PSY2012 Research methodology III: Statistical analysis, design and measurement Exam Spring 2012
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1 Department of psychology PSY2012 Research methodology III: Statistical analysis, design and measurement Exam Spring 2012 Written school exam, Monday 14th of May, 09:00 hrs. (4 hours). o aids are permitted, but a list of relevant formulas is given on the last page. All exercises must be answered. Exercise 1: Multiple choice. Draw a circle around the letter next to the correct answer. Given that the sum of squares for a regression model (SS M ) is 50, and the sum of squares for error (SS R ) is 150 then R 2 (the coefficient of determination) is: a b c d e. 1.0 In a multiple regression model, the error term ε is assumed to be a random variable with a mean of a. -1 b. 0 c. 1 d. σ e. Any normally distributed value In multiple regression analysis, a strong correlation among the independent variables is referred to as a. Homoscedasticity b. Linearity c. Multicollinearity d. Indeterminism e. Adjusted coefficient of determination If the coefficient of determination is 0.8, then the slope of the regression line: a. must be positive b. must be negative c. could be either positive or negative d. is zero e. one of the above In performing a regression analysis involving two numerical variables x and y, we are assuming: a. the variances of x and y are equal b. the variation around the line of regression is the same for each x value c. that x and y are independent d. the variance inflation factor is zero e. all of the above
2 Interaction (moderation) effect is where the relationship between a variable X 1 and another variable Y is: a. zero until a third variable X 2 is controlled for. b. not statistically significant. c. statistically significant but not very strong. d. different for different values of a third variable X 2. e. not interpretable due to the high correlation between X 1 and X 2. If you have a p-value of 0.02, what does this mean? a. That the probability that the null hypothesis is true is 2% b. The probability that the null hypothesis is false is 2% c. The probability of obtaining the observed value on the test statistic due to sampling error if the null hypothesis were true is 2% d. That you cannot reject H 0. e. one of the above. Forward and backward stepwise regression: a. will always yield the best subset of predictors. b. may result in different sets of predictors chosen. c. should never be used when the predictors are uncorrelated. d. allows you to ignore non-normally distributed errors. e. none of the above. Exercise 2 Three individuals are measured on two variables, X and Y, giving you the following dataset. a. What is the covariance between X and Y? X Y b. Pearson correlation is another measure of the degree of linear relationship between two variables. Why is correlation often reported rather than covariance? c. Give an example of two different types of data for which the Pearson correlation coefficient is not suited, and suggest alternative ways to determine the degree of association between these measures. Exercise 3 a. What we mean by the internal consistency of a scale? b. Below is a covariance matrix for four measures of extroversion. In a covariance matrix the variances of the items are placed along the diagonal and the covariances between the items are placed off the diagonal.
3 [ ] i. Show that Cronbach s alpha for the extroversion scale equals (approximately 0.67). ii. Does the scale seem unidimensional? Exercise 4 This exercise concerns a hypothetical study on reading ability among third grade children. You will find SPSS output for different regression models on the subsequent pages. READIG: A score on a standardized test of reading ability, where higher score represents a better reading ability. This variable will be the dependent variable in all analyses. The following predictors (independent variables) are included: BOOKS: umber of books in the child s home. HOURS: umber of hours the child reads per week. GEDER: Males is coded 0, females is coded 1 ICOME: Combined income of parents in 1000 OK. SIBLIGS: umber of siblings the child has. PROGRAM: Each child has participated in one of three different experimental reading improvement programs in school (coded 1,2,3). a) In model 1, a simple (bivariate) regression is performed where BOOKS is used as the only predictor variable. i. Write the regression expression for model 1 using the estimated regression coefficients from the SPSS output. ii. How much of the variance in READIG can be accounted for by this predictor variable? Explain also how you can you use the AOVA table to calculate this value. iii. If a child has 5000 books at home, would you trust model 1 to accurately predict his/her level of reading ability? (justify your answer). iv. Model 1 was fitted by least squares. Explain what we mean by this. b) In model 2, the predictors HOURS and GEDER are added to the regression model. i. Compared to model 1, the estimated effect of BOOKS has now changed. Give a reasonable explanation for why this has happened? ii. What is the predicted reading level based on model 2 for a girl who has 100 books at home and reads 3 hours a week? c) In model 3, ICOME and SIBLIGS are added to the regression model. i. Are either of these two variables significant at a 0.05 level? ii. The variables ICOME and HOURS are measured on different scales. How would you go about comparing the relative importance of these two variables for reading ability?
4 d) Schools in the community are trying out one of three different reading improvement programs which every child attends. These are coded in the variable PROGRAM as values 1, 2, or 3. i. What is wrong with the following model: The variable PROGRAM is recoded into the variables D1 and D2 as indicated in the following table, and in model 4 the predictors D1 and D2 are added. PROGRAM D1 D ii. What is the difference in predicted reading ability for a child how has followed program 3 versus a child who has followed program 1, given that the children are equal on all other measured independent variables? iii. Which of the reading programs would you recommend that the schools should implement? e) When conducting a regression analysis we often make several diagnostic plots. Below the coefficients table you will find a scatter-plot of the standardized residuals of model 1 plotted against the predictor variable BOOKS. i. Why do we make these plots, and what do we look for? ii. Can you see any reason for concern in the given plot?
5
6 Figure 1: Plot of the standardized residuals of model 1 against the variable BOOKS
7 Formulas for use in PSY2012 Mean: X i= X i Variance: i= (X i X ) Standard deviation: Covariance: Y i= (X i X )(Y i Y ) Pearson correlation: r Y s Y Cronbach s alpha ( ) ( ) Regression: b Y b X b (X i X ) (Y i Y ) i= i= (X i X ) cov XY Sums of squares: ( ) ( ) ( ) Coefficient of determination: R SS Model SS Total F-ratio: T-test F t MS M MS R, in a multiple regression analysis distributed F(df 1 =k, df 2 =-k-1) under H 0. b i SE b i, in a multiple regression analysis distributed t(df=-k-1) under H 0.
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