1 Introduction to Biostatistics (BIO/EPI 540) Lecture 11: Hypothesis Testing Acknowledgement: Thanks...

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Introduction to Biostatistics

(BIO/EPI 540)

Lecture 11: Hypothesis Testing

Acknowledgement: Thanks to Professor Pagano (Harvard School of Public Health) for lecture material

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No human investigation can be called true science without passing through mathematical tests.

Leonardo da Vinci (1452-1519) (in Treatise on Painting)

Testing

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Sampling Paradigm

Population

Sample

Inference

μ,

σ

,S

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Inference

•Sample mean is an estimate of

•Sample variance (S) is an estimate of

•Confidence intervals and hypothesis tests are equivalent techniques to quantify uncertainty in sample derived inferences regarding population parameters

μ

σ2

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We know that cholesterol levels inUS men 20-24 yrs are normally distributed with σX 46 mg/100ml.

We obtain a sample of n=25 and want to infer μ.

Confidence Interval - Illustration

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value of μ

Use of C.I. to infer value

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IF 95% C.I.

200 mg/100ml (182,218)

Population mean = 211?

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XIF 95% C.I.

200 mg/100ml (182,218)

190 mg/100ml (172,208)


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IF 95% C.I.

200 mg/100ml (182,218)

190 mg/100ml (172,208)

175 mg/100ml (157,193)


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Alternatively IF= 211 and = 46 and we take a sample of size n=25 from this pop.,then the Central Limit Theorem says that the sample mean is approx. normal with mean = 211 and std. dev. 46/5; i.e.

If true

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Hypothesis Testing

Hypothesis Testing Trial by jury

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Individual on trial. Is he/she innocent?

Evidence Trial

Hypothesis Testing & Trial by jury

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Evidence Trial

Person

Innocent Guilty


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Evidence Trial

JuryPerson

Innocent Guilty

Not Guilty

Guilty


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Evidence Trial

JuryPerson

Innocent Guilty

Not Guilty

Guilty


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Evidence Trial

JuryPerson

Innocent Guilty

Not Guilty x

Guilty x


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Evidence Trial

JuryPerson

Innocent Guilty

Not Guilty x

Guilty x

Test of Hypothesis that = 0? Evidence Trial

Hypothesis Testing

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Trial

JuryPerson

Innocent Guilty

Not Guilty x

Guilty x

Test of Hypothesis that = 0? Sample Trial

Hypothesis Testing

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JuryPerson

Innocent Guilty

Not Guilty x

Guilty x

Test of Hypothesis that = 0? Sample Analysis

Hypothesis Testing

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JuryPopulation

Innocent Guilty

Not Guilty x

Guilty x


Hypothesis Testing

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JuryPopulation

= 0 Guilty

Not Guilty x

Guilty x


Hypothesis Testing

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JuryPopulation

= 0 0

Not Guilty x

Guilty x


Hypothesis Testing

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UsPopulation

= 0 0

Not Guilty x

Guilty x


Hypothesis Testing

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UsPopulation

= 0 0

Not reject x

Guilty x


Hypothesis Testing

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UsPopulation

= 0 0

Not reject x

Reject x


Hypothesis Testing

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UsPopulation

= 0 0

Not reject Type II

Reject Type I


Hypothesis Testing

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Probability of Type I error is i.e. the probability of rejecting the null hypothesis when it is true.

Probability of Type II error is i.e the probability of not rejectingthe null hypothesis when it is false.

1- is the power of the test.

Possible errors in analysis results

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1o Hypothesize a value (0)

2o Take a random sample (n).

3o

Is it likely that the sample came from a population with mean 0

( = 0.05) ?

Hypothesis testing about :

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We know that cholesterol levels inUS men 20-74 yrs are normally

distributed with σX 46 mg/100mland μ = 211. We obtain a random

sample of 12 hypertensive smokers and obtain a sample mean of 217

mg/100ml. We want to test whether their population mean is the same as that of the general population?

2 sided hypothesis test -Illustration

H0 : = 211

HA : 211

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H0 : = 211

HA : 211 = 46 mg/100ml

12 hypertensive smokers have:

217 2110.45

n 46/ 12x

zm

s0- -

= = =/

217 mg/ 100mlx =

2 sided hypothesis test -Illustration

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Some prefer to quote the p-value.The p-value answers the question,“What is the probability of get-ting as large, or larger, a Discrepancy given the null hypothesis is true?”

P-value

Question: Do hypertensive smokers have the same mean as the general population?

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Rejecting the null hypothesis

• Assume a specific threshold of Type I error, α– Typically α = 0.05

• If p value < α Reject null

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Some prefer to quote the p-value.The p-value answers the question,“What is the probability of get-ting as large, or larger, a Discrepancy given the null hypothesis is true?”

P-value

Answer: Do not reject the null hypothesis.No evidence that hypertensive smokers have a different mean than general population

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Decide on statistic:

Determine which values of consonant with the hypothesisthat = 0 and which ones are not.

are

Look at and decide.

Summary

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Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities:

1. Two-sidedH0 : = 0

HA : 0

Alternative hypothesis

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Blood glucose levelof healthy personshas = 9.7 mmol/Land = 2.0 mmol/L

H0 : 9.7

HA : > 9.7

Sample of 64 diabetics yields

Example - One-sided alternative

Do diabetics have blood glucose levels that are higher on averagewhen compared to the general population?

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Blood glucose levelof healthy personshas = 9.7 mmol/Land = 2.0 mmol/L

H0 : 9.7

HA : > 9.7

n = 64

p-value << 0.001

Example - One-sided alternative

Answer: Reject the null hypothesis.Significant evidence that diabetics have a higher mean level of glucose when compared to the general population

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Need to set up 2 hypotheses to cover all possibilities for . Choice of 3 possibilities:

Two-sidedH0 : = 0

HA : 0

One-sidedH0 : 0

HA : < 0

One-sidedH0 : 0

HA : > 0

Alternative hypothesis

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Summary

• Hypothesis testing: – Type I and II errors– Power

• Two sided hypothesis test

• One sided hypothesis test

1 Introduction to Biostatistics (BIO/EPI 540) Lecture 11: Hypothesis Testing Acknowledgement: Thanks...

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