Lecture 8 Quantitative Genetics (T 211) PDF

Quantitative Genetics (T 211) Lecture 8 Separate Variance components( Nested ANOVA) Dr. Shaimaa Ahmed [email protected]  A nested ANOVA is a type of ANOVA (“analysis of variance”) in which at least one...

Quantitative Genetics (T 211) Lecture 8 Separate Variance components( Nested ANOVA) Dr. Shaimaa Ahmed [email protected]  A nested ANOVA is a type of ANOVA (“analysis of variance”) in which at least one factor is nested inside another factor. For The example, suppose we would like to know if three different fertilizers produce different Definition of levels of plant growth. To test this, we have three different technicians sprinkle fertilizer Nested A on four plants each, another three technicians sprinkle fertilizer B on four plants each, and another three technicians analysis of sprinkle fertilizer C on four plants each. In this scenario, the response variable is plant variance growth, and the two factors are technician and fertilizer. It turns out that technician is nested within fertilizer:  The Definition of Nested analysis of variance When we perform a nested ANOVA (using statistical software like R, Excel, SPSS, etc.) , the output will be in the following format: Here’s how to interpret the output: Source: The source of the variance Sum of Squares: The sum of the squared deviations df: The degrees of freedom Mean Square: The mean square, calculated as Sum of We can look at the p-value column to determine whether each factor Squares / df has a statistically significant effect on plant growth. From the table F-Value: The F-value, calculated as Mean Square / Mean above, we can see that fertilizer has a statistically significant effect Square Residuals on plant growth (p-value <.05) but technician does not (p-value = p-value: The p-value that corresponds to the F-value 0.211). This tells us that if we would like to increase plant growth, we should focus on the fertilizer being used rather than the individual technician who is sprinkling the fertilizer. Sib Analysis of Nested ANOVA : S1 S2 S3 𝐷1 𝐷3 𝐷1 𝐷2 𝐷3 𝐷1 𝐷2 𝐷3 𝐷2 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----  How to Separate Variance components from the Nested ANOVA table? The following table that explain the component of analysis of variance for Nested ANOVA SOV df EMS between sires s-1 𝜎2e+n𝜎2D+nd𝜎2s Dams/sires s(d-1) 𝜎2e +n𝜎2D Within Dams s d(n-1) 𝜎2e Total s d n-1 ✓Separate Variance components (𝜎2s, 𝜎2D, 𝜎2e) 14-Dec-24 Quantitative Genetics (T 211) 5  Nested ANOVA Table SOV df MS EMS 2 2 Between Sires S–1 ------------ 𝜎𝑒2 +𝑂 𝜎𝐷 + 𝑂𝐷 𝜎𝑆 2 2 Dam ⊃ Sires S(D–1) ------------ 𝜎𝑒 +𝑂 𝜎𝐷 Offspring ⊃ Dam ⊃ Sires SD (O – 1) ------------ 𝜎𝑒2 ( Within/error ) (SDO – 1) Total N–1 Answer : Nested ANOVA Table : SOV df MS EMS S–1= Between Sires 3–1=2 30 𝜎𝑒2 + 𝑂 𝜎𝐷2 + 𝑂𝐷 𝜎𝑆2 S(D–1) 2 Dam ⊃ Sires 3*(3-1)= 6 20 𝜎𝑒2 +𝑂 𝜎𝐷 Offspring ⊃ SD (O – 1) 2 Dam ⊃ Sires 3*3(3-1) =18 10 𝜎𝑒 (Within/error) (SDO – 1) Total N-1=27-1=26  Separate Variance Components : 𝝈𝟐𝒆 = 10 𝝈𝟐𝒆 + 𝑶 𝝈𝟐𝑫 = 20 𝟏𝟎 + 𝟑 ∗ 𝝈𝟐𝑫 = 20 𝟐 𝟐𝟎 −𝟏𝟎 𝝈𝑫 = = 𝟑. 𝟑𝟑 𝟑 𝝈𝟐𝒆 + 𝑶 𝝈𝟐𝑫 + 𝑶𝑫 𝝈𝟐𝑺 = 30 𝟐𝟎 + 𝑶𝑫 𝝈𝟐𝑺 = 30 𝟐 𝟑𝟎 − 𝟐𝟎 𝝈𝑺 = = 𝟏. 𝟏 𝟑∗𝟑 Sib Analysis : Heritability from both parents and from each one 2 2 2 2 𝜎𝑆 + 𝜎𝐷 ℎ𝑆,𝐷 = 2 2 2 Heritability from both parents 𝜎𝑆 + 𝜎𝐷 + 𝜎𝑒 2 2 4 𝜎𝐷 ℎ𝐷 = 2 2 2 Heritability from mother 𝜎𝑆 + 𝜎𝐷 + 𝜎𝑒 2 2 4 𝜎𝑆 ℎ𝑆 = 2 2 2 Heritability from father 𝜎𝑆 + 𝜎𝐷 + 𝜎𝑒 Sib Analysis : Heritability from both parents and from each one 2 2 𝜎𝑆2 + 𝜎𝐷 2 ℎ𝑆,𝐷 = 𝜎𝑆2 + 𝜎𝐷 2 + 𝜎2 𝑒 2 2 1.1 + 3.3 ℎ𝑆,𝐷 = = 0.61 ( high heritability ) 1.1 + 3.3 + 10 Heritability from both parents Heritability from both parents and from each one : 2 2 4 𝜎𝐷 ℎ𝐷 = 𝜎𝑆2 + 𝜎𝐷 2 + 𝜎2 𝑒 2 4 ∗ 3.3 ℎ𝐷 = = 0.92 ( high heritability ) 1.1 + 3.3 + 10 Heritability from mother Heritability from both parents and from each one : 2 4 𝜎𝑆2 ℎ𝑆 = 𝜎𝑆2 + 𝜎𝐷 2 + 𝜎2 𝑒 2 4 ∗ 1.1 ℎ𝑆 = = 0.31 1.1 + 3.3 + 10 ( intermediate heritability ) Heritability from father  Example: In an S.O.V. df ss MS EMS Sires 4 63209 15802 𝛔 2e+n 𝛔 2D + nd 𝛔 2s nested Dams / sires 10 2 2 88113 8811 𝛔 e +n 𝛔 D 2 distribution of 5 Offspring/ dams / 30 165732 5524 𝛔 e sires 44 317054 sires, 3 dams/sire, Total 3 offspring’s /dams/ sires and 8811−5524 the following is 𝜎2D = = 1095 3 the ANOVA σ2e = 5524 table. 15802−8811 2 σ s= = 776 3x3  Sib analysis of Nested ANOVA  For estimating the best sample size to estimate h2 n= 2/ h2 bop , or n2 = 1−½h2 ½ h2 (1−h2) when h2 is calculated from bop- Relative Covariance Regression (b) or correlation (t) Offspring and one ½ VA b = ½ h2 parat VA b = h2 Offspring and mid ¼ VA r-I =t=¼ h2 parat ½ VA+¼ VD+ t ≥ ½ h2 Half sibs VEC Full sibs  Components of variance in sib analysis and Covariances: Observational component Covariance and estimated component Sires 𝜎2s= cov(HS) =¼VA Dams 𝜎2D= cov(FS) - cov(HS) =¼VA+¼VD+VEC Progeny 𝜎2W= Vp - cov(FS) =½VA+¾VD+VEW 𝜎2T= 𝜎2s+ 𝜎2D+ 𝜎2W= Vp Toral =VA+VD+VEC+VEW 𝜎2s+ 𝜎2D= cov(FS) Sires + Dams =½VA+¼VD+VEC  In an experiment included 3 males, 2 females for each male and every female produced 4 offspring. If you have the next data:- 1.Separate the components of variances. 2.What is the expected value of the heritability coefficient? Source of variance (S.O.V) MS EMS Between Sires 50 Between Dams within Sires 20 Error (within Dams within 10 Sires) Total  lectures of Prof. Dr. Alia Elsoudy. Faculty of Agriculture Ain Shams University https://www.statology.org/nested- anova/#:~:text=A%20nested%20ANOVA%20is%20a,terms%20are%2 0often%20used%20interchangeably. http://www.biostathandbook.com/nestedanova.html https://stats.stackexchange.com/questions/94468/nested-anova- References unequal-sample-sizes-variance-components https://www.jstor.org/stable/1270774 https://www.flutterbys.com.au/stats/tut/tut9.2a.html https://www.itl.nist.gov/div898/handbook/prc/section4/prc44.htm https://www.ibm.com/docs/it/spss-statistics/beta?topic=statistics- variance-components-estimation Question ?