Multiple Regression Analysis Using SPSS Statistics_Abdullahi Abdikarim_18510195

Multiple Regression Analysis using SPSS Statistics

Name: Abdullahi Abdikarim

Nim: 18510195

Mata kuliah: Statistic

Dosen: M. Nanang Choirudidin,SE, MM

Introduction

Multiple regression is an extension of simple linear regression. It is used when we want to predict the value of a variable based on the value of two or more other variables. The variable we want to predict is called the dependent variable (or sometimes, the outcome, target or criterion variable). The variables we are using to predict the value of the dependent variable are called the independent variables (or sometimes, the predictor, explanatory or regressor variables).

For example, you could use multiple regression to understand whether exam performance can be predicted based on revision time, test anxiety, lecture attendance and gender. Alternately, you could use multiple regression to understand whether daily cigarette consumption can be predicted based on smoking duration, age when started smoking, smoker type, income and gender.

Multiple regression also allows you to determine the overall fit (variance explained) of the model and the relative contribution of each of the predictors to the total variance explained. For example, you might want to know how much of the variation in exam performance can be explained by revision time, test anxiety, lecture attendance and gender "as a whole", but also the "relative contribution" of each independent variable in explaining the variance.

Example

A health researcher wants to be able to predict "VO₂max", an indicator of fitness and health. Normally, to perform this procedure requires expensive laboratory equipment and necessitates that an individual exercise to their maximum (i.e., until they can longer continue exercising due to physical exhaustion). This can put off those individuals who are not very active/fit and those individuals who might be at higher risk of ill health (e.g., older unfit subjects). For these reasons, it has been desirable to find a way of predicting an individual's VO₂max based on attributes that can be measured more easily and cheaply. To this end, a researcher recruited 100 participants to perform a maximum VO₂max test, but also recorded their "age", "weight", "heart rate" and "gender". Heart rate is the average of the last 5 minutes of a 20 minute, much easier, lower workload cycling test. The researcher's goal is to be able to predict VO₂max based on these four attributes: age, weight, heart rate and gender.

Setup in SPSS Statistics

In SPSS Statistics, we created six variables: (1) VO₂max, which is the maximal aerobic capacity; (2) age, which is the participant's age; (3) weight, which is the participant's weight (technically, it is their 'mass'); (4) heart_rate, which is the participant's heart rate; (5) gender, which is the participant's gender; and (6) caseno, which is the case number. The caseno variable is used to make it easy for you to eliminate cases (e.g., "significant outliers", "high leverage points" and "highly influential points") that you have identified when checking for assumptions. In our enhanced multiple regression guide, we show you how to correctly enter data in SPSS Statistics to run a multiple regression when you are also checking for assumptions.

1. Click Analyze > Regression > Linear... on the main menu, as shown below:

                       

Published with written permission from SPSS Statistics, IBM Corporation.

Note:

Don't worry that you're selecting Analyze > Regression > Linear... on the main menu or that the dialogue boxes in the steps that follow have the title, Linear Regression. You have not made a mistake. You are in the correct place to carry out the multiple regression procedure. This is just the title that SPSS Statistics gives, even when running a multiple regression procedure.

2. You will be presented with the Linear Regression dialogue box below:

                                                

Published with written permission from SPSS Statistics, IBM Corporation.

3. Transfer the dependent variable, VO₂max, into the Dependent: box and the independent variables, age, weight, heart rate and gender into the Independent(s): box, using the ⤹ buttons, as shown below (all other boxes can be ignored):

                                                      

Published with written permission from SPSS Statistics, IBM Corporation.

Note: For a standard multiple regression you should ignore the Previous and Next buttons as they are for sequential (hierarchical) multiple regression. The Method: option needs to be kept at the default value, which is Enter. If, for whatever reason, Enter is not selected, you need to change Method: back to . The Enter method is the name given by SPSS Statistics to standard regression analysis.

4. Click on the Statistics button. You will be presented with the Linear Regression: Statistics dialogue box, as shown below:

                                       

Published with written permission from SPSS Statistics, IBM Corporation.

5. In addition to the options that are selected by default, select Confidence intervals in the –Regression Coefficients– area leaving the Level(%): option at "95". You will end up with the following screen:

                                                 

Published with written permission from SPSS Statistics, IBM Corporation.

6. Click on the Continue button. You will be returned to the Linear Regression dialogue box.
7. Click on the OK button. This will generate the output. 

Interpreting and Reporting the Output of Multiple Regression Analysis

SPSS Statistics will generate quite a few tables of output for a multiple regression analysis. In this section, we show you only the three main tables required to understand your results from the multiple regression procedure, assuming that no assumptions have been violated. A complete explanation of the output you have to interpret when checking your data for the eight assumptions required to carry out multiple regression is provided in our enhanced guide. This includes relevant scatterplots and partial regression plots, histogram (with superimposed normal curve), Normal P-P Plot and Normal Q-Q Plot, correlation coefficients and Tolerance/VIF values, casewise diagnostics and studentized deleted residuals.

However, in this "quick start" guide, we focus only on the three main tables you need to understand your multiple regression results, assuming that your data has already met the eight assumptions required for multiple regression to give you a valid result:

Determining how well the model fits

The first table of interest is the Model Summary table. This table provides the R, R², adjusted R², and the standard error of the estimate, which can be used to determine how well a regression model fits the data:

                                                

Published with written permission from SPSS Statistics, IBM Corporation.

The "R" column represents the value of R, the multiple correlation coefficient. R can be considered to be one measure of the quality of the prediction of the dependent variable; in this case, VO₂max. A value of 0.760, in this example, indicates a good level of prediction. 

The "R Square" column represents the R² value (also called the coefficient of determination), which is the proportion of variance in the dependent variable that can be explained by the independent variables (technically, it is the proportion of variation accounted for by the regression model above and beyond the mean model). You can see from our value of 0.577 that our independent variables explain 57.7% of the variability of our dependent variable, VO₂max. 

However, you also need to be able to interpret "Adjusted R Square" (adj. R²) to accurately report your data. We explain the reasons for this, as well as the output, in our enhanced multiple regression guide.

Statistical significance

The F-ratio in the ANOVA table (see below) tests whether the overall regression model is a good fit for the data. The table shows that the independent variables statistically significantly predict the dependent variable, F(4, 95) = 32.393, p < .0005 (i.e., the regression model is a good fit of the data).

                                

Published with written permission from SPSS Statistics, IBM Corporation.

Estimated model coefficients

       The general form of the equation to predict VO₂max from age, weight, heart rate, gender, is:

predicted VO₂max = 87.83 – (0.165 x age) – (0.385 x weight) – (0.118 x heart rate) + (13.208 x gender). This is obtained from the Coefficients table, as shown below:

                                   

Published with written permission from SPSS Statistics, IBM Corporation.

Unstandardized coefficients indicate how much the dependent variable varies with an independent variable when all other independent variables are held constant. Consider the effect of age in this example. The unstandardized coefficient, B₁, for age is equal to -0.165 (see Coefficients table). This means that for each one year increase in age, there is a decrease in VO₂max of 0.165 ml/min/kg.

Statistical significance of the independent variables

You can test for the statistical significance of each of the independent variables. This tests whether the unstandardized (or standardized) coefficients are equal to 0 (zero) in the population. If p < .05, you can conclude that the coefficients are statistically significantly different to 0 (zero). The t-value and corresponding p-value are located in the "t" and "Sig." columns, respectively, as highlighted below:

                        

Published with written permission from SPSS Statistics, IBM Corporation.

You can see from the "Sig." column that all independent variable coefficients are statistically significantly different from 0 (zero). Although the intercept, B₀, is tested for statistical significance, this is rarely an important or interesting finding.

Putting it all together

You could write up the results as follows:

• General

A multiple regression was run to predict VO₂max from gender, age, weight and heart rate. These variables statistically significantly predicted VO₂max, F(4, 95) = 32.393, p < .0005, R² = .577. All four variables added statistically significantly to the prediction, p < .05.

REFERENCE

Armstrong, J. S. (2011). Illusions in Regression Analysis. ScholarlyCommons. Retrieved from

http://repository.upenn.edu/marketing_papers/173Acces sed on 13 June 2018.

Dhakal, C.P. (2016). Optimizing multiple regression  model for rice production forecasting in

Nepal.(Doctoral thesis, Central Department of Statistics, Tribhuvan University Nepal).

Dion, P.A. (2008). Interpreting structural equation  modeling results: A reply to Martin and Cullen.  Journal of Business Ethics, 83(3), 365–368. Springer, Stable URL http://www.jstor.org/stable/25482382 Retrieved from  https://www.jstor.org/stable/25482382?seq=1#page_scan_tab_contents Accessed on 15 June 2018.

Example of interpreting and applying a multiple  regression model. (n.d). Retrieved from

http://psych.unl.edu/psycrs/statpage/full_eg.pdfAccessed on 11 June 2018.

Frost, J. (2017). How to interpret R-squared in  regression analysis. Retrieved

fromhttp://statisticsbyjim.com/regression/interpret-r-squared-regression/ Accessed on 02 June 2018.

Grace, B. J., &Bollen, A. K. (2005). Interpreting the results from multiple regression and stru

ctural equation models. Bulletin of the Ecological Society of America, 86(4), 283 – 295. ISSN:0012-9623, EISSN:2327-6096, doi: 10.1890/0012-9623(2005)86[283:ITRFMR]2.0.CO;2

Guthery, F.S., & Bingham, R. (2007). A primer on interpreting regressionmodels. Journal of Wildlife Management, 71(3) 684 – 692. ISSN:0022-541X, EISSN:1937-2817. The Wildlife Society doi:10.2193/2006-285

Interpreting regression output (Without all the statistics theory). (n.d). GraduateTutor.com. Retrieved from http://www.graduatetutor.com/statistics-tutor/interpreting-regression-output/ Accessed on 29 May 2018.

Klees, J. S. (2016). Inferences from regression analysis: Are they valid? University of Maryland, USA. Retrieved from http://www.paecon.net/PAEReview/issue74/Klees74.pdf Accessed on 13 June 2018

Martin, K. G. (2018). Interpreting Regression Coefficients. The Analysis Factor. Retrieved from https://www.theanalysisfactor.com/interpreting-regression-coefficients/Accessed on 13 June 2018.

McCabe, G.P. (1980). The interpretation of regression analysis results in sex and race discrimination problems. The American Statistician 34(4) 212-215. ISSN:0003-1305 EISSN:1537-2731, American Statistical Association,doi: 10.1080/00031305.1980.10483030.

Miler, J.E. (n.d). Interpreting the substantive significance of multivariable regression coefficients., Rutgers University. Retrieved from http://www.statlit.org/pdf/2008MillerASA.pdf Accessed on 13 June 2018. 

.