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# PROBABILITY THEORY I STAT 710

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This 5 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 710 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/229668/stat-710-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.

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Date Created: 10/26/15

TWO Way ANOVA 1 TwoWay ANOVA 2 levels for each with No Inter action The main goal in two way ANOVA is to compare the mean of a certain response variable across di erent levels of two factor e ects Taught BSE No Yes NO 3700 Yer HISTORY Yes 3710 Y The Model that we would like to t is Where our variables for the 2 individual Y1 Perceived bene t of mammography And YM the average perceived bene t of mam mography for history i and BSE j 7 07 N0 History stwryi T 17 History 7 07 Not Taught BSE BSEi 1 Taught BSE What does our model look like when Disease 0 What does our model look like when Disease 1 What does our B0 estimate What does our B1 estimate What does our B2 estimate What is the overall F test testing in terms of our means What will the individual H0 Bl 0 test in terms of our means When is it appropriate to use dummy coding7 contrast coding7 unweighted e ect coding7 and weighted e ect coding 2 TwoWay ANOVA Full Model w Dummy Coding Above we assumed that there was no interaction in the di erent effects However as we saw in our rst example that might not be a good assumption Lets try re tting that model with an interaction term We will be tting the model PBl B0 BlHISTl BngEl BgHISTl gtk BSEl el 2 Using Dummy coding what will our What does our B0 estimate What does our B1 estimate What does our B2 estimate What does our B3 estimate What will the individual H0 B3 0 test Lets try this in SAS proc means by HIST BSE VAR PB PRDC GLM DATAMAMMD MODEL PB HIST BSE HISTBSESDLUTIDN OUTPUT DUTDUT1 RRY PPY PRDC PLOT DATADUT1 PLOT RYPY RUN QUIT Since our interaction term is insigni cant I would re t this model as an additive model no interaction and interpret the means separately How could we use the Type I and Type III sums of squares7 for testing the signi cance of two di erent research factors A with four groups7 B with three groups and A gtlt B When would be the appropriate time to use Type I Type III 3 TwoWay ANOVA Full Model w Effect Coding and Dummy Coding 77As we will see7 designs in which two di erent coding systems are employed require special care to assure proper interpretation of the individual regression coef cients7 Cohen et al pg 365 ie THIS GETS TO BE REALLY CONFUSING Now lets use the same model7 but instead of having 0 as our control we will make 1 our control First lets do this for just ESE7 and we will leave the coding of History to be the same BSE is now i 71 NotTaught BSEi7 17 Taught We now have the same model PBl B0 BlHISTl BngEl BgHISTpkBSEi el 3 But the interpretations have changed So using e ect coding What does our B0 estimate What does our B1 estimate What does our B2 estimate What does our B3 estimate Lets try this in SAS data mammo set TMP1meeXp2 BSE2 1 if BSE 0 then BSE2 1 run PRDC GLM DATAMAMMD MODEL PB HIST BSE2 HISTBSE2SDLUTIDN OUTPUT DUTDUT1 RRY PPY run quit Logistic Regression 1 Introduction The relationship between a dichotomous variable and a set of risk factors becomes more dif cult to explore as the number of factors to be investigated becomes larger Risk models allow the use of the variables without the necessity of categorizing variables One immediate restriction of the traditional risk models is that the outcome must be dichotomous7 ie7 death or survival7 disease or no disease Generalizations are possible7 although we will not discuss them Also7 the investigator must keep in mind that any model imposes a structure of its own upon the possible results7 and that if the model is incorrect bias may result In addition interpretation of the results based on risk models may be less intuitively clear than SLR models The logit linear logistic model i i gzp o 61 60 61X 0 W W 1 log where 15m is the odds PM Note that for a dichotomous independent variable logl1 5 60 and logl1 1ng owl 2 So this means that 1 0 n log i log 3 Raising both sides to the exponential power we get that with OROdds Ratio 5 91 libs OR or 51 1090R 4 1717 0 If we have a continuous independent variable we get that e lA Odds ratio per A units increase in X7 regardless of the baseline value for X Since PwA OR 1A 6191A 5 1X7X 5 11 Where A X 7 X ie two values of the independent variable that we are interested in comparing The odds ratio approximately the relative risk of X compared to reference level X depends only on the distance A and 61 which measures the strength of the association between the risk factor and the distance Relative risk RRpW Ed Risk of an event relative to exposure Pcontrol Example If the probability of developing cancer is 20 for smoders and 10 for non smokers the relative risk of cancer associated with smoking is 2 Here are some Model Options you can specify the following options after a slash 1 CLODDSPL 7 WALD 7 BOTH 2 CLPARMPL 7 WALD 7 BOTH 3 LACKFIT 4 LINKkeyword 5 RSQUARE Lets do an example 2 Interaction Models lnteraction will be done similarly to MLR In the logistic model7 interaction can be rep resented by the product of the risk factor and another adjustment variable In deciding whether to use an interaction term7 remember that the logistic model already speci es a multiplicative relationship between risk factors Here is the model well be working with 10W 7 7 1050 X1 zXz inXz 7171mm 75051X152X253X1X2 or PM 1 mpwo X1 62X2 63X1 3 Here will have for a A change in X17 and a given value of X2 that log OR 6p iA BSAXZ 7

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