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• Many interesting questions in biology involve relationships between

response variables and one or more explanatory variables.

• Biology is complex, and typically, many potential variables, both those

measured and included in an analysis and those not measured, may

influence the response variable of interest.

• A statistical analysis may reveal an association between an

explanatory variable and the response variable.

• It is very difficult to attribute causal effects to observational variables,

because the true causal influence may affect both the response and

explanatory variable.

• However, properly designed experiments can reveal causes of

statistical associations.

• The key idea is to reduce the potential effects of other variables by

designing methods to gather data that reduce bias and sampling

variation.

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Mathematical methods used in the planning of the experiment

Kinds of mathematical methods

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In statistics, the most often used word is ‘variable’ which refers to a characteristic that contains the value, which may vary from one entity to another. The two most common types of variable are dependent variable and independent variable

• Variable – Any factor that can change in a scientific investigation or experiment

Independent variable

As its name suggests, an independent variable is one which remains unaffected by other variables. Alternately known as the predictor variable, explanatory variable, controlled variable

Dependent variable (responding variable)

A dependent variable is a consequence of an independent variable i.e. it is variable that measures the effect of independent variable on the test units. It is also known as the criterion or measured variable. It is something that the experimenter observes during an experiment and is influenced by the experiment. It is expected to change in response to some other factors.

Variable

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• The variable whose values are deliberately changed by the researcher in order to obtain the desired outcome is called independent variable. The variable, which changes its values in order to reciprocate change in the values of the independent variable is called dependent variable.
• The values of the independent variable can be changed as per requirement, by the researcher. Conversely, the value of independent variables is unchangeable.
• Manipulation can be done in the values of independent variable, but the researcher observes the value of a dependent variable during an experiment.
• An independent variable is a presumed cause whereas dependent variable is a measured effect.
• In a simple linear regression, ‘y’ denotes dependent variable while ‘x’ denotes independent variable, which means y depends on x.

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To simplify the communication a few different terms are introduced and defined

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The statistical procedure which separate or split the total variation into different components is known as “ANOVA”

ANOVA

“It is the technique of sorting out the total variation into some known and unknown component of variation from a given set of data”

R.A. Fisher

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It is a mathematical procedure of partitioning the total variation into various recognized source of variance

Recognized source of variance are replication, genotype, error and total.

Variance: The square of the standard deviation.

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ANOVA

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• random errors of observations have a normal distribution;
• factors affect only the change in mean values, and the

variance of observations remains constant;

• experiments are equivalent.

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In what follows we assume that the following suppositions are fulfilled:

• random errors of observations have a normal distribution;
• factors affect only the change in mean values, and the variance of observations remains constant;
• experiments are equivalent.

�Basic methodology

Single-factor analysis of variance

Single-factor analysis of variance

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We will assume that the result of any observation can be represented in the form of a model:

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where

μ –cumulative effect in all experiments

a_i - influence of the factor at the i-th level (i=1,2…,k)

ε_ij – Measurement error at the i-th level.

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The total number of experiments is N:

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ANOVA

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ANOVA

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ANOVA

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k(n-1)=N-k

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Latin Square Design

An order-n Latin square is an n × n square matrix in which n distinct symbols are placed into the n² entries of the square in such a way that every symbol occurs exactly once in each row and column of the matrix.

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When applying the Latin square, it is usually assumed that the effects of interaction between factors are insignificant. Then the experimental results can be represented as a linear model:

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where

μ –cumulative effect in all experiments

a_i - influence of the factor at the i-th level (i=1,2…,k)

βj - is the effect of the j row block,

and

γ_k is the effect of the k-th column block

ε_ij – Measurement error at the i-th level.

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ANOVA

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Latin Square Design

Latin Square of order 3

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The sequence of performance of actions with the use of ANOVA that has been adapted to Latin or Graeco-Latin squares

When conducting an analysis of variance of Latin square without repeated experiments convenient to use the following algorithm for calculating. To do this, are determining:

1. Results by the terms of A_i, columns B_j and Latin letters C_q.

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Sum of squares of all observations

The sum of the squared of overall results by the rows divided by the number of observations in the row

The sum of the squared of overall results by the columns divided by the number of observations in the column

The sum of the squared of overall results by Latin letters Divided by the number of observations corresponding to each letter

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The square of the total results divided by the number of all observations (correcting member)

Sum of squares for a row

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Sum of squares for a column

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The sum of the squares for the letter

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The total sum of squares equal to the difference between the sum of the squares of all observations and the correcting member

Residual sum of squares

The residual sum of squares consists of the variance due to the error of the experiment, and the variance due to the interaction of the factors

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Variance

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Error of variance

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Variance

Variance

Variance

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р – significance level

f₁, f₂ –number of freedom degrees , f₁=n-1; f₂=(n-1)(n-2),

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