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).
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 etal. 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
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y=thedependentvariable(outcome)
x=theindependentvariable(predictor)
b=theslopeoftheline
a=y-intercept(thepointwheretheregressionlineintersectsthey-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 possiblethe line of best fit. The procedure for developing the line of best fit is the method of least squares. The formulas for the beta () and slope () of the regression equation are computed as follows. Note that once the is calculated that value is inserted into the formula for .
=nxyxynx2(x)2
=ybxn
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