-legs compared to the forelegs.

Example H0 Hindleg foreleg Here, T

4.54.5779.5 79.52 51.5 T- 31 4 T

0.05, 10 8 (Table) And P lt0.05 Reject H0

(Hindleg is longer than foreleg)

Note if difference is zero, it is discarded

Analysis of Variance (ANOVA)

- (Parametric test)
- Two means are compared with t-test, if more than

two means need ANOVA - H0 there are no differences among the means
- Comparison depends on purpose and objective or

the experimental design

- Comparisons of five means

Means A B C D E

Freq.

Values

- Experimental designs
- Completely Randomized Design (CRD)
- Randomized Complete Block Design (RCBD)
- Latin Square Design (LSD)
- Factorial Design
- One factor
- Two factors
- Multi-factors

- Experimental designs
- 1. Completely Randomized Design (CRD)
- Assumptions
- all the experimental units are considered uniform

or identical - treatment allocation into experimental units is

completely random

Experimental units

- Hypothesis is tested by comparing the variation,

therefore, called as Analysis of Variance (ANOVA) - between treatments with the variation among

treatments - If variation between treatments (Treatment

effect) is higher than the variation within

treatment (i.e. Random error), there is a

significant difference - Model

Yi ? Ti Ri

Separation of variation

If Ti gt Ri treatment effect is significant

Yi Ri Ti ?

Random errors

Treatment effects

- Also called as
- single factor experiment
- For examples
- Fertilization trials
- - Organic, in-organic and combination
- - 0, 40 and 60 kg N/ha/week etc.
- Crop/vegetable/fruits varieties
- Animal breeds
- Drug efficacy etc.

- Randomization and layout
- Allocation of the treatments and replications is

done by lottery or using random numbers/table

- Determine the total number of experimental units

(n) t x r e.g. to test 6 varieties with 4

replications, you will need 24 plots - Assign plot number to each plot (1 to n)
- Assign treatments to the experimental plots by

using lottery or random table

- Randomization

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18 .? 24

Data analysis 1. Group the data by treatments

and calculate the treatment totals (T) and grand

total (G), the grand mean and the coefficient of

variation (c.v.) etc.2. Using number of

treatments (t) and the number of replications (r)

determine the degree of freedom (d.f.) for each

source of variation3. Construct an

outline/table (next slide) of the analysis of

variance

ANOVA table of a CRD experiment

t number of treatments r number of

replicates per treatment

- 4. Using Xi to represent the measurement of the

ith plot, Ti as the total of the ith treatment,

and n as the total number of experimental plots

i.e. n (r) (t) , calculate the correction

factor (CF) and the various sums of square (SS) - 5. Calculate the mean square (MS) for each source

of variation by dividing SS by their

corresponding d.f. - 6. Calculate the F- value (R.A. Fisher) for

testing significance of the treatment difference

(F MST/MSE) - 7. Enter all the values computed in the ANOVA

table

- 8. Obtain the tabular F values with f1

treatment d.f. (t-1) and f2 error d.f. t

(r-1) and compare as follows

Statistical inference

Example Four different feeds were tested on 20

pigs. Following were the mean final weights (kg)

of 19 pigs (1 pig died). Here, H0 ?1 ?2 ?3

?4

Step 1 Calculate sum squares Correction factor

(C) (Grand total)2 /n (1482.2)2 /19

115,627 Total SS (60.8) 2 (57.0)2 -------

(90.3)2 - C 119,982-115627

4,355 Treatment SS ? (Treatment total)2/n

C (303.1)2 /5 (346.5)2 /5 (401.4)2 /4

(431.2)2 /5 - 115,627 4,226 Error SS Total

SS Treatment SS 4,355 4,226 128

Step 2 Prepare an ANOVA table

Note Numerator df 3 Denominator d.f.

15 Reject H0 which means Treatment (feed) has

effect on pig growth but to compare among feeds

need test for Multiple comparisons

- If ANOVA shows significant difference, we need

posteriori test such as - 1. Comparison between two means e.g. control

verses others - - Students t-test (as before)
- 2. Multiple comparisons or pair-wise comparisons

(compare all the possible combinations

simultaneously or ranking is possible) - - LSD (Least significant difference)
- - DMRT (Duncans multiple range test)
- - Tukeys HSD (Tukeys Honestly Significant

Difference Test) - Note If ANOVA shows no significant difference

there multiple range test are not necessary

- 2. Multiple comparisons or pair-wise comparisons
- Calculate the common value for difference using

pooled variance such as - SE (X1-X2) v (S2 (1/ N1 1/N2)
- v 8.557 (1/51/5) 1.85 g
- t 0.05, 15 df 2.131, 95 CI 1.852.131

3.94 g

Reject H0 - all means are different Results ?1

lt ?2 lt ?4 lt ?3

- 2. Multiple comparisons or Post Hoc Test

Widely accepted

Not suggested

Homogeneous Subsets

Non-significant means are shown in the same

column.

Widely accepted

- 2. Final result presentation tabular
- Table no Mean weights of pigs fed with 4 diets

during the trial.

Values with the same superscripts are not

significantly different at 0.05

- 2. Final result presentation (Graphical)
- Figure no Mean weights (kg ? 95 confidence

intervals) of pigs fed with 4 diets during the

trial.

d

c

b

a

- CRD ANOVA vs Multiple range tests
- Adv.
- High proportion of degree of freedom thus it is

suitable for smaller experiments with fewer

experimental units. - It is stronger than multiple range tests

therefore it is done before multiple range tests - Disad.
- If experimental units are not homogenous, there

will be an increased experimental error - It doesnt compare among the means or does not

locate the differences

Some useful websites related to

ANOVA http//www.physics.csbsju.edu/stats/anova.

html http//www.psychstat.smsu.edu/introbook/sbk2

7.htm

- Thank you!