Simple Linear Regression



Simple Linear Regression

The Simple Linear Regression Model

[pic]

Example

← Let's consider Example 10.2, page 663.

← The following Minitab regression output has all of its essential features labeled.

← It is important that you can understand and interpret this output.

[pic]

Notes about the above output:

Interpretation of [pic]

← The intercept, [pic]= 1.1631, estimates the mean body density (y) when the log of the sum of the skinfold thicknesses (x) is zero. Which in this example, is of no practical use.

← The slope, [pic]= -0.0632, implies that for a unit increase in the log of skinfold thickness (i.e. unit increase in [pic]) there is an associated decrease (because [pic]is negative) in body density ([pic]) of 0.0632.

Residual Plots

← You should always check the assumptions of the model.

← If the assumptions are not valid then the linear model may be incorrect.

← The model assumes that the residuals (also referred to as deviations or errors) are normally distributed, with mean zero and standard deviation [pic].

← Figures 10.6, 10.7 and 10.8 (pp. 667, 668) examine the residuals and Normal probability plots for the body density problem.

← There are no extreme values or patterns appearing in the plots and the residuals appear to conform to a Normal distribution.

Confidence Intervals and Significance Tests about the slope [pic]

In this unit, we will not concern ourselves with inference for [pic]. Quite often, [pic]is of no practical use.

i) The degrees of freedom associated with CI's and significance testing for Linear Regression is n – 2. This is because we are estimating two parameters.

ii) For inference we use the value of the standard error of the estimated coefficient [pic]. This can be found from the computer output, and for our example [pic]is equal to 0.0041. We can confirm that the t-value (-15.41) found in the output is correct using the equation [pic]

iii) For [pic] the [pic]-value is 0.0. Therefore, there is very strong evidence to suggest that the simple linear regression model is useful for predicting body density.

Confidence Interval for a Mean Response

← Look at Figure 10.10 (p. 672). Notice how the confidence intervals widen as the value of [pic]is further from its mean.

← This is another reason to be cautious about extrapolating.

← Remember, in Chapter 2, we found that the linear equation is only valid within the range of observed [pic].

← Even if the relationship holds outside this range, we can see that the confidence intervals are so wide that the predicted value is of little use.

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