1 introduction to biostatistics (pubhlth 540) estimating parameters which estimator is best? study...

1

Introduction to Biostatistics (PUBHLTH 540) Estimating Parameters

• Which estimator is best? • Study possible samples, determine Expected values, bias,

variance, MSE– with replacement example– without replacement example (Exam 1)

• Estimate population mean– point estimator (sample mean)– interval estimator (95% central width)

• Central Limit theorem• Interval estimators based on a sample

– estimating the standard error– determining the multiplier (normal and t-distributions)

2

Sampling with replacement

• Program ejs09b540p19.sas– uses Arrays, Outputs, and Transpose– Select SRS w rep from N=5 with n=3 – Uniform random number generator

• Program ejs09b540p20.sas– Replaces sample size, pop size, and trials with

macro variables (gives flexibility)– Uses functions of arrays to get mean, var, min,

max– Select SRS w rep from N=5 with n=3

3

SRS without Replacement• Program ejs09b540p21.sas

– Process of selecting subjects without replacement

– Do loops, shifting indices etc.

• Program ejs09b540p22.sas– Implementable version with macro

variables

• Program ejs09b540p23.sas– Check that all sample sets have equal

probability– n=3 from N=3 with functions to get sets

4

Which Estimator of Population Median is Best?

• Program ejs09b540p24.sas– Add data from population, and link

response for sample subject sets– Evaluate sample median, mean,

(min+max)/2

• Program ejs09b540p25.sas– Summarize results of samples- using

expected value, variance, MSE of estimators

– Use PROC MEANS options for VARDEF=N, and MAXDEC=2

– Sample mean has smallest MSE– Is this always true?

5

Estimate Pop Median Age in Seasons Study Data

• Program ejs09b540p26.sas– use basev2.sas7bdat with “Age”– include histograms of distribution of

estimator over possible samples– best estimator is not the mean!- BEST

depends on the population…

• Program ejs09b540p27.sas– estimate Pop Mean using sample mean

from SRS w/o rep. of n=25– How does var of sample means relate to

the population variance?

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Relating Population Variance to the Variance of the Sample Means

• Population Variance

• Variance of Sample Mean (without replacement:

• with T=10,000 trials…

2

var 11

N nX

N N n

22

1

1 N

ii

xN

273 25 57.37var 1 2.09

272 273 25X

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Interval Estimate

• idea is to place an interval around an estimate to approximate the width of the estimators sampling distribution

• usually, the width is the central 95% of the estimators sampling distribution

• How wide is this?– measure width in terms of stderr of mean

2

var 11

N nSE X X

N N n

n

8

How good is Approximation?

• Program ejs09b540p28.sas– SRS w/o rep of n=5 to estimate Mean

LDL cholesterol from the Seasons study using the sample mean, 10 samples.

– determine the 2.5th percentile and 97.5th percentile of the distribution of sample means.

– Determine how many multiples of stderror of mean the percentiles are from the population mean

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Example of 95% Width• Program ejs09b540p28.sas

• Change number of samples to 10000• Determine multiples for standard error

– Lower 2.5% multiplier is -1.85– Upper 97.5 multiplier is 2.02– Standard Deviation of sample means =

se(Mean)=15.94

• Program ejs09b540p30.sas– select srs w/o rep of n=5, estimate mean

• sample mean=166.7• Low= 166.7 -1.85(15.94)• High=166.7 + 2.02 (15.94)

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Figure 1a. Histogram of tg for Population of N=291

Source: ejs09b540p31.sas 10/22/2009 by ejs

0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 1260 1320 1380 1440 1500 1560 1620 1680

0

5

10

15

20

25

30

35

40

Pe

rce

nt

triglycerides:* tg

Example of Triglycerides- Seasons Study

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1.5

Take 10,000 SRS w/o replacement of size n=5 (program ejs09b540p31.sas)

Population:

Source Sim Sim Sim Sim

2 142 95.10 100.03 -0.93 2.55

5 144.1 62.07 63.26 -1.1 2.51

10 144.6 45.29 44.73 -1.19 2.95

20 144 30.81 31.63 -1.39 2.6

30 143.7 24.60 25.83 -1.47 2.49

50 143.6 18.10 20.01 -1.63 2.22

Source ejs09b540p31.sas

1

1 T

tt

Y YT

2

1

1

1

T

t tt

sd Y Y YT

n

n

Multiplier of for 2.5 %ile

tsd Y

Multiplier of for 97.5 %ile

tsd Y

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Figure 2. Histogram of sample means of n=10 for tg from Population of N=291


0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

0

5

10

15

20

25

30

35

40

Pe

rce

nt

mn_samp

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0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

0

5

10

15

20

25

30

35

40

Pe

rce

nt

mn_samp

14




0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

0

5

10

15

20

25

30

35

40

Pe

rce

nt

mn_samp

15




0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

0

10

20

30

40

50

Pe

rce

nt

mn_samp

16




0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

0

10

20

30

40

50

Pe

rce

nt

mn_samp

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Sa

mp

le

1

21

41

61

81

101

121

141

161

181

201

221

241

261

281

tg 95% Interval Estimate

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440

Figure 1. Illustration of Point and 95% Interval Estimate for n=50 for tg


id 1

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Conclusions

• With larger sample size, distribution of sample means is more bell shaped (i.e. ‘normal’) (Central Limit Theorem)

• Central 95% of distribution is around + or - 2 standard errors from true population mean

• In practice we don’t know the SE• In practice we don’t know the multiplier• Solution: Estimate SE from sample• Solution: Approximate multipler

assuming a distribution (Normal if known or t-distribution if not known)

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Normal Distribution

• With larger sample sizes, the distribution of SRS means is normal:

2,y yY N 2

2y n

• Standard Normal Distribution

0,1y

YZ N

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Transforming a Random Variable

• Standardization is an example of transforming a random variable.

• Suppose we have a random variable:

Y

• What is the expected value and variance of X=a+bY?

yE Y 2var yY

y

x

E X E a bY

a bE Y

a b

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• Variance of X=a+bY?

2

2

2

22

2 2

var var

y

y

y

y

X a bY

E X E X

E a bY a b

E b Y

b E Y

b

22


• Application for Standardizing

1 1yy

y y y

YZ Y

1y

y

a

1

y

b

Z a bY

1 1

0y yy y

E Z a bE Y

2

2

2

var var

11y

y

Z b Y

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Conclusions- Practical

• Assume Central Limit Theorem holds (usually if n>30)

• Use multiplier based on centered distribution of standard normal (if

is known)

• see Table A3 in Text– central 60% -0.84 to +0.84

– central 80% -1.28 to 1.28

– central 90% -1.64 to 1.64

– central 95% -1.96 to 1.96

– central 99% -2.56 to 2.56

20, 1Z N

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• In practice we don’t know

• Estimate using

• Use a t-distribution with (n-1) degrees of freedom for multiplies (see table A4 in text).– assumes underlying normal

distribution and srs

x

var xX SE Xn

22

1

1

1

n

ii

S X Xn

25


• t-distribution examples for 95% interval estimator (Confidence interval):– n=2 df=1 -4.3 to 4.3– n=5 df=4 -2.776 to 2.776– n=10 df=9 -2.262 to 2.262– n=20 df=19 -2.093 to 2.093– n=30 df=29 -2.045 to 2.045– n=50 df=49 -2.009 to 2.009– n=120df=119 -1.98 to 1.98– n=500df=499 -1.96 to 1.96

1 introduction to biostatistics (pubhlth 540) estimating parameters which estimator is best? study...

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