Statisticssampling distribution and estimation part 1 introduction

Sampling distribution of the mean of any population For infinitely large population For finite population - The finite population correction factor1... Point and interval estimators Poi

2NATIONAL ECONOMICS UNIVERSITY

SCHOOL OF ADVANCED EDUCATION PROGRAMS

-*** -STATISTICS Sampling Distribution and Estimation

(GROUP 1)

Nguyen Phuong Anh

11213233 11210589

Ha Noi, May2023

LIST OF TAB

YTable 1 Baseline characteristics of the patients 9Table 2 Changes in coprimary end points and cardiometabolic risk factors between baselineand week 56 11Table 3 Characteristics of 6712 participants 13Table 4 Adherence to quality indicators, overall and according to type of care and function14Table 5 Adherence to quality indicators, according to mode 14Table 6 Adherence to quality indicators, according to conditions* 16Table 7 Demographics, alcohol advertisment exposure, and market alcohol advertisementexpenditure by mean alcohol use and changes in alcohol use over time 19Table 8 Hierarchical linear modeling parameter estimates predicting alcohol use for the totalsample 20Table 9 Hierarchical linear modeling parameter estimates predicting alcohol use among 15-

to 20-year-olds 23

LIST OF FIGURES

Figure 1 Alcohol use over time by age in markets with high alcohol advertising expenditureper capta 21Figure 2 Alcohol use over time by age in markets with high alcohol advertising expenditureper capta 21Figure 3 Alcohol use by mean advertising exposure, market advertising expenditure percapita, and gender 22Figure 4 Biology scores (2021) 24Figure 5 Physics scores (2021) 24

TABLE OF CONTENTS

PART 1: INTRODUCTION 1

I SAMPLING DISTRIBUTION 1

1 Sampling distribution of a mean 1

1.1 Sampling distributions of 1

1.2 The central limit theorem 1

1.3 Sampling distribution of the mean of any population 1

2 Sampling distribution of a proportion 2

3 Sampling distribution of the difference between 2 means 2

4 T-distribution 3

II ESTIMATION 3

1 Concepts of Estimation 3

1.1 Point and interval estimators 3

1.2 Desirable qualities of estimators 4

2 Confidence interval estimator of μ 5

2.1 General information 5

2.2 The width of the interval 5

3 Application of Estimation 5

3.1 Financial analysis 5

3.2 Quality control 5

3.3 Medical research 6

PART 2: ARTICLE SUMMARY 7

I MAIN ARTICLE 7

1 Background 7

2 Purpose 7

3 Methods 7

4 Results 7

4.1 Trial population 8

4.2 Body weight 9

4.3 Glycemic control 10

4.4 Cardiometabolic variables 10

5 Conclusion 12

II SUB-ARTICLES 12

1 The Quality of Health Care Delivered to Adults in the United States 12

1.1 Background 12

1.2 Purpose 12

1.3 Methods 12

1.4 Results 13

1.5 Conclusion 17

2 Effects of Alcohol Advertising Exposure on Drinking Among Youth 17

2.1 Background 17

2.2 Purpose 17

2.3 Methods 17

2.4 Result 18

2.5 Conclusion 23

PART 3: DATA ANALYSIS 24

I DATASET 24

1 Source 24

2 Descriptive information 24

3 Reasons for choosing the dataset 25

II THEORY APPLICATION 25

1 Central Limit Theorem 25

2 Using sampling distribution for Inference 28

CONCLUSION 31

REFERENCES 32

PART 1: INTRODUCTION

I SAMPLING DISTRIBUTION

+ Draw samples of the same size from a population, calculate the statistics ofinterest, and then use descriptive techniques

+ Use the rules of probability and the laws of expected value and variance toderive the sampling distribution

1 Sampling distribution of a mean

1.1 Sampling distributions of

population from which we’re sampling:

population divided by the sample size:

mean; that is,

1.2 The central limit theorem

random sample drawn from any population is approximately normal for a sufficiently large sample size The larger the sample size, the more closely the sampling distribution of will resemble a normal distribution”.

1.3 Sampling distribution of the mean of any population

For infinitely large population

For finite population

- The finite population correction factor1

2 Sampling distribution of a proportion

5

-(The standard deviation of is called the standard error of the proportion)

3 Sampling distribution of the difference between 2 means

normal populations

independent normal random variables is also normally distributed Thus, thedifference between two sample means is normally distributed if both populations arenormal

variance of the sampling distribution of

and standard deviation (which is the standard error of the difference between two means)

4 T-distribution

General information

than a normal distribution

Notation: Degree of freedom is the number of observations whose value are free tovary after calculating the sample mean

How to use t-tables

II ESTIMATION

1 Concepts of Estimation

The objective of estimation is to determine the approximate value of a populationparameter on the basis of a sample statistics For example, the sample mean which is

1.1 Point and interval estimators

Point estimator

A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point

Drawbacks of using point estimators:

Interval estimator

3

An interval estimator draws inferences about a population by estimating the value of

an unknown parameter using an interval The interval estimator is affected by the samplesize

Applications of estimation:

1.2 Desirable qualities of estimators

Unbiased estimator

population parameter is an estimator whose expected value is equal to that parameter

We want our estimators to be accurate and precise:

Consistency

Another desirable quality is consistency An unbiased estimator is said to beconsistent if the difference between the estimator and the parameter grows smaller as the

2 Confidence interval estimator of μ

2.1 General information

The confidence interval estimator is a probability statement about the sample mean It states that there is 1 - probability that the sample mean will be equal to value such that the interval to will include the population mean

In general, a confidence interval estimator for μ is given by

Notation:

A 95% confidence interval should be interpreted as saying “In repeated sampling, 95% of such intervals created would contain the true population mean”.

2.2 The width of the interval

The width of the confidence interval estimate is a function of the population standarddeviation, the confidence level, and the sample size

Factors influence the width of the interval:

- Vary the confidence level: Decreasing the confidence level narrows the interval,increasing it widens the interval

3 Application of Estimation

3.1 Financial analysis

Estimation techniques are used to:

return for a given level of risk

3.2 Quality control

Estimation techniques are used to:

- Estimate the proportion of defective items in a batch or production run Thisinformation is used to determine whether the batch or run meets the quality standards

or needs to be rejected

- Estimate the process capability index, which is a measure of the ability of amanufacturing process to produce products within specification limits Thisinformation is used to determine whether the process is capable of producing productswithin the required quality standards

5

3.3 Medical research

a medical treatment is effective or not

a population Estimation techniques such as point estimation and interval estimationare used to estimate the prevalence and incidence of diseases and to determinewhether they are statistically significant

PART 2: ARTICLE SUMMARY

3 Methods

The researchers conducted a 56-week, double-blind trial involving 3731 patients whodid not have type 2 diabetes and who had a body-mass index (BMI; the weight in kilogramsdivided by the square of the height in meters) of at least 30 or a BMI of at least 27 if they hadtreated or untreated dyslipidemia (rối loạn lipid máu) or hypertension (tăng huyết áp) The researchers randomly assigned patients in a 2:1 ratio to receive once-dailysubcutaneous injections of liraglutide at a dose of 3.0 mg (2487 patients) or placebo (giảdược) (1244 patients); both groups received counseling on lifestyle modification (cả 2 nhómđều được tư vấn thay đổi lối sống) The coprimary end points were the change in body weightand the proportion of patients losing at least 5% and more than 10% of their initial body

weight The power for the first coprimary endpoint, weight change, was calculated with the use of a two-sided Student’s t-test at a 5% significance level

4 Results

At baseline, the mean (±SD) age of the patients was 45.1±12.0 years, the mean weightwas 106.2±21.4 kg, and the mean BMI was 38.3±6.4; a total of 78.5% of the patients were7

women, and 61.2% had prediabetes At week 56, patients in the liraglutide group had lost amean of 8.4±7.3 kg of body weight, and those in the placebo group had lost a mean of2.8±6.5 kg (a difference of −5.6 kg; 95% confidence interval, −6.0 to −5.1; P<0.001, withlast-observation-carried-forward imputation)

A total of 63.2% of the patients in the liraglutide group as compared with 27.1% in theplacebo group lost at least 5% of their body weight (P<0.001), and 33.1% and 10.6%,respectively, lost more than 10% of their body weight (P<0.001)

The most frequently reported adverse events with liraglutide were mild or moderatenausea and diarrhea Serious events occurred in 6.2% of the patients in the liraglutide groupand in 5.0% of the patients in the placebo group

4.1 Trial population

A total of 3731 patients underwent randomization: 2487 to lifestyle intervention plusliraglutide, at a dose of 3.0 mg once daily, and 1244 to lifestyle intervention plus placebo.The baseline characteristics were similar in the two groups (Table 1) A total of 1789 patients(71.9%) in the liraglutide group, as compared with 801 patients (64.4%) in the placebo group,completed 56 weeks of treatment A larger percentage of patients in the liraglutide group than

in the placebo group withdrew from the trial owing to adverse events (9.9% [246 of 2487patients] vs 3.8% [47 of 1244]); a smaller percentage of patients in the liraglutide groupwithdrew from the trial owing to ineffective therapy (0.9% [23 of 2487] vs 2.9% [36 of1244]) or withdrew their consent (10.6% [264 of 2487] vs 20.0% [249 of 1244])

Table 1 Baseline characteristics of the patients

4.2 Body weight

After 56 weeks, patients in the liraglutide group had lost a mean (±SD) of 8.0±6.7%(8.4±7.3 kg) of their body weight, whereas patients in the placebo group had lost a mean of2.6±5.7% (2.8±6.5 kg) of their body weight (Table 2) Weight loss with liraglutide was9

maintained over 56 weeks and was similar regardless of prediabetes status A greaterproportion of patients in the liraglutide group than in the placebo group lost at least 5% oftheir body weight (63.2% vs 27.1%), more than 10% of their body weight (33.1% vs.10.6%), and more than 15% of their body weight (14.4% vs 3.5%) Overall, approximately92% of the patients in the liraglutide group and approximately 65% of the patients in theplacebo group lost weight The liraglutide group also had a greater reduction than the placebogroup in mean waist circumference and BMI (Table 2).

Several sensitivity analyses confirmed the superiority of liraglutide over placebo with respect

to the coprimary end points Liraglutide appeared to be less effective in patients with a meanBMI of 40 or higher than in patients with a lower BMI

4.3 Glycemic control

There was a greater reduction in glycated hemoglobin, fasting glucose, and fastinginsulin levels in the liraglutide group than in the placebo group (Table 2) Liraglutide wasalso associated with lowering plasma glucose levels and higher insulin and C-peptide levelsrelative to placebo during an oral glucose-tolerance test The effects of liraglutide on glycatedhemoglobin, fasting glucose, and glucose levels during the oral glucose-tolerance test weregreater in patients with prediabetes than in those without (P<0.001) Measures of insulinresistance and beta-cell function also showed improvement with liraglutide as compared withplacebo

The prevalence of prediabetes was significantly lower in the liraglutide group than inthe placebo group at week 56, a finding that was consistent with the improvement inglycemic control with liraglutide Type 2 diabetes developed in more patients in the placebogroup than in the liraglutide group during the course of treatment

4.4 Cardiometabolic variables

By week 56, systolic and diastolic blood pressure decreased more in the liraglutidegroup than in the placebo group (Table 2) All measures of fasting lipid levels (Table 2), aswell as levels of high-sensitivity C-reactive protein, plasminogen activator inhibitor-1, andadiponectin, showed greater improvement in the liraglutide group than in the placebo group

Table 2 Changes in coprimary end points and cardiometabolic risk factors between

baseline and week 56

11

5 Conclusion

In conclusion, 3.0 mg of once-daily subcutaneous liraglutide, as an adjunct (một liềuthuốc hỗ trợ) to diet and exercise, was associated with clinically meaningful weight loss inoverweight or obese patients, with concurrent reductions in glycemic variables and multiplecardiometabolic risk factors, as well as improvements in health-related quality of life

II SUB-ARTICLES

1 The Quality of Health Care Delivered to Adults in the United States

By Elizabeth A McGlynn, Ph.D., Steven M Asch, M.D., M.P.H., John Adams, Ph.D., JoanKeesey, B.A., Jennifer Hicks, M.P.H., Ph.D., Alison DeCristofaro, M.P.H., and Eve A Kerr,M.D., M.P.H

1.1 Background

The degree to which health care in the United States is consistent with basic andlargely unknown quality standards Although previous studies have documented seriousquality deficits, they provide a limited perspective on the issue Few studies from the countryhave only focused on a limited set of topics: preventive care, diabetes, or humanimmunodeficiency virus,… and have accessed health outcomes without a link to specificprocesses involved in care Hence, there would be no comprehensive view of the level ofquality of care given to the average person in the United States

1.2 Purpose

This study aimed to assess the extent to which the recommended processes of medicalcare, one critical dimension of quality, were delivered to a representative sample of the U.S.population for a broad spectrum of conditions

1.3 Methods

The study randomly chose participants by telephoning adults living in 12 metropolitanareas in the United States and asking them about selected healthcare experiences After

- Plus–minus values are means or percentages ±SE

- The number of times a participant is eligible for an indicator is a function of the level

at which the indicator is scored (participant, participant–provider dyad, or episode), the number of participants eligible for the specified process, and the number of indicators in the aggregate-score category

1.4.1 Analysis of care delivered

The three tables below show the number of indicators included in the aggregate score,the number of persons eligible for one or more processes within the category, the number oftimes participants in the sample were eligible for indicators, and the weighted meanproportion (and 95% confidence interval) of recommended processes that were delivered

13

Table 4 Adherence to quality indicators, overall and according to type of care and

function

Overall, according to Table 4, participants received 54.9 percent of recommended care(95 percent confidence interval, 54.3 to 55.5) This level of performance was similar in theareas of preventive care, acute care, and care for chronic conditions The level of performanceaccording to the particular medical function ranged from 52.2 percent (95 percent confidenceinterval, 51.3 to 53.2) for screening to 58.5 percent (95 percent confidence interval, 56.6 to60.4) for follow-up care