8[1].basic stat inference

8/12/2019 8[1].Basic Stat Inference

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Basics of

Statistical Inference

V. Sreenivas

[email protected]


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Basics of Statistical Inference

ImportancePrimary entities in statistical inference

Types of statisticalinference

EstimationconsiderationsMeasure of accuracy of estimates

General framework of testing

Types of errors in statistical testing

Form of a Statistical test

Interpretation of a test result


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In all clinical/epidemiological studies,information collected represents only asample from the target population

Drawing conclusions about thepopulation depends on statistical analysisof data

So the basis of statistical inference isimportant to understand & interpret theresults from the epidemiological studies


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3 primary entities

The target population

Set of characteristics orvariables

Probability distribution of thecharacteristics


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Population

Collection of units of observation that are of

interest & is the target of the investigation

Eg. In studying the prevalence of

osteoporosis in women of a city, all the

women in that city would form the target

population

Essential to identify the population clearly &

precisely


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Variables

Once population is identified, clearly define whatcharacteristics of the units of this population are tobe studied

In the above example, we need to define:

Osteoporosis (reliable & valid method of diagnosis:DEXA/ Ultrasound, normal values of BMD etc.)

Clear & precise methods of measuring thesecharacteristics are essential for the success of thestudy


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Variables

Qualitative: Take a few possible values (Eg.

Sex, Disease status)Quantitative: Can theoretically take any

value within a specified range (Eg.

Blood sugar, Syst. BP)

Type of analysis depends on the type of the

variable


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Probability distribution

Most crucial link between population & its

characteristics

Allows to draw inferences on the population,

based on a sample

Tells what different values can a variable take

How frequently each value can occur in the

population


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Common distributions in health research areBinomial, Poisson & Normal

Eg. Incidence of a relatively common diseasemay be approximated by Binomial distribution

Incidence of a rare disease can be considered

to have a Poisson distribution

Continuous variables are often considered tobe Normally distributed


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Prob. Distribution is characterized by certain quantities

called parametersThese quantities allow us to calculate the probabilities of

various events concerning the variable

Eg. Binomial dist. has 2 parameters nand p.

This distribution occurs when a fixed number (n) of

subjects is observed, the characteristic is dichotomous innature and each subject has the same probability (p) of

having one value and (1-p) of having the other value

The statistical inference then involves finding out the value

of pin the population, based on a carefully selected sample


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Binomial distribution

for n = 10 & p = 0.5

0.25

0.20

0.15

0.10

0.05

0.000 1 2 3 4 5 6 7 8 9 10

Number of successes


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Eg. The Normal distribution is a mathematical curve

represented by two quantities (, )mean andstandard deviation respectively.

Most quantitative characteristics follow this

Symmetric, Bell shaped curve

One half is the mirror image of the other

Mean, median & mode are same and are at center

Mean 1SD covers 68% data, 2SD 95%, 3SD 99%


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X

0Z

68.6%

X-1SD1Z

X-2SD

1.96Z

X+2SD

1.96Z

X+1SD

1Z

95.0% area

X+2.58SD

2.58Z

X-2.58SD

2.58Z

99.0% area under the curve

Empirical properties of a Normal Deviate

X: Variable in original units Z: Standardized variable


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Statistical inference

Estimation

We estimate somecharacteristic of the

population, based

on a sample

Testing

We test some

hypotheses about

the population

parameters


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Descriptive Studies

In these, generally the objective is:To estimate the values of the parameters of

the Prob. dist., based on the sampled

observations

Best guess of the value in the population

and a measure of accuracy of this estimateare obtained


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Estimation

Best guesses

Population mean : Mean of samplePopulation proportion: Sample proportion

Considerations:

Consistency: As the sample size increases, theestimates approach their target values

Unbiased: The average value of the estimatedparameter over a large number of repeated samples ofsame size will be equal to the population value

Maximum likelihood: That value of the parameterwhich maximizes the probability of observing a

sample that has been observed


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Accuracy of estimates

When an estimate (E) of a parameter is obtained,

we need to know how this value (E) wouldchange if another sample is studied

The distribution of values of E over different

repeated samples (under identical conditions) isknown as the sampling distribution of E

This sampling distribution can be determined

empirically or purely based sampling theory

The standard deviation of the estimate E is calledthe Standard Error (SE)


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Accuracy of Estimates

Once the sampling distribution of theestimate is known, it can be answered

How close is my estimate likely to be

the true value of the parameter

Can state with certain confidence thatthe true value will be withincertaininterval (Confidence Interval)


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Confidence Interval

The more the confidence required, morethe width, for a given sample size

Intuitively, more the information wehave (larger sample), the smaller the

width of the interval (the more certain

we are about the result)


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Estimation of parameters from a Normalpopulation - Example

The average Bone Mineral Density (BMD) of 150elder women (60+ years) is 0.678gm/cm2with a SDof 0.12 gm/cm2, what is the 95% C.I of the meanBMD?

Sample size (n) = 150 Mean = 0.678 SD = 0.12

It has been shown that Mean will have a Normal

distribution with Mean as mean itself and theStandard Error as /SE n

0.678 / 150 0.055SE


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Confidence Interval for

BMD in elderly women

We have the Mean = 0.678, SE = 0.055; andwe also know that Mean follows a Normal

Distribution

Using the Normal distribution properties, we know

that Mean 1.96 SE covers 95% of values

0.678 (1.96*0.055) = (0.570 0.786) covers

95% of results if we repeat the study

This interval is called the 95% CI for mean BMD


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Interpretation of CI

Mean BMD = 0.678 and 95% CI: (0.570 0.786)

If we repeat the study 100 times, 95% times we geta mean BMD between a 0.57 and 0.79 gm/cm2

Another interpretation is:

There is 95% chance that these two limits cover thetrue, unknown, but fixed value of BMD in the

elderly women

There is 95% chance that the truth is somewhere in

this interval

Do not get the impression that truth varies from

0.57 to 0.79


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Interpretation of CIMean BMD = 0.678 and

95% CI: (0.570 0.786)

The narrower the interval, the more confidentwe are of the result

Alternatively, the wider this interval, the less

certain we are about the result


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Analytical Studies

Involve testing of hypothesisStudy will have formulated research questions(hypotheses)

Eg. Is treatment A is superior to treatment B ?Based on the observations from the sample, weneed to draw conclusions

Inference is a 2 step process:

- Estimate the parameters

- Test the hypotheses involving these parameters


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Statistical Tests of Hypotheses

Step 1: Identify the Null Hypothesis (H0)

- No additional effect of the new treatment;

- No difference in prevalence rates;

- Relative risk is one etc.

It should be testable

- Possible to identify which parameters

need to be estimated and their sampling

distribution, given the study design


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Statistical tests of Hypotheses

Null hypothesis: cure rate p1= p2

Alternative hypotheses to the null

hypothesis:The cure rates are different

(P1p2 two-tailed alternative)

The cure rate in new method is more

(p2 > p1 one-tailed alternative)


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Step 2: Determine the levels of errorsthat can be acceptable

Decision Truth in the populationH0is true H0is false

Accept H0 No error Type II Error ()

Reject H0 Type I error () No error

Analogous with a laboratory test

: False Positivity: False Negativity

1: Sensitivity (Power of a test)

St 2 D t i th l l f


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Step 2: Determine the levels of errorsthat can be acceptable

Decision

Truth in the population

H0is true H0is false

Accept H0 No error Type II Error ()

Reject H0 Type I error () No error

Impossible to reduce both the errors simultaneously

One decreases when the other increases

Design the study with a desired level of and

minimize the

choice of & is made after determining the

consequences of each of the errors and is made

at the design stage itself


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Step 3: Determine the best Statistical test for

the stated Null hypothesis

Depends on:

Study design (Cross over or Independent

groups, Paired or Unpaired observations etc.)

Type of variable (Qualitative / Quantitative)

The properties of the study variable

(Binomial/Normal distribution, Standard

Error of the estimate etc.)


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Step 3: Determining the best test

Common tests of significancet TEST

Chi-square (2

) testZ test

Non-parametric testsInvolves calculating a critical ratio

that helps to make a decision


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Tests of significance

ParameterCritical Ratio = ----------------------------------(Test Statistic) SE of that parameter

If we are comparing two proportions:

Diff. between

the two proportionsCritical Ratio = Z = ---------------------------------

SE of the differencebetween the two

proportions


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Step 4: Perform the Statistical Test

- Calculate the test statistic (Z / 2/ t etc)

- Using the properties of the distribution of the test

Statistic, obtain the probability of

observing such an estimate of the Statistic

- This probability is the probability of getting the

observed value of the test statistic if the Null

hypothesis is true- If this is small, Null hypothesis is an unlikely

explanation for the resultsReject the Null

hypothesis (Significant result). If not


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- tn-1, 1-/2 0 tn-1, 1-/2

Acceptance region

|t|< tn-1, 1- /2

Rejection region

t< -tn-1, 1- /2Rejection region

t> tn-1, 1- /2

Acceptance & Rejection regions for a paired ttest


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Step 5: If the Null hypothesis is not rejected

at the given level of significance, the

statistical power of the test (1-) should be

computed

Recall that is an error of accepting H0,when it is false. So 1- will be prob. of

rejecting H0, when it is false. If this

quantity is low, we recommend that thestudy be repeated with a larger sample


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Statistical test oh HypothesisAn example

We wish to compare the BMD of Indian elderly

women with Caucasian elderly women

We hypothesize that Indian women will have lower

BMD

Our Null hypothesis: both groups will have equal

BMD level

Our alternative hypothesis is: both groups BMD

will be unequal

We collected data on 150 Indian women and data

on Caucasian women is available from literature


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Indian data:

Caucasian data:

Since the sample sizes are large, we can apply a

test called Z test and Z statistic is calculated as:

Calculations give us a Z value:

0.176/0.0117 = 15.04

1 1 1150 0.678 0.12n x S

2 2 2300 0.854 0.11n x S

1 2

2 2

1 2

1 2

|x xZ

S S

n n


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From the data, we have Z = 15.04

We know that Z follows a Normal distribution

Using the properties of Normal distribution we

realize that the probability of observing this much

value of Z or more extreme in either directionis < 0.0000001 or < one in a million

In other words, if our Null hypothesis is correct, our

chance of finding a Z = 15.04 is so small

We suspect the Null hypothesis and Reject it and

conclude that both groups have statistically different

BMD levels


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Summary

3 entities viz. Population, Variables &Probability distribution of the variables are

important in Statistical Inference

Estimation & Testing are 2 components ofStatistical Inference

Descriptive studies generally deal with

estimation & Analytical studies deal with

testing of hypotheses


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Summary contd.

Estimation is followed by a measure of accuracy

Confidence Interval

2 types of errors can be committed in statisticaltesting

Type I error is nothing but the usual pvalue (FalsePositivity)

The compliment of type II error (False negativity)is called the Power of the test (Sensitivity)

Test estimator is generally of the form of a ratio of

2 quantities


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Summary contd.

The calculated Ratio under given

circumstances follows a known pattern called

its distribution

Using this distribution, we can know theprobability of observing a Ratio of the

magnitude that is observed, by chance alone

If this chance probability is low, chance is

unlikely to explain the observed result and we

Reject the null hypothesis


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Summary Contd..

If the Null hypothesis is rejected, we attribute the

observed difference to the exposure under

consideration

If the null hypothesis is not rejected (accepted), we

should be sure that our data is sensitive enough to

believe the negative result (statistical power

should be calculated)

8[1].basic stat inference

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