1 sta 517 – introduction: distribution and inference discrete data basic data are discretely...

1STA 517 – Introduction: Distribution and InferenceSTA 517 – Introduction: Distribution and Inference

Discrete data

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. We illustrate the theoretical results by data examples. We will use SAS package for this class


Theory

Multivariate analysis of discrete data that is the underlying theory of such analysis

Topics Basic principles of statistical methods Analysis of Poisson counts Cross-classified table of counts (contingency tables) Success/failure records


Problems

Describe and understand the structure of a discrete multivariate distribution

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


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


Chapter 1 Example


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


1.1 CATEGORICAL RESPONSE DATA

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

A categorical variable can be a response variable or independent variable

We consider primarily the CATEGORICAL RESPONSE DATA in this course


1.1.1 Response–Explanatory Variable Distinction

Most statistical analyses distinguish between response (or dependent) variables and explanatory (or independent) variables.

For instance, regression models: selling price of a house = f(square footage, location)

In this book we focus on methods for categorical response variables.

As in ordinary regression, explanatory variables can be of any type.


1.1.2 Nominal–Ordinal Scale Distinction

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. For nominal variables, the order of listing the categories

is irrelevant. The statistical analysis does not depend on that

ordering.


Nominal or Ordinal

Ordinal: ordered categories automobile: subcompact, compact, midsize, large social class: upper, middle, lower political philosophy: liberal, moderate, conservative patient condition: good, fair, serious, critical.

Ordinal variables have ordered categories, but distances between categories are unknown.

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.


Interval variable

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

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


categories are not as clear cut as they sound

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.

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


A variable’s measurement scale determines which statistical methods are appropriate.

In the measurement hierarchy, interval variables are highest, ordinal variables are next, and nominal variables are lowest.

Statistical methods for variables of one type can also be used with variables at higher levels but not at lower levels.

For instance, statistical methods for nominal variables can be used with ordinal variables by ignoring the ordering of categories.

Methods for ordinal variables cannot, however, be used with nominal variables, since their categories have no meaningful ordering.

It is usually best to apply methods appropriate for the actual scale.


1.1.3 Continuous–Discrete Variable Distinction

according to the number of values they can take Actual measurement of all variables occurs in a discrete

manner, due to precision limitations in measuring instruments.

The continuous / discrete classification, in practice, distinguishes between variables that take lots of values and variables that take few values.

Statisticians often treat discrete interval variables having a large number of values, such as test scores, as continuous


This class: Discretely measured responses can be:

Binary (two categories) nominal variables (unordered) ordinal variables (ordered) discrete interval variables having relatively few values,

and continuous variables grouped into a small number of

categories.


1.1.4 Quantitative–Qualitative Variable Distinction

Nominal variables are qualitative distinct categories differ in quality, not in quantity.

Interval variables are quantitative distinct levels have differing amounts of the characteristic of interest.

The position of ordinal variables in the quantitative or qualitative classification is fuzzy.


Analysts often utilize the quantitative nature of ordinal variables by assigning numerical scores to categories or assuming an underlying continuous distribution.

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.


Summary

Continuous variable Ratio Interval Discrete

Categorical Binary Ordinal Nominal


Calculation:

OK to compute.... Nominal Ordinal Interval Ratio

frequency distribution Yes Yes Yes Yes

median and percentiles No Yes Yes Yes

add or subtract No No Yes Yes

mean, standard deviation, standard error of the mean

No No Yes Yes

ratio, or coefficient of variation

No No No Yes


Example1: Grades measured

pass/fail A,B,C,D,F 3.2, 4.1, 5.0, 2.1, …


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

Context is important. The context of the study and corresponding questions are important in specifying what kind of variable we will analyze.


1.2 DISTRIBUTIONS FOR CATEGORICAL DATA

Inferential data analyses require assumptions about the random mechanism that generated the data.

For continuous variable, Normal distribution For categorical variable

Binomial hypergeometric distribution Multinomial Poisson


Overview of probability and inference

The basic problem we study in probability: Given a data generating process, what are the properties of the outcomes?

The basic problem of statistical inference: Given the outcomes (data), what we can say about the process that generated the data?

Observed data

Data generating process

probability

inference


Random variable

A random variable is the outcome of an experiment (i.e. a random process) expressed as a number.

We use capital letters near the end of the alphabet (X, Y , Z, etc.) to denote random variables.

Just like variables, probability distributions can be classified as discrete or continuous.


Continuous Probability Distributions

If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution.

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.


Normal

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.

Normal N(µ, 2) PDF

}2

)(exp{

2

1),;(

2

2

2

2

x

xf


Chi-square distribution, PDF


Discrete random variables

A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........

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.

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.


discrete random variable

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.

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

p1 + p2 + ... + pk = 1.


Example 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

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.

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.

This distribution may also be described by the probability histogram shown to the right


Properties

E(X)= x f(x) var(X)= (x-E(X))2 f(x)

If the distribution depends on unknown parameters we write it as f(x; ) or f(x | )


1.2.0 Bernoulli Distribution

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


1.2.1 Binomial Distribution

Many applications refer to a fixed number n of binary observations.

Let y1 , y2 , . . . , yn denote responses for n independent and identical trials (Bernoulli trials)

Identical trials means that the probability of success is the same for each trial.

Independent trials means that the Yi are independent random variables.


The total number of successes

has the binomial distribution with index n and parameter , denoted by bin(n,)

The probability mass function

where


0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35binomial pdf bin(25, )

=0.10=0.25=0.50


Moments

Because Yi=1 or 0, Yi=Yi2

E(Yi)=E(Yi2)=1 x + 0 x (1-)=

Skewness:


The distribution converges to normality as n increases

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Binomial(25, 0.25)

Normal


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Binomial(5, 0.25)

Normal(1.25,0.96825)


1.2.2 Multinomial DistributionMultiple possible outcomes

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

represents a multinomial trial, with

Let denote the number of trials having outcome in category j.

The counts have the multinomial distribution.


pdf:


1.2.3 Poisson Distribution

count data do not result from a fixed number of trials. y=number of deaths due to automobile accidents on

motorways in Italy y>0 Poisson probability mass function (Poisson 1837)

It satisfies

Skewness


Poisson

0 5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18 =5

=10

=15


Poisson Distribution

used for counts of events that occur randomly over time or space, when outcomes in disjoint periods or regions are independent.

an approximation for the binomial when n is large and is small with µ=n

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


1.2.4 Overdispersion

A key feature of the Poisson distribution is that its variance equals its mean. Sample counts vary more when their mean is

higher. Overdispersion: Count observations often exhibit

variability exceeding that predicted by the binomial or Poisson.


1.2.5 Connection between Poisson and Multinomial Distributions

Example, In Italy this next week, let

y1=# of people who die in automobile accidents

y2=number who die in airplane accidents

y3=number who die in railway accidents

(Y1, Y2, Y3) ~ independent Poisson ( ) The total ~ Poisson ( ) Here n is random variable rather than fixed If n is given, (Y1, Y2, Y3) is no longer independent and

Poisson, WHY


conditional distribution given that

let

~ multinomial distribution


multinomial distributionvs. Poisson distribution Many categorical data analyses assume a multinomial

distribution. Such analyses usually have the same parameter

estimates as those of analyses assuming a Poisson distribution, because of the similarity in the likelihood functions.


1.3 STATISTICAL INFERENCE FOR CATEGORICAL DATA(general)

Once you choose the distribution of the categorical variable, you need to estimate the parameters in the distribution

We first review general method Point estimate Confidence interval

Section 1.4 MLE for binomial Section 1.5 MLE for multinominal


Likelihood

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 .

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.

For example, if we observe x from bin(n; ), the likelihood function is

xnx

x

nxl

)1()|(


Likelihood

The formula for the likelihood looks similar algebraically to the f (x|) but the distinction should be clear!

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 .

Consequently, a graph of the likelihood usually looks different from a graph of the probability distribution.

In most cases, we work with loglikelihood

)|(log)|( xlxL

)1log()(log

)1log()(loglog)|(log)|(

xnx

xnxx

nxlxL


Loglikelihood function bin(5,) and we observe x=0, x=1, and x=2

0 0.5 1-30

-20

-10

0

l (

| x

)

l ( | x )=x log +(n-x) log (1-)

0 0.5 1-40

-30

-20

-10

0

l (

| x

)

l ( | x )=x log +(n-x) log (1-)

0 0.5 1-20

-15

-10

-5

0

l (

| x

)

l ( | x )=x log +(n-x) log (1-)

0 0.2 0.4 0.6 0.8 1-7000

-6000

-5000

-4000

-3000

-2000

-1000

l (

| x

)

l ( | x )=x log +(n-x) log (1-)

bin(842+982,)

x=842 (yes)


Likelihood

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 x1, x2, …, xn from a distribution f (x|), then the overall likelihood is the product of the individual likelihoods:

and loglikelihood is

n

ii

n

iin xlxfxxl

111 )|()|(),,|(

n

ii

n

ii

n

iin

xLxf

xfxxL

11

11

)|()|(log

)|(log),,|(


Log likelihood

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

the loglikelihood for a normal-mean problem is exactly quadratic.

That is, if we observe y1, . . . , yn from a normal population with known variance, the loglikelihood is

or in multi-dimension


MLE (maximum likelihood estimation) 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.

In regular problems, the ML estimate can be found by setting to zero the first derivative(s) of l(θ) with respect to θ.


Transformations of parameters

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) ).

Transformations may help us to improve the shape of the loglikelihood.

If the parameter space for θ has boundaries, we may want to choose a transformation to the entire real space.

For example, consider the binomial loglikelihood,L


binomial loglikelihood

If we apply the logit transformation

whose back-transformation is

the loglikelihood in terms of β is

L


If we observe y = 1 from a binomial with n = 5, the loglikelihood in terms of β looks like this.


Transformations do not affect the location of the maximum-likelihood (ML) estimate.

If l(θ) is maximized at ˆθ, then l(φ) is maximized at ˆφ = g(ˆθ).


score function

A first derivative of L(θ) with respect to θ is called a score function or simply a score.

In a one-parameter problem, the score function from an independent sample y1, . . . , yn is

where

is the score contribution for yi. The ML estimate is usually the solution of the likelihood

equation L’(θ)=0.

L


Mean of the score function.

A well known property of the score is that it has mean zero.

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


For example, in the case of the binomial proportion, we have

which is zero because E(Y ) = n. 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.


Estimating functions.

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.

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.

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


estimating equations

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.

Setting the estimating functions to zero is called the estimating equations.

In the case of the binomial proportion, for example,

Y − nis a mean-zero estimating function, and so is

−1 [Y − n] .


Information and variance estimation.

The variance of the score is known as the Fisher information. In the case of a single parameter, the Fisher information is

If θ has k parameters, the Fisher information is the k x k covariance matrix for scores


Like the score function, the Fisher information is also a function of θ. So we can evaluate it at any given value of θ.

Notice that i(θ) as we defined it is the square of a sum which, in many problems, can be messy.

To actually compute the Fisher information, we usually make use of the well known identity


In the multiparameter case, l(θ) is the k x k matrix of second derivatives

whose (l,m)th element is


why we care about the Fisher information? it provides us with a way (several ways, actually) of

assessing the uncertainty in the ML estimate. 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.


two common ways toapproximate the variance of ˆθ.

The first way is to plug ˆθ into i(θ) and invert,

this is commonly called the “expected information.”

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.”


1.3.2 Likelihood Function and ML Estimate for Binomial Parameter

The binomial log likelihood is

Differentiating with respect to yields

Equating this to 0 gives the likelihood equation, which has solution

the sample proportion of successes for the n trials.


Calculating , taking the expectation, and we get

Thus, the asymptotic variance of is Actually, from E(Y)=n and var(Y)=n (1- ), the

distribution if =Y/n has mean and standard error


Likelihood function and MLE summary

We use maximum likelihood estimate (MLE) asymptotically normal asymptotically consistent asymptotically efficient

Likelihood function probability of those data, treated as a function of

the unknown parameter. maximum likelihood (ML) estimate

parameter value that maximizes this function


MLE and its variance

If y1; y2; … ; yn is a random sample from distribution f(y|), then the score function is

In regular problems, we can find the ML estimate by setting the score function(s) to zero and solving for .

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.

We defined the Fisher information as the variance of the score function and


1.3.3 Wald–Likelihood Ratio–Score Test Triad

Three standard ways exist to use the likelihood function to perform large-sample inference. Wald test Score test Likelihood ratio test

We introduce these for a significance test of a null hypothesis H0: and then discuss their relation to interval estimation.

They all exploit the large-sample normality of ML estimators.


Wald test

With nonnull standard error SE of , the test statistic

has an approximate standard normal distribution when

One- or two-sided P-value by z. Or z2 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).


Wald test

For an .05-level two-side test, we reject H0 if

Equivalently, if

where 3.84 is the 95th percentile of 2(1).

96.1ˆ

0 SE

22

0 96.184.3)ˆvar(

)ˆ(


Wald test

The multivariate extension for the Wald test of

has test statistic

where is the inverse matrix of Information matrix.

W is an asymptotic chi-squared distribution with df = rank of .

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.


uses the likelihood function through the ratio of two maximizations:(1). the maximum over the possible parameter values under H0 (2). the maximum over the larger set of parameter values permitting H0 or an alternative Ha to be true.

The likelihood-ratio test statistic equals

where L0 and L1 denote the maximized log-likelihood functions.

is 2 distribution with df=dim(Ha U H0)-dim(H0)

Reject H0 if

> 2 (=0.05)

Likelihood ratio test


The score test is based on the slope and expected curvature of the log-likelihood function L() at the null value 0.

Score function

The value tends to be larger in absolute value when is farther from 0.

Score statistic

has an approximate standard normal null distribution. The chi-squared form of the score statistic is

Score test

)/)((

)(20

2

0

LE

u

)/)((

)(20

20

2

LE

u


Why is score statistic reasonable?

Recall that the mean of the score is zero and its variance is equal to the Fisher information.

In a large sample, the score will also be approximately normally distributed because it's a sum of iid random variables.

Therefore, it will behave like a squared standard normal [2(1)] if H0 is true.


Wald–Likelihood Ratio–Score Test


The three test statistics - Wald, LR and score are asymptotically equivalent.

The differences among them vanish in large samples if the null hypothesis is true.

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 H0.

Score test does not require to calculate MLE. LR test is scale-invariant. LR statistic uses the most information of the three types

of test statistic and is the most versatile.


1.3.4 Constructing confidence intervals

In practice, it is more informative to construct confidence intervals for parameters than to test hypotheses about their values.

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 H0: has a P-value exceeding 0.05.

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.

Let denote the 100(1-a) percentile of the chi-squared distribution with degrees of freedom df.


Tests and Confidence IntervalsAt significant level ,

reject H0: , if

2/0

ˆ

z

SE

100(1-)%

confidence interval

2/0

ˆ

z

SE

} :{ 0

} :{ 0

0


Confidence Intervals

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.

The likelihood-ratio-based interval is becoming more widely available in software and is preferable for categorical data with small to moderate n.

For the best known statistical model, regression for a normal response, the three types of inference necessarily provide identical results.


1.4 STATISTICAL INFERENCE FOR BINOMIAL PARAMETERS

Recall log likelihood

Score function

MLE

SE=

)1log()(log)|( ynyyL

)1/()(/)( ynyu


1.4.1 Tests about a Binomial Parameter

Since H0 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.

Wald statistic

Evaluating the binomial score and information at 0

The normal form of the score statistic simplifies to


binomial log-likelihood under H0

Under Ha

The likelihood-ratio test statistic

or

has an asymptotic chi-squared distribution with df=1.

)1log()(log 000 ynyL

)ˆ1log()(ˆlog1 ynyL


Test

At significant level , two sided, reject H0, if

(Wald test)

(Score test)

(LR test)


1.4.2 Confidence Intervals for a Binomial Parameter

Inverting the Wald test,

Unfortunately, it performs poorly unless n is very large The actual coverage probability usually falls below the

nominal confidence coefficient, much below when is near 0 or 1.

An adjustment is needed. (Problem 1.24)


Simulation to calculate coverage prob.%let n=1000; %let pi=0.5; %let simuN=10000;

data simu; drop i;

do i=1 to &simuN;

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)/&simuN as coverageprobabilty from res;


it performs poorly if 1) n is small; 2) pi near 0 or 1.

%let n=1000; %let pi=0.5; %let simuN=10000; %let n=20; %let pi=0.5; %let simuN=10000; %let n=20; %let pi=0.1; %let simuN=10000; %let n=20; %let pi=0.9; %let simuN=10000;


An adjustment is needed. (Problem 1.24)


%let n=20; %let pi=0.5; %let simuN=10000;

data simu; drop i;

do i=1 to &simuN;

k=RAND('BINOMIAL',&pi,&n); output;

end;

run;

data res; set simu;

pihat=(k+1.96)/(&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)/&simuN as coverageprobabilty from res;


score confidence interval

The score confidence interval contains 0 values for which

Its endpoints are the 0 solutions to the equations

It is quadratic in 0. This interval is


LR-based confidence interval

The likelihood-ratio-based confidence interval is more complex computationally, but simple in principle.

It is the set of 0 for which the likelihood ratio test has a P-value exceeding .

Equivalently, it is the set of 0 for which double the log likelihood drops by less than from its value at the ML estimate.


1.4.3 Proportion of Vegetarians Example

1 sta 517 – introduction: distribution and inference discrete data basic data are discretely...

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