![C1 = sum of first column
R1 = sum of first row
N = sum of all rows
E(𝑶 𝟏𝟏)=
𝑹 𝟏 𝑪 𝟏
𝑵
, E(𝑶 𝟏𝟐)=
𝑹 𝟏 𝑪 𝟐
𝑵
, E(𝑶 𝟐𝟏)=
𝑹 𝟐 𝑪 𝟏
𝑵
E(𝑶 𝟏𝒏) = R1-[E(𝑶 𝟏𝟏)+ E(𝑶 𝟏𝟐)+…+E(𝑶 𝐧−𝟏)]
E(𝑶 𝐦𝟏) = C1-[E(𝑶 𝟏𝟏)+ E(𝑶 𝟐𝟏)+…+E(𝑶 𝐦−𝟏)]
d.f. = (row-1)(column-1) contd…](https://image.slidesharecdn.com/testofhypothesistestofsignificance-200724112309/85/Test-of-hypothesis-test-of-significance-18-320.jpg)
Test of hypothesis test of significance
- 1. Test of hypothesis / Test of significance Submitted by-: Dr. Jayesh Vyas (M.V.Sc.)
- 2. Test of Hypothesis/ Test of significance – Procedure which enable us to decide whether the accept or reject hypothesis. – The test used to ascertain whether the difference between estimator & parameter or between two estimator are real or due to chance are called test of hypothesis.
- 3. Terms used in testing a hypothesis – Null hypothesis(H0)-: Hypothesis which is tested for possible rejection, under the assumption that is true called H0. – Alternative hypothesis(H1)-: Hypothesis which is complementary to null hypothesis.
- 4. Type of Errors Accept H0 Reject H0 or accept H1 H0 is true Correct decision Type 1 error H0 is false Type 2 error Correct decision
- 5. Level of significance (LOS) – We want to minimize the size of both types of errors, however with fixed size testing procedure, both the errors can’t be minimize simultaneously. – So, we keep the size or probability of commonly type 1 error, fixed at certain level called level of significance.
- 6. – In biological experiments LOS usually employed 5% or 1%. – If LOS is chosen 5% that means probability of accepting a true hypothesis is 95%. – The LOS also called size of rejection region or size of critical region.
- 7. Degree of freedom (df) – It is no. of independent observation used in the making of statistics. – d.f. = total no. of observation (n) - no. of independent constraint (restriction) (k) – λ= n-k
- 8. T-test – Developed by W. S. Gossett. – It is parametric test. – It is small sample test. – Suppose 𝑥1, 𝑥2, … 𝑥 𝑛 be a random sample size 𝑛 drawn from population with mean µ and variance 𝜎2 , then T- statistics defined as – 𝐭 = 𝐗−𝛍 𝐒 𝐧 𝑋= sample mean, S= SD of sample, n= no. of observation
- 9. Applications of t-test 1. Test the significance of sample mean when population variance is unknown. 𝐭 = 𝐗−𝛍 𝐒 𝐧 SD= 𝐗− 𝐗 𝟐 𝐧−𝟏 d.f. =n-1
- 10. 2. Testing the significance of difference between mean of 2 samples / Unpaired t- test-:2 random samples are independent. 𝑥1, 𝑥2, … 𝑥 𝑛 be a random sample size 𝑛1 drawn from population with mean µ1 and variance 𝜎1 2 & another random sample 𝑦1, 𝑦2, … 𝑦𝑛 be a random sample size 𝑛2 with mean µ2 and variance 𝜎2 2 , then 𝒕 = |𝑿− 𝒀| 𝐒 𝟏 𝒏 𝟏 + 𝟏 𝒏 𝟐 S= X−X 2+ Y− 𝑌 2 𝑛1+𝑛2−2 d.f.=n1+n2-2 S= Pooled S.D.
- 11. 3. Paired t- test-: 2 random samples are dependent. (𝑥1 𝑦1), (𝑦2 𝑥2), … (𝑥n 𝑦𝑛) be a random sample size drawn from population n no. of pairs t = 𝑑 S n 𝑑= 𝑑 𝑛 d=x-y SD= d− 𝑑 2 n−1 d.f. =n-1
- 12. 4. Testing the significance of correlation coefficient-: (𝑥1 𝑦1), (𝑦2 𝑥2), … (𝑥n 𝑦𝑛) be a random sample size drawn from bivariate population n no. of pairs then 𝒕 = |𝒓| 𝑺.𝑬.(𝒓) r= correlation coefficient, S.E.= Stand. Error S.E= 𝒓 𝟐 𝒏−𝟐 d.f.=n-2
- 13. 5. Testing the significance of regression coefficient-: (𝑥1 𝑦1), (𝑦2 𝑥2), … (𝑥n 𝑦 𝑛) be a random sample size drawn from bivariate population n no. of pairs then 𝒕 = 𝒃 𝒀𝑿 𝑺.𝑬.(𝒃 𝒀𝑿) S.E.= 𝒀− 𝒀 𝟐−𝒃 𝒀𝑿 𝑿− 𝑿 𝒀− 𝒀 𝒏−𝟐 . 𝑿− 𝑿 𝟐 d.f.=n-2
- 14. Chi-square (𝜒2 )- test – It is non para-metric test. – Easy to compute and used without making assumptions, it is distribution free test. – Magnitude of difference between observed & expected frequency under certain assumption 𝝌 𝟐= 𝐎−𝑬 𝟐 𝑬 ≈ 𝝌 𝟐 𝒏−𝟏 𝒅𝒇 𝑂1, 𝑂2, … 𝑂𝑛 = observed frequency 𝐸1, 𝐸2, … 𝐸 𝑛 = expected frequency
- 15. Applications of 𝜒2 - test 1. Testing the significance of population variance-: 𝜎0 2 is known population variance and n is no. of sample size 𝝌 𝟐= 𝑿− 𝑿 𝟐 𝝈 𝟎 𝟐 d.f. =n-1
- 16. 2. Testing the goodness of fit-: 𝜒2= 𝑂−𝐸 2 𝐸 E= 𝑂 𝑛 d.f.=n-1
- 17. 3. Testing the independence of attribute/contingency test/test for independence-: m rows & n columns = m*n contingency table. A has m mutually exclusive categories 𝐴1, 𝐴2, … 𝐴 𝑚. B has n mutually exclusive categories 𝐵1, 𝐵2, … 𝐵 𝑛. contd... AB 𝑩 𝟏 𝑩 𝟐 𝑩 𝒏 Total 𝐴1 AB 𝑩 𝟏 𝑩 𝟐 𝑩 𝒏 Total 𝐴1 𝑂11 𝑂12 𝑂1𝑛 𝑅1 𝐴2 𝐴2 𝑂21 𝑂22 𝑂2n 𝑅2 𝐴m 𝐴m 𝑂m1 𝑂m2 𝑂mn 𝑅m Total Total 𝐶1 𝐶2 𝐶n N
- 18. C1 = sum of first column R1 = sum of first row N = sum of all rows E(𝑶 𝟏𝟏)= 𝑹 𝟏 𝑪 𝟏 𝑵 , E(𝑶 𝟏𝟐)= 𝑹 𝟏 𝑪 𝟐 𝑵 , E(𝑶 𝟐𝟏)= 𝑹 𝟐 𝑪 𝟏 𝑵 E(𝑶 𝟏𝒏) = R1-[E(𝑶 𝟏𝟏)+ E(𝑶 𝟏𝟐)+…+E(𝑶 𝐧−𝟏)] E(𝑶 𝐦𝟏) = C1-[E(𝑶 𝟏𝟏)+ E(𝑶 𝟐𝟏)+…+E(𝑶 𝐦−𝟏)] d.f. = (row-1)(column-1) contd…
- 19. O E O-E 𝑶 − 𝑬 𝟐 𝐎 − 𝑬 𝟐 𝑬 𝑂11 E(𝑂11) 𝑂12 E(𝑂12) 𝑂mn E(𝑂mn) Total (Value of 𝜒2 )
- 20. 4. Testing the independence of attribute in contingency table-: Only 2 categories = 2 rows*2 columns then Contd…. AB 𝑩 𝟏 𝑩 𝟐 Total 𝐴1 𝑂11(a) 𝑂12(b) 𝑅1 𝐴2 𝑂21(c) 𝑂12(d) 𝑅2 C1 C2 N
- 21. 𝜒2= 𝐚𝐝−𝒃𝒄 𝟐 𝑵 𝑅1 𝑅2 𝐶1 𝐶2 d.f. = (row-1)(column-1) d.f. = 1 always If any observed cell frequency is <5 then we used Yate’s correction 𝝌 𝟐 = |𝐚𝐝−𝒃𝒄|− 𝑵 𝟐 𝟐 𝑵 𝑹 𝟏 𝑹 𝟐 𝑪 𝟏 𝑪 𝟐
- 22. F-Test – The object of F-test is to find out whether the 2 independent estimates of population variance differs significantly. – It is a parametric test. – There are 2 degree of freedoms. 𝐹1 = 𝑺 𝟏 𝟐 𝑺 𝟐 𝟐 , 𝑺 𝟏 𝟐 = 𝑿− 𝑿 𝟐 𝒏 𝟏−𝟏 , 𝑺 𝟐 𝟐 = 𝒀− 𝒀 𝟐 𝒏 𝟐−𝟏
- 23. Applications of F- test 1. Testing of significance of ratio of 2 variances-: 𝐹1 = 𝑺 𝟏 𝟐 𝑺 𝟐 𝟐 d.f. = n1-1 for numerator d.f. = n2-2 for denominator
- 24. 2. Testing the homogeneity of several means-: Significance of difference amongst more than 2 sample means is carried out at the same time and this technique is known as analysis of variance (ANOVA). Contd…
- 25. ANOVA – When observations are classified on the basis of single criteria from K random samples. Sample no. Total 1 𝑌11 𝑌12 𝑌1𝑛1 𝑇1 2 𝑌21 𝑌22 𝑌2𝑛2 𝑇2 𝑘 𝑌𝑘𝑛1 𝑌𝑘𝑛2 𝑌𝑘𝑛 𝑘 𝑇k Total G
- 26. – Correction factor C.F.= 𝐺2 𝑛 , G= Grand total n = n.k – Total sum of sum of square TSS – TSS = 𝛴𝑌ⅈ𝑗 2 − 𝐶. 𝐹. , 𝛴𝑌ⅈ𝑗 2 = Sum of square – Sum of square due to assignable factor S.S.assign – S.S.assign = ( 𝑇1 2 𝑛1 + 𝑇2 2 𝑛2 +…+ 𝑇𝑘 2 𝑛k )- C.F. – Sum of square due to non assignable factor S.S.Error – S.S.Error= TSS- S.S.assign
- 27. – Preparation of ANOVA Table-: – d.f.= k-1 for numerator & n-k for denominator Source of variation d.f. S.S. M.S.S. F-value B/t assign 𝑘 − 1 S1 𝑠1 𝑘−1 = V1 𝑉1 𝑉2 = F-value Error 𝑛 − 𝑘 S2 𝑠1 𝑘−1 =V2 Total 𝑛 − 1 S