# Simple linear regression - Mathematical methods in psychology

## Simple linear regression

Until now, in assessing the statistical relationship, we proceeded from the assumption that both variables under consideration are equal. In practical experimental research, it is important, however, to trace not only the relationship of the two variables to each other, but also how one of the variables affects the other.

Suppose we are interested in whether it is possible to predict the student's assessment in the exam based on the results of the mid-semester test. To do this, we will collect data reflecting the students' assessments obtained in the control work and in the exam. Possible data of this kind are presented in Table. 7.3. It is logical to assume that a student who is better prepared for the control work and received a higher rating, with other things being equal, has a better chance of getting a higher score in the exam. In fact, the correlation coefficient between X (evaluation by test) and Y (assessment in the exam) for this case it is quite large (0.55). However, he does not at all point out that the assessment in the exam is due to an assessment in the control work. In addition, he does not tell us at all about how much the evaluation should change in the examination with a corresponding change in the result of the control work. To assess how Y changes when changing X, to, say, one, linear regression.

Table 7.3

Ratings group of students in general psychology at the control (colloquium) and the exam

 Student Score on the test job ( X ) On the exam ( Y ) A. D. 3 3 B. C. 5 5 B. E. 3 3 F. E. 3 4 F. C. 5 5 F. D. 5 5 F. A. 4 5 E. To. 5 4 H. L. 3 5 To. Yu. 3 4 To. C. 4 5 To. T. 5 4 To. M. 4 4 To. In. 5 4 To. To. 5 4 L. N. 3 3 M. T. 5 4 M. Mr. 4 4 M. To. 3 3 H. T. 4 5 P. A. 3 2 P. And. 4 3 With. N. 5 4 With. A. 3 2 S.-SA 5 5 With. And. 5 4 With. About. 4 4 T. About. 5 4 C. L. 5 4 W. C. 5 5 W. L. 5 5 W. About. 4 3

The meaning of this method is as follows.

If the correlation coefficient between two series of estimates were equal to one, then the score in the exam would simply repeat the evaluation on the test paper. Suppose, however, that the units of measure that the teacher uses for the final and intermediate control of knowledge are different. For example, you can assess the level of current knowledge in the middle of the semester by the number of questions the student gave the correct answer. In this case, a simple correspondence of the estimates of ns will be satisfied. But in any case, there will be a correspondence for the 2-estimates. In other words, if the correlation coefficient between two series of data turns out to be equal to one, the following relation must be satisfied:

If the correlation coefficient turns out to be different from unity, then the expected value of z Y, which can be designated as , and the value of z X should be related by the following relationship, obtained using the methods of differential calculus:

By changing the g values ​​to the original X and Υ, we get the following relation:

Now it's easy to find the expected value of Υ:

(7.10)

Let

Then equation (7.10) can be rewritten as follows:

(7.11)

The coefficients A and In in equation (7.11) is the coefficients of the linear regression . The coefficient B shows the expected change in the dependent variable Y when changing the independent variable X by one unit. In the simple linear regression method, it is called slope. With respect to our data (see Table 7.3), the slope was 0.57. This means that the students who received an assessment for one ball above the test had an average of 0.57 points higher than the rest. The coefficient A in equation (7.11) is called the constant. It shows which expected value of the dependent variable corresponds to the zero value of the independent variable. With respect to our data, this parameter does not carry any semantic information. And this is a fairly common phenomenon in psychological and pedagogical studies.

It should be noted that in the regression analysis, independent X and associates Y variables have special names. For example, the independent variable is usually denoted by the term predictor, and dependent - test

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