Straight Line Model

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In many investigations, two or more variables are observed for each experimental unit in order to determine: 1. Whether the variables are related.
2. How strong the relationships appear to be.
3. Whether one variable of primary interest can be predicted from observations on the others. Regression analysis concerns the study of relationships between quantitative variables with the object of identifying, estimating, and validating the relationship. The estimated relationship can then be used to predict one variable from the value of the other variable(s). In this article, we introduce the subject with specific reference to the straight-line model. Here, we take the additional step of including the omnipresent random variation as an error term in the model. Then, on the basis of the model, we can test whether one variable actually influences the other. Further, we produce confidence interval answers when using the estimated straight line for prediction. The correlation coefficient is shown to measure the strength of the linear relationship. One may be curious about why the study of relationships of variables has been given the rather unusual name “regression.” Historically, the word regression was first used in its present technical context by a British scientist, Sir Francis Galton, who analyzed the heights of sons and the average heights of their parents. From his observations, Galton concluded that sons of very tall (short) parents were generally taller (shorter) than the average but not as tall (short) as their parents. This result was published in 1885 under the title “Regression Toward Mediocrity in Hereditary Stature.” In this context, “regression toward mediocrity” meant that the sons’ heights tended to revert toward the average rather than progress to more extremes. However, in the course of time, the word regression became synonymous with the statistical study of relation among variables. Studies of relation among variables...
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