Running head: MODULE 7 PROBLEM SET Module 7 Problem Set Sharon Cronk-Raby Grand Canyon University: PSY-870 November 18, 2015 1 MODULE 7 PROBLEM SET 2 Module 7 Problem Set: Optimism and Longevity A cancer specialist from the Los Angeles County General Hospital (LACGH) rated patient optimism in 20- to 40-year-old male patients with incurable cancer in 1970. In 1990, the researcher examined hospital records to gather the following data: Socioeconomic status (1-7 rating of occupation; higher ratings indicate higher levels of SES) Age in 1970 Optimism in 1970 (1-100 rating, higher scores indicate higher levels of optimism) Longevity (years lived after the 1970 diagnosis) Using the SPSS data file for Module 7 (located in Topic Materials), calculate a simultaneous multiple regression with SES, age, and optimism as the independent variables and longevity as the dependent variable. 1. Do the independent variables correlate statistically significantly and practically with the dependent variable? The ANOVA table shows that F(3, 240) = 41.684 and is significant at p < .001. This indicates that the combination of the predictor variables of SES, Age, and Optimism significantly predicts the criterion variable Longevity (Leech, Barrett, & Morgan, 2015). ANOVAa Model Sum of Squares df Mean Square Regression 3 554.396 Residual 3192.026 240 13.300 Total 1 1663.187 4855.213 243 a. Dependent Variable: Years Lived after Diagnosis b. Predictors: (Constant), Optimism, Age, Socioeconomic Status F 41.684 Sig. .000b MODULE 7 PROBLEM SET 3 Coefficientsa Model Unstandardized Coefficients Standardized t Sig. Correlations Coefficients B (Constant) 1 Std. Error 3.773 Optimism .164 .175 Age .236 Zero-order 2.131 -.073 Socioeconomic Status Beta Partial Part 1.770 .078 .089 1.441 .151 .369 .093 .075 .044 -.097 -1.675 .095 -.332 -.107 -.088 .023 .485 7.433 .000 .573 .433 .389 a. Dependent Variable: Years Lived after Diagnosis Also, Leech and colleagues (2015) noted the importance of the Coefficients table because \"it provides the standardized beta coefficients, which are interpreted similarly to correlation coefficients or factor weights. The t value and significance opposite each independent variable indicate whether that variable is significantly contributing to the equation for prediction of the dependent variable from the whole set of predictors\" (p. 116). Therefore, it can be observed that the predictor variable of Optimism is statistically significant, whereas the predictor variables of Socioeconomic Status and Age are not. Correlations Years Lived after Diagnosis Years Lived after Diagnosis Socioeconomic Age Optimism Status .369 -.332 .573 .369 1.000 -.290 .520 -.332 -.290 1.000 -.431 .573 .520 -.431 1.000 . .000 .000 .000 Socioeconomic Status .000 . .000 .000 Age .000 .000 . .000 Optimism Pearson Correlation 1.000 .000 .000 .000 . Years Lived after Diagnosis 244 244 244 244 Socioeconomic Status 244 244 244 244 Age 244 244 244 244 Socioeconomic Status Age Optimism Years Lived after Diagnosis Sig. (1-tailed) N MODULE 7 PROBLEM SET 4 Optimism 244 244 244 244 Correlationsb Socioeconomic Age Optimism Status Socioeconomic Status Sig. (2-tailed) Pearson Correlation Age Sig. (2-tailed) Pearson Correlation Optimism 1 Sig. (2-tailed) -.290 ** -.290** .520** .000 Pearson Correlation .000 1 -.431** .000 ** .520 .000 .000 -.431 ** 1 .000 **. Correlation is significant at the 0.01 level (2-tailed). b. Listwise N=244 2. Is collinearity between the independent variables a concern? Accoring toMeyers, Gamst, and Guarino (2013), \"collinearity is a condition that exists when two predictors correlate very strongly; multicollinearity is a condition that exists when more than two predictors correlate very strongly. Note that we are talking about the relationships between the predictor variables only and not about correlations between each of the predictors and the dependent variable\" (p. 363). Leech and colleagues (2015) explained that tolerance and VIF tell whether or not multicollinearity exits. For example, \"if tolerance value is low (< 1 - adjusted R2), then there is a problem with multicollinearity\" (p. 115). Following this guideline: Adjusted R2 = .334 1 - adjusted R2 = .666 Tolerance for SES = .725 (> .666) Tolerance for Age = .809 (> .666) MODULE 7 PROBLEM SET 5 Tolerance for Optimism = .644 (the only value slightly < .666) Therefore, there is no issue with collinearity. Coefficientsa Model Unstandardized Standardized Coefficients Coefficients B Std. Error t Sig. Correlations Collinearity Statistics Beta Zero- Partial Part Tolerance VIF order (Constant) 3.773 .236 1 .164 -.073 .175 Socioeconomic 2.131 1.770 .078 .089 1.441 .151 .369 .093 .075 .725 1.380 .044 -.097 -1.675 .095 -.332 -.107 -.088 .809 1.237 .023 .485 7.433 .000 .573 .433 .389 .644 1.552 Status Age Optimism a. Dependent Variable: Years Lived after Diagnosis Collinearity Diagnosticsa Model Dimension Eigenvalue Condition Index Variance Proportions (Constant) Socioeconomic Age Optimism Status 1 1.000 .00 .01 .00 .00 2 .106 6.040 .01 .38 .12 .01 3 .035 10.449 .01 .61 .14 .46 4 1 3.851 .008 21.866 .99 .00 .74 .53 a. Dependent Variable: Years Lived after Diagnosis 3. What is the R and adjusted R-square for all independent variables entered simultaneously? R = .585 R Square = .343 Adjusted R Square = .334 Model Summary Model R R Square Change Statistics MODULE 7 PROBLEM SET 6 Adjusted R 1 .585 R Square Square a Std. Error of the Estimate Change .343 .334 3.647 F Change .343 df1 41.684 df2 3 Sig. F Change 240 .000 a. Predictors: (Constant), Optimism, Age, Socioeconomic Status What is learned from these values is that the weighted combination of the predictor variables explained approximately 34% of the variance of Longevity. According to Meyers and colleagues (2013), \"the loss of so little strength in computing the Adjusted R Square value is primarily due to our relatively large sample size combined with a relatively small set of predictors. Using the standard regression procedure where all of the predictors were entered simultaneously into the model, R Square Change went from zero before the model was fitted to the data\" (p. 370) to .343 when the variable was entered. Coefficientsa Model Unstandardized Coefficients Standardized t Sig. Correlations Coefficients B (Constant) 1 Std. Error 3.773 Age Optimism .236 .164 .175 Zero-order 2.131 -.073 Socioeconomic Status Beta Partial Part 1.770 .078 .089 1.441 .151 .369 .093 .075 .044 -.097 -1.675 .095 -.332 -.107 -.088 .023 .485 7.433 .000 .573 .433 .389 a. Dependent Variable: Years Lived after Diagnosis 4. What variable(s) provide a significant unique contribution(s)? According to Meyers and colleagues (2013), \"the Partial column under Correlations lists the partial correlations for each predictor as it was evaluated for its weighting in the model (the correlation between the predictor and the dependent variable MODULE 7 PROBLEM SET 7 when the other predictors are treated as covariates)\" (p. 371). Leech and colleagues (2015) further explained, \"The partial correlation values, when they are squared, give us an indication of the amount of unique variance (variance that is not explained by any of the other variables) in the outcome variable [Longevity] predicted by each independent variable [SES, Age, Optimism]\" (p. 116). Therefore, SES = .0932 = .008649 = SES accounts uniquely for about .9% of the variance of Longevity given the other variables in the model. Age = .-.1072 = .011449 = Age accounts uniquely for about 1.1% of the variance of Longevity given the other variables in the model. Optimism = .4332 = .187489 = Optimism accounts uniquely for about 18.7% of the variance of Longevity given the other variables in the model. Additionally, per Meyers and colleagues (2013), \"The raw regression coefficients are partial regression coefficients because their values take into account the other predictor variables in the model; they inform us of the predicted change in the dependent variable for every unit increase in that predictor\" (p. 371). Therefore, SES is associated with a partial regression coefficient of .236 and signifies that for every additional point on the SES measure, a gain of .236 points would be predicted on the Longevity measure. Therefore, Age is associated with a partial regression coefficient of -.073 and signifies that for every additional point on the Age measure, a decrease .073 points would be predicted on the Longevity measure. Therefore, Optimism is associated with a partial regression coefficient of .175 and signifies that for every additional point on the Optimism measure, a gain of .175 points would be predicted on the Longevity measure. Further, Meyers and colleagues (2013) noted that the Y intercept of the raw score model is labeled as the Constant and has a value here of 3.773. Of primary MODULE 7 PROBLEM SET 8 interest here are the raw (B) and standardized (Beta) coefficients, and their significance levels determined by t tests. In this case, Optimism is statistically significant, but SES and Age are not. As can be seen by examining the beta weights, Optimism (.485) is far greater than either Age (-.097) and SES (.089) as making a relatively larger contributions to the prediction model. 5. Compose a results section for this statistical analysis. Socioeconomic Status (SES), Age, and Optimism were used in a standard regression analysis to predict Longevity (years lived after cancer diagnosis) of male patients aged 20 to 40 years old in 1970, all with incurable cancer, using examined data from hospital records in 1990. The correlations of the variables are shown in Table 1. As can be seen, all correlations were statistically significant. The prediction model was statistically significant, F(3, 240) = 41.684, p < .001, and accounted for approximately 34% of the variance of Longevity (R2 = .343, Adjusted R2 = .334). Longevity was primarily predicted by Optimism. The raw and standardized regression coefficients of the predictors together with their correlations with Longevity and their squared semipartial correlations all helped to make this assessment, as shown in Table 2. Optimism received the strongest weight in the model, followed by Age and SES. With the sizeable correlations between the predictors, the unique variance explained by each of the variables indexed by the squared semipartial correlations was much lower for SES and Age than for Optimism. Table 1: Correlations of the Variables in the Analysis (N = 244) MODULE 7 PROBLEM SET Variables 2 3 4 1. Longevity .369 -.332 .573 2. SES --.290 .520 3. Age --.431 4. Optimism -Table 2: Standard Regression Results Model B SE-B Beta Pearson r Constant 3.773 2.131 SES .236 .164 .089 .369 Age -.073 .044 -.097 -.332 Optimism .175 .023 .485 .573 References 9 SR2 .009 .011 .187 Leech, N. L., Barrett, K. C., & Morgan, G. A. (2015). IBM SPSS fo intermediate statistic: Use and interpretation (5th ed.). New York, NY: Routledge. Meyers, L. S., Gamst, G., & Guarino, A.J. (2013). Applied multivariate research (2nd ed.). Thousand Oaks, CA: Sage. Retrieved from http://gcumedia.com/digitalresources/sage/2013/applied-multivariate-research_designandinterpretation_ebook_2e.php College of Doctoral Studies PSY 870: Module 7 Problem Set Optimism and Longevity A cancer specialist from the Los Angeles County General Hospital (LACGH) rated patient optimism in 20- to 40-year-old male patients with incurable cancer in 1970. In 1990, the researcher examined hospital records to gather the following data: Socioeconomic status (1-7 rating of occupation; higher ratings indicate higher levels of SES) Age in 1970 Optimism in 1970 (1-100 rating, higher scores indicate higher levels of optimism) Longevity (years lived after the 1970 diagnosis) Using the SPSS data file for Module 7 (located in Topic Materials), calculate a simultaneous multiple regression with SES, age, and optimism as the independent variables and longevity as the dependent variable. 1. Do the independent variables correlate statistically significantly and practically with the dependent variable? 2. Is collinearity between the independent variables a concern? 3. What is the R and adjusted R-square for all independent variables entered simultaneously? 4. What variable(s) provide a significant unique contribution(s)? 5. Compose a results section for this statistical analysis