### Regression Analysis

Regression analysis is a statistical tool for the investigation of relationships between variables. . In regression analysis data used to describe relationship between variables that are measured on interval scale. Regression allows you to predict variables based on another variable. In statistical modeling, regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (or 'predictors').Usually, the investigator seeks to ascertain the causal effect of one variable upon another the effect as price increase upon demand, The relationship between the two variables is estimated as a linear or straight line relationship, defined by the equation: Y = aX + b where b is the intercept or the constant and a, the slope. The line is mathematically calculated such that the sum of distances from each observation to the line is minimized. By definition, the slope indicates the change in Y as a result of a unit change in X. The straight line is also called the regression line or the fit line and a is referred to as the regression coefficient. The method of calculating the regression coefficient (the slope) is called ordinary least squares, or OLS. OLS estimates the slope by minimizing the sum of squared differences between each predicted (aX + b) and the actual value of Y. One reason for squaring these distances is to ensure that all distances are positive. A positive regression coefficient indicates a positive relationship between the variables; the fit line will be upward sloping). A negative regression coefficient indicates a negative relationship between the variables; the fit line will be downward sloping. The test of significance of the regression slope is a key test of hypothesis regression analysis that tells us whether the slope a is statistically different from 0. To determine whether the slope equals zero, a t-test is performed. As a general rule when observations on the scatter plot lie closely around the fit line, the regression line is more likely to be statistically significant in other words; it will be more likely that the two variables are – positively or negatively – related. Fortunately, SPSS calculates the slope the intercept standard error of the slope, and the level at which the slope is statically significant. The investigator also typically assesses the “statistical significance” of the estimated relationships, that is, the degree of confidence that the true relationship is close to the estimated relationship. Regression techniques have long been central to the field of economic statistics. Increasingly, they have become important to lawyers and legal policy makers as well.
Regression and correlation are actually fundamentally the same statistical procedure. The difference is essentially in what your purpose is - what you're trying to find out. In correlation we are generally looking at the strength of a relationship between two variables, X and Y, where in regression we are specifically concerned with how well we can predict Y from X. Examples of the use of regression in education research include defining and identifying under achievement or specific learning difficulties, for example by determining whether a pupil's reading attainment (Y) is at the level that would be predicted from an IQ test (X). Another example would be screening tests, perhaps to identify children 'at risk' of later educational failure so that they may receive additional support or be involved in 'early intervention' schemes. The chief advantage of regression analysis is that variables relationship is based on mathematical equation. This in turn allows more accurate prediction of one variable from knowledge of other variable which is the objective of regression analysis.

Regression analysis is used when two or more variables are thought to be systematically connected by a linear relationship. In simple regression, we have only two – let us designate them x and y and we suppose that they are related by an expression of the form y = b0 + b1 x + e. We’ll leave aside for a moment the nature of the variable e and focus on the x - y relationship. y = b0 + b1 x is the equation of a straight line; b0 is the intercept (or constant) and b1 is the x coefficient, which represents the slope of the straight line the equation describes. To be concrete, suppose we are talking about the relation between air temperature and the drying time of paint. We know from experience that as x (temperature) increases, y (drying time) decreases, and we might suppose that the relationship is linear. By measuring with a ruler, we could then determine the slope and intercept of this line.