1 Interaction models: Assignment 3

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1 1 Interaction models: Assignment 3 Please answer the following questions in print and deliver it in room 2B13 or send it by to rooijm@fsw.leidenuniv.nl, no later than Tuesday, May 29 before 14:00. All answers should be written in English. Data can be found at and then clicking on the course module. 1 In the file ca.txt there is a data set cross-classifying occupational group with years of schooling. The variable occupational groups has 10 categories and the variables years of schooling six. Occupational groups has categories: (1) Professional; (2) Manager; (3) Clerical; (4) Craftsmen-1; (5) Craftsmen-2; (6) Operatives (Not in Transportation); (7) Operatives; (8) Farmers; (9) Service; (10) Missing Years of schooling has categories: (1) 0-7; (2) 8-11; (3) 12; (4) 13-15; (5) 16; (6) 17+. Fit a correspondence analysis in two dimensions using SPSS (analyze - data reduction - correspondence analysis). Note that the cases should be weighted first by their frequency. Request a row principal normalization and a biplot. Show and Interpret the biplot using distances, projections, inner products,... Maximum number of words is In social psychology a central question is whether people behave Cooperative (C) or Defective (D) under various circumstances. Here we have data where students were classified according to a personality scale in pro-selves and pro-socials. Following they had two make two times a choice between cooperation and defection: The first choice is in an ingroup (defined as a group of similar people); the second in the outgroup. The data are represented in the table below. 1

2 Context Ingroup Outgroup Count Pro-selves C C 6 C D 13 D C 4 C D 20 Pro-socials C C 16 C D 25 D C 0 D D 6 (a) Use McNemar s test to test for marginal homogeneity in both groups (i.e. first for the pro-selves and second for the pro-socials). (b) Make 95%-confidence intervals for the difference of proportions for both groups using the formula on the bottom of Agresti, page 410. (c) What are your conclusions (max 100 words). 3 The following data set cross-classifies 541 subjects first and second purchase choice of instant decaffeinated coffee at two times. Second choice High Taster s Point Choice Sanka Nescafe Brim High Point First Taster s choice Purchase Sanka Nescafe Brim (a) Make a hierarchical lattice from the most restricted model to the least restricted model including the independence model, the quasi-independence model, the quasi-symmetry model, the symmetry model and the saturated model. (b) Fit the symmetry model with LEM (use: examples - general loglinear - prepecified designs - symmetry and change the data and data dimensions) describe chi-squared statistics and use residuals to analyze changes. 2

3 (c) Do a test of marginal homogeneity. (d) Describe in 100 words the results of the test and an interpretation (derived form b and c). 4 There is a sample of 752 people who were scored as having a depression or not at four consecutive time points. The data can be found on the website (file markov.txt). (a) Fit a first order markov chain to these data. Report and discuss the initial and transition probabilities. Discuss and interpret goodness-of-fit. (b) How can the model be generalized to accommodate to the data. (c) Do the analysis proposed in (b) and describe in 200 words the outcome (use fit and transition probabilities) 5 The data from Agresti Table 9.1 are analyzed using a marginal model. The following dummies are made R for race, white =1 G for gender, female=1 S 1 and S 2 for substance type. S 1 = 1, S 2 = 0 for alcohol; S 1 = 0, S 2 = 1 for cigarettes; and S 1 = 0, S 2 = 0 for marijuana. The following model for the probability π of using a particular substance is logit(ˆπ) = α + β 1 S 1 + β 2 S 2 + β 3 R + β 4 G + β 5 G S 1 + β 6 G S 2 (a) What is the interpretation of α? (b) The parameter estimates are β 1 = 1.93 β 2 = 0.86 β 3 = 0.38 β 4 = 0.20 β 5 = 0.37 β 6 =

4 (c) What is, given gender, the factor by which the odds multiplies that a white subject uses a given substance compared to a black subject? (d) What is, given race, the factor by which the odds multiplies that a female has used alcohol compared to that she used marijuana. (e) Provide within 200 words a complete interpretation of this model. 6 For the data from Agresti Table 8.3 let y it = 1 represent the case that subject i uses substance t. The data are analyzed using a logistic normal model. logit [P (Y it = 1 u i )] = u i + β t where u i N(0, σ 2 ). Parameter estimates plus standard errors are β alcohol = 4.22 (SE = 0.18) β cigarettes = 1.62 (SE = 0.12) β marijuana = 0.77 (SE = 0.11) σ = 3.54 (SE = 0.16) (a) Interpret the β s (b) Interpret the value of σ. (c) If a marginal model is fitted to these data, would the β s be similar or the same. Motivate briefly your choice? 7 The following data set gives a cross-classification of 4 categorical variables. Two are white subject s evaluations of surveys and two are interviewers evaluations of these respondents. The first variable asks about the perceived purpose of the surveys, the second about the accuracy of survey results, the third about the perceived cooperation of the respondent and the fourth about the perceived understanding. The data are in the file latclass2.inp 4

5 Cooperation Purpose Accuracy Understanding Interested Cooperative Impatient Good Mostly true Good Fair/poor Not true Good Fair/poor Depends Mostly true Good Fair/poor Not true Good Fair/poor Waste Mostly true Good Fair/poor Not true Good Fair/poor Fit a latent class model to these data using LEM. Determine the number of classes. Show parameter estimates and provide in maximum 200 words an interpretation. 5