STA 517 – Introduction: Distribution and Inference 1 Discrete data q Basic data are discretely measured responses such as § counts, § proportions, § nominal, § ordinal, § discrete variables with a few values, § continuous variables grouped into a small number of categories, etc. q We illustrate theoretical results by data examples. q We will use SAS package for this class
STA 517 – Introduction: Distribution and Inference 2 Theory q Multivariate analysis of discrete data that is the underlying theory of such analysis q Topics § Basic principles of statistical methods § Analysis of Poisson counts § Cross-classified table of counts (contingency tables) § Success/failure records
STA 517 – Introduction: Distribution and Inference 3 Problems q Describe and understand the structure of a discrete multivariate distribution q A sort of “generalization” of regression with a distinction between response and explanatory variables where response is discrete § Predictors can be all discrete, or mixture of discrete and continuous variable § Log-linear model § Logistic regression
STA 517 – Introduction: Distribution and Inference Topics 1. Introduction: Distributions and Inference for Categorical Data 2. Describing Contingency Tables 3. Inference for Contingency Tables 4. Introduction to Generalized Linear Models 5. Logistic Regression 6. Building and Applying Logistic Regression Models 7. Logit Models for Multinomial Responses 4
STA 517 – Introduction: Distribution and Inference Chapter 1 Example 5
STA 517 – Introduction: Distribution and Inference Chapter 1 - Outline 1. 1 Categorical Response Data 1. 2 Distributions for Categorical Data 1. 3 Statistical Inference for Categorical Data 1. 4 Statistical Inference for Binomial Parameters 1. 5 Statistical Inference for Multinomial Parameters 6
STA 517 – Introduction: Distribution and Inference 1. 1 CATEGORICAL RESPONSE DATA q A categorical variable has a measurement scale consisting of a set of categories. § political philosophy: liberal, moderate, or conservative. § brands of a product: brand A, brand B, and brand C q A categorical variable can be a response variable or independent variable q We consider primarily the CATEGORICAL RESPONSE DATA in this course 7
STA 517 – Introduction: Distribution and Inference 8 1. 1. 1 Response–Explanatory Variable Distinction q Most statistical analyses distinguish between response (or dependent) variables and explanatory (or independent) variables. q For instance, regression models: selling price of a house = f(square footage, location) q In this book we focus on methods for categorical response variables. q As in ordinary regression, explanatory variables can be of any type.
STA 517 – Introduction: Distribution and Inference 9 1. 1. 2 Nominal–Ordinal Scale Distinction q Nominal: Variables having categories without a natural ordering § religious affiliation: Catholic, Protestant, Jewish, Muslim, other. § mode of transportation: automobile, bicycle, bus, subway, walk § favorite type of music: classical, country, folk, jazz, rock § choice of residence: apartment, condominium, house, other. q For nominal variables, the order of listing the categories is irrelevant. q The statistical analysis does not depend on that ordering.
STA 517 – Introduction: Distribution and Inference 10 Nominal or Ordinal q Ordinal: ordered categories § automobile: subcompact, midsize, large § social class: upper, middle, lower § political philosophy: liberal, moderate, conservative § patient condition: good, fair, serious, critical. q Ordinal variables have ordered categories, but distances between categories are unknown. q Although a person categorized as moderate is more liberal than a person categorized as conservative, no numerical value describes how much more liberal that person is. Methods for ordinal variables utilize the category ordering.
STA 517 – Introduction: Distribution and Inference 11 Interval variable q An interval variable is one that does have numerical distances between any two values. § blood pressure level § functional life length of television set § length of prison term § annual income q An internal variable is sometimes called a ratio variable if ratios of values are also valid. It has a clear definition of 0: § Height § Weight § enzyme activity
STA 517 – Introduction: Distribution and Inference 12 categories are not as clear cut as they sound q What kind of variable is color? § In a psychological study of perception, different colors would be regarded as nominal. § In a physics study, color is quantified by wavelength, so color would be considered a ratio variable. q What about counts? § If your dependent variable is the number of cells in a certain volume, what kind of variable is that. It has all the properties of a ratio variable, except it must be an integer. § Is that a ratio variable or not? These questions just point out that the classification scheme is appears to be more comprehensive than it is
STA 517 – Introduction: Distribution and Inference 13 q A variable’s measurement scale determines which statistical methods are appropriate. q In the measurement hierarchy, § interval variables are highest, § ordinal variables are next, § and nominal variables are lowest. q Statistical methods for variables of one type can also be used with variables at higher levels but not at lower levels. q For instance, statistical methods for nominal variables can be used with ordinal variables by ignoring the ordering of categories. q Methods for ordinal variables cannot, however, be used with nominal variables, since their categories have no meaningful ordering. q It is usually best to apply methods appropriate for the actual scale.
STA 517 – Introduction: Distribution and Inference 14 1. 1. 3 Continuous–Discrete Variable Distinction q according to the number of values they can take q Actual measurement of all variables occurs in a discrete manner, due to precision limitations in measuring instruments. q The continuous / discrete classification, in practice, distinguishes between variables that take lots of values and variables that take few values. q Statisticians often treat discrete interval variables having a large number of values, such as test scores, as continuous
STA 517 – Introduction: Distribution and Inference 15 This class: Discretely measured responses can be: q Binary (two categories) q nominal variables (unordered) q ordinal variables (ordered) q discrete interval variables having relatively few values, and q continuous variables grouped into a small number of categories.
STA 517 – Introduction: Distribution and Inference 1. 1. 4 Quantitative–Qualitative Variable Distinction q Nominal variables are qualitative distinct categories differ in quality, not in quantity. q Interval variables are quantitative distinct levels have differing amounts of the characteristic of interest. q The position of ordinal variables in the quantitative or qualitative classification is fuzzy. 16
STA 517 – Introduction: Distribution and Inference 17 q Analysts often utilize the quantitative nature of ordinal variables by assigning numerical scores to categories or assuming an underlying continuous distribution. q This requires good judgment and guidance from researchers who use the scale, but it provides benefits in the variety of methods available for data analysis.
STA 517 – Introduction: Distribution and Inference Summary q Continuous variable § Ratio § Interval § Discrete q Categorical § Binary § Ordinal § Nominal 18
STA 517 – Introduction: Distribution and Inference 19 Calculation: OK to compute. . Nominal Ordinal Interval Ratio frequency distribution Yes Yes median and percentiles No Yes Yes add or subtract No No Yes mean, standard deviation, standard error of the mean No No Yes ratio, or coefficient of variation No No No Yes
STA 517 – Introduction: Distribution and Inference Example 1: Grades measured q pass/fail q A, B, C, D, F q 3. 2, 4. 1, 5. 0, 2. 1, … 20
STA 517 – Introduction: Distribution and Inference Example 2 o Did you get a flu? (Yes or No) – is a binary nominal categorical variable o What was the severity of your flu? (low, medium, or high) – is an ordinal categorical variable q Context is important. The context of the study and corresponding questions are important in specifying what kind of variable we will analyze. 21
STA 517 – Introduction: Distribution and Inference 22 1. 2 DISTRIBUTIONS FOR CATEGORICAL DATA q Inferential data analyses require assumptions about the random mechanism that generated the data. q For continuous variable, Normal distribution q For categorical variable § Binomial § hypergeometric distribution § Multinomial § Poisson
STA 517 – Introduction: Distribution and Inference 23 Overview of probability and inference probability Observed Data data generating process inference q The basic problem we study in probability: Given a data generating process, what are the properties of the outcomes? q The basic problem of statistical inference: Given the outcomes (data), what we can say about the process that generated the data?
STA 517 – Introduction: Distribution and Inference 24 Random variable q A random variable is the outcome of an experiment (i. e. a random process) expressed as a number. q We use capital letters near the end of the alphabet (X, Y , Z, etc. ) to denote random variables. q Just like variables, probability distributions can be classified as discrete or continuous.
STA 517 – Introduction: Distribution and Inference 25 Continuous Probability Distributions q If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution. q A continuous probability distribution differs from a discrete probability distribution in several ways. § The probability that a continuous random variable will assume a particular value is zero. § As a result, a continuous probability distribution cannot be expressed in tabular form. § Instead, an equation or formula is used to describe a continuous probability distribution.
STA 517 – Introduction: Distribution and Inference 26 Normal q Most often, the equation used to describe a continuous probability distribution is called a probability density function. Sometimes, it is referred to as a density function, or a PDF. q Normal N(µ, 2) PDF
STA 517 – Introduction: Distribution and Inference Chi-square distribution, PDF 27
STA 517 – Introduction: Distribution and Inference 28 Discrete random variables q A discrete random variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, . . . . q Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete. q Examples: § the number of children in a family § the Friday night attendance at a cinema § the number of patients in a doctor's surgery § the number of defective light bulbs in a box of ten.
STA 517 – Introduction: Distribution and Inference 29 discrete random variable q The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It is also sometimes called the probability function or the probability mass function. q Suppose a random variable X may take k different values, with the probability that X = xi defined to be P(X = xi) = pi. The probabilities pi must satisfy the following: § 0 < pi < 1 for each i § p 1 + p 2 +. . . + pk = 1.
STA 517 – Introduction: Distribution and Inference Example q Suppose a variable X can take the values 1, 2, 3, or 4. The probabilities associated with each outcome are described by the following table: Outcome 1 2 3 4 Probability 0. 1 0. 3 0. 4 0. 2 q The probability that X is equal to 2 or 3 is the sum of the two probabilities: P(X = 2 or X = 3) = P(X = 2) + P(X = 3) = 0. 3 + 0. 4 = 0. 7. q Similarly, the probability that X is greater than 1 is equal to 1 - P(X = 1) = 1 - 0. 1 = 0. 9, by the complement rule. q This distribution may also be described by the probability histogram shown to the right 30
STA 517 – Introduction: Distribution and Inference Properties q E(X)= x f(x) q var(X)= (x-E(X))2 f(x) q If the distribution depends on unknown parameters we write it as f(x; ) or f(x | ) 31
STA 517 – Introduction: Distribution and Inference 32 1. 2. 0 Bernoulli Distribution q the Bernoulli distribution is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability 1 − . So if X is a random variable with this distribution, we have: or write it as Then
STA 517 – Introduction: Distribution and Inference 33 1. 2. 1 Binomial Distribution q Many applications refer to a fixed number n of binary observations. q Let y 1 , y 2 , . . . , yn denote responses for n independent and identical trials (Bernoulli trials) q Identical trials means that the probability of success is the same for each trial. q Independent trials means that the Yi are independent random variables.
STA 517 – Introduction: Distribution and Inference q The total number of successes q has the binomial distribution with index n and parameter , denoted by bin(n, ) q The probability mass function where 34
STA 517 – Introduction: Distribution and Inference 35
STA 517 – Introduction: Distribution and Inference Moments Because Yi=1 or 0, Yi=Yi 2 E(Yi)=E(Yi 2)=1 x + 0 x (1 - )= Skewness: 36
STA 517 – Introduction: Distribution and Inference The distribution converges to normality as n increases 37
STA 517 – Introduction: Distribution and Inference 38
STA 517 – Introduction: Distribution and Inference 39 1. 2. 2 Multinomial Distribution Multiple possible outcomes q Suppose that each of n independent, identical trials can have outcome in any of c categories. if trial i has outcome in category j = 0 otherwise q represents a multinomial trial, with q Let denote the number of trials having outcome in category j. q The counts have the multinomial distribution.
STA 517 – Introduction: Distribution and Inference pdf: 40
STA 517 – Introduction: Distribution and Inference 41 1. 2. 3 Poisson Distribution q count data do not result from a fixed number of trials. § y=number of deaths due to automobile accidents on motorways in Italy q y>0 q Poisson probability mass function (Poisson 1837) q It satisfies q Skewness
STA 517 – Introduction: Distribution and Inference Poisson 42
STA 517 – Introduction: Distribution and Inference 43 Poisson Distribution q used for counts of events that occur randomly over time or space, when outcomes in disjoint periods or regions are independent. q an approximation for the binomial when n is large and is small with µ=n q For example, § n=50 million driving in Italy § death rate/week =0. 000002 § the number of deaths is bin(n, ) § Or approximately Poisson with µ=n =100
STA 517 – Introduction: Distribution and Inference 44 1. 2. 4 Overdispersion q A key feature of the Poisson distribution is that its variance equals its mean. § Sample counts vary more when their mean is higher. q Overdispersion: Count observations often exhibit variability exceeding that predicted by the binomial or Poisson.
STA 517 – Introduction: Distribution and Inference 45 1. 2. 5 Connection between Poisson and Multinomial Distributions q Example, In Italy this next week, let § y 1=# of people who die in automobile accidents § y 2=number who die in airplane accidents § y 3=number who die in railway accidents q (Y 1, Y 2, Y 3) ~ independent Poisson ( ) q The total ~ Poisson ( ) q Here n is random variable rather than fixed q If n is given, (Y 1, Y 2, Y 3) is no longer independent and Poisson, WHY
STA 517 – Introduction: Distribution and Inference conditional distribution given that let ~ multinomial distribution 46
STA 517 – Introduction: Distribution and Inference multinomial distribution vs. Poisson distribution q Many categorical data analyses assume a multinomial distribution. q Such analyses usually have the same parameter estimates as those of analyses assuming a Poisson distribution, because of the similarity in the likelihood functions. 47
STA 517 – Introduction: Distribution and Inference 1. 3 STATISTICAL INFERENCE FOR CATEGORICAL DATA(general) q Once you choose the distribution of the categorical variable, you need to estimate the parameters in the distribution q We first review general method § Point estimate § Confidence interval q Section 1. 4 MLE for binomial q Section 1. 5 MLE for multinominal 48
STA 517 – Introduction: Distribution and Inference 49 Likelihood q Likelihood is a tool for summarizing the data's evidence about unknown parameters. Let us denote the unknown parameter(s) of a distribution generically by . q If we observe a random variable X = x from distribution f (x| ), then the likelihood associated with x, l( |x), is simply the distribution f (x| ) regarded as a function of with x fixed. q For example, if we observe x from bin(n; ), the likelihood function is
STA 517 – Introduction: Distribution and Inference 50 Likelihood q The formula for the likelihood looks similar algebraically to the f (x| ) but the distinction should be clear! q The distribution function is defined over the support of discrete variable x with given, whereas the likelihood is defined over the continuous parameter space for . q Consequently, a graph of the likelihood usually looks different from a graph of the probability distribution. q In most cases, we work with loglikelihood
STA 517 – Introduction: Distribution and Inference Loglikelihood function bin(5, ) and we observe x=0, x=1, and x=2 bin(842+982, ) x=842 (yes) 51
STA 517 – Introduction: Distribution and Inference Likelihood q In many problems of interest, we will derive our loglikelihood from a sample rather than from a single observation. If we observe an independent sample x 1, x 2, …, xn from a distribution f (x| ), then the overall likelihood is the product of the individual likelihoods: and loglikelihood is 52
STA 517 – Introduction: Distribution and Inference 53 Log likelihood q In regular problems, as the total sample size n grows, the loglikelihood function does two things: (a) it becomes more sharply peaked around its maximum, and (b) its shape becomes nearly quadratic q the loglikelihood for a normal-mean problem is exactly quadratic. q That is, if we observe y 1, . . . , yn from a normal population with known variance, the loglikelihood is or in multi-dimension
STA 517 – Introduction: Distribution and Inference MLE (maximum likelihood estimation) 54 q ML estimate for θ is the maximizer of L(θ) or, equivalently, the maximizer of l(θ). This is the parameter value under which the data observed have the highest probability of occurrence. q In regular problems, the ML estimate can be found by setting to zero the first derivative(s) of l(θ) with respect to θ.
STA 517 – Introduction: Distribution and Inference 55 Transformations of parameters q If l(θ) is a likelihood and φ = g(θ) is a one-to-one function of the parameter with back-transformation θ = g− 1(φ), then we can express the likelihood in terms of φ as l( g− 1(φ) ). q Transformations may help us to improve the shape of the loglikelihood. q If the parameter space for θ has boundaries, we may want to choose a transformation to the entire real space. q For example, consider the binomial loglikelihood, L
STA 517 – Introduction: Distribution and Inference binomial loglikelihood q If we apply the logit transformation q whose back-transformation is q the loglikelihood in terms of β is L 56
STA 517 – Introduction: Distribution and Inference q If we observe y = 1 from a binomial with n = 5, the loglikelihood in terms of β looks like this. 57
STA 517 – Introduction: Distribution and Inference 58 q Transformations do not affect the location of the maximum-likelihood (ML) estimate. q If l(θ) is maximized at ˆθ, then l(φ) is maximized at ˆφ = g(ˆθ).
STA 517 – Introduction: Distribution and Inference 59 score function q A first derivative of L(θ) with respect to θ is called a score function or simply a score. q In a one-parameter problem, the score function from an independent sample y 1, . . . , yn is L where is the score contribution for yi. q The ML estimate is usually the solution of the likelihood equation L’(θ)=0.
STA 517 – Introduction: Distribution and Inference 60 Mean of the score function. q A well known property of the score is that it has mean zero. q The score is an expression that involves both the parameter θ and the data Y. Because it involves Y , we can take its expectation with respect to the data distribution f(y|θ). The expected score is no longer a function of Y , but it’s still a function of θ. If we evaluate this expected score at the “true value” of θ— that is, at the same value of θ assumed when we took the expectation—we get zero: If certain differentiability conditions are met, the integral may be rewritten as
STA 517 – Introduction: Distribution and Inference 61 q For example, in the case of the binomial proportion, we have which is zero because E(Y ) = n. q If we apply a one-to-one transformation to the parameter φ = g(θ), then the score function with respect to the new parameter φ also has mean zero.
STA 517 – Introduction: Distribution and Inference 62 Estimating functions. q This property of the score function—that it has an expectation of zero when evaluated at the true parameter θ—is a key to the modern theory of statistical estimation. q In the original theory of likelihood-based estimation, as developed by R. A. Fisher and others, the ML estimate ˆθ is viewed as the value of the parameter that, under the parametric model, that makes the observed data most likely. q statisticians have begun to view ˆθ as the solution the score equation(s). That is, we now often view an ML estimate as the solution to L’(θ)=0
STA 517 – Introduction: Distribution and Inference estimating equations q Any function of the data and the parameters having mean zero at the true θ has this property as well. Functions having the mean-zero property are called estimating functions. q Setting the estimating functions to zero is called the estimating equations. q In the case of the binomial proportion, for example, Y − n is a mean-zero estimating function, and so is − 1 [Y − n ]. 63
STA 517 – Introduction: Distribution and Inference Information and variance estimation. q The variance of the score is known as the Fisher information. In the case of a single parameter, the Fisher information is q If θ has k parameters, the Fisher information is the k x k covariance matrix for scores 64
STA 517 – Introduction: Distribution and Inference 65 q Like the score function, the Fisher information is also a function of θ. So we can evaluate it at any given value of θ. q Notice that i(θ) as we defined it is the square of a sum which, in many problems, can be messy. q To actually compute the Fisher information, we usually make use of the well known identity
STA 517 – Introduction: Distribution and Inference q In the multiparameter case, l(θ) is the k x k matrix of second derivatives whose (l, m)th element is 66
STA 517 – Introduction: Distribution and Inference q why we care about the Fisher information? q it provides us with a way (several ways, actually) of assessing the uncertainty in the ML estimate. q It is well known that, in regular problems, ˆθ is approximately normally distributed about the true θ with variance given by the reciprocal (or, in the multiparameter case, the matrix inverse) of the Fisher information. 67
STA 517 – Introduction: Distribution and Inference 68 two common ways to approximate the variance of ˆθ. q The first way is to plug ˆθ into i(θ) and invert, this is commonly called the “expected information. ” q The second way is to invert (minus one times) the actual second derivative of the loglikelihood at θ = ˆθ, this is commonly called the “observed information. ”
STA 517 – Introduction: Distribution and Inference 1. 3. 2 Likelihood Function and ML Estimate for Binomial Parameter q The binomial log likelihood is q Differentiating with respect to yields q Equating this to 0 gives the likelihood equation, which has solution the sample proportion of successes for the n trials. 69
STA 517 – Introduction: Distribution and Inference q Calculating we get , taking the expectation, and q Thus, the asymptotic variance of is q Actually, from E(Y)=n and var(Y)=n (1 - ), the distribution if =Y/n has mean and standard error 70
STA 517 – Introduction: Distribution and Inference Likelihood function and MLE summary q We use maximum likelihood estimate (MLE) § asymptotically normal § asymptotically consistent § asymptotically efficient q Likelihood function § probability of those data, treated as a function of the unknown parameter. q maximum likelihood (ML) estimate § parameter value that maximizes this function 71
STA 517 – Introduction: Distribution and Inference 72 MLE and its variance q If y 1; y 2; … ; yn is a random sample from distribution f(y| ), then the score function is q In regular problems, we can find the ML estimate by setting the score function(s) to zero and solving for . q The equations L’(θ)=0 are called the score equations. More generally, they can be called estimating equations because their solution is the estimate for θ. q We defined the Fisher information as the variance of the score function and
STA 517 – Introduction: Distribution and Inference 73 1. 3. 3 Wald–Likelihood Ratio–Score Test Triad q Three standard ways exist to use the likelihood function to perform large-sample inference. § Wald test § Score test § Likelihood ratio test q We introduce these for a significance test of a null hypothesis H 0: and then discuss their relation to interval estimation. q They all exploit the large-sample normality of ML estimators.
STA 517 – Introduction: Distribution and Inference 74 Wald test q With nonnull standard error SE of , the test statistic has an approximate standard normal distribution when q One- or two-sided P-value by z. q Or z 2 has a chi-squared null distribution with 1 df The P-value is then the right-tailed chi-squared probability above the observed value This type of statistic, using the nonnull standard error, is called a Wald statistic (Wald 1943).
STA 517 – Introduction: Distribution and Inference Wald test q For an. 05 -level two-side test, we reject H 0 if q Equivalently, if where 3. 84 is the 95 th percentile of 2(1). 75
STA 517 – Introduction: Distribution and Inference 76 Wald test q The multivariate extension for the Wald test of has test statistic where matrix. is the inverse matrix of Information q W is an asymptotic chi-squared distribution with df = rank of. q Wald test is not invariant to transformations. That is, a Wald test on a transformed parameter φ= g( ) may yield a different p-value than a Wald test on the original scale.
STA 517 – Introduction: Distribution and Inference 77 Likelihood ratio test q uses the likelihood function through the ratio of two maximizations: (1). the maximum over the possible parameter values under H 0 (2). the maximum over the larger set of parameter values permitting H 0 or an alternative Ha to be true. q The likelihood-ratio test statistic equals where L 0 and L 1 denote the maximized log-likelihood functions. q is 2 distribution with df=dim(Ha U H 0)-dim(H 0) q Reject H 0 if > 2 ( =0. 05)
STA 517 – Introduction: Distribution and Inference Score test q The score test is based on the slope and expected curvature of the log-likelihood function L( ) at the null value 0. q Score function The value tends to be larger in absolute value when is farther from 0. q Score statistic has an approximate standard normal null distribution. q The chi-squared form of the score statistic is 78
STA 517 – Introduction: Distribution and Inference 79 Why is score statistic reasonable? q Recall that the mean of the score is zero and its variance is equal to the Fisher information. q In a large sample, the score will also be approximately normally distributed because it's a sum of iid random variables. q Therefore, it will behave like a squared standard normal [ 2(1)] if H 0 is true.
STA 517 – Introduction: Distribution and Inference Wald–Likelihood Ratio–Score Test 80
STA 517 – Introduction: Distribution and Inference 81 q The three test statistics - Wald, LR and score asymptotically equivalent. q The differences among them vanish in large samples if the null hypothesis is true. q If the null hypothesis is false, they may take very different values. But in that case, all the test statistics will be large, the p-values will be essentially zero, and they will all lead us to reject H 0. q Score test does not require to calculate MLE. q LR test is scale-invariant. q LR statistic uses the most information of the three types of test statistic and is the most versatile.
STA 517 – Introduction: Distribution and Inference 82 1. 3. 4 Constructing confidence intervals q In practice, it is more informative to construct confidence intervals for parameters than to test hypotheses about their values. q For any of the three test methods, a confidence interval results from inverting the test. For instance, a 95% confidence interval for is the set of 0 for which the test of H 0: has a P-value exceeding 0. 05. q Let denote the z-score from the standard normal distribution having right-tailed probability a; this is the 100(1 -a) percentile of that distribution. q Let denote the 100(1 -a) percentile of the chisquared distribution with degrees of freedom df.
STA 517 – Introduction: Distribution and Inference Tests and Confidence Intervals At significant level , 100(1 - )% reject H 0: confidence interval , if 83
STA 517 – Introduction: Distribution and Inference Confidence Intervals q The Wald confidence interval is most common in practice because it is simple to construct using ML estimates and standard errors reported by statistical software. q The likelihood-ratio-based interval is becoming more widely available in software and is preferable for categorical data with small to moderate n. q For the best known statistical model, regression for a normal response, the three types of inference necessarily provide identical results. 84
STA 517 – Introduction: Distribution and Inference 1. 4 STATISTICAL INFERENCE FOR BINOMIAL PARAMETERS q Recall log likelihood q Score function q MLE SE= 85
STA 517 – Introduction: Distribution and Inference 1. 4. 1 Tests about a Binomial Parameter q Since H 0 has a single parameter, we use the normal rather than chi-squared forms of Wald and score test statistics. They permit tests against one-sided as well as two-sided alternatives. q Wald statistic q Evaluating the binomial score and information at 0 q The normal form of the score statistic simplifies to 86
STA 517 – Introduction: Distribution and Inference q binomial log-likelihood under H 0 q Under Ha q The likelihood-ratio test statistic or has an asymptotic chi-squared distribution with df=1. 87
STA 517 – Introduction: Distribution and Inference Test q At significant level , two sided, reject H 0, if (Wald test) (Score test) (LR test) 88
STA 517 – Introduction: Distribution and Inference 89 1. 4. 2 Confidence Intervals for a Binomial Parameter q Inverting the Wald test, q Unfortunately, it performs poorly unless n is very large q The actual coverage probability usually falls below the nominal confidence coefficient, much below when is near 0 or 1. q An adjustment is needed. (Problem 1. 24)
STA 517 – Introduction: Distribution and Inference Simulation to calculate coverage prob. %let n=1000; %let pi=0. 5; %let simu. N=10000; data simu; drop i; do i=1 to &simu. N; k=RAND('BINOMIAL', &pi, &n); output; end; run; data res; set simu; pihat=k/&n; lci=pihat-1. 96*sqrt(pihat*(1 -pihat)/&n); uci=pihat+1. 96*sqrt(pihat*(1 -pihat)/&n); if lci>&pi or uci<&pi then cover=0; else cover=1; proc sql; select sum(cover)/&simu. N as coverageprobabilty from res; 90
STA 517 – Introduction: Distribution and Inference 91 it performs poorly if 1) n is small; 2) pi near 0 or 1. q %let n=1000; %let pi=0. 5; %let simu. N=10000; n=20; %let pi=0. 1; %let simu. N=10000; n=20; %let pi=0. 9; %let simu. N=10000;
STA 517 – Introduction: Distribution and Inference An adjustment is needed. (Problem 1. 24) 92
STA 517 – Introduction: Distribution and Inference %let n=20; %let pi=0. 5; %let simu. N=10000; data simu; drop i; do i=1 to &simu. N; k=RAND('BINOMIAL', &pi, &n); output; end; run; data res; set simu; pihat=(k+1. 96**2/2)/(&n+1. 96*1. 96); lci=pihat-1. 96*sqrt(pihat*(1 -pihat)/(&n+1. 96*1. 96)); uci=pihat+1. 96*sqrt(pihat*(1 -pihat)/(&n+1. 96*1. 96)); if lci>&pi or uci<&pi then cover=0; else cover=1; proc sql; select sum(cover)/&simu. N as coverageprobabilty from res; 93
STA 517 – Introduction: Distribution and Inference score confidence interval q The score confidence interval contains 0 values for which q Its endpoints are the 0 solutions to the equations q It is quadratic in 0. This interval is 94
STA 517 – Introduction: Distribution and Inference 95 LR-based confidence interval q The likelihood-ratio-based confidence interval is more complex computationally, but simple in principle. q It is the set of 0 for which the likelihood ratio test has a P-value exceeding . q Equivalently, it is the set of 0 for which double the log likelihood drops by less than from its value at the ML estimate.
STA 517 – Introduction: Distribution and Inference 1. 4. 3 Proportion of Vegetarians Example 96
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