For Stanford writer only

lken82

ch._11.docx

Home >Business & Finance homework help >Accounting homework help >For Stanford writer only

Statistics for Business and Economics, Ch. 11

· Multiple Linear Regression The thread has 7 unread messages.

created by LOUIS DAILY

Last updated Mar 07, 2015, 10:40 AM

· Comment on Mar 02, 2015, 11:39 AM

Message collapsed. Message unread Multiple Linear Regression

posted by LOUIS DAILY at Mar 02, 2015, 11:39 AM

Last updated Mar 02, 2015, 11:39 AM

The simple Linear Regression equation is a relationship between one independent variable (X) and one dependent variable (Y). X is used to predict Y. But X might only help in predicting Y, not predict it perfectly. If we had other variables, say, X2 and X3, that also can independently contribute to the prediction, we might have a much better prediction equation. We might have something like: Y = slope X1 + slope X2 + slope X3 + intercept. This would be a multiple regression equation.

Why is this an improvement over a simple linear equation model?

How many variables do you think we would need to make a real good prediction?

What if the relationship between the three independent variables in this equation and the dependent variable is not linear at all, but a curved relationship (curvilinear)?

· Comment on Mar 04, 2015, 1:31 PM

Message collapsed. Message unread Re: Multiple Linear Regression

posted by STEPHANIE RECTOR at Mar 04, 2015, 1:31 PM

Last updated Mar 04, 2015, 1:31 PM

Why is this an improvement over a simple linear equation model?

How many variables do you think we would need to make a real good prediction?

What if the relationship between the three independent variables in this equation and the dependent variable is not linear at all, but a curved relationship (curvilinear)?

Realistically, some research requires the use of multiple variables because some issues include several influential factors. A simple linear equation model will not be able to include those other factors, so we must use a multiple regression model to determine a conclusion. On a side note, multiple regression will not adequately explain the relationship between independent and dependent variables if they are not linear. The steps in multiple regression are basically the same for simple regression. According to csulb.edu, "Regression with only one dependent and one independent variable normally requires a minimum of 30 observations. A good rule of thumb is to add at least an additional 10 observations for each additional independent variable added to the equation. The number of independent variables in the equation should be limited by two factors. First, the independent variables should be included in the equation only if they are based on the researcher's theory about what factors influence the dependent variable. Second, variables that do not contribute very much to explaining the variance in the dependent variable (i.e., to the total R2), should be eliminated" (csulb.edu, 2015). When the relationship between the three independent variables in this equation and the dependent variable is not linear at all, but a curved relationship (curvilinear) then the data points will increase together to a certain point. Then as one continues to increase, the other decreases or the other way around. You basically can see on a scatter plot a line rising to a peak, then declining.

Reference:

http://web.csulb.edu/~msaintg/ppa696/696regmx.htm

· Comment on Mar 05, 2015, 6:51 PM

Message collapsed. Message unread Re: Multiple Linear Regression

posted by LOUIS DAILY at Mar 05, 2015, 6:51 PM

Last updated Mar 05, 2015, 6:51 PM

Stephanie,

Yes, sometimes the relationship between variables is not linear, but curvilinear--we can fit a curve to the data.

thanks

Lou

· Comment on Mar 07, 2015, 10:40 AM

Message collapsed. Message unread Re: Multiple Linear Regression

posted by ERIK SEIDEL at Mar 07, 2015, 10:40 AM

Last updated Mar 07, 2015, 10:40 AM

An example of a curvilinear relationship that quickly came to mind when reading about this was predicting our company's administrative expense ratio for a new line of business. An administrative expense ratio is calculated by simply taking total administrative expenses (sales, customer services, overhead, etc.) as a percentage of total revenue. Our health insurance company has invested in recent years in going into new markets and lines of business. One thing I have learned is that it takes a certain critical mass of membership in order to gain sufficient economies of scale to lower the administrative expense ratio to the point where the line of business is profitable. The slope of this based on my experience is curvilinear. As we gain the first 5,000 to 10,000 members for a particular line of business, the curve moves downward significantly due to being able to spread overhead, marketing, and other fixed and start-up costs over a larger base of membership. After 10,000 members or so, the slope continues downward but at a much slower rate.

· Comment on Mar 06, 2015, 8:26 PM

Message collapsed. Message unread Re: Multiple Linear Regression

posted by CRYSTAL RAMOS at Mar 06, 2015, 8:26 PM

Last updated Mar 06, 2015, 8:26 PM

Multiple Linear Regression

Multiple linear regression attempts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable y. The population regression line for p explanatory variables x1, x2, ... , xp is defined to be y = 0 + 1x1 + 2x2 + ... + pxp. This line describes how the mean response y changes with the explanatory variables. The observed values for y vary about their means y and are assumed to have the same standard deviation . The fitted values b0, b1, ..., bp estimate the parameters 0, 1, ..., p of the population regression line.

Since the observed values for y vary about their means y, the multiple regression 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 + 1x1 + 2x2 + ... pxp. 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 .

Formally, the model for multiple linear regression, given n observations, is yi = 0 + 1xi1 + 2xi2 + ... pxip + i for i = 1,2, ... n.

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, b1, ... bp are usually computed by statistical software.

The values fit by the equation b0 + b1xi1 + ... + bpxip 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.

· Comment on Mar 06, 2015, 11:14 PM

Message collapsed. Message unread Re: Multiple Linear Regression

posted by Jynx Gresser at Mar 06, 2015, 11:14 PM

Last updated Mar 06, 2015, 11:14 PM

According to McClave, Benson, and Sincich (2011), "most practical applications of regression analysis employ models that are more complex than the simple straight-line model" (p. 625). Most probabilities involve one or more independent variables that affect the dependent variable (McClave et al., 2011). This is where multiple linear regression comes into the statistical equation. Princeton University Library (n. d.). found that the simple linear regression displays the relationship between the independent and dependent variable while the multiple linear regression predicts the relationship with multiple independent variables, for example, height, weight, and gender. The predictive value of the relationship between two or more variables can be increased and overall purpose of multiple regressions is to prove a relationship (Princeton University Library, n. d.). As the number of variables increases, the statistical probability of the relationship between the variables is strengthened. What is the best number of relationships to have? I couldn't find any specific numbers to answer this question. Curvilinear relationships can occur in multiple regressions especially when two or more variables stop having affect on one another. The predictive value of these variables starts to decrease; a good example would be friends and happiness (Princeton University Library, n. d.). Happiness can increase to a certain value of friends then start to decrease, thus making the line curvilinear.

References

McClave, J. T., Benson, P. G., & Sincich, T. (2011). Statistics for business and economics (11th ed.). Boston, MA: Prentice Hall. Retrieved from the University of Phoenix eBook Collection database.

Princeton University Library. (n. d.). Data and statistical services. Retrieved from http://dss.princeton.edu/online_help/analysis/regression_intro.htm

Jynx Gresser

· Comment on Mar 07, 2015, 9:29 AM

Message collapsed. Message unread Re: Multiple Linear Regression

posted by KIM DUNLAP at Mar 07, 2015, 9:29 AM

Last updated Mar 07, 2015, 9:29 AM

Hi Professor Lou,

Multiple Linear Regression models are important because in most research there is more than one variable x that affects the outcome y.

The multiple linear regression model allows us to take this into consideration mathematically and allows us to statistically prove the affects of multiple variables. Where is looks like it becomes very complicated is where you can have multiple pairs or options of pairs of variables that have to be ranked in the analysis. While I understand the need to do that and it would be helpful in relation to complete studies involving humans - I have to admit I feel it is nothing that I could accomplish on my own at this point in time.

I never did find any type of information that suggested there is an optimal number of variables to provide us with the best predictability.

Multiple Regression Models The thread has 2 unread messages.

created by BEAU KUSH

Last updated Mar 05, 2015, 6:49 PM

· Comment on Mar 04, 2015, 3:02 PM

Message collapsed. Message unread Multiple Regression Models

posted by BEAU KUSH at Mar 04, 2015, 3:02 PM

Last updated Mar 04, 2015, 3:02 PM

According to McClave, Benson, and Sincich (2011), most practical applications of regression analysis employ models that are more complex than the simple straight-line model. Moreover, the models typically utilize realistic probabilistic models which contain many variables that might be related to one of the factors being analyzed (McClave, Benson, and Sincich 2011). The probabilistic models that include more than one independent variable are called multiple regression models.

Utilizing the independent variables in these models helps researchers make accurate predictions about the outcomes or results. Essentially, researchers will utilize multiple regression models when there is a requirement to predict the value variables based on the value of two or more other variables. Multiple regression allows researchers to make logical predictions based on the data provided. This method saves time and provides a reliable method for obtaining accuracy.

Reference:

McClave, J. T., Benson, P. G., & Sincich, T. (2011). Statistics for Business and Economics (11th ed.). Boston, MA: Prentice Hall.

· Comment on Mar 05, 2015, 6:49 PM

Message collapsed. Message unread Re: Multiple Regression Models

posted by LOUIS DAILY at Mar 05, 2015, 6:49 PM

Last updated Mar 05, 2015, 6:49 PM

Beau,

Yes, multiple regression uses more than one independent variable, or predictor variables. This can lead to a much better predictive model than simple linear regression.

thanks

Lou