Sampling and Confidence Interval

Kenneth Kwan Ho Chui, PhD, MPHDepartment of Public Health and Community Medicine

kenneth.chui@tufts.edu617.636.0853

Epidemiology/Biostatistics

Learning objectives in the syllabus

Understand how a histogram can be read as a probability distribution

Understand the importance of random sampling in statistics

Understand how sample means can have distributions

Explain the behavior (distribution) of sample means and the Central Limit Theorem

Know how to interpret confidence intervals as seen in the medical literature

Know how to calculate a confidence interval for a mean

Population Parameter

Sample statisticsSample

Types of dataHow to summarize data Central tendency VariabilityHow to evaluate graphs

Distribution of sample meansKnow how to interpret and calculate a confidence interval for statistical inference

Assumed knowledge for today

MeanVarianceStandard deviationThe 68-95-99 rule

Central tendency: Mean

Consider a variable with data:1, 2, 3, 3, 4, 4, 4, 5, 5, 6

Variance & Standard deviation

Observation #

Values

2

2

2 2 2

2

Sum them up

Divide by (sample size – 1)

Variance

SD = √Variance

The 68-95-99 rule

68% of sample are within ± 1SD

95% of samples are within ± 2SD

99% of samples are within ± 3SD

50th 84th 97.5th 99.5th16th2.5th0.5thPercentile:

# of SD:

Population Parameter

Sample statisticsThe mean BMI ofa sample fromBoston, Massachusetts

The true mean BMI ofBoston, Massachusetts

Sample

Researcher

?

Sample variation

1, 2, 3, 4, 5, 6

2, 4 4, 6 1, 2 1, 6Samples

3.0 5.0 1.5 3.5Means

Researcher 1 Researcher 2 Researcher 3 Researcher 4Researchers

The whole population

?

Central

limittheorem

Central limit theorem

The means obtained from many samplings from the same population have the following properties

The distribution of the means is always normal if the sample size is big enough (above 120 or so), regardless of the population’s distribution

The mean of the sample means is equal to the population mean

The standard deviation of the sample means, known as the standard error of the mean (SEM) is inversely related to the sample size: if we repeat the experiment with a bigger sample size, the resultant histogram will be “slimmer”

Understanding CLT through simulation

Population size: 10000Possible values: 0 through 9, 1000 eachTrue population mean: 4.50

Simulation scheme

A population of 10000Mean = 4.5

Samplen=500

Sample mean

Frequency

Sample mean

10000

Sample size = 500; # of draws = 10000

Sample means

Fre

qu

ency

4.5

68%

95%

99%SD = 0.13SE

±1 SE:67.95%

±2 SE:95.04%

±3 SE:99.10%

Characteristics for the distribution of means

In the previous slide, the mean 4.5 is the true population parameter, for which we have a Greek name, μ (mu)Similarly, the SD 0.13 is the true population parameter, called σ (sigma) in Greek. We call this SD of means “standard error of means” (SEM) or “standard error” (SE)SE can be estimated using sample SD:

Why bigger sample sizes are often better

Sample size = 500 Sample size = 1000Sample size = 200

Sample means Sample meansSample means

SE = 0.13 SE = 0.08SE = 0.20

Confidenceinterval

I got CLT, so now what?The histogram can be viewed as a “probability distribution”The sample mean from a researcher can be any pixel under the bell curveHow should we define “acceptably close” to the population mean? 95%

The confidence interval

95%

True mean

If we put a CI on every samplemean, about 95% of themwould include thetrue mean.

The two red ones arethe “unlucky”samples whichdo not includethe true mean.

Interpretation of a confidence interval

The mean and 95% confidence interval (CI) of the blood glucose of a sample is: 140 mg/dl (95%CI: 120, 160)We are 95% confident that the interval 120 and 160 mg/dl includes the true population mean. Our best estimate is 140 mg/dl (i.e. the sample mean)Why only 95% certain? Because the sample mean can be, unfortunately, an extreme one beyond ± 2 SE (the blue zones)

Some common CIs and their z-score multipliers

There are two numbers in a confidence interval:the lower and upper confidence limits90%CI:

Mean ± 1.65 SE95%CI:

Mean ± 1.96 SE 2.00 is an approximation, 1.96 is recommended The most commonly used criterion99%CI:

Mean ± 2.58 SEThe more certain we want the interval to include the true mean, the wider the CI becomes

“I am 100% certain that the true mean is between–∞ and ∞.”

How to narrow down confidence interval?

Lower our certainty by opting for, say, a 90%CI instead of a 95%CIDecrease sample standard deviation (for instance, using a more accurate measurement device)Increase sample size

Are confidence intervals always symmetric?

Not in all occasions. CIs for untransformed continuous variables are symmetricHowever, CIs for other statistics such as odds ratios and relative risks are calculated on logarithmic scale. When back-transformed to the ratios, the interval will be asymmetric

“Multivariable analysis revealed a more than 2-fold increase in the risk of total stroke among men with job strain (combination of high job demand and low job control) (hazard ratio, 2.73; 95% confidence interval, 1.17-6.38)”