## Calculating Simple Linear Regression

Simple linear regression is a procedure that provides an estimate of the value of a dependent variable (outcome) based on the value of an independent variable (predictor). Knowing that estimate with some degree of accuracy, we can use regression analysis to predict the value of one variable if we know the value of the other variable (Cohen & Cohen, 1983). The regression equation is a mathematical expression of the influence that a predictor has on a dependent variable, based on some theoretical framework. For example, in Exercise 14, Figure 14-1 illustrates the linear relationship between gestational age and birth weight. As shown in the scatterplot, there is a strong positive relationship between the two variables. Advanced gestational ages predict higher birth weights.

A regression equation can be generated with a data set containing subjects’ x and y values. Once this equation is generated, it can be used to predict future subjects’ y values, given only their x values. In simple or bivariate regression, predictions are made in cases with two variables. The score on variable y (dependent variable, or outcome) is predicted from the same subject’s known score on variable x (independent variable, or predictor).

### Research Designs Appropriate for Simple Linear Regression

Research designs that may utilize simple linear regression include any associational design (Gliner et al., 2009). The variables involved in the design are attributional, meaning the variables are characteristics of the participant, such as health status, blood pressure, gender, diagnosis, or ethnicity. Regardless of the nature of variables, the dependent variable submitted to simple linear regression must be measured as continuous, at the interval or ratio level.

### Statistical Formula and Assumptions

Use of simple linear regression involves the following assumptions (Zar, 2010):

1. Normal distribution of the dependent (y) variable

2. Linear relationship between x and y

3. Independent observations

4. No (or little) multicollinearity

5. Homoscedasticity

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Data that are homoscedastic are evenly dispersed both above and below the regression line, which indicates a linear relationship on a scatterplot. Homoscedasticity reflects equal variance of both variables. In other words, for every value of x, the distribution of y values should have equal variability. If the data for the predictor and dependent variable are not homoscedastic, inferences made during significance testing could be invalid (Cohen & Cohen, 1983; Zar, 2010). Visual examples of homoscedasticity and heteroscedasticity are presented in Exercise 30.

In simple linear regression, the dependent variable is continuous, and the predictor can be any scale of measurement; however, if the predictor is nominal, it must be correctly coded. Once the data are ready, the parameters a and b are computed to obtain a regression equation. To understand the mathematical process, recall the algebraic equation for a straight line:

y=bx+a

where

y=the dependent variable(outcome)

x=the independent variable(predictor)

b=the slope of the line

a=y-intercept(the point where the regression line intersects the y-axis)

No single regression line can be used to predict with complete accuracy every y value from every x value. In fact, you could draw an infinite number of lines through the scattered paired values (Zar, 2010). However, the purpose of the regression equa­tion is to develop the line to allow the highest degree of prediction possible—the line of best fit. The procedure for developing the line of best fit is the method of least squares. The formulas for the beta (β) and slope (α) of the regression equation are computed as follows. Note that once the β is calculated, that value is inserted into the formula for α.

β=n∑xy−∑x∑yn∑x 2 −(∑x) 2

α=∑y−b∑xn