User:Magravat 15



Introduction 

Effect modification P value is a method to determine if there is a condition called homogeneous odds ratio which if present, interaction is possible and must be analyzed. Currently the CMH SAS® command is used to test for whether the odds ratio is homogeneous for large sample sizes with fixed effects. One of the tests currently used is the Breslow -Day test in the PROC FREQ CMH test (SAS). If the P value < alpha, for the new method, you reject the null and say the condition of homogeneous odds ratio is rejected, and that there is effect modification from EM. The new method can be used for large or small sample sizes, meant for random variables with a non- normal distribution, and includes a chi-square test of independence, as well as a test for large sampling approximation from P values of LRT, Score, and Wald tests, and the Durbin Watson statistic test for autocorrelation. A new procedure to transform and fit count data is demonstrated that allows you to use predicted versus expected counts and regression to obtain a P value from regression and residuals to evaluate effect modification P value. The area under the curve, ROC curves, and power is discussed in this paper to support the new method of the author. If the P value is greater than the alpha, then we fail to reject the null and say there is no effect modification. Later the PROC MIXED method will be clear with EM of matrix mathematics. The benefit is also for random effects where errors exist in the distribution. Non-normal data can be analyzed effectively by regression analysis with PROC MIXED. The power is good and standard errors through intercepts are visible in plots (Agravat 2009). Matrices allow for full confidence of the method due to an O statistics which is an expectation of the mean through matrix algebra.



Conditions 

In this type of test of interaction, the problem of effect modification P value is meant for determining if there is a condition that is called homogeneity of odds ratio. The odds ratios must be somewhat equal because that is the question. No one category can have an odds ratio that is much greater than the others. In fact this becomes the null hypothesis, while the alternative becomes that you fail to prove that the odds ratios are not homogeneous which is when the Breslow- Day P value and the new Method have P > α. If the null is rejected, then a condition of effect modification exists. SAS tests for the standard condition of effect modification, are tested for the Breslow-Day test which is coded for by using the PROC FREQ (SAS) command with CMH as the option. Breslow- Day test is meant for linear data that has fixed effects that affect the inference made and is meant for large sample sizes only. If the data is non- normal and covariates are independent of the outcome, then, you can consider that the assumptions of random effects have been met for this new method using PROC UNIVARIATE command and the possibility of non-normal distributions and PROC FREQ Chisq (SAS), and large sample approximation tests from LRT, Score, Wald test, and PROC AUTOREG. The Power for the new method is higher than Breslow- Day’s test (Agravat 2009). The ROC curve shows higher area under the curve for the author’s method than Breslow- Day test. The standard error for small sample sizes is smaller than Breslow –Day test. The P value obtained produces a lower value however the algorithm converges so the results for MLE are valid. There are other points such as maxRsq being good, C statistic being higher, and convergence of algorithm when testing for power (equaling 1) as well as P value that supports the benefits and use of this new method that is shown in SAS outputs.

O Statistics 

Then the development of the O statistics, which have been proved to follow Basu’s theorem for chi square and independence shows the potential and application in a well fashioned test. The matrix has formula to comply with a ‘fit’ an algorithm that is changed made for effect modification as well. The two matrices from data transformed variables zxy and xzy are multiplied. The resultant is another matrix called the expectation of the mean. The formula of the O statistics, and the fit variable as a matrix, is different for confounding and multiplied to the count dataset in the rows. One must refer to the PROC IML algorithm to follow the application in SAS software to facilitate. This method will be demonstrated later. An expectation of the means is the resultant. The term or number is the ‘EM’ or an effect modification term sometimes used as ‘aem’ in literature. EM follows asymptotic chi square which is indented for a large sample approximation.

<span style="font-family:"Arial",sans-serif">This sample of the O statistics is from IHANCE of Hashibe et.al. 2007,an international study conducted in TAMPA, Houston, U.S., Europe, USSR, and more from LYON FRANCE. The data is reading race and no smoking no drinking as the exposure.

<p class="MsoNormal" style="text-indent:.5in"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Obs(hat) =( Obs-  Obs(mean) )2/  (Obs)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Observed O stat Matrix 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">|1  1 |     | 1   1 795 |

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">|0 1 |    |1   1  2586  |

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Observed O stat Matrix

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">| 2  2    3381  |

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">|2   2    2586  |

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Expected O stat Matrix

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">|   1.1   1.1  1916  |

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">|   0.4. 0.4.  1120  |

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Fixed vs. Random Effects: 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> As stated Breslow –Day’s test is a fixed effect test, while the new Method is meant for random

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">effects. To begin with, the importance of understanding the meaning and significance of fixed

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">versus random effects is discussed. Fixed effects are considered non-experimenting controlling

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">or other variables using linear regression. One must include the variables to estimate the effects

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">of the variables. Next, the variable must be measured and chosen for model selection.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Ordinarily, if dependent variables are quantitative, then fixed effects can be implemented

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">through ordinary least squares. Thus, stable characteristics can be controlled eliminating bias.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Fixed effects ignore the between-person variation and focuses on the within-person variation. In

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">a linear regression model: Yij <span style="font-family:Symbol;mso-ascii-font-family: Arial;mso-hansi-font-family:Arial;mso-bidi-font-family:Arial;mso-char-type: symbol;mso-symbol-font-family:Symbol"> = <span style="font-family:"Arial",sans-serif"> <span style="font-family:Symbol;mso-ascii-font-family:Arial;mso-hansi-font-family: Arial;mso-bidi-font-family:Arial;mso-char-type:symbol;mso-symbol-font-family: Symbol">b <span style="font-family:"Arial",sans-serif">0 <span style="font-family:Symbol; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-bidi-font-family: Arial;mso-char-type:symbol;mso-symbol-font-family:Symbol">+ <span style="font-family:"Arial",sans-serif"> <span style="font-family:Symbol; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-bidi-font-family: Arial;mso-char-type:symbol;mso-symbol-font-family:Symbol">b <span style="font-family:"Arial",sans-serif">1xij <span style="font-family: Symbol;mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-bidi-font-family: Arial;mso-char-type:symbol;mso-symbol-font-family:Symbol">+ <span style="font-family:Symbol;mso-ascii-font-family:Arial;mso-hansi-font-family: Arial;mso-bidi-font-family:Arial;mso-char-type:symbol;mso-symbol-font-family: Symbol">a <span style="font-family:"Arial",sans-serif">i <span style="font-family:Symbol; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-bidi-font-family: Arial;mso-char-type:symbol;mso-symbol-font-family:Symbol">+ <span style="font-family:"Arial",sans-serif"> <span style="font-family:Symbol; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-bidi-font-family: Arial;mso-char-type:symbol;mso-symbol-font-family:Symbol">e <span style="font-family:"Arial",sans-serif">ij Β1*x is fixed and all x’s are measured while

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">β1 is fixed. ε or the error term is defined as a random variable with a probability distribution that

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">is normal and mean 0 and variance sigma2. Thus the models can be both  fixed and random.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Fixed effects are also considered as pertaining to treatment parameters where there

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">is only one variable of interest. They are used to generalize results for within effects. In a

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">random effects model, the ε can exist and do not have to be zero and be considered. Random

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">effects exist when the variable is drawn from a probability distribution. Blocking, controls, and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">repeated measures belong to random effects. Random effects are involved in determining

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">possible effects and confidence intervals. Unbalanced data may cause problems in inference

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">about treatments. Random effects are involved in clinical trials and in making causal inferences.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Random effects do not measure variables that are stable and unmeasured characteristics. Thus

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the α’s is uncorrelated with the measured 3 parameters. Random effects model can be used to

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">generalize effects for all the variables that belongs in the same population.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height: normal"><span style="font-family:"Arial",sans-serif">Data Transformations 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The author calculates, a new eﬀect modiﬁcation, P value statistic, from transformed count data.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The beta estimates are calculated from survival analysis representing interaction with

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">explanatory variable, depicting interaction with outcome and eﬀect modiﬁer, (Z is a variable for

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the eﬀect modiﬁer). Zx represents confounder interaction with explanatory variable. Yz

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">represents interaction with outcome and confounder. Z represents the confounder. The model of

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the (TM) SAS code class level variables is chosen sometimes dependent on whether the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">outcome converges with all the variables in the model. In other words, one variable may be the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">same as another and can be left out. The format is same for each level, the outcome comes ﬁrst

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">with y =1 positive for lung cancer then follows with ’1’ for a ﬁt variable. This row’s ’1’ means not

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">ﬁtted values. Next, there is a 1 for the eﬀect modiﬁer and 1 for the explanatory variable Agravat

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">(2008). Then, the count or n is from the data set directly. For the next row, the outcome will be

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">same (y = 1) and ones for modiﬁer and explanatory variable followed by the raw count. The next

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">two rows will have the ﬁt values, therefore, both ﬁt variables will be 0 in each row, and the ﬁt

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">values will come from the sequence shown: for the z, or the eﬀect modiﬁcation variable, the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">adjusted for count is the: observed(count)/ | βzx <span style="font-family:"CambriaMath",serif; mso-bidi-font-family:"CambriaMath"">∗ <span style="font-family:"Arial",sans-serif">βz | as designed by the author.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">For the explanatory variable, the new count estimate is: observed(count)/|βyz |. The value of

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">count comes from the observed count, and this method is used to calculate a new count. One

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">must alternate it until the symmetric count dataset is created and you must use the absolute

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">value of the beta estimates to adjust the count data. If there is a 0 in the count make it adjusted

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 in the count or ’n’ data column. The beta estimates are obtained from using the original count

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">data unadjusted. If the P value is greater than the alpha, then we fail to reject the null and say

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">there is no eﬀect modiﬁcation. The P values are used to choose which distribution is better for

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">beta; hence, this step is parametric. Interaction terms are used to measure beta for instance βzx

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">for the slope of vector of βzx the interaction between βz and βx Agravat (2009)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal" style="line-height:normal"><span style="font-family:"Arial",sans-serif">  

<p class="MsoNormal" style="line-height:normal"><span style="font-family:"Arial",sans-serif">TOOL 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">SAS ( TM) as a tool and a means for analysis is very helpful for analysis of this problem of

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Effect Modification P value when the previous method is meant for large datasets and only fixed

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">effects. The variability of this new Method will allows SAS ( TM) to produce more meaningful

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">results with higher power and more area under the curve for large and small sample sizes with

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">non-normal distribution, independence, and random effects.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoListParagraphCxSpFirst" style="text-indent:-.25in;mso-list:l0level1lfo1"> ·<span style="font-variant-numeric:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"TimesNewRoman";">         <span style="font-family:"Arial",sans-serif">Testing for Non-Normal Distribution

<p class="MsoListParagraphCxSpMiddle" style="text-indent:-.25in;mso-list:l0level1lfo1"> ·<span style="font-variant-numeric:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"TimesNewRoman";">         <span style="font-family:"Arial",sans-serif">Independence

<p class="MsoListParagraphCxSpMiddle" style="text-indent:-.25in;mso-list:l0level1lfo1"> ·<span style="font-variant-numeric:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"TimesNewRoman";">         <span style="font-family:"Arial",sans-serif">Power Analysis of RANTBL

<p class="MsoListParagraphCxSpMiddle" style="text-indent:-.25in;mso-list:l0level1lfo1"> ·<span style="font-variant-numeric:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"TimesNewRoman";">         <span style="font-family:"Arial",sans-serif">Autocorrelation and Randomness

<p class="MsoListParagraphCxSpMiddle" style="text-indent:-.25in;mso-list:l0level1lfo1"> ·<span style="font-variant-numeric:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"TimesNewRoman";">         <span style="font-family:"Arial",sans-serif">ROC Curve

<p class="MsoListParagraphCxSpLast" style="text-indent:-.25in;mso-list:l0level1lfo1"> ·<span style="font-variant-numeric:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"TimesNewRoman";">         <span style="font-family:"Arial",sans-serif">Regression

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">BETA ESTIMATES 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Beta Estimates are obtained from R software in the manner they are shown. Potentially they

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">can be from AML matrix method. The confounder is the z term or zxy. The explanatory variable

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">is x term or xzy. One must follow the other variables such as cases as y term or the outcome for

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the algorithm not to be confusing. Data originally written this way to comply with R software.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Thebetaestimatesareobtainedfromthemethodshown.

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">library(survival)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">control<c(1,1,0,0,1,1,0,0,1,1,0,0)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">cases<c(1,0,1,0,1,0,1,0,1,0,1,0)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">race<c(1,1,1,1,2,2,2,2,3,3,3,3)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333;letter-spacing:.35pt">count<c(795,2586,763,4397,111,233,62,238,40,152,45,170)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">dataRace<data.frame(control,cases,race,count)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">survreg( Surv(count,control)cases,dist="weibull",data=dataRace,)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Call:survreg(formula=Surv(count,control)cases,data=dataRace,dist="weibull")

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Coefficients:(Intercept)cases7.8838841.419907

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333;letter-spacing:.35pt">Scale=1.308425

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Loglik(model)=48.4Loglik(interceptonly)= 49.1Chisq=1.45on1degreesoffreedom,p=0.23n=12

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">survreg( Surv(count,control)race,dist="weibull",data=dataRace,)

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Call:survreg(formula=Surv(count,control)race,data=dataRace,dist="weibull")

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Coefficients:(Intercept)race9.5751851.531427

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333;letter-spacing:.35pt">Scale=0.760424

<p class="MsoNormal" style="margin-bottom:0.0001pt;line-height:0%;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-attachment:initial;background-origin:initial;background-clip:initial;"><span style="font-size:22.5pt;font-family:"Arial",sans-serif; mso-fareast-font-family:"TimesNewRoman";color:#333333">Loglik(model)=45.2Loglik(interceptonly)= 49.1Chisq=7.79on1degreesoffreedom,p=0.0053n=12

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">library(survival)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">control<-c(1,1,0,0,1,1,0,0,1,1,0,0)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">cases<-c(1,0,1,0,1,0,1,0,1,0,1,0)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">race<-c(1,1,1,1,2,2,2,2,3,3,3,3)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">count<-c(795,2586,763,4397,111,233,62,238,40,152,45,170)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">dataRace<-data.frame(control,cases,race,count)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">survreg( Surv(count,control)~cases, dist="weibull",data=dataRace,)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Call: survreg(formula=Surv(count,control)  cases,data = dataRace,dist ="weibull")

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Coefficients (Intercept) cases 7.883884   1.419907

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Scale= 1.308425

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Loglik(model)= 48.4  Loglik(intercept  only)=

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">49.1Chisq= 1.45 on  1 degrees of freedom, p= 0.23 n= 12

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">survreg( Surv(count,control) race, dist="weibull",data=dataRace,)

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Call: survreg(formula = Surv(count, control)race, data=data Race, dist ="weibull")

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Coefficients: (Intercept) race 9.575185 1.531427

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Scale=0.760424

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Loglik(model)=  45.2 Loglik(intercept only)= 49.1 Chisq= 7.79on 1 degrees of freedom, p= 0.0053 n= 12

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">A SAS HYPERGEOMTRIC CODE 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">data micasesHoaemnewD;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">input n age fit fitrawz lcwocz locusex total lem ;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">label fitrawz = 'fit with the raw count for confounder z'

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> cwoc = 'cases with oc use'

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">lcwocz = log of 'cases with oc use adjusted for confounder'

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> locusex =log of '# in age stratum using ocs

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> adjusted for explanatory'

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">n = '# of cases in age stratum'

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">ltotal =log of 'sample size in this age stratum' laem='log(em)';

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> ; datalines; 6 0 1 2 1 1 292 .72

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 21 0 1 9 1 1 444 1

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 37 1 0 4 5.77 4.06 393 1

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 71 1 0 6 5.95 3.75 442 1

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">99 1 1 6 1 1 405 5.56 ;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">run;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">proc mixed data=micasesHoaemnewD;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">weight total;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">class lcwocz ;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">model age= lcwocz lem /solution ddfm=satterth covb chisq ;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">run;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">OUTPUT 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Type 3 Tests of Fixed Effects   

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Effect    Num   DF   Den DF   Chi-Square F Value     Pr > ChiSq      Pr > F

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">lcwocz      2                     1         604.47         302.23     <0.0001       <0.0406

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">lem         1                     1         412.74          412.74     <0.0001          <0.0313

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Hyper geometric Distribution Code 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> Agravat’s method for hyper-geometric distribution works well for the typical groups of data that involves time and age. Hyper-geometric distribution is involved with the number of successes of drawing from a population without replacement. Often the studies involve the term total, referring to total number in sample size. One category is count and may involve the number in time or age stratum like this example of myocardial infarction with exposures including birth control. The grouping of data to analyze variables still involves outcome variables first include the frequency of that outcome. The outcome variable for this code with PROC MIXED involves bivariate outcomes for grouped data that can be set by the individual and used to test per selected category for effect modification with “lAEM” variable log of “aem”. Fit variable is still used with the same procedure described for lung cancer data (Agravat 2009). There is also a fit variable called “fitrawz” where the raw count is kept for the effect modifier level group. There may be a variable for count of number of cases with exposure “-z” or effect modifier variable and number with exposure including age or time strata. When the weight is ’n’ the result is that there is a significant difference in risk for outcome of age category for exposure birth control use which includes the number in level of birth control adjusted for cases involving that category because P <0.0001. ) <span style="font-family:"Arial",sans-serif">There is asymptotic chi- square with “1” for age being a categorical variable which is selected by the study designer is between 35-49 years old. FOR lem and there is a statistically significant relationship for chi square and F statistics P<.0001 and P <0.0313.Both the chi square and F statistic converge due to large sample. “Generalized Linear Models”, (Nelder and McCullough 1989), stated that in large samples give the approximate distribution for χ2. With normality, there may be exact results. As n approaches <span style="font-family:Symbol; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-bidi-font-family: Arial;mso-char-type:symbol;mso-symbol-font-family:Symbol">¥ <span style="font-family:"Arial",sans-serif">, the degrees of freedom, in the denominator approaches infinity, and the F-statistic is equivalent to χ2. However, the author’s method is able to produce this convergence with non-normal data and small sample size, as in hyper-geometric data set regarding myocardial infarction, as well as in large samples as in the case of head/neck cancer. The author’s methods with the matrices, algorithm, aem, and PROC IML with PROC MIXED produces, asymptotic chi-squares in the case of effect modification.

<p class="MsoNormal">

<p class="MsoNormal">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">EM CODE with PROC MIXED 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">data smtobAEMmixed;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> input cases fit zxy xzy aem count;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> datalines;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 1 1 1 1470.29  795

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 0 1 1 1 1 2586

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 0 332 569 1 763

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 0 0 1913 3279  1 4397

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 1 1 1 1 140.5678 111

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 0 1 1 1 1 233

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 0 27 46 1 62

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 0 0 104 178 1 238

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 1 1 1 283.71789 40

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 0 1 1 1 1 152

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 0 20 34 1 45

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> 0 0 74 127 1 170 ;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> run;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">proc mixed data=smtobAEMmixed;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">weight count; class zxy ; model cases= zxy aem /solution ddfm=satterth covb chisq ;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif"> run;

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Type 3 Tests of Fixed Effects 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Effect Num DF Den DF Chi-Square F Value     Pr > ChiSq    Pr > F

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Zxy                6       4            27.19           4.53     0.0001     0.0826

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Em                1      4             21.56         21.56    <0.001     0.0097

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The PROC MIXED and PROC IML code shows evidence again for effect modification by the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">significant chi- square for “EM”. Also the F-statistic, F < 0 for the data set indicating statistically

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">significant evidence to reject the null of homogeneous odds for head/neck cancer due to no

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">drinking for level of race.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">  

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The O Stat Method and EM 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Chi-square for Eﬀect Modiﬁcation for Head Neck Cancer in INHANCE The O Stat Method works

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">by utilizing the data transformed values that come from the data transformation shown in the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">SAS code and making matrices from ”A New Eﬀect Modiﬁcation P Value Test Demonstrated” by

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the author. The ﬁrst two rows and 2x2 matrixes are multiplied by the 2x3 matrix for the next

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">columns. This yields an observed table 3a and expected table is calculated by the O statistic.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Next, calculate the expected matrix by ﬁrst estimating the expected 2x3 table. The observed is

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">obtained by getting the ﬁrst two rows and all columns of data with variables ”cases”, ”ﬁt”, ”zxy”,

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">and ”xzy”. Then the 2x3 table is multiplied by the 2x2 table giving a 2x3 table. The new

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">expected table is a type of mean estimate of observed values calculated by multiplying the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">observed value by row total, then divided by sum total. The O statistic is calculated obtaining

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">through matrices. Sum the O statistic of each set of these products of matrices for ﬁrst step has

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">DF 2. The above matrix gives a 2x3 matrix. The same procedure is repeated for the next set

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">and this is shown in tables for rows 3 and 4 of the data transformation. This step yields several

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">0 values for the observed and the expected values for the 2x3 table for both the observed and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the expected hence this may indicate characteristics of singularity. This procedure is repeated in

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">SAS/IML (TM) ( see Formulas Calculating Risk Estimates and testing for Effect Modification and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Confounding for PROC IML code).

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The matrix calculation is depicted below. This pattern continues in pairs of rows of the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">data transformational method’s procedure. The above matrices are from the head/neck cancer

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">example (INHANCE study Hashibe et al.  2007).The properties of matrices include commutative

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">properties regard ing rings and for multiplication. If E1 is nonsingular, then k*E1 is non-singular

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">For a square matrix, which is also possible, for the matrices the commutative property of multi

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">plication shows that if E1 or E2 are singular when 0 is a possible determinant when dealing

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">with the O statistics determinants of certain elements, then if E1 and E2 are singular then

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">concentric rings are possible which may be related to how the matrices shown demonstrates the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">involvement of complex and real numbers which shows properties of being both nonsingular

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">and singular because of nonzero values and being square and having 0 values. Thus the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">allowance of complex numbers allows more calculations with than without them.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">O Stat Matrix Values Observed 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">2x2 Observed 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Col1 Col2 Col3 Total

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">2 2 3381 3385

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1 1 2586 2588

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Total 3 3 6967 6973

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">2x2 Mean 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Col1 Col2 Col3 Total

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">1.1 1.1 1916 1918.2

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">.4 .4 1120 1120.8

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Total 1.5 1.5 3036 3039

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The columns of the O stat are column totals from the PROC IML code. The sum of the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">O stat from the columns of Each matrix multiplication is summed for each pair of rows of the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">data transformational code and summed with a P-value outputted by the SAS code in PROC

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">IML and the O stat total can be calculated to give the chi square P-value of the ”aem” or “EM”

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">which is the from the O stat calculated by the formula above. The mean O stat is calculated by

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the value times the row total divided by overall total. The alternating matrix calculations give O

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">statistics that are used as ”em” for the PROC MIXED calculations. Each set of calculations

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">will give a number value followed by a 0 for the ”0” level ﬁt variable which were adjusted for

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">by beta estimates. The number 1 is put for the column of ﬁrst O statistic and ”em” value

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">that includes the ”0” ﬁt variable continued throughout. Deter01 represents observed values and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">deter1 represents observed mean in the PROC IML SAS code. ”EM” is another name for the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">O stat for that matrix calculation. The zeros are ignored in the matrix calculation. The sum

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">of the ”em” or the O stats after all the matrix calculations give 1894.5757 with 12 degrees

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">of freedom for the data with a P-value from PROC IML of 0 and P < 0.0001 for chi-square for

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">”em” from PROC MIXED for the same variable ”em” that are statistically signiﬁcant. You

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">may then reject the null of homogeneous odds. One may then reject the null of homogeneous

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">odds for head/neck cancer and no drinking as exposure and the level of races (Non-Hispanic,

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">black, and Hispanic) of the INHANCE study population. One may generalize that there may

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">be a diﬀerence by race for head/neck cancer for no drinking as exposure. The interaction may

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">be less due to protection which is possible. The hazard ratio is .213 which indicates less harm

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">for the outcome head/neck cancer and exposure no drinking based on level of race. The PROC

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">MIXED SAS code shows that the corresponding ”EM” values in the program created by the

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">author is signiﬁcant for chi-square and P-value of 21.56 and P < 0.0001. The ”em” is also

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">signiﬁcant by the F-statistic, which is for multivariate analysis, rejecting the diﬀerence for all

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">groups 21.56 and P < .0001. Hence the SAS code for the data transformational method follows

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">an asymptotic chi square statistic. The” zxy” variable has a signiﬁcant P < 0.0001 that is

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">chi-square. ”ZXY” is the eﬀect modiﬁed transformed variable. You may conclude therefore

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">that, the eﬀect modiﬁer variable is giving signiﬁcant evidence to reject eﬀect modiﬁcation null

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">of homogeneous odds (Causal Inference and Proofs of Bio-statistics and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Probability).

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Matrices and PROC MIXED 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">To start this procedure the, 2x2 by 2x3 matrices are multiplied as shown in ”Formulas

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Calculating Risk Estimates and Testing for Eﬀect Modiﬁcation and Confounding Agravat

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The means are also calculated in the same way for tables of observed and mean values. Next

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">using the formulas of the  O(mean) statistic, calculate the output, through the PROC IML code,

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">calculate the ”EM” variable for the SAS algorithm intended for evaluating for confounding with

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">PROC MIXED. The program is from the author Agravat (2011), and if  ”EM” is signiﬁcant one

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">May conclude that the null of homogeneous null is rejected concluding eﬀect modiﬁcation exists.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The matrix formulas are shown here in the PROC IML code as well as the O statistics. In

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the ”New Eﬀect Modiﬁcation P Value Test Demonstrated” Agravat (2009), the cases variable

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">is used in 1,0 1, 0 sequence. This algorithm for eﬀect modiﬁcation has ”ﬁt” set to 1,0,1, and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">0. In the eﬀect modiﬁcation algorithm, the technique using O statistics and matrices utilize

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the observed products form matrix multiplications and mean matrices and the same method of

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">count data transformation. The rest of the count data has to follow the data transfomration

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">method.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">INFERENCE 

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Eﬀect Modiﬁcation analysis of this study, with PROC IML, shows signiﬁcant P < 0 with

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">alpha=.05 hence, the null of homogeneous odds is rejected and calculates values needed for

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the PROC MIXED algorithm in SAS. PROC MIXED for eﬀect modiﬁcation Agravat (2011) has

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">P < 0.0001 for chi-square and P < 0.01 for F-statistic (a multivariate statistic) for ”em” indi

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">cating that the null is rejected and eﬀect modiﬁcation exists. One may conclude that there are

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">statistically diﬀerent risks for head neck cancer for exposure nonsmoking per levels of race.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Since -2LL is 21.5 there is a good model ﬁt with the ”em” method using O statistics Agravat

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">(2011). The conclusion is that per level of race (non-Hispanic, black, and Hispanic), the result

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">is diﬀerent for nonsmoking vs. nondrinkers hence the homogeneous odds null is rejected and

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">eﬀect modiﬁcation exists. In support of this new eﬀect modiﬁcation method is that the power

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">of ”em” is 100 percent by exposure non-smoking.

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">

<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">CONCLUSION 

<p class="MsoNormal" style="line-height:200%"><span style="font-family:"Arial",sans-serif">The new effect modification P value result in Formulas calculation risk estimates and testing for effect modification and confounding using my new Method is α=.05 hence one concludes that there is effect modification and that the null of homogeneous odds ratios is rejected of no difference from passive smoke exposure for countries: United States, Great Britain, and Japan (Blot and Fraumeni 1986) P <0.0001. Since this method is indicated for random variables with non-normal data that have independent covariates, the inference will deal with differences in levels into design by chance not design that is found in variables of the same type of variable in the 12 same population. The relative difference among levels of interest chosen while the author’s method tells of the variable not typically chosen for their unique personal attributes, while the new method collectively represents random variables with non-normal distributions that are independent. From the data, the PROC UNIVARIATE (SAS TM) test show that P< .0003 hence there is evidence to reject normality. The chi square independence test of PROC FREQ (SAS) shows that P< .0001 which support independence between outcome of cases and the variable zxy.

<p class="MsoNormal" style="text-indent:.5in;line-height:200%"><span style="font-family:"Arial",sans-serif">  <span style="font-family:"Arial",sans-serif">  New Method for Effect Modification of Head/Neck Cancer Data: Results of Head Neck Cancer Study INHANCE and the category of never drinkers vs. never smokers were previously too small to be valid statistically according to Mia Hashibe. Thus, a larger category of this pool was added to study head neck cancer which is normally casually linked to cigarette smoking and drinking alcohol 75 percent [Mia Hashibe, et. al., 2007]. Per the check on assumptions for data distribution of head cancer from nondrinkers/nonsmokers (comprising 15.6 and 26.6 percent respectively of cases and controls for non-drinkers versus 10.5 and 37.9 percent of cases and controls for non-smokers from the pool from the study) and per race coded: 1 for non-Hispanic, 2 for black, and 3 for Hispanic, the chi-square is P <0.0001. The PROC AUTOREG (SAS) shows that the Durbin Watson Statistic is 3.23 indicating negative autocorrelation, hence the data is non-normal and involves random effects model. Heterogeneity is expected in this study, and regression may not be fixed or linear as a result. Effect modification is expected to exist for the outcome head/neck cancer for the level of race from the exposure of never drinking/never smoking based on P <0.0001. The risks for head/neck cancer for the three races (non-Hispanic, black, and Hispanic) vary by more than 10 percent for never drinkers vs. never smokers. The C statistic is .799 indicating a very good confidence for the results. The data is fairly large, over 11,500. The algorithm for effect modification program converged as indicated.

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