Some basic formula in Statistics
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Transcript of Some basic formula in Statistics
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Formula related toSTAT310
2012
FORMULA AND OTHER PRACTICAL THINGS TO KNOWRAJU RIMAL
NORWEGIAN UNIVERSITY OF LIFE SCIENCES | s, Norway
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For tesng the dierences in
means:Hypothesis:
: :
Test Stasc:
1 + 1
Where,
1 + 1 + 2 The condence interval for the dierencebetween two treatment means is given as,
. . ,2 Fundamental Decomposion: + Where,
( ..) =
=
...
=
( . )== Tesng a two-way ANOVA modelThe model is given as,
+ + + + Where,
1,2, , For Factor A 1,2, , For Factor B 1,2, , Replication The esmate for the unknown factors , andresidual is given as,
Esmate .. . .
. ... .+
ANOVA Table
Source of Error .. Sum of SquareTotal 1
=
=
=
Factor A 1 1
. .
=
Factor B 1 1 .. = Interacon 1 1 1 .
=
=
Error 1
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Post-hoc Test
Mulple Tesng
While tesng the dierent levels of factors, if we
use the mulple paired t-test, we can have
problem of falsely rejecng hypothesis. For
instance, if a factor has 4 levels then we can
have 6 pairwise t-test, so that,
at least one false rejecon 1 1 0.05 0.264So that assumpons of independence will be
violated, this is adjusted by Tukeys post-hoc
test.
Tukeys HSD Test
Based on,
| | The two means are declared signicantly
dierent if,
| | > . .,
Where,
= Number of groups= Degree of Freedom.,= From TableThe condence interval or is given as,
. . ,
2 1
+ 1
ContrastContrast is dened as,
= , 0
= For example, while tesng the treatment totals,
the contrast can be constructed as,
.=
The variance of is,var =
For tesng the contrast hypothesis : 0,the test stasc is constructed as, . =
However is unknown it is replaced by .Alternave approach is to construct the -Stasc as,
. = ResidualThe residual is given by,
Standardize Residual
The standardize Residual is given as,
1
Outliers
With common rule of thumb the standardized
residual greater than 3 or smaller than -3 are
considered as outliers.
Normality
Normality is checked primarily with the graphs.
The scaered points in Residuals VS Fied graphshould be random and should not follow any
kind of paern.
Further, in Normal Q-Q plot (Theorecal
Quinles VS Standardized Residuals) the points
should lie close to the standard line. The points
that are far from the line are considered as an
outliers.
Condence interval for
The condence interval for is given as,
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[ , ]
Power of Test
The power of test is the probability of rejecngNull Hypothesis when it is false.
In any test, the possible outcomes are,
Accept Reject is true Correct1 Type error is false Type error Correct1 It is given as,
> ( > )Here,
Solving for
, we get,
( )
Paral F-TestFor tesng the eect of a factor in an
experiment, we reduce the model and compare
it with the full model. For reducing the model
for tesng a factor, it is removed along with all
of its interacons. The reduced model is then
ed. For Example, if we are tesng the eect
of factor C then the hypothesis is set as,
: 0The Test stasc is,
(Reduced Full) Full This is distributed with with and errordegree of freedom
. Where
is the
number of parameter in , i.e. the dierence
in degree of freedom of error for Full and
Reduced Model.
Lan Square DesignThis is special case with two or more factors
regarded as blocks and doesnt have enough
observaons to do completely randomized
block experiment.
In this design, each treatment level are tested
exactly once in each lock of the rst blocking
factor (Row) and exactly once in each block of
second blocking factor (Columns).
Example:
A B CB C AC A B
2Factorial DesignThe full factorial design with 3 factors is wrien
as,
+ + + + + + + + The eect and standard error of the eect,
Eect Contrast2 Sum of Square Contrast2
Before ng the full model, we can check
which factors and their interacons have
signicant eect on the model. This can also be
performed only with one replicaon using the
Normal Probability Plot.
In Normal Probability Plot, the negligible eects
tends to fall along the line and the signicant
eects will have non-zero mean and fall o the
lines.
Non-signicant eects are considered to be
removed from the model.
Fraconal Factorial Design
A Full design with factor requires 2 experiments per replicaon which will increase
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signicantly for large and only few degree offreedom are used in esmaon by main and
lower degree interacon eects.
For Instance, in
2 experiment,
64runs are
needed and the main eects use only 6 degreeof freedom and two factor interacon use 15.Other 42 degree of freedom are associated withhigher order interacon which might have
insignicant eect on response. If we can run
the fracon of full model experiment, it can
save a lot of work and cost of experiment.
Aliases
While running fraconal factorial design, some
factor are confounded with other. For instance,
if BC is confounded with A then when
esmang A, we are actually esmang A+BC.
Thus A and BC are aliases.
Design Resoluon
The resoluon of a design is equal to the
smallest number of eects in the dening
relaon.
Random Eect Model
When a factor in a model is considered asrandom then a restricon that it follows an
independent and idencal normal distribuon.
In the model,
+ + It is assumed that,
0, 0, The term
and
are called variance
components. Thus,
var() var( + + ) + Also, the covariance is given as,
cov( , ) cov( + + , + + ) cov , + cov , + cov( , ) + cov( , ) + 0 + 0 + 0 Thus, the correlaon between the two is givenas,
cor( , ) cov( , )var() var
+
Nested DesignThe design discussed so far are all cross-
seconal design. The design where the levels of
a factor is nested under the levels of another
factor is called Nested Design.
A Two stage Nested Model is,
+ + + Here, the factor with levels is nested underfactor with level . Interacon is not possibleunder Nested Design.
The Sum of square for a Nested Design from a
cross-seconal model with interacon can be
obtained as,
+ Expected Sum of Square
Cross-seconal Designs
Two Factor, both Fixed
+
1
() + 1
Sire 1
Herd 1
Cow 1
Cow 2
Herd 2
Cow 3
Cow 4
Sire 2
Herd 3
Cow 5
Cow 6
Herd 4
Cow 7
Cow 8
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() + 1 1 Two Factor, both random
+ + () + + () + Two Factor, one random
+ + 1 () +
()
+
Nested DesignThree Stage Nested- A and B Fixed C Random
+ + 1 () + + 1 () +
ANCOVAANCOVA is a combinaon of regression and a
linear model without covariate as independent
factor. An ANCOVA model might contain both
categorical and connuous variable in the samemodel.
From example, the weight of a person can
depend on the sex and height. The model
including these variables along with their
interacon is,
+ + + . + Here, 1 for female and 1 formale.
is the measurement for the height.
Then the separate regression line for female and
male can be obtained as,
For Female:
+ + + + For Male:
+ +