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# Linear regression calculator: the definition of regression

Have you ever wanted to know what is a linear regression calculator? Let us provide you with the simplest possible explanation of regression, which is one of the essential concepts in statistics. Consider two continuous variables x = (x1, x2,.., xn), y = (y1, y2,..., yn). We will place some points on a two-dimensional scatter plot and say that we have a linear relationship, in case the data can be approximated by a straight line. If we assume that y depends on x, and that any changes of y are caused by changes in x, we can determine the regression line (regression of y on x), which best describes the linear relationship between two variables. The use of the word regression in its statistical meaning is based on the phenomenon known as regression to the mean and is attributed to Sir Francis Galton (1889). It was him who showed that, although tall fathers tend to have taller sons, the average height of male children is less than that of their fathers. The average height of sons regressed and moved backwards to the average height of all the fathers in the population. Thus, on the average, tall fathers have shorter (but still comparatively tall) sons and short fathers have sons who are a bit taller (but still fairly short).

What exactly represents a regression line that is given as a result by a linear regression calculator? Here is the mathematical equation that evaluates a line of simple (double) linear regression: Y = a + bx, where the members are as follows:

• x is an independent variable, or a predictor;
• Y is a dependent variable or, in other words, a response variable. The latter is a value, which we expect for y to obtain (at the average), if we know the value of x, i.e. that is a predicted value of y;
• a is a free term (a crossing) of the line of assessment; it represents the value of Y, when x = 0;
• b is an angular coefficient or a gradient of the line of assessment; it represents the amount by which Y increases at the average, if we increase x by one unit;
• a and b are called regression coefficients of the line of assessment, although this term is often used for b only.

Any linear regression calculator allows you to expand simple linear regression in order to include more than one independent variable; in this case, it is known as multiple regression. How can the least square method be combined with a linear regression calculator? Whenever we perform regression analysis we use a sample of observations where a and b are sample estimates of true (general) parameters, namely α and β, which determine the line of linear regression in the population. Thus, the simplest method for determining the coefficients a and b is the method of least squares (OLS).

## Linear regression calculator: assessment of model fitting

As long as the logistic regression model is not related to any distributional assumptions, it is exclusively the assumptions of linearity and additivity that need to be obligatorily verified. Usually, this is done in course with the usual assumptions concerning the independence of observations and the necessity of inclusions of important covariables. An ordinary linear regression calculator does not feature global test for lack of model fit unless there are tools allowing you to replicate observations at various settings of x. Such peculiarity is caused by the fact that ordinary regression entails estimation of a separate variance parameter σ2. Specialists use global tests for goodness of fit for logistic regression. However, quite often the most frequently used methods are totally inappropriate. For instance, the simplest method for validating the consistency of data with the no-interaction linear model rarely requires using a linear regression calculator, but rather involves stratifying the sample by X1 and quantile groups (e.g., deciles) of X2. As a rule, the proportion of responses Pˆ within each stratum is computed and the log odds calculated from log[Pˆ/(1 − Pˆ)]. At the same time, the subgrouping method always requires relatively capacious samples and, predominantly, does not use continuous factors effectively. The number of quantile groups should be such that there are at least 20 (and perhaps many more) subjects in each X1 × X2 group. Otherwise, probabilities cannot be estimated precisely enough to allow trends to be seen above noise in the data. Since at least 3 X2 groups must be formed to allow assessment of linearity, the total sample size must be at least 2 × 3 × 20 = 120 for this method to work at all.

Model fitting is estimated by considering the remains (the vertical distance of each point on the line, for example, the residue amounts to the value of y observed minus the value of y predicted). The line of best fit is selected so that the sum of squared residuals is minimal – and we discuss the specifics of this operation in this part in detail. It is common to see a deviance test of goodness of fit based on the residual log likelihood, where P-values are obtained from a χ2 distribution with n − p d.f. This P-value is inappropriate since the deviance does not have an asymptotic χ2 distribution, due to the facts that the number of parameters estimated is increasing at the same rate as n and the expected cell frequencies are far below five (by definition).

### Linear regression calculator: linear regression assumptions, abnormal values (emissions) and impact points

So, for each observed value of x the residue is tantamount to the difference of y and the corresponding predicted value of y. Primarily, each residue can be positive or negative. You can use the residues to verify the following assumptions underlying the linear regression:

• Between x and y there is a linear relation, namely: for any pair (x; y) the data must be approximated by a straight line. Drawing the remains on a two-dimensional graph, we must observe a random scattering of points rather than any systematic pattern;
• The residues have the same variability (constant variance) for all of the predicted values of y: we must observe a random scattering of points calculating the residues of the predicted values of Y from y. If the graph of the residue scattering increases or decreases with the increasing of Y then the aforementioned assumption is not satisfied;
• The residues are normally distributed with zero mean.

If the assumptions of linearity, normality, and/or constant variance are questionable, we always can convert x or y and calculate a new regression line, for which these assumptions are satisfied (for example, we can use a logarithmic transformation or the like). Now, let us discuss the importance of abnormal values (emissions) and impact points for interpreting results of a linear regression calculator. An influential observation, if it is omitted, changes one or more estimates of the model parameters (i.e. an angular coefficient or an absolute term). An overshoot (in other words, an observation, which is the contrary to the majority of values in a dataset) can be an influential observation and may well be detected visually when viewed from a two-dimensional scatterplot graph or residues. Thus, both for overshoots and influential observations (points) specialists use models either with their inclusion or without them calculating regression coefficients with especial precision and scrupulousness. While conducting an analysis it is not necessary to discard overshoots or impact points automatically, because a simple ignoring may affect the results obtained. Using a linear regression calculator students should always study the causes of these emissions and analyze them. If the model contains two continuous predictors, they may both be expanded with spline functions in order either to test linearity or describe nonlinear relationships. Of course, testing interaction is more difficult here. For instance, if X1 is continuous, one might temporarily group X1 into quartile groups as it allows one to test whether a factor or set of factors is related to the response.

Finally, how to evaluate the quality of the linear regression and determination the coefficient R2 for an effective use of a linear regression calculator? Because of the linear relationship of x and y, we expect that y changes with the changes of x and, respectively, we call this a variation, which is caused or explained by the regression. The residual variation must be as small as possible. In case the latter requirement has been met, most of the variation of y will be explained by the regression, whereas the points will lie close to the regression line, i.e. line will fit the data well. The proportion of the total variance that is explained by the regression is called coefficient of determination, and it is usually expressed in terms of percentage and designate as R2. The latter is designated as the value r2 for simple linear regression (the square of correlation coefficient), and it allows you to subjectively evaluate the quality of the regression equation. The difference is the percentage of variance that cannot be explained by the regression. Unfortunately, there is no formal test for the evaluation, therefore, we have to rely on the subjective judgment in order to determine the quality of the fit of the regression line.

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