Multiple Regression Analysis - essentiavitae.com

RHensonMultipleRegression 1 

Module 3 Assignment 3: Multiple Regression 

Robin Henson 

August 8, 2009 

NSG 6163 Health Outcomes 

Texas Woman’s University


Assign 3: Multiple Regression 

Use the sleep.sav data set and the multiple correlation/regression 

procedures that you have learned to use in SPSS to generate the statistics 

necessary to complete the following tasks. Place your answers in a word 

file and post by the Assignment Function under Module 3 Assignments by 

August 8, 2009- @2400hrs. 

PART 1 

Run a standard multiple regression procedure to explore how factors such 

as gender (sex), age (age), physical fitness (fitrate) and depression 

(depress) impact the level of daytime sleepiness (totSAS dep variable). 

Provide relevant computer output to support your check of the 

assumptions, outliers and multicollinearity. Present and interpret your 

findings. Copy and paste relevant output from your SPSS run into your 

word file. 

Descriptive Statistics: 

Sleepy & Associated sensations (totSAS) is ordinal data that is treated as continuous 

data. Scores 5=low and 50= extreme sleepiness. There are 251 valid cases and the 

mean score is 26 (SD 10.5). 

Gender (sex) is dichotomous data. 0=females and 1=males. The mean is .45 (SD .498) 

and contains 271 valid cases. There are slightly more female participants than males. 

Age (age) is continuous data. The mean age is 44 (SD 12.7) and contains 248 valid 

cases. 

Physical fitness (fitrate) is ordinal data that is treated as continuous data. Scores 

1=very poor and 10=very good. The mean score is 6.42 (SD 1.7) and contains 266 valid 

cases. These individuals perceive themselves to have an above average level of 

fitness. 

Depression (HADS Depression) is ordinal data that is treated as continuous data. 

Scores 0=no anxiety and 21= severe anxiety. The mean score is 3.5 (SD 2.99) and 

contains 269 valid cases. These individuals have lower average reported levels of 

anxiety. 

sleepy & assoc sensations 

scale 

Descriptive Statistics 

Mean Std. Deviation N 

26.04 10.520 251 

Sex .45 .498 271 

Age 43.87 12.684 248 

physical fitness 6.42 1.717 266 

HADS Depression 3.50 2.993 269


Pearson 

Correlation 

Sig. (1-tailed) 

N 

Sleepy & assoc 

sensations scale 

Correlations 

sleepy & 

assoc 

sensations 

scale sex age 

physical 

fitness 

HADS 

Depression 

1.000 -.199 -.141 -.267 .482 

Sex -.199 1.000 -.017 .110 -.071 

Age -.141 -.017 1.000 -.039 -.004 

physical fitness -.267 .110 -.039 1.000 -.314 

HADS Depression .482 -.071 -.004 -.314 1.000 



. .001 .017 .000 .000 

Sex .001 . .393 .037 .124 

Age .017 .393 . .271 .473 

physical fitness .000 .037 .271 . .000 

HADS Depression .000 .124 .473 .000 . 



251 251 230 247 249 

Sex 251 271 248 266 269 

Age 230 248 248 243 246 

physical fitness 247 266 243 266 265 

HADS Depression 249 269 246 265 269 

Model 

Variables Entered/Removed b 

Variables 

Entered 

Variables 

Removed 

Method


1 HADS 

Depression, 

age, sex, 

physical fitness a 

a. All requested variables entered. 

. Enter 

b. Dependent Variable: sleepy & assoc sensations 

scale 

Model R R Square 

Model Summary b 

Adjusted R 

Square 

Std. Error of the 

Estimate 

1 .541 a .293 .280 8.927 

a. Predictors: (Constant), HADS Depression, age, sex, physical 

fitness 

b. Dependent Variable: sleepy & assoc sensations scale 

ANOVA b 

Model Sum of Squares df Mean Square F Sig. 

1 Regression 7413.343 4 1853.336 23.258 .000 a 

Residual 17929.187 225 79.685 

Total 25342.530 229 

a. Predictors: (Constant), HADS Depression, age, sex, physical fitness 


Coefficients a 

Standar 

dized 

95.0% 

Unstandardized 

Coefficie 

Confidence 

Collinearity 

Coefficients 

nts 

Interval for B 

Correlations 

Statistics 

Model 

B 

Std. 

Error Beta t Sig. 

Lower 

Bound 

Upper 

Bound 

Zeroorder 

Partia 

l 

Part 

Tolera 

nce 

VIF 

1 (Constant) 32.211 3.481 9.255 .000 25.352 39.069


Sex -3.323 1.193 -.157 - 

2.786 

Age -.121 .047 -.146 - 

2.604 

.006 -5.673 -.973 -.199 -.183 -.156 .986 1.014 

.010 -.213 -.029 -.141 -.171 -.146 .998 1.002 

physical 

fitness 

-.731 .364 -.119 - 

2.008 

.046 -1.447 -.014 -.267 -.133 -.113 .892 1.121 

HADS 

Depressio 

n 

1.522 .208 .433 7.329 .000 1.113 1.932 .482 .439 .411 .900 1.111 

a. Dependent Variable: sleepy & assoc sensations scale 

Model 

1 

Dime 

nsion 

Eigenvalu 

e 

Condition 

Index 

Collinearity Diagnostics a 

(Constant 

) sex age 

Variance Proportions 

physical 

fitness 

HADS 

Depression 

1 4.027 1.000 .00 .02 .00 .00 .02 

2 .539 2.734 .00 .69 .00 .00 .21 

3 .343 3.424 .00 .28 .02 .03 .58 

4 .071 7.509 .00 .01 .62 .34 .03 

5 .020 14.235 .99 .00 .35 .63 .16 


Residuals Statistics a 

Minimum Maximum Mean Std. Deviation N 

Predicted Value 15.04 42.28 26.15 5.653 242 

Std. Predicted Value -1.933 2.853 .018 .993 242 

Standard Error of Predicted 

Value 

.816 2.269 1.281 .283 242 

Adjusted Predicted Value 15.04 41.95 26.27 5.577 225 

Residual -24.781 19.438 .178 8.839 225 

Std. Residual -2.776 2.178 .020 .990 225 

Stud. Residual -2.810 2.207 .020 1.002 225


Deleted Residual -25.393 19.980 .172 9.050 225 

Stud. Deleted Residual -2.854 2.226 .019 1.006 225 

Mahal. Distance .919 13.804 3.949 2.314 242 

Cook's Distance .000 .075 .005 .009 225 

Centered Leverage Value .004 .060 .017 .010 242 

a. Dependent Variable: sleepy & assoc sensations scale


State 2 applicable research questions (Hint: use questions on page 151 of 

SPSS survival manual as an example) 

1. How well do the variables gender, age, fitness, and depression predict totSAS? 

How much variance in totSAS can be explained by scores on these 

variables/scales? 

2. Which is the best predictor in totSAS: gender, age, fitness, or depression? 

Were the assumptions met? Yes, and are described below: 

A. Sample Size: Based on the equation N>50 + 8 (# of independent variables), 

this sample size is adequate (248> 50 + 8(4)= 248> 82). The assumption for 

sample size is not violated. 

B. Multicollinearity: 

Correlation of independent variables- The independent variables sex (r= - 

.199), age (r=-.141), fitrate (r=-.267) have a small negative correlation with 

totSAS. The independent variable, HADS Depression has a moderate positive 

correlation with totSAS. All 4 variables are


There is an apparent linear relationship between the independent variables and totSAS. 

There is no apparent curvature. This supports the proposition that there is no 

violation of linearity. 

G. Outliers 

Scatterplot-On the scatterplot there are no potential outliers. All cases fall within 

-3.3 to 3.3. 

Mahalonobis Distances (MAH-1)-Outliers can also be checked through the use 

of MAH-1. The critical chi-square value for 4 independent variables is 18.47. The 

maximum value for this data file is 13.804, which does not exceed the critical value. If 

the MAH-1 exceeded this critical value, the researcher would need to investigate the 

cases of concern on SPSS by looking up the MAH-1 value for each potential outlier. If 

the case(s) in question exceed the MAH-1 number, the researcher would need to 

consider if that particular case (cases) would need to be removed. In this instance, no 

cases should be considered for removal. There is no major violation of the 

assumption of multicollinearity. 

Cook’s Distance-The maximum value for Cook’s Distance is .086. Since it is 1, the SPSS data file 

must be checked for all cases that have a COO_1 value >1. The researcher would need 

to consider removing high value cases. 

Evaluating the model: 

29% (R square=.293) of the variance in totSAS is explained by the model (gender, age, 

physical fitness, and depression). The adjusted R square (.280) corrects the value of R 2 

when the study sample is small, which is not a problem in this case because the sample 

size is adequate. The ANOVA table indicates that this model reaches statistical 

significance [F(4,225)= 23.268, p


explaining totSAS (p


Run a hierarchical multiple regression procedure using the same variables 

you used in regression number one. Control for gender and age then 

examine the effects of physical fitness and depression on daytime 

sleepiness. Present and interpret your findings. Copy and paste relevant 

computer output that displays results and provide a narrative summary of 

the output. 

State your research question. 

Research question: 

1. If we control for the possible effect of gender and age, is our set of variables 

(physical fitness and depression) still able to predict a significant amount of the 

variance in perceived stress? 

sleepy & assoc sensations 

scale 

Descriptive Statistics 

Mean Std. Deviation N 

26.04 10.520 251 

Sex .45 .498 271 

Age 43.87 12.684 248 

physical fitness 6.42 1.717 266 

HADS Depression 3.50 2.993 269


Pearson 

Correlation 

Sig. (1-tailed) 

N 

sleepy & assoc 


Correlations 

sleepy & 

assoc 

sensations 

scale Sex age 

physical 

fitness 

HADS 

Depression 

1.000 -.199 -.141 -.267 .482 

Sex -.199 1.000 -.017 .110 -.071 

Age -.141 -.017 1.000 -.039 -.004 

physical fitness -.267 .110 -.039 1.000 -.314 

HADS Depression .482 -.071 -.004 -.314 1.000 



. .001 .017 .000 .000 

Sex .001 . .393 .037 .124 

Age .017 .393 . .271 .473 

physical fitness .000 .037 .271 . .000 

HADS Depression .000 .124 .473 .000 . 



251 251 230 247 249 

Sex 251 271 248 266 269 

Age 230 248 248 243 246 

physical fitness 247 266 243 266 265 

HADS Depression 249 269 246 265 269 

Model 

Variables Entered/Removed b 

Variables 

Entered 

Variables 

Removed 

1 age, sex a . Enter 

2 HADS 

Depression, 

physical fitness a 

Method 

. Enter


a. All requested variables entered. 

b. Dependent Variable: sleepy & assoc sensations 

scale 

Mod 

el 

R 

R 

Square 

Adjusted R 

Square 

Model Summary c 

Std. Error 

of the 

Estimate 

R Square 

Change 

Change Statistics 

F 

Change df1 df2 

Sig. F 

Change 

1 .245 a .060 .052 10.243 .060 7.267 2 227 .001 

2 .541 b .293 .280 8.927 .232 36.948 2 225 .000 

a. Predictors: (Constant), age, sex 

b. Predictors: (Constant), age, sex, HADS Depression, physical fitness 

c. Dependent Variable: sleepy & assoc sensations scale 

ANOVA c 

Model Sum of Squares Df Mean Square F Sig. 

1 

2 

Regression 1524.909 2 762.455 7.267 .001 a 

Residual 23817.621 227 104.923 

Total 25342.530 229 

Regression 7413.343 4 1853.336 23.258 .000 b 

Residual 17929.187 225 79.685 

Total 25342.530 229 

a. Predictors: (Constant), age, sex 

b. Predictors: (Constant), age, sex, HADS Depression, physical fitness 

c. Dependent Variable: sleepy & assoc sensations scale


Coefficients a 

Standardized 

Unstandardized Coefficients 

Coefficients Correlations Collinearity Statistics 

Model 

B Std. Error Beta 

T 

Sig. 

Zero-order Partial Part Tolerance VIF 

1 

(Constant) 33.184 2.521 13.162 .000 

Sex -4.246 1.359 -.201 -3.123 .002 -.199 -.203 -.201 1.000 1.000 

Age -.120 .053 -.144 -2.240 .026 -.141 -.147 -.144 1.000 1.000 

2 

(Constant) 32.211 3.481 9.255 .000 

Sex -3.323 1.193 -.157 -2.786 .006 -.199 -.183 -.156 .986 1.014 

Age -.121 .047 -.146 -2.604 .010 -.141 -.171 -.146 .998 1.002 

physical fitness -.731 .364 -.119 -2.008 .046 -.267 -.133 -.113 .892 1.121 

HADS Depression 1.522 .208 .433 7.329 .000 .482 .439 .411 .900 1.111 


Excluded Variables b 

Collinearity Statistics 

Model Beta In T Sig. Partial Correlation 

Tolerance VIF Minimum Tolerance 

1 

physical fitness -.254 a -4.046 .000 -.260 .987 1.014 .987 

HADS Depression .470 a 8.303 .000 .483 .995 1.005 .995 

a. Predictors in the Model: (Constant), age, sex 


Collinearity Diagnostics a 

Model 

Dimensio 

n Eigenvalue Condition Index 

Variance Proportions 

(Constant) Sex Age physical fitness HADS Depression 

1 

1 2.522 1.000 .01 .06 .01 

2 .440 2.395 .02 .92 .03 

3 .038 8.112 .97 .02 .96 

2 1 4.027 1.000 .00 .02 .00 .00 .02


2 .539 2.734 .00 .69 .00 .00 .21 

3 .343 3.424 .00 .28 .02 .03 .58 

4 .071 7.509 .00 .01 .62 .34 .03 

5 .020 14.235 .99 .00 .35 .63 .16 


Residuals Statistics a 

Minimum Maximum Mean Std. Deviation N 

Predicted Value 15.04 42.28 26.15 5.653 242 

Std. Predicted Value -1.933 2.853 .018 .993 242 

Standard Error of Predicted 

Value 

.816 2.269 1.281 .283 242 

Adjusted Predicted Value 15.04 41.95 26.27 5.577 225 

Residual -24.781 19.438 .178 8.839 225 

Std. Residual -2.776 2.178 .020 .990 225 

Stud. Residual -2.810 2.207 .020 1.002 225 

Deleted Residual -25.393 19.980 .172 9.050 225 

Stud. Deleted Residual -2.854 2.226 .019 1.006 225 

Mahal. Distance .919 13.804 3.949 2.314 242 

Cook's Distance .000 .075 .005 .009 225 

Centered Leverage Value .004 .060 .017 .010 242 


STEP 1: Evaluating the model: 

After the variables in Block 1 (age & sex) have been entered, the overall model explains 

5.2% of the variance (.052 X 100). After Block 2 (fitrate & HADS Depression) has been 

included, the model as a whole explains 28% (.28 X 100). 

How much of this overall variance is explained by the variables fitrate and HADS after 

the effects of age and gender are removed? The R square change for Model 2 is .232. 

Fitrate and HADS explain an additional 23% of the variance of totSAS, even when the 

effects of gender and age are statistically controlled for. This is a statistically significant 

contribution (Sig. F change is .000). The ANOVA table indicates that the model as a 

whole is statistically significant [F(4,225)= 23.26, p


(beta=-.157), age (beta=-.146), and fitness (beta=-.119).The relationship between sex (-.199), 

age (-.141), and physical fitness (-.267) show a low negative relationship with totSAS. HADS 

Depression has a positive moderate correlation (.482) with totSAS. 

Analysis conducted to assure that there was no violation of the assumption of sample 

size, multicollinearity, normality, homoscedasticity, and linearity can be found in the first 

section of this assignment. 

Collinearity diagnostics: Tolerance results for each variable are large (.892-.998) 

indicating that multiple correlation with other variables is low and suggests low 

probability of multicollinearity. The VIF scores for each variable (1.002-1.111) are less 

than 10, indicating low probability of multicollinearity. The Tolerance and VIF values 

meet the criteria for multicollinearity. 

Outliers: The critical chi-square value for 2 independent variables is 13.82 and the 

MAH-1 value obtained (13.804) does not exceed this value. Because these values are 

very close, the researcher might consider examining the MAH-1 data values on SPSS 

data view window and consider removing large value cases. To determine if this outlier 

has an influence on the results the researcher would check the Cook’s distance. If the 

value was >1, the SPSS data file must be checked for all cases that have a COO_1 

value >1. The researcher would need to consider removing high value cases. In this 

model, the Cook’s distance is .075 and suggests that there are no problems with 

outliers. There is no major violation of the assumption of multicollinearity. 

Be sure to answer your research question: 

1. If we control for the possible effect of gender and age, is our set of 

variables (physical fitness and depression) still able to predict a significant 

amount of the variance in perceived stress? Yes. When controlling for gender 

and age, physical fitness and depression share 23% of the variance of daytime 

sleepiness. 

Narrative Summary: 

Hierarchical multiple regression was used to assess the ability to control two 

measures (fitrate & HADS Depression) to predict sleepiness and associated 

symptoms (totSAS). After controlling for age and gender, preliminary analysis 

were conducted to assure that there was no violation of the assumption of 

normality, linearity, multicollinearity, and homoscedasticity. Gender and age were 

entered at Step 1, explaining 5.2% of the variance in totSAS. After entry of fitrate 

and HADS Depression scale at Step 2, the total variance explained by the model 

as a whole is 28%, [F(4,225)=23.26, p


(beta=.433, p

Multiple Regression Analysis - essentiavitae.com

Create successful ePaper yourself

Delete template?

Save as template?