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Notes 5 - $ ST3241 Categorical Data Analysis I Generalized...

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& $ % ST3241 Categorical Data Analysis I Generalized Linear Models Introduction and Some Examples 1
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& $ % Introduction We have discussed methods for analyzing associations in two-way and three-way tables. Now we will use models as the basis of such analysis. Models can handle more complicated situations than discussed so far. We can also estimate the parameters, which describe the effects in a more informative way. 2
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& $ % Example: Challenger O-ring For the 23 space shuttle flights that occurred before the Challenger mission disaster in 1986, the following table shows the temperature at the time of flight and whether at least one primary O-ring suffered thermal distress. 3
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& $ % The Data Ft Temp TD Ft Temp TD Ft Temp TD 1 66 0 9 57 1 17 70 0 2 70 1 10 63 1 18 81 0 3 69 0 11 70 1 19 76 0 4 68 0 12 78 0 20 79 0 5 67 0 13 67 0 21 75 1 6 72 0 14 53 1 22 76 0 7 73 0 15 67 0 23 58 1 8 70 0 16 75 0 Is there any association between Temperature and thermal distress? 4
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& $ % Fit From Linear Regression 5
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& $ % Fit From Logistic Regression 6
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& $ % Example: Horseshoe Crabs Each female horseshoe crab in the crab in the study had a male crab attached to her in her nest. The study investigated factors that affect whether the female crab had any other males, called satellites , residing nearby her. Explanatory variables included the female crab’s color, spine condition, weight , and carapace width . The response outcome for each female crab is her number of satellites . 7
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& $ % Example Continued We consider the width alone as a predictor. To obtain a clearer picture, we grouped the female crabs into a set of width categories 23.25, 23.25-24.25, 24.25-25.25, 25.25-26.25, 26.25-27.25, 27.25-28.25,28.25-29.25, > 29.25 Calculated sample mean number of satellites for female crabs in each category. 8
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& $ % 9
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& $ % 10
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& $ % Components of A GLM Random component Identifies the response variable Y and assumes a probability distribution for it Systematic component Specifies the explanatory variables used as predictors in the model Link Describes the functional relation between the systematic component and expected value of the random component 11
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& $ % Random Component Let Y 1 , · · · , Y N denote the N observations on the response variables Y . The random component specifies a probability distribution for Y 1 , · · · , Y N . If the potential outcome for each observation Y i are binary such as ” success ” or ” failure ”; or, more generally, each Y i might be number of ”successes” out of a certain fixed number of trials, we can assume a binomial distribution for the random component. If each response observation is a non-negative count, such as cell count in a contingency table, then we may assume a Poisson distribution for the random component. 12
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& $ % Systematic Component The systematic component specifies the explanatory variables.
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