Normality test

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Normality test
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• Non-parametric tests
• Kolmogorov-Smirnov test (1-sample or 2-sample
  tests, normality tests)
• Mann-Whitney test (two samples)
• Wilcoxons signed rank test (paired samples)
• Kruskal-Wallis test (ANOVA multiple
  comparisons)
• Friedman test (RCBD)
• Spearmans Rank Correlation
• Kendalls Coefficients of concordance
  (association/agreements among many)

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• Kruskal-Wallis test (H-test)
• Similar to analysis of variance therefore, also
  called ANOVA by ranks
• Does multiple comparisons - more than two
  groups/samples
• Samples are not from normal distribution/populatio
  n or the variances are heterogeneous

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• Kruskal-Wallis test (Example)

H0 pH of four ponds are not different H1
Four ponds differ in pH Notes pH is not direct
numbers, dont calculate average directly.
Here, pond is treatment not the block
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• Kruskal-Wallis test (Example)

Total no. of data (N) 84 32, Number of
treatments (n) 4 Rank totals R1 (Pond 1)
123.53.58101017 55 Similarly, R2
132.5, R3 176.5, R4 164 H 12/N(N1)
?R2/n -3(N1) 12/(3233) 552132.52145216
42/4 3(321) 12.69 Correction factor (C)
1 - ?T /(N3-1) T- Tied group, check
no. of tied groups 3.5 (3,4), 6 (5,6,7), 10
(9,10,11), 13.5 (12,13,14,15) 20 (19,20,21),
23.5 (23,24), 26 (25,26,27), 31.5 (31,32) Sum of
tied groups (?T) ?(t3-t) (23-2)
(33-3)(33-3)(43-4) (33-3) (23-2) (33-3)
(23-2) 174
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• Kruskal-Wallis test (Example)

Correction factor (C) 1 - ?T /(N3-1)
1-174/(323-1) 0.995 Hc H/C 12.69/0.995
12.76 Check the table of Chi-square Tabulated
value (?2 0.05, 3 ) 7.815 Calculated
value is higher, therefore, Reject H0
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• Kruskal-Wallis test multiple comparisons

It means there is significant difference (Plt0.05)
in pH among the four ponds. But it does not
locate the differences which pairs differ? Need
to go for multiple comparisons Basic principle
A B / SE SEp v N(N1)/12 - ?T /12(N-1)
2/n v 32(33)/12174/(1231) 2/8
4.68
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• Kruskal-Wallis test (Example)

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Friedman test for RCBD
Note ranking within the block
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Friedman test for RCBD Treatment effect
12 SRi2 ? 2 3b(a1) ba
(a1) Where, a treatment, b block, ? 2
12/54(41) 12.52 102 102 182 9.52 -
35(41) 12.6 v a-1 4-1 3 ? 2 0.05,
3 7.815 (From table) Reject null hypothesis
(Plt0.05) which means weight gain of fish was
significantly different or diet has sig. effect
then Proceed to multiple comparisons
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Friedman test for RCBD
SE ba(a1) 12 5
x 4 x (41) / 12 2.89 From table, Q
0.05, 3 2.639 Therefore, critical difference
(d) 2.89 x 2.394 6.9 Multiple comparison
results
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Friedman test for RCBD for Block effect
Note ranking within the treatment
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Friedman test for RCBD for Block effect
12 SRi2 ? 2 3a(b1) ab
(b1) Where, a treatment, b block, ? 2
12/54(41) 12.521021029.52182-35(41)
17.46 v a-1 5-1 4 ? 2 0.05, 4 9.49
(From table) Reject null hypothesis (Plt0.05)
which means the block has significant effect,
then gt Proceed to multiple comparisons
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Friedman test for RCBD
SE ab(b1) 12 4
x 5 x (51) / 12 3.16 From table, Q
0.05, 4 2.639 Therefore, critical difference
(d) 3.16 x 2.639 8.3 Multiple comparison
results
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Miscellaneous topics
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Miscellaneous topics

• 1. Sample size
• Use formula n N / (1N x e2)
• Where,
• n sample size, NTotal population
• e ß error (normally 10)
• If you have 400 fish in a tank, then the number
  of fish to be sampled (n) 400 / (14000.102)
  80
• It means you have to calculate n for each
  experimental unit
• It is valid even if the population is very large
  maximum number comes 400 but ß should be
  considered 5.

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Miscellaneous topics

• 1. Sample size
• For more information
• Calculation of sample size (on-line)
• http//www.berrie.dds.nl/calcss.htm
• http//www.raosoft.com/samplesize.html
• http//www.tufts.edu/gdallal/SIZECALC.HTM

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• Miscellaneous topics
• Power of the statistical test 1- ?
• Where, ? is the probability of committing Type II
  error
• Therefore, power is the probability of detecting
  the significant difference, or correctly
  rejecting a false null hypothesis

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• Power of the tests and sample size
• One-Sample t-test
• Two-sample t-test

n s2 / d2 ( t?, df t ?, df )2
n 2sp2 / d2 ( t?, df t ?, df )2
Where, n no. of samples s standard deviation
of the sample (normally obtained from similar
studies d minimum detectable difference/meaningf
ul difference df degree of freedom t ?, df -
significance level (e.g. 0.05) t ?, df power of
the statistical test (e.g. 90)
Exercise in the lab session!
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• Power of one-sample t-test

Example, Find the power of the test if d1.0 g,
n12, and s2 1.5682 g2.
n s2 / d2 ( t?, df, t ?, df )2 Or t ?, df
d / vs2/n t ?, df 1 / v 1.5682/12 -
2.201 0.57 From the t table, 0.57
corresponds to about 0.25 which is the ?,
therefore, power 1- ? 1-0.25 0.75, which
means there is 75 chances of detecting the
significant difference.
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• Power of the two-sample t-test

n 2sp2 / d2 ( t?, df t ?, df )2
Sample size and power of ANOVA
F (k-1) (Treatment MS s2) nd2 /
2ks2 ks2
Where, F statistics based on which to see
probability from F table k no. of
treatments/factors d minimum detectable
difference s2 variance
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More information Non-parametric
tests http//www.statsoftinc.com/textbook/stnonpa
r.html http//www.math.niu.edu/NPAR/ Power and
sample size http//www.surveysystem.com/sscalc.ht
m http//www.health.ucalgary.ca/rollin/stats/ssiz
e/ http//www.dssresearch.com/toolkit/default.asp
http//calculators.stat.ucla.edu/powercalc/ http/
/www.dssresearch.com/toolkit/default.asp http//ww
w.raosoft.com/samplesize.html http//home.ubalt.ed
u/ntsbarsh/Business-stat/otherapplets/SampleSize.h
tm
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See you in lab session !

• Thank you!