Example contd'

1
Example contd.

• Regression Model Interpretation( k independent
  variables) - AOV

Model yi ?0 xi ?i , ?i
N ID(0, s2) Partition Variation Due to Regn.
Variation About Regn. Total Variation
Explained Unexplained
(Error or Residual)
Source d.f. SSQ MSQ
FRegression k SSR MSR MSR/MSE
(again, upper tail test) Error n-k-1
SSE MSE Total n -1 SST -
- Note Here k independent variables.
If k 1, F-test ? t-test on n-k-1 dof.
2
Examples Different Designs What are the Mean
Squares Estimating /Testing?

• Factors, Type of Effects
• 1-Way Source dof
  MSQ EMS
• Between k groups k-1
  SSB /k-1 ?2 n??2
• Within groups
  k(n-1) SSW / k(n-1) ?2
• Total
  nk-1
• 2-Way-A,B AB Fixed Random
  Mixed
• EMS A ?2 nb?2 A ?2 n?2AB
  nb?2A ?2 n?2AB nb?2A
• EMS B ?2 na?2B ?2 n?2AB
  na?2B ?2 na?2B
• EMS AB ?2 n?2AB ?2 n?2AB
  ?2 n?2AB
• EMS Error ?2 ?2
  ?2
• Model here is
• Many-way

3
Nested Designs

• Model
• Design p Batches (A)
• ? ? ?
  ?
• Trays (B) 1 2 3 4
  .q
• Replicates ??? ??? .r per
  tray
• ANOVA skeleton dof EMS
• Between Batches p-1
  ?2r?2B rq?2A
• Between Trays p(q-1)
  ?2r?2B
• Within Batches
• Between replicates pq(r-1) ?2
• Within Trays
• Total pqr-1

4
Examples in Genomics and Trait Models

• Genetic traits may be controlled by No. of genes
  - usually unknown
• Taking this genetic effect as one
  genotypic term, a simple model
• 
  for
• where the yij is the trait value for
  genotype i in replication j, ? is the mean, Gi
  is the genetic effect for genotype i and ?ij the
  errors.
• If assume Normality (and interested in Random
  effects)
• and zero covariance between genetic effects
  and error
• and if the same genotype is replicated b
  times in an experiment, with phenotypic means
  used, the error variance is averaged over b.

5
Example - Trait Models contd.

• What about Environment and G?E interactions?
  Extension to Simple Model.
• ANOVA Table Randomized Blocks
• Source dof
  Expected MSQ
• Environment e-1
• Blocks (b-1)e
• Genotypes g-1
• G?E (g-1)(e-1)
• Error (b-1)(g-1)e

6
Example contd.

• HERITABILITY Ratio genotypic to phenotypic
  variance
• Depending on relationship among genotypes,
  interpretation of genotypic variance differs. May
  contain additive, dominance, epistatic
  interactions, variances (Above
  broad sense heritability).
• For some experimental or mating schemes, an
  additive genetic variance may be calculated.
  Narrow sense heritability then
• Again, if phenotypic means used, can obtain a
  mean-based heritability for b replications.

7
Extended Example- Two related traits

• Have
• where 1 and 2 denote traits, i the gene and j an
  individual in population.
• Then y is the trait value, ? overall mean, G
  genetic effect, ? random error.
• To quantify relationship between the two traits,
  the variance covariance matrices for phenotypic,
  ? p genetic ? g and environmental effects ?e
• So correlations between traits in terms of
  phenotypic, genetic and environmental effects
  are
• e.g. due to linkage of controlling genes, same
  gene controlling both traits

8
Linear (Regression) Models
RegressionSuppose modelling relationship
between markers and putative genes ENV 3
5 4 5 6 7 MARKER 0 1
2 3 4 5 Want straight line Y m X
c that best approximates the data. Best in
this case is the line minimising the sum of
squares of vertical deviations of points from the
lineSSQ S ( Yi - mXi c ) 2 Setting
partial derivatives of SSQ w.r.t. m and c to zero
? Normal Equations X.X X X.Y
Y Y.Y 0 0 0 3
9 1 1 5 5 25 4
2 8 4 16 9 3
15 5 25 16 4 24
6 36 25 5 35 7
49 55 15 87 30 160
Y
Yi
m Xi c
X
0
Xi
Env
10
5
0
5
Marker
9
Example contd.

• Model Assumptions - as for ANOVA (also a Linear
  Model)
• Calculations give
• X.X X X.Y Y Y.Y 0 0
  0 3 9 1 1 5
  5 25 4 2 8 4
  16 9 3 15 5 25
  16 4 24 6 36 25 5
  35 7 49 55 15 87
  30 160

So Normal equations are 30 15 m 6 c gt
150 75 m 30 c 87 55 m 15 c gt
174 110 m 30 cgt 24 35 m gt 30 15
(24 / 35) 6 c gt c 23/7
10
Yi
Y
Example contd.
Y
Y

• Thus the regression line of Y on X is Y
  (24/35) X (23/7)and to plot the line we need
  two points, so
• X 0 gt Y 23/7 and X 5 gt Y (24/35) 5
  23/7 47/7. It is easy to see that ( X, Y )
  satisfies the normal equations, so that the
  regression line of Y on X passes through the
  Centre of Gravity of the data. By expanding
  terms, we also get
• Total Sum ErrorSum Regression Sumof
  Squares of Squares of Squares SST
  SSE SSRX is the
  independent, Y the dependent variable and above
  can be represented in ANOVA table

X
11
LEAST SQUARES ESTIMATION in general

• Suppose want to find relationship between group
  of markers and phenotype of a trait
• Y is an N?1
  vector of observed trait values for
• N
  individuals in a mapping population, X is an N?k
  matrix of re-coded marker data, ? is a k?1 vector
  of unknown parameters and ? is an N?1 vector of
  residual errors, expectation 0.
• The Error SSQ is then
• all terms in matrix/vector form
• The Least Squares estimates of the unknown
  parameters ? is
• which minimises ?T? . Differentiating this
  SSQ w.r.t. ?s and setting 0 gives the normal
  equations

12
LSE in general contd.

• So
• so L.S.E.
• Hypothesis tests for parameters - use F-statistic
  - tests H0 ? 0 on k and N-k-1 dof (assuming
  Total SSQ corrected for the mean)
• Hypothesis tests for sub-sets of Xs, use
  F-statistic ratio between residual SSQ for the
  reduced model and the full model.
• 
  has N-k dof, so to test H0 ?i 0 use
• 
  with dimensions (k-1)? 1 and N ? (k-1) for ?
  
• 
  and X reduced, so SSEreduced has N-k1 dof
• 
  tests that subset of Xs adequate

13
Prediction, Tolerance, Residuals

• PredictionGiven value (s) of X(s), line (plane)
  substitute to predict Y
• Both point and interval estimates - C.I.
  for mean response line,
• prediction limits for new individual value
  (wider since Ynew? ? )
• General form same
• Tolerance - bands s.t. 95 of C.I. contain mean
• Residuals Observed - Fitted (or
  Expected) values
• Measures of goodness of fit, influence of
  outlying values of Y used to investigate
  assumptions underlying regression, e.g. through
  plots.

14
Correlation, Determination, Collinearity

• Coefficient of Determination r2 (or R2) (0? R2 ?
  1) proportion of total variation that is
  associated with the regression. (Goodness of Fit)
• r2 SSR/ SST 1 - SSE / SST
• Coefficient of correlation, r or R (0? R ? 1) is
  degree of association of X and Y (strength of
  linear relationship). Mathematically
• Suppose rXY close to 1, X is a function of Z and
  Y is a function of Z also. It does not follow
  that rXY makes sense, as relation with Z may be
  hidden. Recognising hidden dependencies
  (collinearity) between distributions is
  difficult. A high r between heart disease deaths
  now and No. of cigarettes consumed twenty years
  earlier does not establish a cause-and-effect
  relationship.

r 1
r 0