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Conditional Test Statistics

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Suppose that we are considering two Log- linear models and that Model 2 is a special case of Model 1. That is the parameters of Model 2 are a subset of the parameters of Model 1. Also assume that Model 1 has been shown to adequately fit the data.

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In this case one is interested in testing if the differences in the expected frequencies between Model 1 and Model 2 is simply due to random variation] The likelihood ratio chi-square statistic that achieves this goal is:

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Example

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Goodness of Fit test for the all k-factor models Conditional tests for zero k-factor interactions

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Conclusions 1.The four factor interaction is not significant G 2 (3|4) = 0.7 (p = 0.705) 2.The all three factor model provides a significant fit G 2 (3) = 0.7 (p = 0.705) 3.All the three factor interactions are not significantly different from 0, G 2 (2|3) = 9.2 (p = 0.239). 4.The all two factor model provides a significant fit G 2 (2) = 9.9 (p = 0.359) 5.There are significant 2 factor interactions G 2 (1|2) = 33.0 (p = 0.00083. Conclude that the model should contain main effects and some two-factor interactions

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There also be a natural sequence of progressively complicated models that one might want to identify. In the laundry detergent example the variables are: 1.Softness of Laundry Used 2.Previous use of Brand M 3.Temperature of laundry water used 4.Preference of brand X over brand M

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A natural order for increasingly complex models which should be considered might be: 1.[1][2][3][4] 2.[1][3][24] 3.[1][34][24] 4.[13][34][24] 5.[13][234] 6.[134][234] The all-Main effects model Independence amongst all four variables Since previous use of brand M may be highly related to preference for brand M][ add first the 2-4 interaction Brand M is recommended for hot water add 2 nd the 3-4 interaction brand M is also recommended for Soft laundry add 3 rd the 1-3 interaction Add finally some possible 3- factor interactions

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Models d.f.G2G2 [1][3][24]1722.4 [1][24][34]1618 [13][24][34]1411.9 [13][23][24][34]1311.2 [12][13][23][24][34]1110.1 [1][234]1414.5 [134][24]1012.2 [13][234]128.4 [24][34][123]98.4 [123][234]85.6 Likelihood Ratio G 2 for various models

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Stepwise selection procedures Forward Selection Backward Elimination

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Forward Selection: Starting with a model that under fits the data, log-linear parameters that are not in the model are added step by step until a model that does fit is achieved. At each step the log-linear parameter that is most significant is added to the model: To determine the significance of a parameter added we use the statistic: G 2 (2|1) = G 2 (2) – G 2 (1) Model 1 contains the parameter. Model 2 does not contain the parameter

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Backward Selection: Starting with a model that over fits the data, log-linear parameters that are in the model are deleted step by step until a model that continues to fit the model and has the smallest number of significant parameters is achieved. At each step the log-linear parameter that is least significant is deleted from the model: To determine the significance of a parameter deleted we use the statistic: G 2 (2|1) = G 2 (2) – G 2 (1) Model 1 contains the parameter. Model 2 does not contain the parameter

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K = knowledge N = Newspaper R = Radio S = Reading L = Lectures

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Continuing after 10 steps

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The final step

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The best model was found a the previous step [LN][KLS][KR][KN][LR][NR][NS]

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Logit Models To date we have not worried whether any of the variables were dependent of independent variables. The logit model is used when we have a single binary dependent variable.

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The variables 1.Type of seedling (T) a.Longleaf seedling b.Slash seedling 2.Depth of planting (D) a.Too low. b.Too high 3.Mortality (M) (the dependent variable) a.Dead b.Alive

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The Log-linear Model Note: ij1 = # dead when T = i and D = j. ij2 = # alive when T = i and D = j. = mortality ratio when T = i and D = j.

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Hence since

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The logit model: where

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Thus corresponding to a loglinear model there is logit model predicting log ratio of expected frequencies of the two categories of the independent variable. Also k +1 factor interactions with the dependent variable in the loglinear model determine k factor interactions in the logit model k + 1 = 1 constant term in logit model k + 1 = 2, main effects in logit model

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1 = Depth, 2 = Mort, 3 = Type

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Log-Linear parameters for Model: [TM][TD][DM]

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Logit Model for predicting the Mortality or

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The best model was found by forward selection was [LN][KLS][KR][KN][LR][NR][NS] To fit a logit model to predict K (Knowledge) we need to fit a loglinear model with important interactions with K (knowledge), namely [LNRS][KLS][KR][KN] The logit model will contain Main effects for L (Lectures), N (Newspapers), R (Radio), and S (Reading) Two factor interaction effect for L and S

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The Logit Parameters for the Model : LNSR, KLS, KR, KN ( Multiplicative effects are given in brackets, Logit Parameters = 2 Loglinear parameters) The Constant term: -0.226 (0.798) The Main effects on Knowledge: LecturesLect0.268 (1.307) None-0.268 (0.765) NewspaperNews0.324 (1.383) None-0.324 (0.723) ReadingSolid0.340 (1.405) Not-0.340 (0.712) RadioRadio0.150 (1.162) None-0.150 (0.861) The Two-factor interaction Effect of Reading and Lectures on Knowledge

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