Multiple Regression Model

The term multiple regression was used for the first time in the year 1908 by Pearson. Multiple regression is to learn in detail about the relationship between several predictor or independent variables and a dependent or criterion variable. If we take a real estate agent as example, he might record the size of the house (in square feet) for each listing, the number of bedrooms, the average income in the respective neighborhood according to census data, and a subjective rating of appeal of the house. Once all the information has been compiled for various houses, the measures taken relate to the price for which a house is sold. This so-called regression line, once has been determined, the analyst can now easily construct a graph of the expected.

The Regression Equation: A line in a two-variable or two dimensional space is defined by the equation Y=a+b*X; in full text: the Y variable can be expressed in terms of a constant ‘a’ and a slope ‘b’ times the X variable. The constant ‘a’ can also referred to as intercept, and the slope as regression coefficient or B coefficient. In the case of multivariate, when there is more than one independent variable, the regression line cannot be visualized in the two dimensional space, but can be computed easily. Generally, the multiple regression procedures will estimate the linear equation of the form:

Y = a + b1*X1 + b2*X2 + ... + bp*Xp

In the above equation, the regression coefficients (or B coefficients) represent the independent contributions of each independent variable to the prediction of the dependent variable. Another way to express this fact is to say that, the variable X1 is correlated with the Y variable, after controlling for all other independent variables.


Below is an example of a quadratic regression problem. The table below lists the total estimated numbers of Flu cases, by year of diagnosis from 1999 to 2003 in the Congo.

Year

AIDS Cases
1999

41,356
2000

41,267
2001

40,833
2002

41,289
2003

43,171



Let us approach with following step:

plot the data, letting x = 0 correspond to the year 1998,
find a quadratic function that models the data,
plot the function on the graph with the data and determine how well the graph fits the data,
use the model to predict the cumulative number of Flu cases for the year 2006.


First we will plot the data using a table. Since 1998 corresponds to x = 0, the year 1999 will represent x = 1, 2000 will represent x = 2, etc.

After entering the data into the table, graph the data. Your screen should look like the following:


Next we want to find a quadratic equation that best fits the data we have plotted. According to the Table, the equation is the following:

Between, if you have problem on these topics Percentage Error, please browse expert math related websites for more help on Quotient Rule Integration.

The function is the following:

Based on the graph and equation information listed above, it is clear that a quadratic is not a perfect function for representing this data. We know that R= 0.903486496, so . Remember that a graph is a perfect fit for data when . Based on the graph, our function is a fit for the given data.

Using our model  to predict the cumulative number of Flu cases for the year 2006, we find that we expect that there will be approximately 51,347 cumulative Flu cases diagnosed in the year 2006.