Module 3 Assignment 3: Multiple Regression. Robin Henson. August 8, NSG 6163 Health Outcomes. Texas Woman s University

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1 RHensonMultipleRegression 1 Module 3 Assignment 3: Multiple Regression Robin Henson August 8, 2009 NSG 6163 Health Outcomes Texas Woman s University

2 RHensonMultipleRegression 2 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, 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 Sex Age physical fitness HADS Depression

3 RHensonMultipleRegression 3 Pearson Correlation Sig. (1-tailed) N Sleepy & assoc sensations scale Correlations sleepy & assoc sensations scale sex age physical fitness HADS Depression Sex Age physical fitness HADS Depression Sleepy & assoc sensations scale Sex Age physical fitness HADS Depression Sleepy & assoc sensations scale Sex Age physical fitness HADS Depression Model Variables Entered/Removed b Variables Entered Variables Removed Method

4 RHensonMultipleRegression 4 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 a 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 a Residual Total a. Predictors: (Constant), HADS Depression, age, sex, physical fitness b. Dependent Variable: sleepy & assoc sensations scale 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)

5 RHensonMultipleRegression 5 Sex Age physical fitness HADS Depressio n 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 a. Dependent Variable: sleepy & assoc sensations scale Residuals Statistics a Minimum Maximum Mean Std. Deviation N Predicted Value Std. Predicted Value Standard Error of Predicted Value Adjusted Predicted Value Residual Std. Residual Stud. Residual

6 RHensonMultipleRegression 6 Deleted Residual Stud. Deleted Residual Mahal. Distance Cook's Distance Centered Leverage Value a. Dependent Variable: sleepy & assoc sensations scale

7 RHensonMultipleRegression 7 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> (# of independent variables), this sample size is adequate (248> (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 <.7. The assumption for correlation of independent variables is not violated. C. Collinearity Diagnostics Tolerance (1-R squared)- For each of the independent variables, tolerance values are all greater than.10 ( ). Tolerance values meet the criteria for multicollinearity. Variance Inflation Factor (1/Tolerance)- For each of the independent variables, VIFs are below 10 ( ). VIF values meet the criteria for multicollinearity. D. Normality To determine if residuals are distributed normally, an inspection of the normal probability plot (P-P) of the Regression Standardized Residual is observed for a reasonably straight line. The Normal P-P Plot demonstrates a fairly straight line and proposition residuals are distributed normally without deviations. Assumption of normality is met. E. Homoscedasticity A visual examination of the standardized residuals by the regression standardized predictive value on the scatterplot reveals a slightly rectangular shape. All residuals fall between 3.3 and This supports the proposition that there are no outliers and no great violation in homoscedasticity. F. Linearity

8 RHensonMultipleRegression 8 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 The maximum value for this data file is , 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, this suggests that there are no major problems with cases influencing this data model. There is no major violation of the assumption of multicollinearity. Did you discover any outliers? No. Did you spot any multicollinearity? No. If there were problems what did you decide to do? If any outliers had been identified on the scatterplot then the researcher could identify the case number and determine if the number of outliers identified warrants removing them. If there is an exceeded critical value for the MAH-1, then the researcher would need to find the case(s) with the largest MAH-1 value in the 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. 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)= , p<.0005). Evaluating the independent variables: HADS: For this study, the independent variable, HADS Depression has the greatest standardized beta coefficient (.433) and makes the greatest unique contribution to

9 RHensonMultipleRegression 9 explaining totsas (p<.0001). The part correlation coefficient is.411. When squared (.169), depression uniquely explains 17% of the variance in totsas scores. GENDER: Gender makes the next greatest contribution and has a standardized beta coefficient of (p<.006) and makes a small unique contribution to totsas. The part correlation coefficient is and when squared (.024), gender uniquely explains 2.4% of the variance of totsas scores. AGE: Age makes less of a contribution and has a standardized beta coefficient of (p<.01) and makes a small unique contribution to totsas. The part correlation coefficient is When squared (.021), age uniquely explains 2.1% of the variance of totsas scores. PHYSICAL FITNESS: Physical fitness makes the least unique contribution, noted by the standardized beta coefficient of (p<.046). The part correlation coefficient is When squared (.013), physical fitness uniquely explains 1.3% of the variance of totsas scores. Be sure to answer your research questions. The Research Questions: 1. How well do the variables gender, age, fitness, and depression predict totsas? All four independent variables contribute to predicting totsas. Depression make the greatest unique contribution (beta=.433, p<.0001), followed by gender (beta=-.157, p<.006), age (beta=-.146, p<.01), and fitness (-.119, p<.046) How much variance in totsas can be explained by scores on these variables/scales? Our model explains 29% of the variance in totsas. 2. Which is the best predictor in totsas: gender, age, fitness, or depression? Of the four variables, depression makes the largest unique contribution (beta=.433, p<.0001) and predicts 17% of the variance in daytime sleepiness scores. Provide a narrative summary of the output. Narrative Summary: Multiple standard regression was used to assess the impact of gender, age, physical fitness and depression on daytime sleepiness symptoms. There is a small negative correlation (-.141 to -.267) between sex, age, physical fitness with daytime sleepiness symptoms. There is a positive moderate correlation (.482) between depression and daytime sleepiness symptoms. Preliminary analysis was conducted to assure that there was no violation of the assumption of sample size, multicollinearity, normality, homoscedascticity, and linearity. 29% of the variance in daytime sleepiness is explained by the four variables [F(4,225)=23.268, p<.0005]. All four variables are statistically significant, with the HADS Depression scale recording the highest beta value (beta=.433, p=<.0001) and predicts 17% of the variance in daytime sleepiness scores. PART 2

10 RHensonMultipleRegression 10 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 Sex Age physical fitness HADS Depression

11 RHensonMultipleRegression 11 Pearson Correlation Sig. (1-tailed) N sleepy & assoc sensations scale Correlations sleepy & assoc sensations scale Sex age physical fitness HADS Depression Sex Age physical fitness HADS Depression sleepy & assoc sensations scale Sex Age physical fitness HADS Depression sleepy & assoc sensations scale Sex Age physical fitness HADS Depression Model Variables Entered/Removed b Variables Entered Variables Removed 1 age, sex a. Enter 2 HADS Depression, physical fitness a Method. Enter

12 RHensonMultipleRegression 12 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 a b 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 a Residual Total Regression b Residual Total a. Predictors: (Constant), age, sex b. Predictors: (Constant), age, sex, HADS Depression, physical fitness c. Dependent Variable: sleepy & assoc sensations scale

13 RHensonMultipleRegression 13 Coefficients a Standardized Unstandardized Coefficients Coefficients Correlations Collinearity Statistics Model B Std. Error Beta T Sig. Zero-order Partial Part Tolerance VIF 1 (Constant) Sex Age (Constant) Sex Age physical fitness HADS Depression a. Dependent Variable: sleepy & assoc sensations scale Excluded Variables b Collinearity Statistics Model Beta In T Sig. Partial Correlation Tolerance VIF Minimum Tolerance 1 physical fitness a HADS Depression.470 a a. Predictors in the Model: (Constant), age, sex b. Dependent Variable: sleepy & assoc sensations scale Collinearity Diagnostics a Model Dimensio n Eigenvalue Condition Index Variance Proportions (Constant) Sex Age physical fitness HADS Depression

14 RHensonMultipleRegression a. Dependent Variable: sleepy & assoc sensations scale Residuals Statistics a Minimum Maximum Mean Std. Deviation N Predicted Value Std. Predicted Value Standard Error of Predicted Value Adjusted Predicted Value Residual Std. Residual Stud. Residual Deleted Residual Stud. Deleted Residual Mahal. Distance Cook's Distance Centered Leverage Value a. Dependent Variable: sleepy & assoc sensations scale 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<.0005]. STEP 2: Evaluating each of the independent variables: In the coefficient table for Module 2 (Sig. column) all 4 variables make a significant contribution (p<.05). Their contribution, in order of importance: HADS Depression (beta=.433), sex

15 RHensonMultipleRegression 15 (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 ( ) indicating that multiple correlation with other variables is low and suggests low probability of multicollinearity. The VIF scores for each variable ( ) 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 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<.0005]. The 2 control measures explained an additional 23% of the variance of sleepiness and associated symptoms (totsas), after controlling for gender and age, [R squared change=.23, F change (2,225)=36.95, p<.0005]. In the final model, all 4 variables were statistically significant, with the HADS Depression scale recording a higher beta value

16 RHensonMultipleRegression 16 (beta=.433, p<.0005). When controlling for gender and age, physical fitness and depression scores predict daytime sleepiness 23% of the time.

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