Since the observed values for y vary about their means y, the statistical model includes a term for this variation. In words, the model is expressed as DATA = FIT + RESIDUAL, where the "FIT" term represents the expression 0 + 1x. The "RESIDUAL" term represents the deviations of the observed values y from their means y, which are normally distributed with mean 0 and variance . The notation for the model deviations is .
In formal terms, the model for linear regression is the following:
Given n pairs of observations (x1, y1), (x2, y2), ... , (xn, yn), the observed response is yi = 0 + 1xi + i.
In the least-squares model, the best-fitting line for the observed data is calculated by
minimizing the sum of the squares of the
vertical deviations from each data point to the line (if a point lies on the fitted line exactly,
then its vertical deviation is 0). Because the deviations are first squared, then summed, there
are no cancellations between positive and negative values. The least-squares estimates
b0 and b1 are usually computed by statistical software. They
are expressed by the following equations:
The computed values for b0 and b1 are unbiased estimators of 0 and 1, and are normally distributed with standard deviations that may be estimated from the data.
The values fit by the equation b0 + b1xi are denoted i, and the residuals ei are equal to yi - i, the difference between the observed and fitted values. The sum of the residuals is equal to zero.
The variance ² may be estimated by s² =
, also known as the mean-squared error (or MSE).
The estimate of the standard error s is the square root of the MSE.
Using the MINITAB "REGRESS" command with "sugar" as an explanatory variable and "rating" as the dependent variable gives the following result:
Regression Analysis The regression equation is Rating = 59.3 - 2.40 SugarsA plot of the data with the regression line added is shown to the right:
After fitting the regression line, it is important to investigate the residuals to determine whether or not they appear to fit the assumption of a normal distribution. A plot of the residuals y - on the vertical axis with the corresponding explanatory values on the horizontal axis is shown to the left. The residuals do not seem to deviate from a random sample from a normal distribution in any systematic manner, so we may retain the assumption of normality.
The MINITAB output provides a great deal of information. Under the equation for the regression line, the output provides the least-squares estimate for the constant b0 and the slope b1. Since b1 is the coefficient of the explanatory variable "Sugars," it is listed under that name. The calculated standard deviations for the intercept and slope are provided in the second column.
Predictor Coef StDev T P Constant 59.284 1.948 30.43 0.000 Sugars -2.4008 0.2373 -10.12 0.000 S = 9.196 R-Sq = 57.7% R-Sq(adj) = 57.1%
The test statistic t is equal to b1/sb1,
the slope parameter estimate
divided by its standard deviation. This value follows a t(n-2) distribution.
In the example above, the slope parameter estimate is -2.4008 with standard deviation 0.2373. The test statistic is t = -2.4008/0.2373 = -10.12, provided in the "T" column of the MINITAB output. For a two-sided test, the probability of interest is 2P(T>|-10.12|) for the t(77-2) = t(75) distribution, which is an extremely small value. The "P" column of the MINITAB output provides the P-value associated with the two-sided test.
In the example above, a 95% confidence interval for the slope parameter 1 is computed to be (-2.4008 + 2.000*0.2373) = (-2.4008 - 0.4746, -2.4008 + 0.4746) = (-2.8754, -1.9262).
The value for "S" printed in the MINITAB output provides the estimate for the standard deviation , and the "R-Sq" value is the square of the correlation r written as a percentage value. This indicates the 57.7% of the variability in the cereal ratings may be explained by the "sugars" variable.
The MINITAB "BRIEF 3" command expands the output provided by the "REGRESS" command to include the observed values of x and y, the fitted values y, the standard deviation of the fitted values (StDev Fit), the residual values, and the standardized residual values. The table below shows this output for the first 10 observations.
Obs Sugars Rating Fit StDev Fit Residual St Resid 1 6.0 68.40 44.88 1.07 23.52 2.58R 2 8.0 33.98 40.08 1.08 -6.09 -0.67 3 5.0 59.43 47.28 1.14 12.15 1.33 4 0.0 93.70 59.28 1.95 34.42 3.83R 5 8.0 34.38 40.08 1.08 -5.69 -0.62 6 10.0 29.51 35.28 1.28 -5.77 -0.63 7 14.0 33.17 25.67 1.98 7.50 0.84 8 8.0 37.04 40.08 1.08 -3.04 -0.33 9 6.0 49.12 44.88 1.07 4.24 0.46 10 5.0 53.31 47.28 1.14 6.03 0.66To compute a confidence interval for the mean response of an observation, first choose a critical value from the appropriate t distribution. For a 95% confidence interval, the t(75) critical value is approximately 2.000. For the second observation in the table above, a 95% confidence interval for the mean response is computed to be (40.08 + 2.000*1.08) = (40.08 + 2.16) = (37.92, 42.24).
Note:The standard error associated with a prediction interval is larger than the standard deviation for the mean response, since the standard error for a predicted value must account for added variability.
The MINITAB "PREDICT" subcommand computes the predicted response variable and provides 95% confidence limits. Suppose we are interested in predicting the rating for a cereal with a sugar level of 5.5. MINITAB produces the following output:
Fit StDev Fit 95.0% CI 95.0% PI 46.08 1.10 ( 43.89, 48.27) ( 27.63, 64.53)The fitted value 46.08 is simply the value computed when 5.5 is substituted into the equation for the regression line: 59.28 - (5.5*2.40) = 59.28 - 13.20 = 46.08. The value given in the 95.0% CI column is the confidence interval for the mean response, while the value given in the 95.0% PI column is the prediction interval for a future observation.
For additional tests and a continuation of this example, see ANOVA for Regression.
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