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Winter, 2014 Wednesday, Feb. 12Stat 418 – Day 21Multicategory Logit Models (Ch. 6)- Don’t forget about initial (graphical) explorations of the data- Prediction tables (Confusion matrix) can be a useful way to summarize the predictive power of a logistic regression model, correct classification rateo May want to try difference choices of cut-off valueo May want to balance sensitivity (probability of correct prediction of success) and specificity (probability of correct prediction of failure)- ROC curves summarizes predictive power for all possible cut-off valueso Area under curve = concordance index = what proportion of pairs with one success and one failure estimated a higher probability for the success Example 1: Suppose a business wants to predict a firm’s commitment of resources to total quality management (large, moderate, small) from size of firm, type of industry, and several other explanatory variables. Or a researcher wants to predict the severity of disease (mild, moderate, severe) based on age of patient, gender of patient, and other explanatory variables. Or you want to predict students’ mode of transportation to school: car, walking, bicycle, bus and how this choice is related to age, sex, distance, etc. (a) What is the main difference in these research questions and what we have examined before?(b) What is a key distinction between the last two examples?So we want a model of the form: P(Yi = k ) = eX- / [1+eX-] for k = 1, …, JOne approach is rather than only comparing P(Yi = k) to 1-P(Yi = k), we can compare P(Y = k) to P(Y = m). If we choose one response category as the baseline, then we can set up J – 1 logit functions:...log22110xxiiiJiYou could of course just fit pairs of categories separately, but fitting the multicategory logits simultaneously will generally have smaller standard errors and allows for analyses to “adjust” for the effects of the other variables. Also you will be able to test whether a predictor or set of predictors is useful across the levels of the response rather than just in a specific pair.Example 2: Hosmer and Lemeshow (2000) present data from a study of assessment factors associated with women’s knowledge, attitude, and behavior toward mammography. The variables include: mammography experience (never, within one year, over one year ago), agreement with “You do not needa mammography until you develop symptoms” (strongly agree, agree, disagree, strongly disagree), perceived benefit on a scale of 5-20 (low values indicate higher perception of benefit), mother or sister 1Winter, 2014 Wednesday, Feb. 12with history (yes, no), whether believe mammogram would detect new case (not likely, somewhat likely,very likely).(a) Open the mammography.jmp data file. Produce a two-way table of mammography experience and family history. Calculate the odds of having a mammogram over one year ago vs. never comparing those with family history to those without. Hint: What is the 2 × 2 table?(b) Now find the odds of having a mammogram within one year vs. never, comparing those with family history to those without.(c) In JMP, highlight the mammography experience column and select Column Info. Then click on the Column Properties and select Value Ordering. Move “Never” to be last in the list - that will be our reference event. Now fit a nomial logistic regression, entering mammography experience as the responseand history in the Model Effects box. What are the prediction equations? How do the odds ratios compare to what you have just computed? [Hint: We aren’t using indicator variables.](d) Run the model just using benefit as a quantitative predictor. Interpret the first odds ratio.To convert from odds to probability: hxxjhhjjee......110110ˆ Note, the denominator is the same for each probability and the numerators sum to the denominator.(e) Use JMP to save the Probability Formula.(f) Now use the graph builder putting benefit on the x-axis and dragging the three estimated probability columns into the graph. (You should be able to get all 3 to appear at once.) What is true about the sums of the probabilities of each response category at a particular value of x?(f) Summarize the relationship. In particular, if the benefit scale value is low, what do you predict they will do? What if the benefit scale is high?2Winter, 2014 Wednesday, Feb. 12(g) How does this graph compare to the Logistic Plot in the Nominal Fit?The Logistic Plot is showing you cumulative probabilities. The lowest curve is the predicted probabilities for getting a mammogram over a year ago. The second curve is the predicted probability ofgetting a mammogram within one year or over one year ago (at all). The individual probabilities can be recovered by looking at distances between the curves.(h) From the Nominal Fit pull-down menu, select Profiler. At the bottom of the output window, you willsee the Prediction Profiler, which will display the predicted probabilities at each x value. Click on the vertical dashed red line and move to the left and right. How do the predicted probabilities in each category change as you change the benefit variable? (i) From the Logistic Plot and/or the Profiler, for what benefit scores is the probability of getting a mammogram within one year the largest? How does the probability of never getting a mammogram change with the benefit score?(j) Highlight the mammography experience column and select Rows > Color or Mark by Column. (I had already chosen Columns > Column Info > Column Properties > Value Colors.) Now examine the Logistic Plot. Does the model appear to fit the data well? How are you deciding? How often with the model predict “over one year”?Example 3: One way to predict authorship of works of literature is to identify characteristics of an author’s work. A common quantification is based on the frequencies of each word in a text, the lengths of words, vocabulary richness, and sentence length. The famous study of The Federalist papers focused on frequencies of “function” words (words with little contextual meaning), thinking authors wouldn’t think much about them and therefore use them consistently. The data in authorship.jmp compare word counts (from blocks of text that each contain 1700 total words) from several sets of works by Jack London (1876-1916), Jane Austen (1775-1817), John Milton (1608-1674), and William Shakespeare


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Cal Poly STAT 418 - Multicategory Logit Models

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