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Statistical Modeling - Princeton University

chapter 1 Statistical Statistical ModelsExample 1: ( sampling inspection). A lot containsNproducts with defective rate . Take a sample without replacement ofnproducts and getxdefective are the defective rates?Possible outcomes: GGDGGGDD , realization of do we connect the sample with the population?Modelling think of data as a realization of a the random 524: Statistical Modeling : Illustration of the sampling that a D = is large,a G = is Law: Under this physical experimentP(X=x) =(N x)(N N n x)(Nn),for max(0,n N(1 ))6x6min(n,N ). Convention:(n0)= 1,(nm)= 0 ifm > example,X/n and n(X/n ) N(0, (1 )).ORF 524: Statistical Modeling : unknown, space : the possible value of : ={0/N,1/N, ,N/N}or[0,1].For this specific example, the model comes from physical experiment.

Chapter 1 Statistical Modeling 1.1 Statistical Models Example 1: (Sampling inspection). A lot contains Nproducts with defective rate θ. Take a sample without replacement of nproducts and get xdefective products.

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Transcription of Statistical Modeling - Princeton University

1 chapter 1 Statistical Statistical ModelsExample 1: ( sampling inspection). A lot containsNproducts with defective rate . Take a sample without replacement ofnproducts and getxdefective are the defective rates?Possible outcomes: GGDGGGDD , realization of do we connect the sample with the population?Modelling think of data as a realization of a the random 524: Statistical Modeling : Illustration of the sampling that a D = is large,a G = is Law: Under this physical experimentP(X=x) =(N x)(N N n x)(Nn),for max(0,n N(1 ))6x6min(n,N ). Convention:(n0)= 1,(nm)= 0 ifm > example,X/n and n(X/n ) N(0, (1 )).ORF 524: Statistical Modeling : unknown, space : the possible value of : ={0/N,1/N, ,N/N}or[0,1].For this specific example, the model comes from physical experiment.

2 Now sup-pose thatN= 10,000,n= 100 andx= 2. Our problem becomes an inverseproblem: What is the value of ?Logically, if = 1%, it is possible to getx= 2. If = 2%, it is also possibleto getx= 2. If = , it is also possible to getx= 2. So, givenx= 2,we can not tell exactly which it is. Our conclusion can not be drawn withoutuncertainty. However, we do know some are more likely than the others and thedegree of uncertainty gets smaller, asngets large, : Statisticians think data as realizations from a stochastic model; this connectsORF 524: Statistical Modeling sample and parameters. Statistical conclusions can not be drawn without uncertainty, as we have only afinite sample. Probability is from a box to sample, while statistics is from a sample to a 2: A measurement model ( molecular weight, RNA/protein expres-sion level, fat-free weight).

3 An object is weighedntimes, with outcomesx1, , be the true weight. We think the observed data as realizations of randomvariablesX1, ,Xn, modeled asXi= + iwhere iis error of measurement ) iis independent of .ii) i,i= 1,2, ,nare 524: Statistical Modeling : Illustration of the idea of ) i,i= 1,2, ,nare identically ) the distribution of is continuous, withE( ) = 0; or specifically symmetricabout 0:f(y) =f( y) for , we assume further that i N(0, 2). Parameters in the model =( , 2), where 2is a nuisance a realizationx= (x1, ,xn) ofX= (X1, ,Xn), what is the value of ?ORF 524: Statistical Modeling , if = 100, it is possible to observex. If = 1, it is also possible toobservex. So we can not absolutely tell what value of is. But from the square-rootlaw:var( X) =E( X )2= , xis likely close to whennis : Distributions of individual observation versus that of averageExample 3: Drug evaluation (Hypertension drug)Drug A m patiets Drug B n patietsORF 524: Statistical Modeling : blood eliminate confounding factors, use randomized controlled experiment.

4 Hereare the hypothetical outcomes:Drug ADrug B150 110 160 187 153120 140 160 180 133 136x1x2x3x4x5y1y2y3y4y5y6To model the outcomes, a possible idealization is the following : Illustration of a two-sample problemDrug ADrug Brandom outcomesX1, ,XmY1 ,Ynrealizationsx1, ,xmy1, ,ynORF 524: Statistical Modeling , we might assume thatX1, , N( A, 2A)Y1, , N( B, 2B).We sometimes assume further A= B= .Parameters in the model: = ( A, B, A, B).Parameters of interest: = A Band possibly .Connection sample with population: data are realizations from a population,whose distribution depends on .Model diagnostics: Statistical models are idealizations, postulated by statisti-cians needed to be verified. For example, the data histograms should look liketheoretical distributions.

5 Two sample variances are about the same, formulationData:x= (x1, ,xn) are thought of the realization of a random vectorX=(X1, ,Xn).ORF 524: Statistical Modeling : The distribution ofXis assumed inP={P : }, is the : Inferences about . In Example 1:P (x) =(N x)(N N n x)(Nn),where ={0,1/N, ,N/N}or [0,1]. In Example 2:P (x) = ni=1 1 (xi )where ( ) is the normal density, ={( , ), >0, >0}. In Example 3:P (x) = mi=1 1A (xi A A) ni=1 1B (yi B B),where ( ) is the normal density, ={( A, B, A, B) : A, B, A, B>0}.ORF 524: Statistical Modeling Dataxor its random variableXcan include bothx- parameter doesn t have to be inRk. In Example 2, without the normalityassumption,P (x) = ni=1f(xi ),assuming that{ i,i= 1, ,n}are random variables with densityf.

6 Then, ={( ,f) : >0,fis symmetric}.Since no form offhas been imposed, not been parameterized, theparameter space is called nonparametric or assumption: Throughout this class, we will assume that(i) Continuous variables: AllP are continuous with densitiesp(x, ) or(ii) Discrete variable:AllP are discrete with frequency functionsp(x, ). Further,there exists a set{x1,x2, ,}such that i=1p(xi, ) = 1,wherexiis independent of .ORF 524: Statistical Modeling convenience, we will callp(x, ) as density in both of parameters: There are sometimes more than one way ofparameterization. In Example 3: writeX1, , N( + 1, 2)Y1, , N( + 2, 2). = ( , 1, 2, ). Hence,p (x,y, ) = mi=1 1 (xi 1 ) ni=1 1 (yi 2 ),If 1= (0,1,2,1) and 2= ( , , ,1), thenP 1=P 2. Thus, the parameters are not : The model{P , }is identifiable if 16= 2impliesP 16=P 4: (Regression Problem).

7 Suppose a sample of data{(xi1, ,xip,yi)}ni=1are collected ,x1=age,x2= year of experience,x3= job grade,x4= gender,x5= PC 524: Statistical Modeling wish to study the association betweenYandX1, ,Xp. How to predictYbased onX? Any gender discrimination? (Note: the dataxin the generalformulation now include all{(xi1, ,xip,yi)}ni=1). Model I: linear modelY= 0+ 1X1+ 2X2+ + 5X5+ , G,where is the part that can not be explained byX. Thus the parameter spaceis ={( 0, 1, , 5,G)}. Model II: semiparametric modelY= (X1,X2,X3) + 4X4+ 5X5+ .The parameter space is ={( ( ), 4, 5,G)}. Model III: nonparametric modelY= (X1, ,X5) + .ORF 524: Statistical Modeling parameter space is ={( ( ),G)}. Modeling : Data are thought of a realization from (Y,X1, ,X5) with the rela-tionship betweenXandYdescribed this example, the model is a convenient assumption made by data , Statistical models are frequently useful fictions.

8 There are trade-offs amongthe choice of Statistical models:larger model reducing model biases increasing estimation decision depends also available sample : a function of data only, + +Xnn, X1, X21+ X22+X23+ 3,butX1+ ,X+ are 524: Statistical Modeling : an estimating procedure for certain parameters, .Estimate: numerical value of an estimator when data are observed, 3,x=2 + 6 + 43= for all potential realizations, estimate for a realized : An estimator is an estimating procedure. The performance criteria for amethod is based on estimator, while Statistical decisions are based on estimate inreal Bayesian ModelsProbability: Two view points: long run relative frequency Frequentistprior knowledge w/brief BayesianORF 524: Statistical Modeling far, we have assumed no information about beyond that provided by , we can have some (vague) knowledge about.

9 For example, defective rate is 1% the distribution of DNA nucleotides is uniform, the intensity of an image is locally 1. (Continued) Based on past records, one can construct a distributionof defective rate ( ):P( =i/N) = i, i= 1,2, , provides as a prior distribution. The defective rate 0of the current lot isthought of as a realization from ( ). Given 0,P(X=x| 0) =(N 0x)(N N 0n x)(Nn),Basic element of Baysian modelsORF 524: Statistical Modeling : Bayesian Framework(i) The knowledge about is summarized by ( ) prior dist.(ii) A realization from ( ) serves as the parameter ofX.(iii) Given , the observed dataxare a realization ofp . The joint density of ( ,X)is ( )p(x| ).(iv) The goal of the Bayesian analysis is to modify the prior of after observingx: ( |X=x) = ( )p(x| ) ( )p(x| )d , continuous, ( )p(x| ) ( )p(x| ), summarizing the distribution by posterior mean, median and SD, 524: Statistical Modeling : Prior versus Posterior distributionsExample 5(Quality inspection) Suppose that from the past experience, the de-fective rate is about 10%.

10 Suppose that a lot consists of 100 products, whose qualityis independent of each 524: Statistical Modeling : Prior knowledge of the defectsThe prior distribution about the lot s defective rate is ( i) =P( = i) =(100i) i, i= mean and variance areE =EX100= ( ) =11002var(X) =100 ,SD( ) = 524: Statistical Modeling suppose thatn= 19 products are sampled andx= 10 are defective. Then ( i|X= 10) =P( = i,X= 10)P(X= 10)= ( i)P(X= 10| = i) j ( j)P(X= 10| = j). ( > |X= 10) =P(100 X>10|X= 10) 1 (10 81 81 ) 30%.(100 Xis the number of defective left after 19 draws, having distribution Bernoulli(81, )). Compared with the prior probabilityP( > ) =P(100 >20)= 1 (20 100 100 ) ,where 100 Bernoulli(100, ).Example 6. Suppose thatX1, ,Xnare random variables with Bernoulli( )and has a prior distribution ( ).


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