<p>Multiple Linear Regression Example (Simultaneous Entry) – KEY</p><p>A resilience researcher conducted a study with an at-risk preschool population in a continuing effort to determine why many children growing up in potentially damaging environments come through relatively unharmed and well adjusted whereas other children succumb to the stress. This preschool population was deemed to be at risk for educational and developmental delays due to a number of risk factors, including low social status, parental psychiatric disorders or criminality, and overcrowding/large family size. Because past resilience research suggested that family social support, child biological dispositions (temperament), and characteristics of home environment are associated with the likelihood of future adaptive outcomes, the researcher decided to investigate these variables to determine whether they would be powerful predictors of resilience in a sample of 30 at-risk preschool children. The children were tested for age- appropriate developmental abilities (verbal, motor, cognitive, self-help, emotional, and social), and an aggregate score for these abilities was used as the dependent measure of resilience. (Higher scores indicate greater overall developmental abilities.) Families were asked to fill out a social support questionnaire to measure the helpfulness of sources of support to families raising a young child. (Higher scores indicate greater support.) Teachers were asked to rate aspects of each child’s home environment (such as emotional and verbal responsiveness to parents, learning and language stimulation, availability of play materials, and so on) during regular home visits. (Higher scores indicate a more positive home environment.) Teachers also completed a child temperament scale. (Higher scores indicate an easy, adaptive, flexible temperament style, and lower scores indicate an intense, difficult, less malleable temperamental style.)</p><p>NOTE: For this scenario, the child’s developmental ability (DEVELOP) is the dependent variable, Y – and the following three variables are the independent variables, X’s:  (SUPPORT) – Social Support  (TEMPER) – Temperament  (HOMEENV) – Home Environment</p><p>Using the above scenario and the data provided in the SPSS output, answer the following questions (where  = .05):</p><p>1. First – check for multicollinearity. Conduct the appropriate diagnostic analysis and indicate your findings.</p><p>Based on the Variance Inflation Factor</p><p>All VIF values were well under 10.0, and as such indicate no concerns with multicollinearity for this set of data.</p><p>Social Support’s VIF 2.141 Temperament’s VIF 1.356 Home Environment’s VIF 2.174 2. Regress developmental ability (DEVELOP) on social support (SUPPORT), temperament (TEMPER), and home environment (HOMEENV).</p><p> a. What proportion of the variance in developmental ability is explained by SUPPORT, TEMPER, and HOMEENV?</p><p>Approximately 87%  R2 = 0.865</p><p> b. Does the set of variables significantly predict developmental ability? Indicate how you came to this conclusion.</p><p>Yes  (From the ANOVA Table) F(3, 26) = 55.342, p < .001</p><p> c. Which of these three variables have a significant influence on developmental ability? Indicate how you came to this conclusion.</p><p>Social Support Yes t = 6.284, p (.000) < α (0.05) Temperament No t = 1.967, p (.060) > α (0.05) Home Environment Yes t = 2.080, p (.048) < α (0.05)</p><p> d. Which of these three variables have the greatest influence (relative importance) on developmental ability? Indicate how you came to this conclusion.</p><p>In order of importance (from greatest to least):</p><p>Social Support β = .663 significant t, p (.000) < α (0.05) Home Environment β = .221 significant t, p(.048) < α (0.05)</p><p>(Temperament was not significant, and as such is not identified as being significantly influential for this sample)</p><p> e. Choose any one of the three predictor (independent) variables and explain its beta value as it relates to the criterion (dependent) variable.</p><p>Social Support Higher levels of a child’s Social Support (controlling for the effects of Temperament and Home Environment) are associated with higher levels of Developmental Ability to a significant degree (β = . 663)</p><p>MLR – KEY Page 2 Home Environment Higher levels of a child’s Home Environment (controlling for the effects of Social Support and Temperament) are associated with higher levels of Developmental Ability to a significant degree (β = .221)</p><p>(Temperament would not be discussed here as it was not found to be significant for this sample.)</p><p>3. Based on the results of this study, would you suggest to future researchers to remove any of the predictor (independent) variables that were used in this analysis? If so, which one(s) and why, or if not, why not.</p><p>I would not suggest removing any of the predictor variables from this model (so as to not miss specify the model) because all of the betas (β) were above .05 (i.e., Social Support β = .663, Temperament β = .165, and Home Environment β = .221).</p><p>4. What is the degree of error for this regression analysis?</p><p>Standard Error of the Estimate = 5.664</p><p>MLR – KEY Page 3 5. Complete Tables 1 and 2 using three (3) decimal places. Don’t forget to put asterisks where applicable on Table 2 (showing significance).</p><p>Table 1</p><p>Means, Standard Deviations, and Correlations for Regression of Developmental Ability</p><p>1 2 3 4</p><p>1. Developmental Ability 1.000 .899 .582 .776 2. Social Support 1.000 .468 .716 3. Temperament 1.000 .481 4. Home Environment 1.000</p><p>Means 39.400 54.230 33.100 24.530 Standard Deviations 14.576 21.614 8.285 5.728 N = 30</p><p>Table 2</p><p>Results of Regression of Developmental Ability</p><p>Independent Variables B  t</p><p>Social Support .447 .663 6.284*** Temperament .291 .165 1.967 Home Environment .563 .221 2.080* Note. R2 = .865, p < .001 *p < .05, ***p < .001</p><p>MLR – KEY Page 4</p>