A Linear Regression Model

A Linear regression model with one explanatory variable is called a Simple linear regression, that is it involves 2 points: single predictor / explanatory variable and the response variable, which is the x and y coordinates in a Cartesian plane and finds a linear function a non-vertical straight line that, as precisely as possible it predicts the dependent variable values as a function of the independent variables.
The term simple refers to the fact that the response variable is related to one predictor. The regression model is given as Y=?0+?1 + ? and they are two parameters that are used estimate the slope of the line ?1 and the y- intercept of the line ?0. ? is the random error term.
Background

Regression analysis is a vital statistical method for the analysis of medical data. It makes it possible for the identification and characterization of relationships among multiple factors. It also enables the identification of prognostically relevant risk factors and the calculation of risk scores for individual prognostication, this was made possible by English scientist Sir Francis Galton (1822–1911), a cousin of Charles Darwin, made significant contributions to both genetics and psychology.
He is the one that came up with regression and a pioneer in using statistics to biology. In his study One of the data sets that he considered consisted was the heights of fathers and first sons. He wanted to find out whether he can predict the height of a son based on the father height. Looking at the scatterplots of these heights, Galton saw that the was relationship which was linear and increasing.
After fitting a line to these data using the statistical techniques, he observed that for fathers whose heights were taller than the average, the regression line predicted that taller fathers tended to have shorter sons and shorter fathers tended to have taller sons.
Purposes
Simple linear regression could be for example be purposefully when we Consider a relationship between weight Y (in kilograms) and height X(in centimeters), where the mean weight at a given height is ?(X) = 2X/4 – 45 for X > 100.
Because of biological variability, the weight will vary for example, it might be normally distributed with a fixed ? = 4. The difference between an observed weight and mean weight at a given height is referred to as the error for that weight. To discover the relationship which is linear, we could take the weight of three individuals at each height and apply linear regression to model the mean weight as a function of height using a straight line, ?(X) = ?0 + ?1X .
The most popular way to estimate the parameters, intercept ?0 and slope ?1 is the least squares estimator, which is derived by differentiating the regression with respect to ?0 and ?1 and solving, Let (xi , y i ) be the Ith pair of X and Y values. The least squares estimator, estimates ?0 and ?1 by minimizing the residual sum of squared errors, SSE = ?(y i – ? i)2, where y i are the observed value and ?i = b0 + b1xi are the estimated regression line points and are called the fitted, predicted or “hat” values.
The estimates are given by b0 =¯y – b1 ¯x and b1 = SSXX / SSYY, and where ¯Xand ¯Y are the means of samples X and Y, SSXX and SSYY being their standard deviation values and r = r(X,Y) being their Pearson correlation coefficient. It is also referred to as Pearson’s r, the Pearson product-moment correlation coefficient, is a measure of the linear between two variables X and Y Where X is the independent variable and Y being the Dependant variable as stated above.
The Pearson correlation coefficient, r can take a range of values from -1 to +1. A value of 0 suggests that there is no association between the two variables X and Y. A value greater than 0 indicates a positive association that is, as the value of one variable increases, so does the value of the other variable.
Before using simple linear regression analysis it is always vital to follow these few steps: Choose an independent variable that is likely to cause the change in the dependent variable Be certain that the past amounts for the independent variable occur in the exact same period as the amount of the dependent variable Plot the observations on a graph using the y-axis for the dependant variable and the x-axis for the independent variable review the plotted observations for a linear pattern and for any outliers keep in mind that there can be correlation without cause and effect.
Importances
Simple linear regression is considered to be extensively useful in many practical applications and methodologies. Simple linear regression functions by assuming that the variables x and y have a relationship which is linear within the given set of data. As assumptions are and results are interpreted, persons handling the analysing role in a such data will have to be more critical because it has been studied before that there are some variables which inhibit marginal changes to occur while others will not consider being held at a fixed point.
Although the concept of linear regression is one complex subject, it still remains to be one of the most vital statistical approaches being used till date. Simple linear regression is important because it has be wildly being used in many biological, behavioural , environmental as well as social sciences.
Because of its ability to describe possible relationships between identified variables independent and dependent , it has assisted the fields of epidemiology, finance, economics and trend line in describing significant data that proves to be of essence in the identified fields. More so, simple linear regression is important because it provides an idea of what needs to be anticipated, more specially in controlling and regulating functions involved on some disciplines.
Despite the complexity of simple linear aggression, it has proven to be adequately useful in many daily applications of life.
References

Fahrmeir L, Kneib T, Lang S. 2nd edition. Berlin, Heidelberg: Springer; 2009.
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Karl Pearson (20 June 1895) “Notes on regression and inheritance in the case of two parents,” Proceedings of the Royal Society of London, 58 : 240–242.{ https://en.wikipedia.org/wiki/Pearson_correlation_coefficient}
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Williams, M. N; Grajales, C. A. G; Kurkiewicz, D (2013). “Assumptions of multiple regression: Correcting two misconceptions”. Practical Assessment, Research ; Evaluation. 18 (11).{ https://en.wikipedia.org/wiki/Ordinary_least_squares}