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Completely randomised design (CRD)
Model π¦ππ = ππ + πππ
i=1,2,β¦t, j= 1, 2,β¦r
Assumptions β’ Data are independent (study
design)
β’ The residuals are normally
distributed (check histogram,
normal probability plot).
β’ The residuals have equal
variances (boxplots, test of equal
variances)
Estimated
parameters
Standard
error
π = βπππΈ
95% CI
Contrasts
πΆ = β πππππ‘π=1 , β ππ = 0π‘
π
β’ ki indicates the coefficient of the contrast.
π = β πποΏ½Μ οΏ½π.π‘π=1 Var (c) = π2 β
ππ2
πππ
π‘ππππ‘ =πβπΆ
βπππΈΓβππ
2
πππ
, v= N-t (df of SSE)
Sum of squares: πππΆ =π2
βππ
2
πππ
β πππΆ = πππππ‘ππππ‘ ππ
Orthogonal contrast
Contrasts that convey independent information
If πΆ1 = π1π1 + π2π2 + β― + ππ‘ππ‘
πΆ2 = π1π1 + π2π2 + β― + ππ‘ππ‘
C & D are orthogonal if
Complete set of orthogonal contrasts
β’ t- 1 mutually orthogonal contrast
β’ Each pair of contrasts is orthogonal
Multiple comparisons
1. Bonferroni
β’ suitable for ad hoc comparisons
β’ small number of tests
πΌπ = ππ£πππππ π πππππππππππ πππ£ππ
πΌπ = πππππ£πππ’ππ =πΌπ
π, π = π‘(π‘ β 1)= no. of comparisons
95% CI:
π ππππππ2 =
(π1 β 1)π 12 + (π2 β 1)π 2
2
π1 + π2 β 2
2. Tukey
β’ suitable for balanced study
β’ less conservative
π»0: π1 = π2 = β― = ππ
k=t treatment, v = df of SSE
To find individual differences between means
β’ Find threshold value ππΌ,π,π£ Γ βπππΈ/π
β’ If |π¦π.Μ β π¦π.Μ Μ Μ | exceeds threshold value, then reject H0
3. Scheffe
β’ suitable for post hoc comparisons
β’ many tests
β’ can only conduct two-tailed tests as we are using F-
values.
π = β(π‘ β 1)πΉπΌ,π‘β1,πβπ‘
π π = βπππΈ Γ βππ
2
πππ
CI: cΒ±S Γ sc
If data is non-normal
β’ Transformation: log or sqrt
β’ Non-parametric test (Kruskal-Wallis)
Test the different between treatment medians
Assumption: all groups have similar shape
If variances are not equal
β’ Use Leveneβs test (F distribution) or Barlettβs test
(chi-square distribution) to see if variances are equal
Randomised complete block design (RBD)
β’ Suitable when we have homogenous groups
Model π¦ππ = π + ππ + ππ + πππ
i=1,2,β¦t, j= 1, 2,β¦r
Assumption
Parameter
constraint
β’ πππ~ππΌπ·(0, π2)ππππππππ‘ππ¦
β’
Variance
Relative
efficiency π πΈ =
(π1+1)(π2+3)π 22
(π1+3)(π2+1)π 12=
πΆπ π·
π π΅π·
If RE=2 => RBD is twice efficient. CRD
need a sample size 2 times greater to
achieve the same precision.
If assumptions fail:
β’ Transformation
β’ Non-parametric (Friedmanβs test)
Assumptions:
- Each block contains t random variables or
ranking
- The blocks are independent
- Within each block, observations can be arranged
in increasing order (not too many ties)
H0: Each ranking of the random variables within a
block is equally likely
H1: At least one treatment has larger observed values.
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R(yij): rank from 1 to t assigned to yij within block j
Friedman test statistics S (chi-square distribution)
Factorial experiment
β’ Examine multiple factors at the same time
β’ Examine interaction first
(1) Significant: Need to carry out multiple
comparisons on the levels of one factor at each
level of the other factor
(2) Insignificant: Remove interaction term.
Model π¦πππ = Β΅ + πΌπ + π½π + (πΌπ½)ππ + ππππ
i=1,2,β¦a, j= 1, 2,β¦b, k= 1, 2,..r
Assumptions β’ πππ~ππΌπ·(0, π2)ππππππππ‘ππ¦
β’
β’ Main effect
Main effect of A
Two-factor
interaction effect
(a2b2 -a1b2) β (a2b1 -a1b1)
Estimated
parameters
NOTE: Var(nX)=n2Var(X)
Var (X+Y)= Var(X) + Var(Y)
2n Factorial Design
πππ (οΏ½ΜοΏ½)=π2
π2πβ2 SST= π(.)2
2ππ
Yatesβ Algorithm
Partial confounding: Confounding different effects in each rep
Fractional replication: (1) find identity relation, (2) find the
effect subgroup, (3) Decide fractional replicate and its aliases
Random effects model
β’ Treatments which are drawn at random from a
population of treatments
Model π¦ππ = Β΅ + πΌπ + πππ
i=1,2,β¦t, j= 1, 2,β¦r
Assumption
β’ πππ~ππΌπ·(0, π2)ππππππππ‘ππ¦
β’ πΌπ~ππΌπ·(0, π2)ππππππππ‘ππ¦
Hypothesis π»0: Οa2 = 0 (no treatment effects)
π»1: Οa2 > 0 (there is treatment difference)
Estimated
parameters
The variance among X accounts for x% of
the variation and the variance within X
accounts for the other (1-x)%.
CI
Intraclass
correlation
coefficient
The analysis of covariance
β’ Examine one factor, but also take into account
extraneous (continuous) variables
β’ We can only measure covariate during the
experiment
β’ The influence of covariate on the response is
unknown
Model
π¦ππ:(π)πππ ππ‘β π π’πππππ‘ ππ ππ‘β π‘ππππ‘ππππ‘
Β΅: ππ£πππππ ππππ (π)
ππ: ππππππ‘ ππ π‘βπ ππ‘β π‘ππππ‘ππππ‘ ππ (π)
Ξ²: coefficient for the linear regression of yij on
xij
xij: covariate for jth subject in ith treatment
π₯..Μ : overall covariate mean
Assumption
Parameter
constraint
β’ πππ~ππΌπ·(0, π2)ππππππππ‘ππ¦
β’ β’ Common slope Ξ² (not significant)
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Adjusted
mean of Y π¦ππππΜ Μ Μ Μ Μ Μ = π¦π.Μ Μ Μ + οΏ½ΜοΏ½(π₯..Μ β π₯π.Μ Μ Μ )
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Survey design
Sampling frame: a list of sampling units
Sampling unit: non-overlapping units for sampling
Unit: A group of elements
Element: an object on which measures are taken
NOTE: unit can be element.
Probability concepts
πππ(π) = β (π¦π β π)2
ππππ=1 = E(Y2)-[E(Y)]2
Simple random sampling
β’ Sampling is done without replacement.
β’ Simpliest, appropriate with no prior information.
Sampling size
To estimate pop. total within D of true value
To estimate pop. proportion with a specified margin of error B
at siginifcance level Ξ±
If no p is given, use p=0.5 (conversative + large sample size)
Stratified random sampling
β’ Use to reduce variance
β’ Obtain best results when within-stratum differences
is small, and large differences between stratum
means.
Sample size
Design effect
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If deff <<1, then stratified random sampling is better than
SRS.
Cluster sampling
β’ Easy to implement
β’ Clusters are generally geographical entites
β’ Useful when there is a large within-cluster variation
but small between-cluster variation.
Systematic sampling
β’ Simple, save time and effort
β’ Useful when we do not have a list of population
β’ If the population is period, DO NOT USE.
Method A: When N/k is an integer, choose a unit at random
from the first kth unit. Take every kth unit from the starting
unit.
Method B: Choose a unit at random from the population.
Starting point depends on remainder.
When N/k is not an integer, use Method B to ensure an
unbiased estimator of Β΅
Ratio estimator
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Regression estimator
MSE=