statistics sampling distribution and estimation

Sampling distribution of a mean - The standard deviation of the sampling distribution is called the standard error of the mean; that is, 1.2.. - Central Limit Theorem states that: “7he s

2NATIONAL ECONOMICS UNIVERSITY

SCHOOL OF ADVANCED EDUCATION PROGRAMS

STATISTICS Sampling Distribution and Estimation

(GROUP 1)

Ha Noi, May2023

LIST OF TAB

Y

Table 1 Baseline characterisflcs of the DatI€TES ch 2H HH HH1 1 1 ray 9 Table 2 Changes in coprimary end points and cardiometabolic risk factors between baseline

and WeEeK 56 nố ốốố 11

Table 3 Characteristics of 6712 parHICIDATIES - 1L H1 H1 H111 011811011111 1111111 kg 13 Table 4 Adherence to quality indicators, overall and according to type of care and function 14 Table 5 Adherence to quality indicators, according to MOdC ccc ecccetecseeeetetseeneetseees 14 Table 6 Adherence to quality indicators, according to condItIOnSẼ co co ve 16 Table 7 Demographics, alcohol advertisment exposure, and market alcohol advertisement expenditure by mean alcohol use and changes 1n alcohol use over time 19

Table 8 Hierarchical linear modeling parameter estimates predicting alcohol use for the total k1 a4 20

Table 9 Hierarchical linear modeling parameter estimates predicting alcohol use among 15- i02) vn ND 23

LIST OF FIGURES Figure 1 Alcohol use over time by age in markets with high alcohol advertising expenditure POT CAPA 21

Figure 2 Alcohol use over time by age in markets with high alcohol advertising expenditure POT CAPA 21

Figure 3 Alcohol use by mean advertising exposure, market advertising expenditure per 6201811569030 22

lai số s00 0v 1/2/2000 ea 24

Eigure 5 Physics seores (202 Ï) c cc tt 2n 0H 1H11 1111101111 11101111011 1111111 TH Hết 24 Figure 6 Sampling distribution of of Physics scores [n = 40, 100 samples] 25

Figure 7 Sampling distribution of of Physics scores [n = 100, 100 samples]| 26

Figure 8 Sampling distribution of of Biology scores [n = 40, 100 samples] 27

Figure 9 Sampling distribution of of Biology scores [n = 100, 100 samples] 27

Figure 10 Sampling distribution of of Biology scores [n = 300, 100 samples] 27 Figure I1 Biology scores of 40 random sfudenfs ác St S29 121212111111 8e, 29

TABLE OF CONTENTS

PART 1: INTRODUCTION

I, SAMPLING DISTRIBUTION

- The distribution formed by all the possible values for sample statistics obtained for every possible different sample of a given size is called the sampling distribution

- Two ways to create a sampling distribution:

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

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

- The primary function of the sampling distribution is statistical inference

1 Sampling distribution of a mean

- The standard deviation of the sampling distribution is called the standard error of the

mean; that is,

1.2 The central limit theorem

- Asn gets larger, the sampling distribution of becomes increasingly bell shaped

- Central Limit Theorem states that: “7he sampling distribution of the mean of a 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

- The mean of sampling distribution is always equal to the mean of the population

- The standard error is equal to

For finite population

Notation: - The population size

- The finite population correction factor

2 Sampling distribution of a proportion

is approximately normally distributed provided that and are greater than or equal to

5

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

3 Sampling distribution of the difference between 2 means

The sampling plan calls for independent random samples drawn from each of two normal populations

Based on the central limit theorem, it has been shown that the difference between two

independent normal random variables is also normally distributed Thus, the difference between two sample means is normally distributed if both populations are normal

By using the laws of expected value and variance we derive the expected value and variance of the sampling distribution of

The sampling distribution of is normal with mean

4 T-distribution

General information

- The t-distribution is a type of normal distribution that is used for smaller sample sizes

- It is symmetric around 0, mound-shaped (like a normal), but has a higher variance than a normal distribution

How to use t-tables

- Bottom row has = infinite, this is the standard normal probabilities

- Ifdfis very large, use Z tables even if is unknown

II ESTIMATION

1 Concepts of Estimation

The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistics For example, the sample mean which is employed to estimate the population mean is referred as the estimator of the population mean Once the sample mean has been computed, its value is called the estimate

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:

- The estimate will be wrong

- We often need to know how close the estimator is to the parameter

- Point estimators don’t have the capacity to reflect the effects of larger sample sizes

Interval estimator

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 sample

Size

Applications of estimation:

- Calculate the proportion of television viewers who are tuned in to a network

- Calculate the mean income of university graduates

1.2 Desirable qualities of estimators

Unbiased estimator

One desirable quality of an estimator is unbiasedness An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter

We want our estimators to be accurate and precise:

- Accurate: On average, our estimator is getting towards the true value

- Precise: Our estimates are close together

Relative efficiency

A third desirable quality is relative efficiency If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to have relative efficiency

Error of estimation

The sampling error is defined as the difference between an estimator and a parameter

We can also define this difference as the error of estimation

2 Confidence interval estimator of p

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 iis given by

Notation:

© The probability of is called the confidence level

ois called the lower confidence limit (LCL)

o is called the upper confidence limit (UCL)

A 95% confidence interval should be interpreted as saying “Jn 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 standard

deviation, the confidence level, and the sample size

Factors influence the width of the interval:

- Vary the sample size: As the sample size gets bigger, the interval gets narrower

- 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:

- Estimate the value of financial assets

- Estimate the risk of financial assets to develop risk management strategies

- Optimize portfolios by selecting a combination of assets that maximizes the expected 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 This information 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 a manufacturing process to produce products within specification limits This information is used to determine whether the process is capable of producing products within the required quality standards

3.3 Medical research

Estimation techniques are used to estimate treatment effects and to determine whether

a medical treatment is effective or not

Estimation techniques are used to estimate the prevalence and incidence of diseases in

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

PART 2: ARTICLE SUMMARY

I MAIN ARTICLE

A Randomized, Controlled Trial of 3.0 mg of Liraglutide in Weight Management (kiém

soat can nang)

By Xavier Pi-Sunyer, M.D., Ame Astrup, M.D., D.M.Sc., Ken Fujioka, M.D., Frank Greenway, M.D., Alfredo Halpern, M.D., Michel Krempf, M.D., Ph.D., David C.W Lau, M.D., Ph.D., Carel W le Roux, F.R.C.P., Ph.D., Rafael Violante Ortiz, M.D., Christine Bjorn Jensen, M.D., Ph.D., and John P.H Wilding, D.M., for the SCALE Obesity and Prediabetes

3 Methods

The researchers conducted a 56-week, double-blind trial involving 3731 patients who did not have type 2 diabetes and who had a body-mass index (BMI, the weight in kilograms divided by the square of the height in meters) of at least 30 or a BMI of at least 27 if they had treated or untreated dyslipidemia (r6i loan lipid mau) or hypertension (tang huyết áp) The researchers randomly assigned patients in a 2:1 ratio to receive once-daily subcutaneous injections of liraglutide at a dose of 3.0 mg (2487 patients) or placebo (gia dugc) (1244 patients); both groups received counseling on lifestyle modification (ca 2 nhom đều được tư vấn thay đối lối sống) The coprimary end points were the change in body weight and 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 weight

was 106.2+21.4 kg, and the mean BMI was 38.3+6.4; a total of 78.5% of the patients were

7

women, and 61.2% had prediabetes At week 56, patients in the liraglutide group had lost a mean of 8.4+7.3 kg of body weight, and those in the placebo group had lost a mean of

2.8+6.5 kg (a difference of —5.6 kg; 95% confidence interval, —6.0 to —5.1; P<0.001, with

last-observation-carried-forward imputation)

A total of 63.2% of the patients in the liraglutide group as compared with 27.1% in the placebo 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 moderate nausea and diarrhea Serious events occurred in 6.2% of the patients in the liraglutide group and 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 plus liraglutide, 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 2487 patients] vs 3.8% [47 of 1244]); a smaller percentage of patients in the liraglutide group withdrew from the trial owing to ineffective therapy (0.9% [23 of 2487] vs 2.9% [36 of 1244]) or withdrew their consent (10.6% [264 of 2487] vs 20.0% [249 of 1244])

Table 1 Baseline Characteristics of the Patients.*

Characteristic Liraglutide (N=2487) Placebo (N = 1244) Sex — no, (%)

American Indian or Alaska Native 5 (0.2) 4 (0.3)

Native Hawaiian or other Pacific Islander 2 (<0.1) 2 (0.2)

+ Race and ethnic group were self-reported Patients from France did not report race or ethnic group

+ The body-mass index is the weight in kilograms divided by the square of the height in meters

§ The reference range is 3.0 to 25.0 plU/mL for both sexes and all ages

4 Prediabetes was defined according to American Diabetes Association 2010 criteria.’®

| The diagnoses of dyslipidemia and hypertension were based on self-reported medical history

maintained over 56 weeks and was similar regardless of prediabetes status A greater proportion of patients in the liraglutide group than in the placebo group lost at least 5% of their 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, approximately 92% of the patients in the liraglutide group and approximately 65% of the patients in the placebo group lost weight The liraglutide group also had a greater reduction than the placebo group 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 mean BMI 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 fasting insulin levels in the liraglutide group than in the placebo group (Table 2) Liraglutide was also associated with lowering plasma glucose levels and higher insulin and C-peptide levels relative to placebo during an oral glucose-tolerance test The effects of liraglutide on glycated hemoglobin, fasting glucose, and glucose levels during the oral glucose-tolerance test were greater in patients with prediabetes than in those without (P<0.001) Measures of insulin resistance and beta-cell function also showed improvement with liraglutide as compared with placebo

The prevalence of prediabetes was significantly lower in the liraglutide group than in the placebo group at week 56, a finding that was consistent with the improvement in glycemic control with liraglutide Type 2 diabetes developed in more patients in the placebo group 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 liraglutide group than in the placebo group (Table 2) All measures of fasting lipid levels (Table 2), as well as levels of high-sensitivity C-reactive protein, plasminogen activator inhibitor-1, and adiponectin, showed greater improvement in the liraglutide group than in the placebo group

End Point (N=2437) (N=1225) vs Placebo (95% CI)† P Value

Coprimary end points

Change in body weight

Body weight-related end points

the safety-analysis set, which included all patients who were randomly assigned to a study group and had exposure to a

study drug Data for fasting insulin, fasting C-peptide, and fasting lipids were log-transformed for analysis and are pre- sented as relative treatment differences

+ Loss of at least 5% and more than 10% of body weight were analyzed by logistic regression with data from the full-analysis set, with LOCF imputation, and are presented as the proportions of patients (%) and odds ratios

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

baseline and week 56

5 Conclusion

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

H 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., Joan Keesey, B.A., Jennifer Hicks, M.P.H., Ph.D., Alison DeCristofaro, M.P.H., and Eve A Kerr,

M.D., M.PH

1.1 Background

The degree to which health care in the United States is consistent with basic and largely unknown quality standards Although previous studies have documented serious quality deficits, they provide a limited perspective on the issue Few studies from the country have only focused on a limited set of topics: preventive care, diabetes, or human immunodeficiency virus, and have accessed health outcomes without a link to specific processes involved in care Hence, there would be no comprehensive view of the level of quality 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 medical care, 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 metropolitan areas in the United States and asking them about selected healthcare experiences After several stages, among the 17,937 eligible adults from the initial sample of 20,028; only 6712 participants returned their consent forms and sent at least one medical record This information could be used to evaluate the performance on 439 indicators of quality care The indicators in this research were chosen based on the measures of all phases of care or medical functions (screening, diagnosing, treatment and follow-up) In order to produce aggregate scores, all instances were divided in which recommended care was delivered by the number

of times participants were eligible for indicators in the category The proportions of recommended care received was calculated with the use of two-sided Student’s t-test at

a 5% significance level

No of conditions and preventive care for

which participants were eligible Mean 2.540.02

No of times participants eligible for

indicatorst Mean 15.840.17 Range 2-304

Table 3 Characteristics of 6712 participants Notation:

- 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 of times participants in the sample were eligible for indicators, and the weighted mean proportion (and 95% confidence interval) of recommended processes that were delivered

areas of preventive care, acute care, and care for chronic conditions The level of performance

according to the particular medical function ranged from 52.2 percent (95 percent confidence interval, 51.3 to 53.2) for screening to 58.5 percent (95 percent confidence interval, 56.6 to 60.4) for follow-up care