some basic formula in statistics

7/28/2019 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:

+ +

some basic formula in statistics

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