1. Theme 6. Linear regression
1. Introduction.
2. The equation of the line.
3. The least squares criterion.
4. Graphical representation.
5. Standardized regression coefficients.
6. The coefficient of determination.
7. Introduction to multiple regression.

2. Introduction
The establishment of a correlation between two variables is important, but this is considered a first step when predicting one variable from the other. (Or others, in the case of multiple regression; ie., multiple predictors.)
Of course, if we know that the variable X is closely related to Y, this means that we can predict Y from X. We are now in the field of prediction. (Obviously if X is unrelated to Y, X does not serve as predictor of Y.)
Note: We will use the terms "regression" and "prediction" as almost synonymous. (The reason for the use of the term "regression" is old, and has remained as such.)

3. Introduction
To simplify, we will focus (for simplicity) in the case that the relationship between X and Y is linear.
Performance
Of course, the issue now is how to get the "best" line that connects the dots. We need a criterion. While there are other criteria, the most commonly used, and we will see here, it is the least squares criterion. IQ
Least squares criterion: it minimizes the squared distances of the points on the line.

4. Review of the equation of a line
Y=A+BX
A is the intercept (This is where the short straight the Y axis)
B is the slope (observe that in the case of positive relationships, B is positive, in the case of the negative ratio, B is negative, if there is no relation, B will be approximately 0)
PErformance IQ
If we want to predict Y from X, we need to calculate (in the case of linear relationship) the regression of Y on (from) X.

5. Calculation of the linear regression equation (Y on X)
The least squares criterion gives us a value of A and one of B, such that Y’
Performance (Y)
Is minimal
IQ (X)

6. Calculation of the linear regression equation (Y on X)
IQ (X) Performance (Y)

7. Calculation of the linear regression equation (Y on X)
The least-squares line is:
Y’=-8’5+0’15X
Is minimal
It is 11.5 in this case
Important notes:
-Each Unit in IQ increases the grde in 0’15.
Although in this case, the following does not make sense, a person with IQ of 0, would be predicted a grade of -8.5

8. Calculation of the linear regression equation (Y on X)
Formulae... In direct scores
intercept
slope
Note: Both A and B can be easily obtained in any calculator with "LR" option (Linear Regression)

9. Calculation of the linear regression equation (Y on X)
then
Y’=-8’5+0’15X

10. Calculation of the linear regression equation (Y on X)
The formulas in differential scores
Notice that the average of X and Y will mean in typical score 0
intercept
IMPORTANT: B = b
That is, the slope differential scores is the same as in direct scores
slope
Therefore, the regression in differential scores is in our case:
y’=0’15x

11. Calculation of the linear regression equation (Y on X)
Formulas in standarized scores
As in differential scores
intercept
This is actually Pearson’s coefficient
slope
Therefore, the regression line in standard scores is in our case: zy’ =0’703zx

12. Calculation of the linear regression equation (Y on X)
OUTPUT from SPSS
Intercept and slow
(stand.scores)
Ord. y pendiente (punt.directas)
Note: in standardized scores, the slope matches Pearsons’ r coefficiente.

13. Calculation of the linear regression equation (Y on X)
We know that y
And from Theme 5
And also
therefore

14. Calculation of the linear regression equation (Y on X)
Therefore,
and

15. Prediction errors in the regression line of Y on X
Observed scores
Predicted scores
Prediction errors with the equation
The question now is how much variance is reduced by using the regression of Y on X (ie, having X as a predictor) compared to the case where we did not have the regression line

16. Prediction errors in the regression of Y on X
If we had no predictors, what could score for scores of Y?
In this case, since the least squares criterion, whether we are in Y and
  lack data on X, our best estimate of Y will be your average
Recall that the average minimizes the sum of the differences
quadratic
es minimal
If we use the average as a predictor, the variance of the predictions will be

17. Prediction errors in the regression of Y on X
But if we have a predictor X, the variance will be
This is the variance of Y not explained by X
It can be proven that
And as a result

18. How good is the prediction of the regression line
How good is the prediction of the regression line? The coefficient of determination as an index of the goodness of fit of our model (the regression line)
We just showed that
This is called coefficient of determination. It indicates how good the fit of the regression line (or in general linear model). It is bounded between 0 and 1.
If all the points in the scatterplot are on the line (with a slope different from 0), then it will be 0, and the coefficient of determination is 1

19. The coefficient of determination and the proportion of variance associated / explained / Common (2)
Let's start with a tautology
This expression indicates that the observed score for the ith subject is equal to the predicted score for said subject over a prediction error.
It can be shown that the predicted scores and the prediction errors are independent, so we can indicate the following
Total variance of Y
Variance of the predicted data in Y
Variance of the residuals (errors) when using the equation to predict Y

20. The coefficient of determination and the proportion of variance associated / explained / Common (2)
From the previous slide, we have
As we know that
therefore
In short, the coefficient of determination measures the proportion of the variance of Y that is associated / explained by the predictor X

21. Introduction to multiple linear regression (1)
We have seen the case of a predictor (X) and a variable predicted (Y), and obtained the regression of Y on X by the least squares method.
Given the nature of human behavior, in which each observed behavior can be influenced by different variables, We can have several predictors X1, X2, to predict Y (or if you prefer, several predictors, X2, X3, ...., to predict X1). This is the case of multiple regression.
So far we has
Dependent variable
But now we will have k predictors:
Predictors (independent) variabless

22. Introduction to multiple linear regression (2)
Recta regresión
Introduction to multiple linear regression (2)
It is important that you realize that the B2, B3, ..., weights are similar to those we saw in the case of the regression line.
For instance
Such coefficients represent how important predictor respective variable in the regression equation.
As was the case in the regression line (notice that the case of 1 predictor is a particular case of multiple regression), A represents where the multiple regression hyperplane short axis of the predicted variable.
For simplicity, the whole process is usually done by computer, so we will not see the formulas...

23. Introduction to multiple linear regression (3)
In raw scores, the regression equation is that we know
In differential scores, remember that A is 0 in the regression; the same applies in the regression equation. .
And applying the same logic, the value of the weights is the same as we had in direct scores
etcétera

24. Introduction to multiple linear regression (4)
Data(N=5)
Rendim Ansied Neurot
As in the case of one predictor

25. The general linear model
The general linear model underlies much of the statistical tests that are conducted in psychology and other social sciences.
To say a few:
Regression analyses (already seen)
Analysis of Variance (2nd semester will be)
T-Test (2nd term)
Analysis of Covariance
Analysis of clusters (cluster analysis)
-Factorial analysis
-Discriminant analysis ….

26. The general linear model (2)
Clearly, the regression analyses we have seen is a particular case of the general linear model or
Observed = Predicted + prediction error
In general terms

27. The general linear model
The general expression is
Y: Dependent Variable
X1, X2, ..., independent variables (predictors of Y)
e: random error
B1, B2, ..., they are the weights that determine the contribution of each independent variable.