Tutorial Linear Model


Linear Model


   


Regression involves the study of equations. First we talk about some simple equations or linear models. The simplest mathematical model or equation is the equation of straight line.

Example: Suppose a shop keeper is selling pencils. He sells one pencil for 2 cents. Table as shown gives the number of pencils sold and the sale price of the pencils.

Number of pencils sold

Sales Prices (Cents)


Let us examine the two variables given in table. For the sake of our convenience, we can give some names to the variables given in the table. Let denote the number of pencils sold and ( for sale) denote the amount realized by selling pencils. Thus,

The information written above can be presented in some other forms as well. For example we can write an equation describing the above relation between and. It is very simple to write the equation. The algebraic equation connecting and is..
It is called mathematical equation or mathematical model in which depends upon. Here is called independent variable and is called dependent variable. Cent . Neither less than nor more than. The above model is called deterministic mathematical model because we can determine the value of without any error by putting the value of in the equation. The sale is said to be function of. This statement in symbolic form is written as: .
It is read as “ is function of”. It means that depends uponand only and no other element. The data in the table can be presented in the form of a graph as shown in the figure.

The main features of the graph in the figure are:

  1. The graph lies in the first quadrant because all the values of and are positive.
  2. It is an exact straight line. But all graphs are not in the form of a straight line. It could be some curve also.
  3. All the points (pair of and) lies on the straight line.
  4. The line passes through the origin.
  5. Take any point on the line and draw a perpendicular line which joins with the X-axis. Let us find the ratio . Here units and units. Thus units.

It is called the slope of the line and in general it is denoted by “”. The slope of the line is the same at all points on the line. The slope “” is equal to the change in for a unit change in. The relation is also called linear equation between and .

Example: Suppose a carpenter wants to make some wooden toys for the small children. He has purchased some wood and some other material for $. The cost of making each toy is $. Table gives the information about the number of toys made and cost of the toys.

Number of Toys

Cost of Toys

Let denote the number of toys and denote the cost of the toys. What is the algebraic relation between and. When, . This is called fixed or starting cost and it may be denoted by “”. For each additional toy, the cost is $. Thus and are connected through the following equation:
It is called equation of straight line. It is also mathematical model of deterministic nature. Let us make the graph of the data in given table. Figure as shown is the graph of the data in table.

Let us note some important features of the graph obtained in figure.

  1. The line does not pass through the origin. It passes through the point on Y-axis. The distance between and the origin is called the intercept and is usually denoted by “”.
  2. Take any point on the line and complete a triangle as shown in the figure. Let us find the ratio between the perpendicular and the base of this triangle. The ratio is, units.

This ratio is denoted by “” in the equation of straight line. Thus the equation of straight line has the intercept and slope. In general, when the values of intercept and slope are not known, we write the equation of straight line as. It is also called linear equation betweenand, and the relation between andis called linear. The equation may also be called exact linear model between and or simply linear model between and. The value of can be determined completely when is given. The relation is therefore, called the deterministic linear model between and. In statistics, when we shall use the term “Linear Model”, we shall not mean a mathematical model as described above.



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