Predictions based on regression analysis / Nooreha Husain - Husain, Nooreha (Universiti Teknologi MARA Cawangan Pahang, 1990)
The use of statistical methods is becoming increasingly important in all fields of everyday task. In fact it is quite impossible to name an activity which does not employ its own particular statistics as an aid to influencing human behaviour. But the usefulness of any statistical enquiry depends entirely on the competence of those who attempt to interpret the results they obtain. Tomorrow's high temperature will be between 75°F and 78°F; The results of ITM students based on their GPA is forecasted to increase between 10% and 20% during the next semester; About 4 million motorists will take to the Karak Highway during this holiday weekend. The three sets of statements above refer to something that might happen in the future it is not an estimate, because it has not yet happened. These three statements are forecasts or PREDICTIONS. Here, we will be concerned about the two questions: How do you prepare predictions? What do they mean? One of the statistical tools that can be use to make prediction is the linear regression analysis. The prediction can be done graphically by reading off from the graph the y-value corresponding to the given x value or by substituting the given x-value into the regression equation of y on x, that reveals the relationship between x and y; and calculating the value of y. (NOTE: This equation should not be used to predict x for a given value of y. If that kind of prediction is required the regression of x on y must be calculated and used). If in the case of the y-value being predicted for an x-value within the range of x values in the original data, this is called interpolation. The particular x-value used in this procedure is almost equal to the mean of the x-values in the original data and thus interpolation can be said to be a very respectable procedure which will lead to sensible predictions. If appropriate distributional assumptions hold concerning the data and we can proceed to find confidence limits on the regression parameter estimates β and β and on the predictions from the regression, the limits on an interpolated predition will be tight. In linear regression analysis, two types of estimates or predictions of values of the dependent variable are made. The first type of estimate involves predicting an individual value of the dependent variable Y. For example suppose we want to estimate the Grade-Point Average (GPA) of a particular student, based on a linear regression equation between the dependent variable y, the GPA and the independent variable x, the first exam score. The second is an estimate of a conditional mean, estimating the mean of the Y population for a specified X. Again the single estimated value is simply Y but an internal of reasonable values is different than if we are interested in a single Y value. For example, we might be interested in the final mean score of the population of students scoring 70% on the first examination. this is obviously different than being interested in the final score of a single student scoring 70% on the first examination, and the intervals are different. Another problem is to give intervals for the mean of all Y population simultaneously. This is equivalent to specifying a region in which the entire true regression line lies with reasonable assurance.