Estimation of Various Population Parameters

Point Estimation and Confidence Intervals

Dr. M. H. RahbarProfessor of Biostatistics

Department of Epidemiology

Director, Data Coordinating Center

College of Human Medicine

Michigan State University

Important Statistical Terms

• Population: A set which includes all measurements of interest to the researcher

• Sample: Any subset of the population

• Parameter of interest: Characteristics of interest in a population are called population parameters

• (e.g, mean, standard deviation, proportion)

Inferential Statistics

• Estimation includes point and interval estimation of population parameters

• Mean = • Median

• Standard deviation = • Proportion = p

• Incidence

Central Limit Theorem

• When large samples, (n>30), are selected from a population with mean and standard deviation , then

• the sampling distribution of the sample mean is approximately normal with the same mean ,

and the standard error, SE= n

Estimation of Population Mean

A 95% confidence interval for based on a large sample, when n>30, is approximately,

2( )xn

Estimation of Population Mean

A (1- )100% confidence interval for based on a large sample, when n>30, is approximately,

/ 2

/ 2

/ 2

( )

0.05, 1.96

0.10, 1.28

Sx Zn

Z

Z

Example # 1Suppose we want to estimate the average birth-weight of children born to non-smoking women in community X. Assume the standard deviation of birth-weights is 0.5 Kg and the sample mean birth-weight is 3.0 Kg. Find a 95% confidence interval for .

0.53.0 2( )100

3.0 0.1

(2.9,3.1)

Interpretation of CI’s

What do we mean by (2.9 Kg, 3.1 Kg) is a 95% CI for the mean birth-weight of children born to non-smoking women in community X?

Does interval (2.9 , 3.1) contain ?

Confidence Interval for P Suppose we want to estimate the prevalence of hypertension among adults in community X. Assume in a random sample of 100 adults from this community the prevalence of hypertension is 0.50. Find a 95% confidence interval for P.

0.50 2( (0.50)(1 0.50) 100)

0.50 0.10

(0.40,0.60)

Probability vs Odds of an Event • An experiment is

repeated for a large number of times. Then the probability of occurrence of an event E is the percentage of the time that that particular event has occurred in repeated experiments.

• Suppose P(E)=0.40

• Odds of an event E

=P(E)/{1-P(E)}.

Example,

If P(E)=0.40, then

Odds of E=2/3.

In the following contingency table estimate the proportion and odds of drinkers among those who develop Lung Cancer and those without the disease?

Lung Cancer Total Case Control Drinker Yes A=33 B=27 60 No C=1667 D= 2273 3940

P1=33/1700 P2=27/2300 Odds1=33/1667 Odds2=27/2273

Odds Ratio as a Measure of Association

• QUESTION: Is there a difference in the proportion of drinkers among those who develop Lung Cancer and those without the disease?

• Odds Ratio=ad/bc

• OR=1.67

• This means that the odds of drinking among cases is 67% more than the odds of drinking among controls.

Relative Risk as Measure of Association

• QUESTION: Are persons exposed to “factor X” more likely to develop the disease than those not so exposed?

Risk among exposed

= a/(a+b)

Risk among non-exposed

= c/(c+d)

Relative Risk

= Risk among exposed relative to risk among non-exposed

QUESTION: Are persons exposed to “factor X” more likely to develop the disease than those not so exposed?

Disease TotalYes No

Factor X Yes A=33 B=27 60 (fixed)No C=1667 D=2273 3940 (fixed)

Risk among exposed = a/(a+b)=33/60=0.55Risk among non-exposed = c/(c+d)=1667/3940=0.42

Relative Risk = Risk among exposed relative to risk among non-exposed RR=1.31

95% Confidence Intervals for Odds Ratios and Relative Risk

• NOTE:• Confidence Intervals

for Odds Ratios and Relative Risk are not symmetrical

• Odds Ratio=ad/bc• OR=1.67 • E.g., 95% CI for OR

(1.2, 2.9)

• If the 95% CI for OR does not include one then we conclude a difference in the odds of drinking between the two groups

• We use Statistical packages to calculate the CI’s for OR or RR

QUESTION: Estimate the difference between the proportions of drinkers among Lung Cancer cases and controls!

Lung Cancer Total Yes No Drinker Yes A=33 B=27 60 No C=1667 D= 2273 3940

P1- P2 = 33/1700 – 27/2300

QUESTION: Estimate the difference between the mean blood levels of cases and controls?

G rou p 1D isease

M ean B P

G rou p 2N o D isease

M ean B P

Factors influencing the width of the CI’s

Sample size

Variance

Confidence level = (1-)100%

Width of the CI’s

If Sample size =n , then Width If Variance , then Width

If Confidence level , then Width

Choose ‘n’ to achieve the desired width for the CI’s